Topics in MHD Turbulence Citation Lithwick, Yoram (2002) Topics in MHD Turbulence. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/FP3Q-EZ29. https://resolver.caltech.edu/CaltechTHESIS:01302012-162107660 Abstract I consider two topics in MHD turbulence. First, I work out the theory of compressible MHD turbulence, including kinetic effects. I use this theory to understand features of interstellar scintillation that have hitherto been unexplained. Second, I work out the theory of imbalanced weak MHD turbulence. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Physics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Goldreich, Peter Martin (advisor) Readhead, Anthony C. S. (advisor) Group: TAPIR, Astronomy Department Thesis Committee: Goldreich, Peter Martin (chair) Blandford, Roger D. Phinney, E. Sterl Scoville, Nicholas Zabriskie (co-chair) Readhead, Anthony C. S. Defense Date: 3 May 2002 Record Number: CaltechTHESIS:01302012-162107660 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:01302012-162107660 DOI: 10.7907/FP3Q-EZ29 ORCID: Author ORCID Lithwick, Yoram 0000-0003-4450-0528 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6796 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 31 Jan 2012 16:09 Last Modified: 05 Nov 2021 22:39 Thesis Files Preview PDF - Final Version See Usage Policy. 28MB Repository Staff Only: item control page

Topics In MHD Turbulence Thesis by Yoram Lithwick In Partial Fulfillment of the Requirements for t he Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2002 (Defended May 3, 2002) 11 © 2002 Yoram Lithwick All Rights Reserved III Acknowledgements I thank my advisor, Peter Goldreich, for showing me how enjoya ble physics can be. He is not only a great physicist, but a great advisor as well. IV Abstract I consider two topics in MHD turbulence. First, I work out the theory of compressible MHD t urbulence, including kinetic effects. I use this theory to understand features of interst ella r scint illation t hat have hitherto been unexplained . Second , I work out the theory of imbala nced weak MHD t urbulen ce. v Contents Acknow ledgements 111 Abstract iv 1 Compressible MHD Turbulence in Interstellar Plasmas 1 1.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Compressible MHD Turbulence for Explaining Scintillation 2 1.1.2 Diffractive Interstellar Scintillation (DISS) . . . . . 3 1.1.3 Basic Theory of DISS and the Scattering Measure 3 1.1.4 Examination of Assumptions 5 1.1.5 Observational Overview . . . 6 1.1.6 Two Constraints on the Turbulence 7 1.1.7 SM From Turbulence Behind Shocks 10 Incompressible MHD Turbulence 11 1.2.1 Alfven Wave Spectrum . 12 1.2.2 Eddies . . . . . . . . . . 13 1.2.3 Passive Scalar Spectrum 15 1.2.4 Slow Wave Spectrum. 18 1.2.5 Numerical Simulations 19 1.3 Compressible Turbulence: Overview 20 1.4 The Outer Scale and t he MHD Scale 22 1.5 The Cooling Scale 24 1.2 1.6 1.7 ..... . ... . 1.5.1 Density F luctuations Below the Cooling Scale: 1 < (3 < Lout/Cstcoo l 25 1.5.2 Density Fluctuations Below the Cooling Scale: (3 > L out / cstcool > 1 27 The Collision less Scale of the Neutrals 28 1.6.1 Frequency Change . . . . . . . 29 1.6.2 Effect of Neutrals on the Turbulent Cascade. 32 1.6.3 Neutral Hydrogen Atoms 34 1.6.4 Neutral Helium Atoms .. 34 1.6.5 If Neutrals Damp the Cascade 35 The Collision less Scale of the Ions 1.7.1 The E lectron Diffusion Scale . 36 37 Vl l.7.2 The Electron-Proton Equilibration Scale. 38 l.7.3 The Proton Diffusion Scale: Death of the Slow Mode and Entropy Mode. 38 l. 7.4 Density Spectrum Below the Proton Diffusion Scale 40 l.8 The End: the Proton Gyroscale . 40 l.9 Summary . . . . . . . . . . . . . 40 l.10 Comparison with Higdon's Work 41 l.11 Compressible Turbulence When (3 < 1 42 l.12 Future Work 43 ............. . l.13 Appendix: Waves in Compressible MHD 43 44 l.13.1 Alfven Mode . 45 l.13.3 Entropy Mode 45 l.13.2 Slow Mode Bibliography 47 2 Imbalanced Weak MHD Thrbulence 49 2.1 Introduction . . . 49 2.2 Basic Equations. 51 2.3 Weak Turbulence: Heuristic Discussion. 53 2.3.1 Scaling Relation 53 2.3.2 Insufficiency of Scaling Arguments for the Imbalanced Cascade 55 2.3.3 Dynamics at the Dissipation Scale: Pinned Spectra . 55 2.3.4 Two Peripheral Issues 56 Steady State Energy Spectra 58 2.4.1 Fixed Energy at the Outer Scale 59 2.4.2 Fixed Flux . . . ....... . 59 2.5 Kinetic Equations in Weak Turbulence 60 2.6 Numerical Simulations of Kinetic Equations 62 2.6.1 Fixed Energy at the Outer Scale 63 2.6 .2 Fixed Flux . . . . . . 65 2.6.3 Decaying Turbulence. 67 2.4 2.7 ........ . A Model of the Kinetic Equations: Coupled Diffusion Equations 67 2.7.1 Derivation of Coupled Diffusion Equations . . . . . . . . 69 2.7.2 Numerical Simulations of Coupled Diffusion Equations 70 2.8 The Bottleneck Effect 73 2.9 Discussion . . . . . . . 73 2.10 Appendix: Kinetic Equation in Weak Turbulence 75 Vll 2.10.1 Preliminaries . . . . . . . . . . . . . . .. 75 2.10.2 A Toy Problem: the Linear R andom Oscillator 76 2.10.3 Derivation of Kinetic Equation 79 2.10.4 Steady State Fluxes . . . . . . 81 Bibliography 83 Vlll List of Figures 1.1 Wandering of Magnetic Fieldlines 14 l.2 Three-Dimensional Spectrum .. 16 2.1 Simulation of Kinetic Equations with Fixed Energies at the Outer Scale 64 2.2 Simulation of Kinetic Equations with Fixed Fluxes 66 2.3 Decaying Simulation of Kinetic Equations . . . . . 68 2.4 Simulation of Diffusion Equations with Fixed Energies at the Outer Scale 71 2.5 Simulation of Diffusion Equations with Fixed F luxes 72 2.6 The Bottleneck Effect 74 ...... ... ....... . IX List of Tables 1.1 Summary of Lengthscales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22 1 Chapter 1 Compressible MHD Turbulence in Interstellar Plasmas Written with P eter Goldreich ; published in t he Astrophysical Journal, 562 , 279 (2001 ). (Introduction has been expanded for t hesis.) ABST RA CT R adio-wave scintillation observations reveal a nearly Kolmogorov spectrum of density flu ctuations in t he ionized interstellar medium . Although this density spectrum is suggestive of turbulence, no theory relevant to its interpretation exists. We calculate the density spectrum in turbulent magnetized plasmas by extending the t heory of incompressible magnetohydrodynamic (MHD ) turbulence given by Goldreich and Sridhar (1995) to include t he effects of compressibility and particle transport . Our most important results are as follows. (1) Density flu ctuations are due to t he slow mode and t he entropy mode. Both modes are passively mixed by the cascade of shear Alfven waves . Since t he shear Alfv en waves have a Kolmogorov spectrum , so do the density fluctu ations. (2) Observed density fluctuation amplitudes constrain the nature of MHD turbulence in the interstellar medium. Slow mode density flu ctuations are suppressed when t he magnetic pressure is less t han t he gas p ressure. Entropy mode density fluctuations are suppressed by cooling when the cascade t imescale is longer t han t he cooling timescale. These constraints imply eit her that t he magnetic and gas pressures are comparable, or t hat the outer scale of t he t urbulence is very small. (3) A high degree of ionization is required for the cascade to survive damping by neutrals and thereby to extend to small lengthscales. Regions that are insufficiently ionized produce density flu ctuations only on lengths cales larger than the neutral damping scale. These regions may account for t he excess of power that is found on large scales. (4) Provided t hat the thermal pressure exceeds the magnetic pressure, both t he entropy mode and t he slow mode are damped on lengthscales below t hat at which protons can diffuse across an eddy during t he eddy 's turnover time. Consequently, eddies whose extents along the magnetic fi eld are smaller t han the proton collisional mean free path do not contribute to t he density spectrum . However , in MHD t urbulence eddies are highly elongated along t he magnetic field . From an observational perspective, the relevant lengthscale is that transverse to t he magnetic field . Thus t he cu toff lengthscale for density flu ctuations is significantly smaller than the proton mean free path . (5) The Alfven mode is critically damped at t he transverse lengthscale of the proton gyroradius, and thus cascades to smaller lengths cales than either t he slow mode or t he entropy mode. 2 1.1 Introduction 1.1.1 Compressible MHD Turbulence for Explaining Scintillation Diffractive scintillations of small a ngular-diam eter radio sources indicate that the interstellar electron density sp ectrum on lengths cales 10 8 - 10 10 cm is nearly Kolmogorov; i.e., r.m .s. density fluctuations across a lengthscale ,\ a re nearly proportional t o ,\1 /3. Density fluctuations that obey the Kolmogorov scaling occur in homogeneous subsonic hydrody- namic turbulence. They are due to the entropy mode, a zero-frequency isobaric m ode whose density fluctuations are offset by temperature fluctuations. Since subsonic turb ulence is nearly incompressible , the velocity fluctuations follow Kolmogorov 's scaling. To a good approximation , the entropy mode is passively mixed by the velocity field , so it also conforms to t he Kolmogorov spectrum 1 Density fluctuations in Earth 's atmosphere, which cause stars to twinkle, obey t he Kolmogorov scaling. They a rise from the passive mixing of t he entropy mode. The electron density spectrum in the interstellar medium cannot be explained by hydrodynamic turbulence.2 Because the medium is ionized, magnetic effects must be accounted for. This is evident since the lengthscales probed by diffractive scintillations a re smaller than the collisional mean free paths of both electrons and protons. If t he magnetic field were negligible, freely strea ming plasma would wipe out density fluctuations at diffractive scales . In the presence of a magnetic field, electrons and protons a re tied to fieldlines at the scale of their gyroradii. For typical interstellar field strengths, these gyroradii a re smaller than the diffractive scales. A magnetic field thus impedes the plasma from streaming across fieldlines, and allows the turbulent cascade and the associated density fluctu ations to reach very small scales across the fieldlines before dissipating. Therefore, a theory for compressible turbulence in magnetized plasmas is required to explain the observed density spectra. Our objective is to develop this theory. Until now, t he only description of density fluctuations in interstellar plasmas was by Higdon (1984, 1986). These papers, while prescient, preceded a theory for incompressible MHD turbulence , and therefore did not account for t he full dynamics of the cascade. We compare Higdon's theory with ours in §1. 1O. Before presenting our theory of compressible MHD turbulence, we give a n overview of t he observations. 1 Density fluctuations due to the Reyno lds stress scale as )..2/3 . In addition, th e d iss ipat ion of turbu lent kinetic energy y ields entropy fluctuations. The ratio of the correspond in g density fluctuations to t he mean density is comparab le to the square of the Mach number at t he lengthscale of int erest; hence these density fluctuations a re a lso proportional to )..2/3. 2Charge neutrality is maintained on d iffractive scales, so electron density fluctuations include compensating flu ctuations in the density of positive ions . 3 1.1.2 Diffractive Interstellar Scintillation (DISS) Rickett (1977, 1990) and Armstrong, Rickett, and Spangler (1995) review the observations of diffractive interstellar scint illation (DISS) and their interpretation . They also discuss refr active scintillations and dispersion m easure flu ctuations. However, we focu s primarily on diffractive m easurements because they are much more sensitive. Radio waves that propagate through ionized gas are affected by plasma disp ersion: their phase speed dep ends on the density of free electrons. Since free electrons are distributed in homogeneously in the interstellar medium , wavefronts in the ISM are distorted. One consequence is that point-like radio sources are observed to be scatter-broadened, i. e., their images have a fin ite a ngular extent, Bscatt. The scattering disks of a few dozen sources have been imaged- primarily extra-galactic radio sources, as well as a few Galactic sources such as the Galactic center and Galactic OH-IR stars. A much more common method for detecting scattering disks is through t heir effects on pulsars. The scattering disk of a pulsar is usually too small a nd too weak to be directly imaged. However, rays t hat travel through the center of the scattering disk arrive at Earth faster t han those that travel through the edges. If the distance between t he Earth and the region where scattering occurs is D , then the difference in travel time is Tscatt ::: (D / c)( 1 - cos Bscatt) ::: (D / c)(Bscatt )2/2 (assuming that the distance from the scattering region to the pulsar is » D). Therefore a pulse that is emitted for an infinitesimal time is observed at Earth to b e broadened in time by Tscatt. Pulse broadening has b een observed directly in many pulsars. There are also many pulsars whose pulse-broadenings, while not directly observable, can be inferred in frequency-space; i.e. , b ecause of pulse broadening, the observed pulsar signal is correlated with itself only over a na rrow ba ndwidth , 6.vsca tt "-' I/Tsca tt ; this is called t he decorrelation bandwidth. 1.1.3 Basic Theory of DISS and the Scattering Measure If a pla ne wave is incident on a localized region of scattering material, then fronts of constant phase are distorted. In this subsection, we calculate the typical angle by which phase fronts are b ent. This angle is closely related to the observed angular size of a point-like radio source that is located at an infinite distance behind the scattering region. We will see that the angular size dep ends on a single combination of parameters within the scattering region: the scattering measure. Because of plasma disp ersion, a radio wave that travels through ionized plasma has phase speed c(1 - (v p /v)2)-1/2, where c is the speed of light, v is the radio-wave frequency and vI' is the p lasma frequency (e.g., Spitzer 1978). Neglecting factors of order unity t hroughout , (v p /v)2 "-' nrej,2 , where n is the electron number density, re is the classical electron radius, and j, is the radio-wavelength (we use a tilde to differentiate radio-wavelength from lengthscale, which we denote by A in this thesis); for the observations of interest, nre5.2 « 1. As a result , a wave t hat travels through a region of 4 plasma for a distance A, has its phase increased by (1.1 ) relative to the phase it would h ave had if it had travelled through vacuum. Let u s consider the scattering of a plane wave by flu ctuations in the scattering region t h at have length scale A. 3 We label the r.m.s. electron density fluctuation on th is lengthscale n A . After a plan e wave p enetrates the scattering region a distance A along t he lin e sig ht, it becomes distor ted; the r.m.s. value of its phase is ~ nA A[re5:.]. As t he pla ne wave penetrates fur ther, the phase ch a nge along a ny line of sight due to plasma dispersion cha nges randomly ever y step of length A, as in a random walk. The r.m. s. phase thus increases as the squ are root of t he number of steps . If the scattering region has a n extent L turb along t he line of sight (where L t llrb » A), then t he r.m .s phase of the wavefront after L tllrb is (1.2) Therefore a front of constant phase becomes corrugated; t he r.m.s. bending a ngle is (1. 3) As long as cPA » 1, eA is t he angle by which rays are deflected. Let u s next consider the scattering when density fluctuations exist on a range of lengthscales. We assume a Kolmogorov density spectrum on lengthscales of interest, n A ex A1 / 3 . If we insert this n A into equation (1.2), we find cPA ex A5 / 6 ; we then see from equ ation (1.3) t hat Larger values of eA ex A- 1 / 6 . eA result from smaller lengthscales. The smallest relevant length scale is t h at at which cPA ~ 1. It is called t he diffractive lengthscale, A d iff . On sm aller scales, cPA < 1, so two rays separated by A < Adiff at the scattering region can only partially interfere. As a result, the observed angular size of the scattering disk is eA diff" Before calculating Adiff and e Adirr , it is convenient to introduce the scattering m easure (SM), (1.4) which is indep endent of lengthscale. Equation (1.2) becomes (1. 5) 3We assume isotropy, so oX refers to lengt hscales both transverse and parallel to the lin e of s ight. A lthou gh MI-ID turbu lence is anisotropic on small scales, the angle-ave raged spectru m is nearly isot ropi c. 5 At the diffractive scale ¢), ~ 1, so (1.6) Typically ),diff ~ 10 8 _10 10 cm. Inserting ),diff into equ ation (1.3) , we h ave for the observed angular size of t he scattering disk: Bscatt"""" BAdiff rv /\'/ A' d iff r-...J SM 3 / 5 A ,1l /5re6/5 ' (1.7) It is set by SM. Since Tscatt ~ l/b.vsca tt ~ (D/c)(Bscatt)2, all DISS measurements are characterized by SM (in addition to D and t he spectral slope). 1.1.4 Examination of Assumptions One of our reasons for repeating the above well-known calculation of Bscatt is to cla rify what assumptions have been made. We do t h at in t his subsection. First, we assumed that the amplit ude of t he Kolmogorov density spectrum , n),/ ),1/3, is constant along t he line of sight. If it varies, t hen it is apparent from our derivation t hat SM should simply b e defined as the integral of (n),)2 / ),2/3 along the line of sight (instead of eq. [1.4]) ; i.e., SM is an extensive quantity. Second , we assumed that n), ex ),1/3 in t he vicinity of ),c1iff. More generally, if n), ex ),a where a < 1/2, t hen instead of equation (1.7) we derive +1) ) Bscatt rv [SM a A,2a+3r2]1/(2a e (1.8) where SMa is the generalized scattering measure, SMa ~ (nV),2a) Ltur b.4 The exp onent on '\, i.e., (2a + 3)/( 2a + 1), is observable from the scalings of DISS measurements with radio-wavelength. While it is often observed to be near 11 /5 (implying t he Kolmogorov value a = 1/3), there are a number of lines of sight where it differs substantially from t his value. Third , we assumed that the radio source is at infinity. To consider t he more genera l case, we first define D1 as the Earth-to-scattering region distance and D2 as t he scattering region-to-radio source distance . Then, as long as D1 < D 2, the Bscatt given by equation (1.7) is approximately t h e angula r size of the scattering spot (viewed from Earth). If t he converse is true (D1 > D 2)' then t he Bscatt given by equation (1.7) corresponds to the angula r spot size at the scattering screen as viewed from the source; from Earth, the observed spot size would b e Bobs ~ (D d DdBscatt < Bscatt. When 4Cordes et al. (1991) define their SM as fo ll ows. They ass um e that the density power spect rum is given by Inkl2 = C~k-"' , w here nk is the Fourier transform of the d ensity fie ld n(x). They then define SM as the integra l of C~ a long the lin e of s ight . So their C~ is nearly equal to o ur nV)..2n, aside from an order unity multipli cat ive factor that can eas ily be calc ul ated by perform ing the Fourier transform; their SM is nearl y equal to o ur SMa; a nd Q = 2a + 3. 6 inferring SM from observations of a ngula r broadening, one often assumes t hat Dl < D 2 , which allows t he distance correction to be ignored. However, this could underestimate SM . The corresponding problem exists also for measurements of either temporal broadening or decorrelation bandwidth, which form t he bulk of DISS observations . For arbitrary Dl and D 2 , Tscatt ~ (min{D l, D2}/C)( escatt)2, where escatt is given by equation (l.7), and similarly for .6.vsca tt (i .e., .6.vsca tt ~ l/Tsca tt). Typically, the Earth-to-pulsar distance is estimated from the observed dispersion measure together with a model of the electron density distribution within the Galaxy. The scattering region is assumed to like halfway to the pulsar , so t he inferred 8M is typically an underestimate. 1.1.5 Observational Overview Within a kiloparsec of the galactic plane, t he contributions to C; ~ (n>Y / )..2 / 3 appear to be of two kinds. There is a mean background with C; ~ 1O-3.5 m -2o/3, together with localized regions, mostly close to t he galactic plane, in which C; is considerable larger (Cordes, Weisberg, and Boriakoff 1985). Along one line of sight through t he HII region complex NGC 6334, e ~ 3" at radio-frequency ~ l.5 GHz, making this the strongest scattered source yet observed (Moran et al. 1990). Equation (l.7) implies that SM~ 2 x 103m -2o/3kpc. The authors assume t hat the thickness of the scattering screen is ~ 1 pc , which means that C; ~ 2 x 106m-2o/3. By contrast, in the local bubble of hot gas surrounding the sun , C~ ~ 1O - 4.5m-2o/3 (Phillips and Clegg 1992) . The background value of C; is contributed by the diffuse, ionized, interstellar medium (DIM). The DIM has mean (local) electron density n ~ 0.1 cm -3 , and temp erature T ~ 10 4 K. With a filling factor of about 0.2 in the galactic plane, which some evidence suggests approaches unity at Izl 2: kpc, it probably accounts for most of the electron density deduced from pulsar dispersion measures (Reynolds 1991). The DIM appears to be ionized by 0 stars (Mathis 1986) . This requires that about 0 .14 of t he ionizing photons from 0 stars manage to escape being absorb ed by neutral gas near t he galactic plane. A detailed model of the ionization structure produced by 0 stars within 2.5 kpc of t he sun shows good agreement with observational determinations of both electron density and emission m easure (Miller and Cox 1993). A useful diagnostic of turbulence in the Galaxy is the relation of a pulsar's SM to its dispersion measure, DM . The disp ersion m easure is defined as the integral of n along the line of sight. It can be used as a surrogate for distance. Cordes et al. (1991) plot the 8M 's and DM 's of over a hundred pulsars. They find that for nearby pulsars (with small DM ), SM increases roughly linearly with DM . This is consistent with a constant level of C;, contributed by t he DIM . However, 8M ex. DM3 when DM2: 100 pc cm- 3 , i. e., for dista nces greater than a few kpc. Scattering measures that increase faster t han distance can arise if scattering occurs in localized structures, such as HII regions, supernova shocks , or stellar wind bubbles. To be specific, let us 7 consider HII regions. The denser an HII region , t he more it contributes to SM. The integrated SM through the Galaxy will increase faster t h an distance if high-density HI! regions are sufficiently rare. Quantitatively, we denote the number of HII regions p er unit volume that have electron densities b etween nand n + dn by N (n) dn. If the distance to the source of radio waves is D , then t he "optical depth" for t he line of sight to intersect HI! regions with electron density b etween nand n + dn is (N(n)dn) . A(n) . D , where A(n) is the cross-section of t hese HII regions. So the expected value of SM is (SM) ~ J dnN (n) . A(n) . D . SMreg(n) , (1.9) where SMreg(n) is t he SM through a single region with density n. To determine the limits of this int egral, we ass ume that most of the contribution to (SM) is from regions with la rge n, i.e., that nN . A . SM reg is a n increasing function of n for the n's of interest; otherwise, we wou ld have (SM) ex D. For (SM) not to diverge, t he optical depth, nN· A· D , must decrease with increasing n. The la rgest n which contributes to (SM) is t h at at which t he optical depth is unity. Therefore, (1.10) where n max is given by (1.11 ) Thus far we h ave kep t our discussion sufficiently general t hat it can b e ap plied to other objects, such as supernova shocks or extreme density flu ctu ations in a t urbulent cascade (i .e., Levy flights , see Boldyrev and Gwinn 2002); n could represent any prop erty t hat varies from obj ect to object , such as the age of supernova remnants . Let us now compu te t he distribution function of HI! regions, assuming that HI! regions a re responsible for the scattering, and t hat (SM) ex D 3. We take the distribution function to b e given by (1.12) Our goal is to calculate b. The radius of an HII region (the Stromgren radius) is R s ~ 70( cm- 3 /n)2/3 p c (e.g., Spitzer 1978) . Its cross-sectional area is A ~ R ; ex n - 4 / 3 . If t he outer scale of t he turb ulent density sp ectrum is ~ R s , t hen SM reg ~ n 2 R !/3 ex n 16 / 9 (eq . [1.4]). From equation (1.11 ), with nNr eg given by equation (1.12), n max ex D 3 /( 4 -3 b). Substit uting t his into equation (1.10), we h ave (SM) ex D 16 /( 12-9b). For t his to be proportional to D 3 , we must have b = (4/3) - (16/27) ~ 0 .74. 1.1.6 Two Constraints on the Turbulence We consider two general constraints on the turbulence. 8 Constraint From EM /SM The first constraint comes from comparing observations of SM with observations of EM , the emission measure. This constraint has b een considered by other authors, e.g., by Cordes et al. (1991). Assuming that the density spectrum is Kolmogorov in the vicinity of the diffraction scale a nd that the scattering material is homogeneous, SM ~ (n>. 2/ ,\2/3) L turb , where L tllrb is the path-length through the turbulence along the line of sight (eq. 1.4). We define the dimensionless amplitude E, (1.13) where n is the number density of ionized particles and Lout is the outer scale, i. e., the lengthscale a t which the turbulence is stirred ; it satisfies Lout < Ltur b . If the Kolmogorov sp ectrum extends to Lalit' then E = nLou,!n . In terms of E, we have 2 2L SM ~ E n turb 2/3 Lalit (1.14) The emission m easure is defined as the integral of n 2 along the line of sight. It h as b een measured along m any lines of sight, e.g., from free-free emission or from Ra. The turbulent gas contributes n2Lturb to EM. If a fraction f < 1 of the ionized gas along the line of sight contributes to SM , then (1.15) We divide equation (1.15) by (1.14), which yields a n expression for the outer scale: 3 3/2 (EM) 3/2 Lalit ~ f. f SM = E3f 3/2 L E M / SM , (1.16) where we define L EM / SM == (EM/SM) 3/2 . Since f < 1 a nd , typically, f. < 1, the outer scale is smaller than L EM / SM ' Lines of sight with large SM tend to have surprisingly small EM; L 8M / SM is often much sm aller than a pa rsec. It is a challenge to find a site in the ISM that gives such sm all outer scales. One of the main goals of the b ody of this cha pter is to calculate E in a compressible MRD cascade. We summarize our result as follow s (neglecting neutral damping a nd ass uming t h at the proton damping scale is smaller than the diffraction scale): = E m ax _1_} {M 5/2 ( Cs teoOI) 1/2 L '(31/2' out (1.17) where M is the Mach number of the turbulence, teool is the cooling time, a nd (3 is the ratio of t hermal to magnetic pressure. In deriving this result , we ass ume that M Cs t eoo l < Lalit, M < 1, and 9 f3 > 1, so E < 1. If the magnetic field is a mplified by the turbulence, the smallest f3 achievable is f3m in ~ M - 2 . Constraint From Energetics The second constraint on L out is based on energetics. In all of the sources that we shall con sider , we exp ect that the kinetic energy that is required to stir the turbulence is less than the thermal energy that is r adiated away by the gas. If this were not true, then the turbulence would heat the gas faster than it could cool. Furthermore, most astrophysical sources a re ineffi cient in converting their energy output into kinetic energy- most of the energy is radiated away. To b e quantitative, kinetic energy must b e put into the turbulence at the rate Bkin ~ (mpvVt,\) (nLturbA), where mp is the proton mass, v,\ is the velocity of the gas on lengthscale A, h is the cascade time on this scale, and A is t he a rea occupied by the turbulence transverse to the line of sight. In both hydro a nd strong MHD turbulence, t,\ ~ A/V,\. Since t he energy cascade rate must be independent of lengthscale in steady state, this cascade time implies that v,\ oc A1 / 3. We denote t he Mach number of t he t urbulen ce as follows _ v,\/ A1 / 3 M = 1 /3 ' cs/ L o ut (1.18) where C s is the sound speed. In terms of M , we have (1.19) We wish to compare this with Beoo), the rate at which energy is radiated away by t he ionized gas along the line of sight (both turbulent and quiescent gas) : (1.20) where t eoo l is the cooling time. Bkin So Bkin/Beool ~ M 3j(csteoo l / L o u t), and our requirement that < Beool can be rewritten as follows: L out> M 3 j Cs teool . (1.21 ) Interstella r gas in photoionization equilibrium typically has a temperature of ~ 10 4 1{ a nd Csteoo l ~ 0.2(cm - 3 / n) pc. This Cs t eoo l probably charact erizes much of the interstellar gas that is responsible for DISS . If the turbulence is trans-sonic (M ~ 1) a nd j ~ I , then Lou t should be la rger than Csteoo l. 10 COIllbining the Two Constraints Inequality (1.21) can b e combined with our other constraint (eq . [1.16]). For given EM / SM, tcool, M , a nd E, this leads to a lower limit on f and hence a lower limit on Lout ; e.g., if E ~ I/ VTJ < 1 and M ~ 1, > {33 (C s t coo l )3 L (L EM / sM )2 out (l.22) Alternatively, we can address the following question : given EM , what is the la rgest possible SM? From equations (l.14) a nd (l.15), SM ~ E2 fEM/L~~~. So we should m aximize E and f (i .e. , E ~ 1 and f ~ 1) and minimize Lout . The smallest Lout t hat can b e achieved without overheating the m edium is given by equ ation (1.21); so the m aximum SM is 1 EM SMmax ~ M 2 (C t cool )2/3 (1.23) s For sufficiently small M , SMmax can be m ade as large as desired. However , we sh all see t h at m a ny astrophysical sites a re exp ected to have M 1.1.7 ~ 1. SM From Turbulence Behind Shocks R adio waves that pass through a shock front in the ISM a re scattered both by the density discontinuity across the shock and by density flu ctuations in the turbulence behind the sho ck. In this subsection, we show that scattering in the turbulence a lways dominates. Let us consider the scattering across a single adia batic shock, where the density jump across t he shock is ~ n, with n t he post-shock density. We assume that the post-shock gas is fu lly ionized. To calculate the amount of scattering, we first calculate the difference between t he accumulated phases along two lines of sight separa ted by A (see §1.1.3). Associated with turbulence b ehind the shock, from equation (1.2) , 1/ 6 [1' 5:] ,./,tur b ~ n A A(L out / A)1 /2 [1' e 5:] ~ nA 5 / 6 L out e , 'f' A (1.24) where we have assumed that the extent of t he turbulence is compa ra ble to t he outer scale (L turb ~ Lout) a nd that n A ~ n(A/L out )1/3 . Associated with the discontinuity across the front , (1.25) since the column densities along the two lines of sight differ by ~ nA. Therefore (1.26) a nd as long as Lout is la rger t ha n the diffractive scale, ¢t sc is negligible. We expect Lout» Ad iff ~ 11 10 8 - 1010 cm , SO the signature of the shock front is always smothered by t he turb ulence behind it. If we erroneously neglect t he t urbulence, we find from using equation (1.25) tha t 61scatt ex ).2 , which is consistent with observations along certain lines of sight. (Recall that the Kolmogorov result is 61scatt ex )'11 /5, eq. [1.7].) Therefore it has b een frequently claimed in the literature that th e ).2 scaling can b e explained by shocks (e.g., Lambert a nd Rickett 2000). However , shocks give )'11 /5 because of the turbulence behind them. A more pla usible expla nation for the ).2 scaling is a t urbulent cascade with a n inner scale t hat is larger than the diffractive scale (e.g., if t he cascade is cu t off at a relatively large scale by damping caused by neutral atom s, see §1.6. 5). 1.2 Incompressible MHD Turbulence Our compressible theory extends the theory of incompressible MHD turbulence given by Goldreich and Sridhar (1995) by including a slightly compressible slow mode and a p assive ent ropy mode. We also consider kinetic effects: on sufficiently short lengthscales, t he mean free paths of the p a rticles a re significant , and the equations of compressible MHD must b e modified . This is especially important for da mping. Before considering compressible t urbulence, we discuss incompressible MHD t urbulen ce, focu sing on issues that a re important for the compressible case. Goldreich and Sridhar (1995, 1997) propose a picture of the dynamics of incompressible strong MHD turbulence and describe t he power spectra of Alfven waves, slow waves, and passive scala rs. We extend their picture to cover additional features such as the parallel cascades of both slow waves a nd passive scala rs. Unless explicitly stated otherwise, t hroughout this ch a pter "parallel" a nd "transverse" refer to the orientation relative to t he local mean magnetic field , which is t he m agnetic field averaged over the scale of interest. Our discussion of incompressible MHD turb ulen ce, while somewhat lengthy, is imp ortant for understanding the extension to compressible turbulence t hat follows. Consider a uniform unperturb ed plasma with an embedded magnetic fi eld. Turbulence is excited at t he MHD outer scale, L MHD , by random and statistically isotropic forcing, with r .m. s. velocity fluctu ations a nd r.m. s. magnetic field fluctuations (in velocity units) which a re compara ble to the Alfven speed , VA. 5 As t he turbulence cascades from t he MHD outer scale to sm aller scales, power concentrates in modes with increasingly transverse wave vectors. The inertial ra nge velocity sp ectrum a pplies to lengthscales below LMH D but a bove the dissipation scale. It is anisotropic a nd is characterized by A.L ) 1/3 LMHD ' VA ( - - (1.27) 5The forcing flu ctuations may a lso be less VA, in whi ch case LMHD wo uld be d efi ned as the lengt hscale at whi ch t he flu ct uations ext rapo la te to VA . 12 \ 2/3 1/3 /\ 1- L M HD (1.28) the inertial range m agnetic field sp ectrum is identical. Here A1- is t he lengthscale tra nsverse to the local mean magnetic field, vA.L is the r.m.s. velocity flu ctuation across A1-, and A ll is the lengthscale parallel to the local mean magnetic field across which the velocity fluctuation is vA.L' We interpret A ll as t he elongation along the magnetic field of an eddy which h as a size A1- tra nsverse to the magnetic field; it is not a n independent varia ble, but is a function of A1- . Deep within the inertial range, where >- 1- « L M H D, eddies are highly elongated along the m agnetic field: A ll » >' 1- . In t he following subsections, we explain the physics underl ying the sp ectrum , and consider some of t he im plications. 1.2.1 Alfven Wave Spectrum Arbitrary disturban ces can be decomposed into Alfv E'!11 waves a nd slow waves. The Appendix summarizes t he properties of t hese waves in t he more general case of compressible MHD . In incompressible MHD , Alfven waves a nd slow waves are usually referred to as shear-Alfven waves and pseudo-Alfven waves, but the former designation is more convenient for making the connection wit h compressible M HD. Our understa nding of the MHD turbulence is based on two facts: (i) MHD wave- p ackets propagate at t he Alfven sp eed either parallel or antiparallel to t he local m ean magnetic field ; and (ii) nonlinear interactions a re restricted to collisions between oppositely directed wave-packets. These facts imply that in encounters between oppositely directed wave-packets, each wave-packet is distorted as it follows fi eld lines p erturbed by its collision partner. A wave-packet cascades when the fieldlines that it is propagating along have spread by a distance comparable to its transverse size. Alfven waves h ave quasi two-dimensional velocity a nd magnetic field fluctuations which are confined to planes p erp endicular to the local mean magnetic field . As their more complete na me shear-Alfven implies, they dominate the shear of t he mapping of pla nes t ra nsverse to the local mean magnetic field produced by field line wander. Thus, Alfven waves control t he dyn a mics of MHD cascades; slow waves may be ignored when considering t he dyna mics of Alfven waves. In strong M HD t urbulence the cascade time of a n Alfven wave-packet is comparable to its travel time across t he pa rallel length of a single oppositely directed Alfven wave-packet of similar size. Goldreich and Sridha r (1995) refer to t his balance of times cales as critical b ala nce. It relates the parallel size of a wave-packet, All , to its transverse size, A1- . Wave-packets of t ra nsverse size A1cascade when the fieldlines they follow wander relative to each other by a transverse distan ce A1-. Critical bala nce implies t hat t his occurs over a pa rallel distance A ll ' The Alfven wave spectrum is given by equations (1.27) and (1.28) , with vA.L referring to t h e velocity flu ctuations of the Alfven waves. It is deduced from two scaling arguments: (i) Kolmogorov 's 13 argument that the cascade time tA L ~ ).. l../v AL leads to a n energy cascade rate, Vt/tA L ~ v~ L / ).. l.. ' which is independent of lengthscale; and (ii) the critical bala nce assertion that the linear wave p eriod which characterizes the Alfven waves in a wave-packet is comparable to the nonlinear cascade time of tha t wave-packet, i. e., vAl All ~ l /t AL · Before considering slow waves in MHD turbulence, we discuss two topics that are governed by the dynamics of Alfven waves only: eddies and passive scalars. 1.2.2 Eddies Because of their transverse polarization, Alfven waves are responsible for the wandering of m agnetic fieldlines. A snapshot of wandering fieldlines is shown in Figure 1.1. Each of these field lines passes through a localized region of size )..l.. in one plane transverse to the mean magnetic field . Away from this plane the bundle of fieldlines diverges due to the differential wandering of the individual lines. At a second plane, the bundle's cross sectional a rea has approximately doubled. Critical bala nce implies that the distance to this second plane is comparable to the parallel wavelength which cha racterizes t he bundle, A ll ' As the bundle spreads, other fieldlines, not depicted , enter from its sides. In general, the neighboring fieldlines of a ny individual field line within a region of transverse size ).. l.. change substantially over a parallel dista nce of order All' It is natural to think of A ll as the pa rallel size of a n eddy that has transverse size ).. l... Two eddies with the same transverse lengthscale that a re sep a rated by a parallel distan ce greater than their A ll incorporate different fieldlines, and hence are statistically independent. Eddies a re distinct from wave-packets. The former a re rooted in the fluid whereas the latter propagate up a nd down magnetic fieldlines at the Alfven sp eed . Aside from their anisotropy, eddies in MHD turbulence a re similar to those in hydrod ynamic turbulence. They are spatially localized structures with cha racteristic velocity flu ctu a tions and lifetimes. The r.m.s. velocity difference between two points is determined by the sm allest eddy that contains both. Different eddies of a given size are statistically independent. The three-dimensional sp ectrum for r.m.s. velocity fluctuations across transverse lengthscales ).. l.. a nd parallel lengthscales ).. 11 is for ).. 11 « A ll (1.29) for ).. 11 » A ll There is negligible addit ional power within an eddy on parallel lengths cales smaller than A ll, so for ).. 11 « A ll ()..l..) , VAL,AII = VAL ' For).. 11 » A II().. l..), the smallest eddy that contains both ).. l.. and ).. 11 h as a transverse length scale ).. ~ which satisfies A ll ()..~) = ).. 11 ' The velo city flu ctuation of t his edd y is obtained by solving this equation for )..~ (eq. [1.28]) , a nd inserting this ).. ~ in equation (1.27). Contours of the three-dimensional spectrum are plotted in Figure 1.2. Each contour represents 14 Figure 1.1: Wandering of Magnetic Fieldlines A fieldline bundle of transverse size ).1- diverges after a parallel distance A ll, where A ll is t he parallel size of an eddy (eq. [1.28]) as determined by critical balance. 15 eddies of a cha racteristic size. Maron and Goldreich (2001) give the three-dimensional sp ectrum in Fourier-space. Since eddies that a re separated by more than A ll are statistically independent, the power spectrum at a fixed transverse wavenumber k 1. is indep endent of the parallel wavenumb er kll in the corresponding region of Fourier-space , i.e ., where kill 2: A ll W1: l). The turbulent cascade is generally viewed as proceeding from larger eddies to smaller eddies as t his is the direction of energy transfer. However , smaller eddies cascade many times in the time t hat a la rge eddy cascades. This is pa rticularly important in t urbulent mixing. Consider the evolution of two fluid elem ents whose init ial sepa ration is larger tha n the dissipation scale. On cascade timescale tAl.' their transverse separation will random walk a distance A1. as the result of the cascade of sm aller eddies. Therefore, on a timescale comparable to an eddy 's cascade time, the transverse lo cations of its component fluid elements- whose sizes m ay be considered to b e compa rable to the dissipation scale- are completely ra ndomized . Moreover , since mixing a t the dissipation scale causes neighbouring fluid elements to be rapidly homogenized , transverse smoothing of the edd y occurs on t he timescale t hat it cascades. Ra pid transverse mixing in MHD turbulence is simila r to the more fa milia r isotropic mixing in hydrodynamic turbulence. 1.2.3 Passive Scalar Spectrum A passive scalar, a , satisfies the continuity equation , (a/at + v . V )a = 0 , and does not affect the fluid 's evolution. It could represent , for example, the concentra tion of a contamin a nt. Vve consider the sp ectrum of a passive scalar mixed by the Alfven wave cascade. These considerations a re important for our subsequent investigation of compressible turbulence. They are also helpful for understanding the slow wave spectrum. We discuss the passive scalar spectrum both in the inertial range and also below the scale at which the Alfven wave sp ectrum is cut off. Passive Scalar Spectrull1 in the Inertial Range As we show in this subsection , the transverse spectrum of the passive scala r in the inertial range is (1.30) where A1. is the lengthscale transverse to the local mean m agnetic field a nd a Al. is the r.m.s. flu ctua tion in the passive scala r across A1.. The passive scalar p arallel spectrum is the same as the Alfven wave p arallel spectrum given in equa tion (1.28) , where All is now to b e interpreted as the lengthscale parallel to t he local m ean magnetic field across which the passive scalar flu ctuation is aA l. · Mix ing of the passive scala r is due to Alfven waves. Slow wave mixing is negligible . This is because transverse velocity gradients a re much la rger than pa rallel ones in MHD turbulence. Thus 16 I I I I I I I I I- 1 - / / / / / / / / / / 0.8 I-- - / / / r // Au/LYHD - / / - / / / CI i - / 0.6 I- / ....J / "'-== / / .< / / / 1 / 0.4 I- / - / r / 3/4 / / t- / I 0.2 I-- - I I I 1/2 / / o ~1/4 o I I 0.2 I I 1 0.4 1 1 0.8 1 Figure 1.2: Three-Dimensional Spectrum Contours of constant VAl.,AII (eq. [1.29]), labelled by the value of VAl.,AII/VA. Along the A.l-axis, vAl.,a/VA = (A.l/LMHD)l/3; along the All-axis, va,AII/VA = (AIi/LMHD)1/2. 17 Alfven waves, whose velocity fluctuations are p erpendicular to the magnetic field , are much more effective at mixing than slow waves, whose velocity flu ctuations are nearl y parallel to the magnetic field. The transverse cascade arises from the shuffling of fieldlines as Alfven waves propagate through the fluid. The transverse spectrum (eq. [1.30]) follows from t he Kolmogorov-like hypothesis that the cascade rate of the "energy" in the scalar field is independent of lengthscale; i.e., (Tt it),l. is constant , where t),l. is the p assive scalar cascade time, which is assumed to be proportional to t he cascade time of Alfven waves. Comparing this with the constancy of the kinetic energy cascade rate, vt it),l. , we conclude that (T),l. ex v)'l.' which implies equ ation (1.30). A similar a rgument holds for the cascade of a passive scalar in hydrodynamic turbulence (e.g., Tennekes and Lumley 1972). The parallel cascade of a passive scalar is more subtle. It might appear that a passive scalar cannot cascade along fieldlines since, neglecting dissipation, both t he scalar and the magnetic field are frozen to the fluid , and thus the scalar must be fro zen to fieldlines. In that case there certainly could not be a parallel cascade. If the scalar were inj ected on large scales, t hen flu ctuations with smaller wavelengths along the magnetic field would not be generated. However, dissipation cannot be neglected. It is an essential part of MHD turbulence , as it is of hydrodynamic turbulence. For example, the description of turbulent mixing in §1.2.2 depends crucially upon small scale dissipation. Perhaps the b est way to understand the pa rallel cascade is to consider mixing on t he transverse lengthscale A1. within two planes which are perpendicular to the local mean magnetic field , and which are separated by a parallel distance larger than A ll' Velocity fluctuations within the two planes a re statistically independent. This is evident because a bundle of fieldlines cannot be localized wit hin a transverse distance A1. over a parallel separation greater t han A ll ' Even a pair of fluid elements, one in each plane, t hat are initially on the same fieldline are mixed into two regions with different valu es of passive scalar concentration. It follows t hat t he parallel cascade of the passive scalar also o b eys equ ation (1. 28). Passive Scalar Spectrum Below Alfven Wave Cutoff A passive scala r cascade may extend below t he transverse scale at which t he MHD cascade is cut off. Mixing on t hese scales is driven by fluid motions at Aclltoff which results in a scale-independent mixing time equal to the cascade time at Aclltoff. This yields (T), l. = constant , A1. < Ac utoff . (1.3 1) A similar argument applies in hydrodynamic t urbulence. Tennekes a nd Lumley (1972) call t his regime in hydrodynamic turbulence the viscous-convective subrange . 18 1.2.4 Slow Wave Spectrum The slow wave spectrum is the sam e as that of the Alfven waves. It is given by equ ations (1.27) a nd (1.28), with vA.l referring to the velocity flu ctuations of the slow waves. This is a consequence of the similar kinematics of slow waves and shear Alfven waves and the fact that both a re cascaded by shear Alfven waves. Slow waves obey the same linear wave equation as Alfven waves, a nd to lowest nonlinear order they travel up and down the local mean magnetic field lines at the Alfven speed just as Alfven waves do. However , the dynamics of the MHD cascade is controlled by the Alfven waves (Goldreich a nd Sridhar 1997, Maron and Goldreich 2001) because their velocity a nd m agnetic field flu ctu ations are perpendicular to the local mean m agnetic field , whereas those of the slow waves are nearly p a rallel to it . Since perpendicular gradients a re much la rger th an parallel ones in the MHD cascade, Alfven waves are much more effective at mixing than a re slow waves. Hen ce, Alfven waves cascade both t hemselves and slow waves, whereas slow waves cascade neither. 6 The transverse mixing of t he slow waves by Alfven waves is an alogous to t he mixing of a passive scalar. As discussed in §1.2.3, a passive scalar assumes t he same inertial ra nge spectrum as t hat of t he velocity field which is responsible for its mixing . Thus, equation (1.27) is also a pplicable to the velocity fluctuations of the slow waves. Similarly, the parallel cascade of slow waves is an alogous to the parallel cascade of a passive scalar. Since Alfven waves cascade in the time they move a distance A ll, slow waves sep a rated by t his distance are independently mixed. Thus, equation (1.28) also applies to slow waves . There is, however , a conceptual difference b etween the p a rallel cascades of t he passive scala r a nd of the slow mode. In the absence of dissipation a passive scalar is frozen to fieldlines, whereas slow mode wave-packets travel along them at the Alfven speed . A passive scalar has a parallel cascade b ecause Alfvenic fluctuation s a re statistically indep endent within two transverse pla nes frozen in the fluid and separat ed by All' The pa rallel cascade of slow waves occurs b ecause two transverse planes sepa rated by All a nd travelling at the Alfven speed in the sam e direction experien ce uncorrelated sequences of distortions suffered as a result of interactions with oppositely directed Alfven waves . Nevertheless, these two requirements a re both satisfied in t he MHD cascade when t he transverse p la nes are separated by a distance great er than A ll' a nd so the passive scala r a nd t he slow mode have the same parallel sp ectrum. Whereas passive scala r mixing is due to eddies, slow wave mixing m ay be t hought of as due to travelling eddies, i.e., eddies which travel up and down the m agnetic field at the Alfven sp eed . 6 We a re ass umin g t hat Alfve n and s low waves have co mparable stren gt hs at a give n le ngt hscale . 19 1.2 .5 N umerical S im ulatio n s Numerical simulations offer some support for the above description of incompressible MHD turbulen ce. Those by Cho a nd Vishniac (2000b) support both equations (1.27) and (1.28) , a nd those by Muller and Biskamp (2000) support equa tion (1.27). However , although the simulations of Maron a nd Goldreich (2001) support equa tion (1.28), they yield vAl. ex A~t instead of equ ation (1.27) . Because the simulations of Maron a nd Goldreich (2001) a re stirred high ly a nisotropicaliy, whereas t hose of Cho and Vishniac (2000b) and Mulier a nd Biskamp (2000) a re stirred isotropicaliy, it is not clear whether t h ese disparate results conflict . Maron a nd Goldreich (2001 ) speculate that the discrep a ncy between their spectrum and the scaling prediction of Goldreich a nd Sridha r (1995) results from intermittency. In any case, we expect tha t the physical picture of a critically b alan ced cascade, which underlies Goldreich and Sridhar's description of MHD turbulence, rem ains valid. Even if the spectrum is proportional to A~4, we expect that the results of this investigation- which assumes a spectrum proportional to A~3 - would not be significantly altered. The simulations of Maron and Goldreich (2001 ) confirm that a passive scala r h as t he same transverse sp ectrum as that of Alfven waves, although both a re proport ional to A~4 . They a lso indicate that the parallel cascade of a passive scala r conforms to equation (1.28). Maron and Goldreich (2001) present results from simulations of the interaction between oppositely directed slow a nd Alfven waves; the slow waves cascade whereas the Alfv en waves do not. They also compute the spectrum of slow waves in a simulation of MHD turbulence a nd find t hat both its transverse a nd longitudinal b ehavior m atches t hat of the Alfven wave spectrum. Kolrnogor ov Constants Scaling arguments do not yield values for the Kolmogorov constants, the order-unity multiplicative constants of the sp ectrum. However , they can be obtained from simulations. We define them su ch that equ ation (1.27) rem ains valid; i. e., we take L M HO to be the separation at which the r.m.s. velocity difference is equal to , or extra polates to, VA. Two Kolmogorov constants, Mil and M t , are needed in t his chapter: VA - VAl. A.L => A _ M-1 A2/3 L 1/3 -A II II .L MHO II vA tA l. (1.32) (1.33) A ll In these definitions , A.L a nd All are the inverses of t he wavenumbers transverse and parallel to the local mean magnetic field, and hl. is the cascade time of waves with transverse wavenumber A1: 1 . 20 From t he num erical simulations of Maron and Goldreich (2001), M il ~ 3.4 and Mi ~ 1.4 . (1.34) i'3, Because these simulations yield a transverse spectrum which is proportional to A~t instead of A the resulting Kolmogorov constants a re not truly consta nt . 1.3 Compressible Turbulence: Overview Our primary concern is interstella r scintillation, which is affected by electron density flu ctuations on very small scales, t ypically 108 - 10 10 cm for diffractive scintillation. In the rem ainder of this ch a pter , we calculate the sp ectrum of density fluctuations which results from compressible turbulence in magnetized plasmas with parameters appropriate to the interstellar medium . Throughout, we consider plasmas that have more ions t han neu trals, a nd t hat h ave 1 ;S {3 < 00, where {3 is the ratio of t he thermal pressure to t he mean m agnetic pressure: {3 == 87rpl B 2 = 24Iv~. Here, CT == (pi p)1 / 2 is t he isothermal sound speed, VA is the Alfven speed , p is the thermal pressure, B is the m agnetic field strength, and p is the m ass density.7 The incompressible limit corresponds to {3 = 00. On the lengthscales that we consider , compressible MHD is a good approximat ion for the dyn amics of the ionized interstella r medium . Kinetic effects, where importa nt, m ay b e accounted for by simple modifications to t he MHD equations. Therefore, we turn our attention to turbulence in compressible MHD. The turbulent velocity sp ectrum in compressible MHD is a pproximately the same as the turbulent velocity spectrum in incompressible MHD , because the Alfven mode rem ains incompressible in a compressible medium , and t he slow mode is only slightly compressible. Thus, the velocity sp ectrum for both of these modes is given by equations (1.27) and (1.28). The Appendix summa rizes the properties of the relevant modes in compressible MHD. There are two additional modes in compressible MHD which are not present in in compressible MHD. One of these is the fast mode. However, as long as {3 .2: a few, the fast m ode is essentially a sound wave. Its phase sp eed is larger than the phase speed of either the Alfven m ode or the slow mode, a nd so the fast mode does not couple to them. Thus, we ignore the fast mode. An an alogous approxima tion is often made in subsonic hydrodyna mic t urbulence, where sound waves h ave little influen ce on the turbulent cascade. Some supp ort for our neglect of the fast m ode is provided by numerical simulations. Balsara a nd Pouquet (1999) show that in simulations of turbulen ce with Mach number of order unity, the compressible component of the velocity is significantly sm aller than the solenoidal component; they argue that this is because shocks dissipate much of t he irrotationa l 7We briefly di scuss plasmas wit h f3 < 1 in §l.ll. 21 component of the velocity. The second mode t hat is present only in compressible MHD is the entropy mode. When turbulent motions a re adiabatic, the entropy mode is a zero-frequency mode with unperturbed pressure, and with density perturbation offset by temperature perturbation (see Appendix). In hydrodynamics, simple scaling arguments show that the entropy mode does not affect t he dynamics of t h e fluid- i. e., the fluid obeys the incompressible equations of motion- provided that fluid motions are subsonic and that density fluctuations are sm aller than the mean density; these arguments carryover directly to MHD (e.g., Higdon 1986, a nd references therein). Furthermore, since the entropy of any fluid element is conserved in the inertial range of adiabatic turbulence, the mixing of the entropy mode is identical to the mixing of a passive scalar (see eq. [1.30]) , yielding the t ransverse spectrum (1.35) where s)..1. is the r.m .s. entropy flu ctu ation across A.L. The parallel sp ectrum is given by equation (1.28). Analogously, in hydrodyn amic turbulence the effect of the entropy mode on turbulent motions can often be neglected, and the entropy mode is mixed as a passive scalar (Monin a nd Yaglom 1971). Based on the above discussion , there a re two sources of density flu ctuations on sm all scales: the slow mode and the entropy mode. Since the slow mode den sity perturbation is proportional to its velocity perturbation, and since the entropy mode density perturbation is proportional to its entropy perturbation , both yield a Kolmogorov sp ectrum of density perturbations: (1. 36) In t he remainder of t his ch apter, we investigate t he density spectrum in more detail. We calculate the density spectrum on diffractive lengths cales for given values of the number density, n, a nd outer scale, Lout. Typically, in regions of the interstella r medium which are relevant for scintillation , 1 cm -3 < n < 100 cm -3. Values of Lout are more uncertain , though it is likely that Lout is typically within a few orders of magnitude of 1 p c. It is also plausible that the value of Lout is related to t hat of n . In a future pap er , we discuss in much more detail the values of these p ar am eters in turbulent regions of the ionized interstellar medium, e.g., in supernova shocks, HII regions, a nd stellar winds. A naive guess for the resulting density flu ctuation at the lengthscale A.L is n)..1. = n(A.L/L out )1/ 3, which we refer to as the fiducial spectrum . However , there a re a number of physical effects which suppress the small-scale density spectrum in the interstellar medium relative to the fiducial spectrum. When considering scintillation observations, it is the transverse- not p a rallel- Iengthscale which is relevant. E ach line of sight crosses regions with different orientations of th e local m ean magnetic field . The observational effect s of the parallellengthscale are washed out if the orientation varies by a n a ngle greater than a tiny ratio: A.L/ A ll at the diffractive scale. Since the variation in a ngle due 22 Symbol Lout LMHD LcooJ LH LHe L mfp Led Leq Lpd L (1-) pd L p,gyr Table l.1 : Summa ry of Lengthscales Description lengthscale tra nsverse to the mean magnetic field ; it is comparable to the observed lengthscale. lengthscale pa rallel to the mean magnetic field parallel size of an eddy; it is a function of A1- (eq. [l.28]) . A1- = Lout a t the outer scale A1- = LMHD at the MHD scale A1- = LcooJ a t the cooling scale A1- = LH a t the collisionless scale of hydrogen atoms A1- = L He at t he collisionless scale of helium atoms mean free pa th of protons and of electrons All = Led at the electron diffusion scale A ll = Leq a t t he electron-proton equilibra tion scale All = Lpd at the proton diffusion scale A1- = L~~) a t t he proton diffusion scale A1- = Lp ,gyr at t he proton gyro scale i'3, it increases wit h scale, so t hese to eddies on scales larger t h an t he diffrative scale is A1-/ All ex A eddies render the p arallellengthscale unobservable. As a result , we frequently refer to t he t ra nsverse lengthscale as, simply, t he length scale. Because of the multitude of speciallengthscales involved , we organize the discussion by decreasing lengthscale. We b egin at t he outer scale, and proceed down to the sm allest scales relevant to interstella r scintillation , while considering the effects of t he Alfve n mode, the slow m ode, a nd the ent ropy mode simultaneously. Since most of t he relevant lengthscales in t he interstella r medium have a similar dep endence on t he background density- t hey decrease with increasing density- t he ordering of lengthscales is fairly universal. The lengthscales which we consider a re t he outer scale, the MHD scale, the cooling scale, the collision less scale of the neutrals, the collisionless scale of the ions, a nd the proton gyro scale. We conclude with a summa ry of the most importa nt effects . Ta ble l.3 lists the lengthscales t ha t a re used in this ch apter . 1.4 The Outer Scale and the MHD Scale As a model for the excita tion of t he t urbulence, we consider pl asma which is stirred on a n outer scale Lout with velocity fluctu a tions tha t are of order the sound speed: VL o u t ~ C s . The generalization from t his case of Mach 1 turbulence to subsonic turbulence with a rbitra ry Mach number M < 1 is trivia l: Lout would be interpreted as a n effective outer scale at which velocity flu ctua tions extrap ola te to Cs · However , we fo cus on t he Mach 1 case b ecause it is proba bly t he most releva nt for interstella r scintilla tion . 23 If, initially, the strength of t he magnetic field is negligible, then random fieldline stretching amplifies the mean field (eho and Vishniac 2000a). It is uncertain both how quickly the magnetic field is amplified, and whether its energy density is amplified until it a pproaches equipartition with the turbulent kinetic energy density. If it does reach equipartition within a few outer-scale crossing times, then fJ ~ 1 would be appropriate for Mach 1 turbulence. Recall that fJ is the ratio of thermal pressure to magnetic pressure, or equivalently, fJ = 2c}/v~. We generally assume t h at the mean magnetic field can approach equipartition with the gas pressure, so we think of fJ as close to but a li ttle larger than uni ty, although we leave its exact value unsp ecified. Provided fJ is larger than unity and M ~ 1 on scale Lout, the turbulent kinetic energy dom- inates the mean m agnetic energy on scales just below Lout. Thus the cascade is hydrodynamic on these lengthscales, a nd the velocity fluctu ations are given by Kolmogorov 's isotropic scaling: VA.L ~ c s (')'.L/Lout )1/3. The kinetic energy density decreases towards smaller scales. At a critical scale, which we denote LMHO , the kinetic energy density is sufficiently small that it is comparable to the mean magnetic energy density: Vl MHD ~ v~, which implies that (1.37) Below this scale, the turbulent kinetic energy density is smaller than the magnetic energy density, and the theory of compressible MHD turbulence is applicable. Note that the effects of large-scale velocity fields can be neglected when considering fluctuations on smaller scales, because a la rge-scale velocity field can be eliminated by a Galilean transformation. However , since large-scale magnetic fields cannot be transformed away, they affect the dynamics of small-scale eddies. Hydrodynamic turbulent motions on scales slightly larger than L MH O have speeds ~ VA. Thus they couple to, and efficiently excite, Alfven waves and slow waves whose phase velocities w / k ~ vA(kz/k) are of comparable magnitude. Provided fJ > 1 a nd M ::; 1 , fast waves, which have w/k ~ Cs > VA, a re weakly excited at L MHO . In what follows, we neglect fast waves. Alfven waves excited at LMHO cascade to smaller scales, a nd these cascading Alfven waves in turn cascade slow waves. The spectra of both of these cascades are given by equations (1.27) and (1.28), so that VA.L ~ C s A.L ( -L )1/3 (1.38) out for both modes, even on scales smaller than L MHO. We have not yet discussed density flu ctuations on the scales which have been considered in t his section. It is more convenient to do so in the following section. 24 1.5 The Cooling Scale On the len gthscales in the interstellar medium t hat we h ave considered thus far, t he radiative cooling time is shorter than the eddy turnover time. 8 Consequently, in photoionized regions, turbulence on t hese lengthscales is expected to b e isothermal (Sridhar 1998; Goldreich 1998; Higdon a nd Conley 1998). This h as only a m a rginal effect on the turbulent dyna mics described in t he previous section , b ecause isothermal Alfven waves a re identical to adia batic Alfven waves, a nd isothermal slow wa ves a re only slightly different from adia batic slow waves. However , t he entropy mode is rapidly damped in isothermal turbulence. As a consequence, sm all-scale density flu ctuat ions may b e significant ly suppressed. There a re two possible solutions to t his "cooling catastrophe": eit her (i) t he ou ter scale is extrem ely sm all , sm all enough t ha t the turbulence at the ou ter scale is nearly adiab atic; or (ii) t here a re significant density flu ctuations associated with t he slow mode. However , in the latter case, t he mean m agnetic field must be a mplified almost to equipa rtition wit h t he gas pressure, so t hat (3 ~ 1. Eit her of t hese two solut ions would place stringent constraints on the n ature of t he t urbulence which is responsible for observed density flu ctua tions. In the following, we consider t he cooling scale in more detail. There is a critical scale for the t urbulence, which we call t he cooling scale, L eoo l ' Above this scale the t urbulence is isothermal, a nd b elow it the turbulen ce is adia ba tic. We ass ume that L eool < LMHD throughout this chapter , except in §1.5.2 where we consider the case in which this inequality is reversed. The cooling scale is where t he edd y t urnover time, Al. /VAJ. ' is equ al to t he cooling time, t eoo l ; i.e., with VAl. ~ Cs ( Al./ L ou t )1/3, L eoo l L out ~ ( Cs teoo l ) 3/2 . (1.39) L ou t As we will discuss in detail in a future p aper, most pla usible astrophysical sources of interstellar scintillation have Cs t eoo l ;S L ou t, and thus L eoo l ;S L ou t . In general, t he kinetic power p er unit mass t h at is dissipated in Mach 1 t urbulence is ~ C;/ tL out ~ c~/ Lout, where tLout is t he edd y turnover t ime at t he outer scale. The t hermal power per unit mass required to keep t he gas hot is ~ c;/teoo l. Hen ce, only if the energy that goes into turbulent motions is as large as the total energy t h at goes into heating can the ou ter scale b e as small as the cooling scale. Nonetheless, in plasmas t h at are t hermally unstable, a nd in isotherm al shocks (assuming t hat t he outer scale is compa rable to t he scale across which the density doubles), t he two powers are comparable, a nd the cooling scale is comparable to the outer scale. As a n example, we consider a n HII region. Photoionized plasma is thermally stable . Heating is primarily due to photoionizing photons, and cooling is prima rily due to electron impact excitation BFo r d efi ni te ness, we co ns ider pl asma t hat is t herm a ll y stabl e; dev iat io ns of th e temperat ure from its eq uilibriu m valu e d ecay in a coo lin g t im e, tcooJ. 25 of metal line t ransitions (e.g., Spitzer 1978).9 A cha racteristic temperature for plasma photoionized by hot stars is T ~ 8,000K , which implies that the sp eed of sound is Cs ~ 10 km /sec . The cooling t ime is teool ~ 20, OOO ( cm - 3 In) years, where n is the numb er density of electrons. Therefore, cm Csteool ~ 0.2 ( -n- 3 ) pc . (1.40) A plausible value for Lout might b e the radius a n HII region (the St rom gren radius), which is ~ 70 ( cm - 3 / n)2/3 pc (Spitzer 1978). Therefore, wit h typi cal values of n in HII regions (1 cm - 3 < n < 100 cm - 3 ), Lout is significant ly la rger t h an Cs teool, and so t he cooling scale is significantly sm aller t ha n the ou ter scale. The fact tha t the turbulen ce is isothermal on large scales has imp ortant implications. Ha d cooling been ignored , i.e., had it been implicit ly assumed t h at t he cooling time is infinite , t hen one would have calculated t he density spectrum as follows: t here should be density fluctuations of order unity on t he outer scale, implying excita tion of entropy modes on t he outer scale which are passively m ixed by t h e Alfvenic t urbulence to sm all scales . This would yield t he fidu cial density spectrum , nAl. ::::: n( A.L/L out )1 /3 . However , since t he t urb ulen ce is isothermal on large scales, t he a bove calculation is incorrect. The entropy mode is not p assively mixed on these scales, since it is damped by cooling b efore it can b e m ixed. In t he following, we consider separately two cases: fir st, t he 10w-,8 case (,8 < L oll t / cs teool) , a nd t hen t he high-,8 case (,8 > LOllt/Cs teool)' In each case, we show t h at t he sm all-scale density flu ctu ations a re subst antially sm aller t ha n those predicted by t he fi ducial sp ectrum unless a relatively stringent condition holds: either ,8 ~ 1 or LOllt ~ Cs teool. 1.5.1 D ensity Fluctuations Below the Cooling Scale: 1 < (3 < L out/Cstcool With regards t o interstellar scintillation , the crucial difference between high-,8 a nd 10w-,8 t urbulen ce lies in t he compressibility of t he slow mode. Since t he sum of t hermal pressure a nd magnetic pressure vanishes for t he slow mode, t he mode's density pert urbation satisfi es nAl. /n ~ PA l. /p ~ ,8-1BAl./ B ~ ,8- 1VAl./VA ~ ,8-l/2VAJ./Cs (see Appendix) . So, assuming t h at t he energy in slow waves is compara ble to t h at in AlfvEm waves, (1.41) 9I n t hi s case, t here is a not her lengthscale, t he photo ioni zat ion scale, whi ch we do not co ns id er because its effects a re unimportant for scin ti ll at ions. T hi s is t he scale at whi ch the edd y t urn over t ime is co m parab le to t he recomb inat ion time. It is sli ghtly larger t han t he coo ling scale because the recom bin at ion t ime is approx im a tely five t im es la rger t h an t he cooling time (S pi tzer 1978). On scales larger t han t he photoioni zat ion scale, t he tu r b ul ence is in ph oto ionization eq u ili briu m, whereas o n scales s m a ll er t ha n t hi s, t he ion izat ion fr act ion of a flui d elem ent is co nse r ved . No net heless, t he temp erature of t he gas is on ly weak ly dependent on ioni zat ion fract ion: metal- lin e coolin g by free electro ns is ex p o ne nt ia lly depend ent o n te mperature, a nd t hus acts as a th erm ostat whi ch is onl y s li ght ly a ffect ed by t he io n or ne ut ra l density. T hu s, t he ph otoion izat ion lengt hscaie does not pl ay a n im portant ro le in t he d ens ity spectru m. 26 from the slow mode, both above and below the cooling scale. Thus , slow waves can produce density flu ctua tions which are not much smaller than the fiducial sp ectrum if fJ is not much larger tha n unity. Comparing with equation (1.47) b elow, the cont ribution of t he slow mode t o the density flu ctua tions a t L eoo l exceeds the contribution from Reynolds stresses provided fJ < L o u t/Cs t eoo l. The compressibility of the slow mode also yields density fluctuations associa t ed with the entropy m ode. The reason is as follows. Entropy fluctua tions are associa ted with isothermal compressible waves. Those due to isothermal slow modes are p assively mixed by Alfven waves. Mixing due to Alfven waves with wavelengths smaller than the cooling scale takes place in less than t he cooling time. It produces a spectrum of entropy modes for >'1- < L eoo l tha t gives rise to a density sp ectrum simila r to the one given in equa tion (1.41 ). Nonetheless, there is some d amping of entropy fluctua tions when they a re mixed from scales la rger than the cooling scale t o scales sm aller t h a n the cooling scale. We expect that the damping is of order unity. Since, as we show in the following section , the da mping of the slow mode is not very large, we expect that the amplitudes of t he slow a nd entropy modes b elow the cooling scale are roughly comparable. In this cha pter, however , we do n ot quantify the amount of damping of the entropy mode more precisely, b ecause it is a difficult calculation. We h ave a ttempted to quantify this damping by p erforming numerical simulations. However , the limited dyn a mical ra nge made the interpret ation of the results difficult. Nevert heless, t he relative a mplitude of slow a nd entropy waves on sm all scales does not a ppear to be observationally importa nt; as we show in §1. 7.3 b elow , both slow a nd entropy waves damp a t the same scale. Slow Mode Damping at the Cooling Scale Although there is negligible damping of the slow mode on lengthscales tha t are either much larger or much sm aller than the cooling scale, there is some damping on lengthscales tha t a re compa ra ble to the cooling scale. As we show in this section , t he da mping of the slow wave cascade is sm all , b ecause the slow mode is not very compressible. In t he limit tha t fJ ~ 00, the slow mode is incompressible, a nd the da mping disappears. In t he following, we calculate the damping to first order in l /fJ . First , we calcula t e the damping ra te of a slow wave of a fixed wavenumber. The damping rate is obtained by substituting c (given in the Appendix , eq. [1.95]) into the slow mode disp ersion relation (eq . [1.99]) ; noting tha t c} == p/ p = fJv~ /2 , this gives, t o lowest order in 1/ fJ, (1.42) where Wi is the imaginar y pa rt of w, and k z from the Appendix is here rela b elled kll ' Only on len gthscales where vA k ll teool ~ 1 is this ra tio non-negligible. To obtain the total decrement in the a mplitude of t he slow mode through the cooling scale, we solve a kinetic equa tion obtained by b ala ncing the slow mode k-space energy flu x with t he loss-ra t e 27 of slow mode k-space energy density due to damping ; i.e., (1.43) where A:L 1 == kJ.. is the transverse wavenumber , tA.L is t he cascade time of a slow wave with this wavenumber , and VA.L is the velocity perturbation of the slow mode; since kJ.. ::' k, we drop the J.. subscript on k . We rewrite the kinetic equation as follows : d V~ Wi In _.L_ = 2Wih.L = 2Mt - -- , d ln k tA.L VA k II (1.44) where M t == vAkll tA.L ::' 1.4 is a Kolmogorov constant (see §1.2.5) . Substituting equation (1.42) into equation (1.44) and integrating, using kll <X k 2 / 3 (eq. [1.28]) and tA .L <X k- 2 / 3 , yields the net damping through the cooling scale: [VA .L / A~3 lIA.L «: L coo l [VA.L / A~3lIA.L :» L coo l exp ( - 37rMt ) 10,8 (1.45) 1-1.3/,8, to lowest order in 1/,8. Thus for ,8 slightly larger than unity, slow mode damping can be ignored. 1.5.2 Density Fluctuations Below the Cooling Scale: f3 > Loudcstcool > 1 In this subsection only, we assume that ,8 > L out / Cstcool , which implies L MHD < L coo l . In this case, the isothermal hydrodynamic turbulent cascade extends from the outer scale down to the cooling scale. Because the entropy mode is rapidly damped under isothermal conditions and hydrodynamic turbulence is incompressible to order VA.L/CS, to this order there are no density fluctuations on lengthscales larger than L coo l ' However, pressure fluctuations due to Reynolds stresses, PA.L ~ pvL, create second order density and entropy fluctuations, nA.L = PA .L = S'A.L ~ ( VA.L ) 2, n P Cs L coo l < 1\.1. \ < L out· (1.46) These entropy fluctuations couple to the entropy mode at the cooling scale. On scales below the cooling scale, the entropy mode is mixed as a passive scalar yielding nA.L ~ vL 2 n Cs 0 0 I (~) 1 /3 ~ (LcOOI) 2/3 (~) 1 /3 ~ ( Cs t COOI ) 1/ 2 ( AJ.. ) 1/ 3 , AJ.. < L coo l. (1.47) L cool L ou t L cool L ou t L out 28 Hydrodynamic turbulence mixes the entropy mode down to L MHD. At t his scale, t he hydrodynamic motions turn into Alfven and slow waves, so the entropy mode cont inues to be mixed below L MHD, and equation (1.47) remains valid. Comparing with equation (1. 41), t he density fluctuations associated with the entropy mode exceed those associated wit h the slow mode when fJ > L ou t / cs t cool. The entropy mode's density spectrum (eq. [1.47]) is smaller than the fidu cial spectrum by the small factor (cs t cooI/Lout) 1/2. As will be seen in a fu t ure paper , it yields density flu ctuations t hat are too small to explain much of the observed interstellar scintillation for most plausible values of L out, such as those given by St romgren radii of HII regions. There are two alternatives; either L out is not much larger t han Cstcool , or fJ < L out/ Cs t coo l. 1.6 The Collisionless Scale of the Neutrals In the remainder of this chapter , we consider lengths cales which are comparable to, or smaller than , the mean free paths of the various particles; thus, kinetic effects must be considered . In t his section , we calculate the damping of the turbulence by neutrals. We assume tha t the neutral density is smaller t han t he electron density, i. e., nN ~ n, and t hat most of t he ions are prot ons. The neutral particles which we consider are hydrogen atoms and helium atoms. T he relevant lengthscales are the hydrogen and helium mean free paths for collisions wit h protons: LH = 5 x 10 13 ( cm - 3 In ) cm and LH e = 1. 5 x 10 15 ( cm - 3 I n) cm (Banks 1966). The hydrogen mean free path is significant ly smaller t han the helium mean free path because its collisions wit h protons are due to resonant charge exchange. There are two main results for this section . • (i) Only if the neutral fraction is sufficiently small (eq. [1.75], below) can t he cascade survive on lengthscales smaller than LH . If the neut ral fraction does not satisfy t his condit ion, then all three modes- Alfven , slow, and entropy- are damped at the same lengthscale . • (ii) Regions where the cascade does not survive cont ribute to an excess of density fluctuations on large scales. This might explain observations which detect an excess of power in large-scale density fluctuations. The organization of t he calculation is as follows : first, the change in wave frequency due to neutrals is calculated. Second , we consider the effect of the frequency change on t he cascade. T hird , we consider the cases in which the neutrals are hydrogen at oms and in which they are helium atoms. Alt hough helium atoms have a larger mean free path , we consider t hem after neut ral hydrogen atom s because they are of lesser importance. Finally, we consider regions where neut ral damping t erminat es the cascade. 29 A similar, though less detailed calculation is performed by Goldreich and Sridhar (1995) for t he case in which the neutrals are hydrogen atoms . 1.6.1 Frequency Change Consider an incompressible wave, either an Alfven wave or a slow wave in the incompressible limit . Although the slow wave is, in fact, slightly compressible, this does not have a large effect on the final result. We calculate the frequency change of an incompressible wave with fixed k: kll along the mean magnetic field and kJ. transverse to the mean magnetic field, where kll « kJ. ~ k. We define v to be t he mean velocity of the protons averaged over the proton distribution function. Thus v satisfies the equation of motion for the Alfven wave or for the slow wave derived in the Appendix. We then calculate the force that these protons exert on the neutrals. Since this force is the same as the force of the neutrals on the protons, inserting it into t he equation of motion for the protons yields the change in frequency. We denote the perturbed distribution function for the neutrals, i. e., perturbed to first-order in VN, after being Fourier-transformed in space and time, by IS!N( k ,w, v N), where VN is the velocity of the neutrals. The evolution of the neutrals is determined by the Boltzmann equation: -iwlS!N + i( k · v N)IS!N = C , (1.48) where C is the Fourier-transformed, perturbed, collision integral of t he neutrals with the protons. Neutral-neutral collisions are negligible relative to neutral-proton collisions. The collision integral is simplified by making the approximation that, on the timescale that a neutral collides with protons, IS!N is driven towards a Maxwellian with mean velocity v: (1.49) where the first term in t he square brackets is generated by expanding a Maxwellian distribution function with mean velocity v to linear order in v and retaining only the perturbation, and I/N ,p is the frequency with which a given neutral atom collides with protons. With the above approximation to the collision integral, the solution to the Boltzmann equation reads IS! = (v · VN) nNexp(-mNvJ./2T) I/N ,p N TlmN (27rTlmN)3 / 2 I/N,p-iw+i k ' VN Next we verify that the perturbed neutral number density vanishes, (1.50) 30 x [ / dVNIIVNII exp(-mNV~Il/2T)] (1.51) x (1.52) 0, since the first square-bracketed integral vanishes by antisymmetry. For the purpose of evaluating the above integrals, we use 1.. and II for the perpendicular and parallel projections of v N relative to v , and not relative to the magnetic field as in the rest of this chapter; i.e., v N11 == (v· VN)V/V 2 and v N.J.. == V N - v Nil . We also use the incompressibility relation k . v = 0 , which implies that k· v NIl = O. With the neut ral distribution function given by equation (1.50), the for ce per unit volume of t he protons on the neutrals may be calculated as 3 mN / vN(-iw JfN+ i( k · VN)Jf N)d v N F (1.53) mNnNvvN,p (271")3/2(T / mN )5 /2 x [/ dVNIlV~1l exp(-mNv~II/2T)] x (1.54) x (1. 55) (1.56) The second equality follows after replacing the overall multiplicative v N (= v Nil + v N.J..) by v Nil, because the v N.J.. term vanishes by antisymmetry of the integral over v Nil. In t he second equality, the first bracketed integral is equal to (271")1/2 (T /m N )3 / 2. The second bracketed integral is a double integral; the integral over the component of v N.J.. which is perpendicular to k is equal to (271"T /mN )1 /2 . The remaining integral is over the component of v N.J.. which is parallel to k ; we label this component s in t he third equality. Finally, the fourth equality follows from the change of variables t == (ks - W) /VN,p, and from the following definitions _k_( 2T )1 /2 = vAk (31/2 ( mp )1 /2 VN,p mN VN,p mN w VA k il =-- (1.57) (1. 58) where mp is the proton mass . The dimensionless parameter II I is the number of wavelengths a 31 neutral crosses before colliding, and the dimensionless parameter Ih is the number of waveperiods a neutral travels before colliding. Since k » kll and f3 2:. 1, III » II2 in the inertial range of the MHD cascade. Performing the integral for the real and imaginary parts of F to lowest order in II2/IIl yields (1.59) where (1.60) (1.6 1) Since -F is equal to the force of the neutrals on the protons , we insert -F into the equation of momentum conservation for the protons. More precisely, we add -iF/(mpn) to the right-hand side of equation (1.90) in the Appendix. We label the resulting change of frequency caused by the presence of neutrals as 6.w. Assuming that 6.w « VA kll ' it follows that (1.62) Discussion of the Frequency Change The physical interpretation of this frequency change is straightforward. Recall that k » kll' so that the wavelength, 27f/k, is nearly identical to the wavelength transverse to the magnetic field , 27f/k.l. It is convenient to define the neutral mean free path, LN, as follows: (1.63) where CT = (p/ p)1 /2 is the isothermal sound speed. The imaginary part of 6.w is the negative of the damping rate, and is given by (1.64) for kLN « 1 for kLN » 1 (1.65) The above expression may be explained as follows (Goldreich and Sridhar 1995). The neutrals are nearly locked to the protons only if both III < 1 and II2 < 1. These conditions hold for wavelengths larger than the neutral mean free path (k L N < 1) . So on these scales damping is due to the viscosity 32 of the neutrals . For wavelengths smaller than the neutral mean free path , the neutral atoms are decoupled from t he ions, and hence are effectively at rest. The damping rate is thus the ion-neutral collision frequency. The real part of t he frequency change, .0.wr , is for kL N « 1 for kLN » 1 (1.66) Its physical significance is apparent. For small wavelengths the neutral atoms are effectively freely streaming, t he motion of the protons nearly decouples from the motion of the neutrals, and t here is negligible real frequency change associat ed with the presence of the neutrals. For large wavelengths the neutrals are locked to the protons, so the mass density of the fluid which participates in t he waves is larger than that of the protons. Since the Alfven speed, and hence the wave frequency, is inversely proportional to t he square root of the mass density, i.e., W ex: VA ex: p- l / 2, this results in a decrease of the frequency by a fractional amount which is equal to one-half of the mass ratio of neutrals to protons . 1.6.2 Effect of Neutrals on the Turbulent Cascade Effect of Wi on the Cascade The turbulent cascade is quenched if the damping rate exceeds the eddy cascade rate or, equivalently, if IWi l/vAkll > 1. From equation (1.65) and kll ex: k 2 / 3 , it is seen that for small wavenumbers IWil/VAkll ex: k 4 / 3 , and for large wavenumbers IWil / vAkll ex: k- 2 / 3 . Therefore IWi l/VAkll is a m aximum for transverse wavelengths, k- 1 , comparable to the neutral mean free path , and decreases for both larger and smaller wavelengths. The requirement that the cascade survive damping by neutrals is then that IWi l/vAk ll < 1 for k- 1 comparable to the neutral mean free path , or equivalently, nNvN ,p/n ;S VA kil at this scale (eq. [1.65]) . Since nNvN ,p/n is the rate at which a given proton collides with neutrals, the cascade survives to small scales if the wave frequency, and hence the cascade rate , at the scale of the neutral mean free path is faster than the rate at which a proton collides with neutrals. To obtain t he total decrement in the amplitude of both the Alfven mode and the slow mode t hrough the damping scale, we solve the kinetic equation for the cascade.lO As in §1.5.1 , the kinetic equation is obtained by balancing t he k-space energy flux with the loss-rat e of k-space energy density laThe calculation of the amplitude decrement in this sect ion is similar to that given in §1.5 .1 for slow m ode damping at t he coolin g scale. The prin cipal difference, aside from t he fact that the im aginary part of t he frequency is different , is that here we must account for t he decrease in the cascade rate, t Al. ' caused by damping. This is not necessary wh en treati ng slow mode damping at the cooling scale because there the A lfven modes w hich control t he cascad e rate are un damped. 33 due to damping: d v~A.L dk h.L 2 v~A.L = wiT' (1.67) where AJ:l == k.l. :::' k is the transverse wavenumber, and tA.L is the cascade time. The kinetic equation can be rewritten as follows: (1.68) where M t is a Kolmogorov constant (see §1.2.5). Before integrating this equation, we re-express the damping frequency in terms of the relevant lengthscales, Lout and LN: 1 mNnN g(kLN) - - -- - VN 2 mpn ,p vAkil (1.69) 1 mNnN g(kLN) ------vN 2MII mpn ,p vA.Lk (1. 70) _ _1_(mN)1 /2nN (Lout)1 /3 g(kLN) V2MII mp n LN (kLN )2/3 (1. 71) The first equality above follows from equation (1.64). The second equality follows from the Kolmogorov constant Mil = vAklllvAl.k (see §1.2.5). The third equality follows from the velocity spectrum, VAl. = cT(kL out )-1/3 (eq. [1.38]) , and from the definition of the neutral mean free path (eq. [1.63]). For convenience, we define Lout in this section such that VAl. = CT when k- 1 = Lout · Integrating the kinetic equation (eq. [1.68]), and using tA l. ~ A.l. lvA.L ' yields the net decrement due to damping by collisions with neutrals: [VAl. I A~3lIA.L « LN [VA.L I A~3lIAl. »LN (1. 72) after substituting from equation (1.71). The numerical prefactor in the second equality, 0.2, follows after inserting the values of the Kolmogorov constants (eq. [1.34]) and the value of the integral (1. 73) which was integrated numerically; see equation (1.60) for the definition of g(x) . For the cascade to continue to small scales, the right-hand side of equation (1.72) cannot be very small. This can be viewed as an upper limit on the neutral fraction: nN In ;S (LN IL out )1 /3 . R ecall that the condition for the cascade to reach small scales is that the cascade rate at LN be faster than the rate at which a proton collides with neutrals. If the neutral fraction is too large, then so is the proton collision rate , and the cascade is quenched . Moreover, for a fixed L N , a large value for 34 Lout implies that the cascade time at LN is large, and hence that the cascade is more susceptible to damping by neutrals. We postpone consideration of the damping due to hydrogen and helium atoms until after we evaluate the effect of 6.w r on the cascade. Effect of 6.w r on the Cascade We can picture the cascade as proceeding from large scales to small scales. As it crosses the scale of the neutral mean free path (eq. [1.66]), the effective Alfven speed, and hence the real part of . wave frequency, increases by the fraction mNnN /2m pn. Consequently, the cascade time decreases by the same amount. To the extent that the flux of energy in Alfven waves from large scales to small scales is constant in the turbulent cascade, vt /tAl. = constant, where tAl. is the cascade time. Therefore, a fractional decrease of mNnN /2mpn in the cascade time causes a decrement in the smallscale Alfvenic velocity perturbation by the fractional amount mNnN /4mpn . This decrement is in addition to that due to damping. Moreover, it applies to the slow mode and the entropy mode as well as to the Alfven mode. Because nN /n must be small for the cascade to pass through the scale of neutral damping, this decrement is also small, and we ignore it from here on. 1.6.3 Neutral Hydrogen Atoms When the neutrals are hydrogen atoms we set nN = nH , VN ,p = VH,p , LN = L H , and mN = mp. Collisions between hydrogen atoms and protons are due to resonant charge exchange. From equation (1.63), with the value of VH,p taken from Banks (1966), -3 LH = 5 X 10 13 ( C: ) (1.74) cm at 8 x 10 3 K; the temperature dependence of LH is very weak. For the cascade to survive damping by neutral hydrogen atoms, the right-hand side of equation (1. 72) cannot be very small. This sets an upper limit on the neutral fraction of nH < 5 ( -LH ) 1/3 ,...., n""" Lout (6 x 10 15 cm ) 1/3 (n/cm- 3 )L out · (1. 75) The real part of w/vAk ll is larger on scales below LH than it is on scales above LH by the fractional value nH /2n. 1.6.4 Neutral Helium Atoms Although helium has a lower abundance than hydrogen, it has a higher ionization potential. Therefore, in some regions helium might comprise the majority of neutrals. When the neutrals are helium 35 at oms we set n N = n He, vN,p = VHe,p, LN = L H e , and mN = 4m p. From equation (1.63), with t he value of VHe ,p t aken from Banks (1966), L He = 1.5 X 10 15 ( c: -3 ) cm (1. 76 ) at 8 x 103 K , and LH e ex: T 1/ 2. To place an upper bound on damping by helium atoms, we assume that most of the helium is neutr al, i.e., n H e ::::: O.l n. Then the decrement is given by the right-hand side of equation (1.72): (1. 77) Only if L ou t .2: 2 x 10 19 ( cm - 3 /n ) cm could the cascade be terminat ed at t he scale of t he helium mean free path . If there are not many neutral helium atoms, or if the outer scale is not sufficient ly large, then damping at L H e may be neglected , and the cascade extends at least to the scale of the hydrogen mean free path. Collisions of neutral helium atoms with singly ionized helium ions might also be signifi cant . Alt hough t here are fewer helium atoms than protons, Heo -He+ collisions have a larger cross-section than Heo-proton collisions because they are due to resonant charge exchange. Nonetheless, given the cosmic abundance of helium, t he mean free pa th of neutral helium due to Heo-He+ collisions cannot be significantly smaller than t hat due to Heo-prot on collisions regardless of t he ionization fraction of helium. The real part of w/vA k ll increases by 2n He/n below the scale LH e , or 0.2 if most of the helium is neut ral. 1.6.5 If N e u t r a ls D a mp t h e Cascade In subsequent sections, we consider regions in t he interstellar medium where there are too few neutrals to damp the cascade. However , t here are almost certainly many regions where t he neutrals do damp t he cascade. We discuss these regions here. Suppose both Alfven waves and slow waves are damped by neutrals. What happens to t he entropy waves? If undamped , t hey would be mixed down to smaller scales by Alfven waves at t he neut ral damping scale. The resulting density spectrum would be n>..1. = constant (see eq . [1. 31]). Because it is flatter than n >..1. ex: >.. ~3, regions in which damping by neutrals truncated the Alfven cascade might be important contributors to small scale density fluctuations. However , t he fact t hat they would contribute a density spectrum different from t hat which is observed suggest s that t he ent ropy wave cascade is not more resist ant than the Alfven wave cascade to damping by neutrals. Indeed that is the case. Recall that the condition for the truncation of Alfven and slow wave cascades 36 is that each proton collide with at least one neutral atom during one wave period at the neutral damping scale. Under this condition, the neutrals would damp the entropy waves by conducting heat across them. Although regions in which the cascade is damped by neutrals do not contribute small scale density fluctuations, they may still be significant. Observational evidence indicates that there is more power in density flu ctuations on large scales, 10 13 _10 1 4 cm, than would be predicted by extrapolating from small scales, 108 - 10 10 cm, with the Kolmogorov scaling. See, for example, Lamb ert and Rickett (2000) for a review of the observations. Perhaps this excess arises in regions where the cascade is damped by neutrals. We complete this section by briefly considering and then rejecting the possibility that a turbulent cascade truncated at the neutral damping scale might be regenerated on a much smaller scale due to stirring by eddies at the damping scale. Although the ratio of the damping rate to the wave frequency decreases below the scale of the neutral mean free path, the absolute damping rate approaches a constant value. Provided the cascade is truncated by neutral damping, this rate is larger than the stirring rate and th e cascade cannot be regenerated. 1.7 The Collisionless Scale of the Ions If the cascade survives the neutral collision less scales, then, pro ceeding to smaller scales, the next scale of importance is the ion collisionless scale. This scale is set by the mean fr ee path of protons to collide with other protons: 3 cm- ) L mfp = 6 x 10 11 ( -n- cm (1.78) at a temperature of 8,OOOK (Braginskii 1965). Since the electron and proton gyroradii (see eq. [1.88) for protons) are very small compared to the proton mean free path, the electrons and protons are tied to magnetic fieldlines. Therefore, when considering the collisionless effects of electrons and protons, the relevant lengthscales are those parallel to the magnetic field. Since turbulent eddies are highly elongated along the magnetic field, their transverse lengthscales are much smaller than their parallel ones (eq. [1.28]). It is the transverse lengthscale which is relevant when considering observations of the density spectrum, because each line of sight averages over regions with different magnetic field orientations. In this section, we show that both the slow mode and the entropy mode are cut off at the lengthscale where the parallel eddy size is comparable to the proton mean free path. The lengthscale where the density spectrum is observed to cut off, i.e., the transverse lengthscale, is therefore significantly smaller than the mean free path. Before discussing the damping of the slow mode and the entropy mode , we consider two larger lengthscales, which are set by the effects of t he electrons. Although the behaviour of the cascade at these lengths cales is interesting, it is shown to be unimportant for our purposes. 37 Throughout this section , we neglect numerical factors of order unity, such as the factors which are associated with kinetic corrections to the fluid equations (given in Braginskii 1965) and t he Kolmogorov constants. However , we retain the dependences on (3, which is assumed to satisfy (3 > 1, and on the ratio of the proton mass to the electron mass: mp/me = 1840. 1.7.1 The Electron Diffusion Scale The electrons have the same mean free path as the protons, but they are faster than the protons by the square root of the mass ratio: Cs,e::::; ( m ) m: 1/ 2 Cs , (1. 79) where cs,e is the electron thermal speed and Cs is the sound speed, which is comparable to the proton thermal sp eed . Because of charge neutrality, the electrons have the same density, both perturbed and unperturbed , as the protons. Viscous damping caused by the electrons may be neglected : since the dynamic viscosity of the electrons is smaller than that of the protons by the square root of their m ass ratio, electron viscous damping is always subdominant . As we now show , electrons are important for conducting heat on parallel lengthscales slightly larger than the proton mean free path. Electrons diffuse parallel to the m agnetic field across a distance All in the time (Lmfp/ cs,e)(AII/Lmfp) 2. This is equal to the cascade time of an eddy with parallellengthscale All, i.e., it is equal to A ll / VA, at the electron diffusion scale, given by All ~ Led , where Cs,e Led::::; L mfp ::::; (3 1/2 (m - p) 1 /2 Lmfp . VA me (1.80) In eddies with parallel lengths smaller than this, the electrons diffuse across the parallel lengths of the eddies, thereby conducting heat, and the electrons are isothermal; in eddies with parallel lengths larger than this , conduction is unimportant. We must also consider the effect of electron conduction on the protons. Electrons and proton temperatures approach a common value on the timescale that there are mp/me collisions per particle. Since electrons are faster , the collision time is set by the electron speed , and the time for electron and proton t emperatures to equilibrate is mp Lmfp 1 1 (m p) 1/2 Teq ::::; ---::::; - (31/2 Lmfp' m e Cs,e VA me (1.81) This time is smaller, by a factor (3, than the cascade time of eddies with parallel size Led. Thus in eddies of this parallel size, and in those which are slightly smaller , protons are at t he same temperature as the electrons. And, since the electrons are isothermal, so are the protons. As a result of the above considerations, when the parallel cascade crosses the electron diffusion scale, the cascade becomes isothermal. This is similar to the crossing of the cooling scale, discussed 38 in §1. 5.1, t hough backwards , and similar considerations apply. In particular , the AlfvEln mode is incompressible, and hence unaffected. The slow mode is nearly incompressible, and is only slightly affected: it suffers some damping to first order in 1/13. The entropy mode, however , is rapidly damped under isothermal conditions; as the electron diffusion scale is crossed, entropy waves are converted into slow waves, and the density fluctuations which had been associated with entropy waves now become associated with slow waves. 1. 7.2 The Electron-Proton Equilibration Scale Continuing to slightly smaller scales, the isothermal cascade reaches the equilibration scale, where the cascade time, All/VA, is comparable to the electron-proton equilibration time (eq. [1.81]). At this scale, All ~ L eq , where 1 L eq ~ 13 1 / 2 (mm :)1/2 L mfp . (1.82) On larger scales, the protons are thermally coupled to the electrons , and hence they are isothermal on slightly larger scales; on smaller scales, the protons are thermally independent of the electrons, and hence adiabatic. The transition through the equilibration scale is nearly identical to the transition through the cooling scale (§1.5. 1): the Alfven waves are unaffected , and they mix larger-scale isothermal slow waves into smaller-scale adiabatic slow waves and into smaller-scale entropy waves , although with some damping. Therefore, below the equilibration scale, the entropy mode reappears. The result of the calculations in both this subsection and the previous subsection is that, in crossing the electron diffusion scale and the equilibration scale, density fluctuations which were associated with the entropy mode on large scales are transferred from the entropy mode to the slow mode and t hen back to the entropy mode. Density fluctuations which were associated with t he slow mode on large scales are unaffectedY Therefore, for our purposes, t hese two lengthscales have little net effect on the density spectrum. Although there is some damping, the amount of damping is comparable to the amount at the cooling scale, and hence is not very significant. 1. 7.3 The Proton Diffusion Scale: Death of the Slow Mode and Entropy Mode Entropy waves and slow waves with parallel wavelengths smaller than the proton mean free path both damp on the timescale that protons stream across their parallel wavelengths (B arnes 1966). However, because we consider 13 > I , there is a scale slightly larger than the proton mean free path 11 However , if the density flu ctu at ions in t he large-scale slow mode were less than t he dens ity flu ct uatio ns in the large-scale entropy mode, then, on small scales, the slow mode wo uld be boosted so t hat its density fluctuations would be com parable to those of t he entropy m ode: the slow mod e can steal approxim ately half of t he ent ropy mode . Note that, since the behaviour of the cascade at the electron-proton equ ili brat ion scale is nearly identical to t he the behaviour at the coolin g scale, similar uncertainties apply; see disc ussion in §1.5.1. In part icula r , we are unab le to qu a nt ify the dampin g of the entropy fluctuat ions , although we exp ect t ha t this damping is not so la rge as to render the entropy mode negli gible relative to the slow mode on sm a ll scales. 39 at which these waves damp in the turbulent cascade. This is the proton diffusion scale, which is t he scale at which protons can diffuse across an eddy within a cascade time. Below the proton diffusion scale, proton viscosity kills the slow waves and proton heat conduction kills the entropy waves. Alfven waves are unaffected by either of these effects. The density spectrum therefore cuts off below the proton diffusion scale. To evaluate t he proton diffusion scale, we equate the time for protons to diffuse across an eddy of parallel lengthscale All, i.e., (Lmfp/cs)(AII/Lmfp)2, with the cascade time, All/VA. This gives All ,...., Lpd , where L pd ~ -Cs L mfp ~ (31/2 L mfp . (1.83) VA However, from an observational point of view, it is the transverse size of a damped eddy, A1., which would be observed. This is related to the parallel size through equation (1.28). Alternatively, we use the Kolmogorov constant Mil == VAA1. / VA.LAII (see §1.2.5).12 Then, the transverse scale of a damped eddy, i.e., an eddy with parallel size All ,...., L pd , is A1. (1.84) which we solve for the cutoff lengthscale: A1. ,...., L~~) , where /2 M II3/ 2(33/2 L mfp (LmfP)1 L (1.85) PC ) 1/2 ( 2 X 10 9 ( - - (1.86) out L out Below this scale, the density spectrum is cut off. (3 ) 3/2 cm. 3 n/cm- For plausible values of (3, Lou t, and n, this lengthscale is significantly smaller than L mfp . It is also larger than the proton gyroradius (eq. [1. 88], below). Armstrong, Rickett, and Spangler (1995) summarize the observations of the density spectrum cutoff. There is considerable evidence that the cutoff scale is smaller than about 10 10 cm along many directions. There is weaker evidence, from refractive scintillation, that along some lines of sight the cutoff scale is larger than around 10 9 cm. Our theory might have implications for these observations. We conclude this section with two remarks. First, we re-emphasize the importance of the fact that t he cascade is anisotropic. It is this fact which allows the density spectrum to reach lengthscales which are significantly smaller than the proton mean free path. Second, we note an important 12 A lthough we drop other factors of order unity, we keep the dependence on t his one because of the importa nce of the proton diffusion scale. 40 consequence of the parallel cascades of the slow mode and of the passive scalar; these parallel cascades were explained in great detail when considering incompressible MHD turbulence (§1.2). Had there been no parallel cascade, then neither the slow mode nor the entropy mode would damp at the proton diffusion scale. The wavelength along the magnetic field would be effectively infinite, and so the ions could not diffuse across wavelengths. Rather, the slow mode and the entropy mode would be mixed down to the proton gyro scale, where the Alfvenic cascade is cut off. 1.7.4 Density Spectrum Below the Proton Diffusion Scale On scales smaller than L~~), density fluctuations are wiped out. Since on these scales protons diffuse across the lengths of many eddies before the eddies cascade, the density within neighbouring eddies is homogenized. Homogenization occurs not only in the direction parallel to the local mean magnetic field- as might have been expected since protons are tied to fieldlines- but also in the transverse direction. This is because eddies that are adjacent in the parallel direction incorporate substantially different fieldlines, so proton diffusion also wipes out density differences amongst eddies with transverse separations. The result is that the density spectrum on scales smaller than L~~) is determined by density fluctuations at L~~); equivalently, (.L) for A.L < Lpd 1.8 (1.87) The End: the Proton Gyroscale The Alfven mode is undamped in a collisionless medium (Barnes 1966). Thus, the Alfven wave cascade survives below the proton diffusion scale, without the accompaniment of the slow waves and entropy waves. The Alfvenic cascade is damped at the scale of the proton gyroradius: m c Lp ,gyr = .../2CT e~ = 2.5 x 10 7 (n/cm(3 )1/2 cm . 3 (1.88) At this scale, Alfven waves are converted into whistlers. The whistlers cascade to smaller scales, where they are damped by the collisionless effects of the electrons (Quataert 1998, viz. the curve in his Fig. 1b that corresponds to equal electron and proton temperatures). 1.9 Summary The primary goal of this chapter has been to calculate the small-scale density spectrum in turbulent interstellar plasmas. Our theory of compressible MHD turbulence is based upon the incompressible theory of Goldreich and Sridhar (1995), for which there is growing support from numerical simula- 41 tions. We hypothesize that t he compressible theory is similar to t he incompressible t heory of MHD t urbulence, but with two main modifications: a compressible slow mode and an ent ropy mode which is passively advected. While we believe that this hypothesis is plausible, it can , and should , be test ed with numerical simulations of compressible MHD . Because of the multitude of special lengths cales encountered when discussing t he t urbulent cascade in the interst ellar medium , a recapitulation might be useful. In t he following, we list the most important lengthscales and summarize their significance to t he turbulent cascade. Th e Outer S cale (Lout): This is t he lengthscale at which t he t urbulent motions are stirred . On slight ly smaller scales, there is isothermal hydrodynamic t urbulence for most plausible astrophysical sources. Th e MHD S cale (LMHO): At this lengthscale there is a transition from hydrodynamic to MHD t urbulence. Larger-scale hydrodynamic motions couple to smaller-scale Alfven waves and slow waves. The Cooling S cale (Leool): At this lengthscale t here is a t ransit ion from isothermal to adiabatic turbulence. In high-,B turbulence, where small scale density flu ctuations are due to the ent ropy mode, entropy- and hence density- flu ctuations are suppressed by cooling. In low-,B t urbulence, i.e., 1 .:s ,B < Lout / Cs teool, density fluctuations due t o the slow mode are important, and cooling has a negligible effect on the small-scale density spectrum . Th e Collisionless S cale of the N eutrals (L H; LHe): Neut rals decouple from ions across these scales . If the neutral fr action is not sufficiently small , then all three modes- Alfven , slow, and entropy- are damped . The Collisionless S cale of the Io ns (A ll = Led; A ll = Leq ; All = Lpd ¢:> >'1- = L~~)): Across t hese lengthscales t here is a gradual t ransit ion from fluid behaviour to collisionless plasma behaviour . The first two of these lengthscales, t he electron diffusion scale and the equilibration scale, are set by t he electrons. These lengthscales have only a small net effect on the cascade. The proton diffusion scale, however, is critically important for density fluctuations: both the slow mode and the entropy mode are cut off at this scale, and hence there are no density flu ctuations below L~~) . Th e P roton Gyro S cale (Lp ,gyr): Alfven waves are cut off at this scale. However , t his scale has lit tle importance for the density spectrum , because there are no density flu ctuations below the proton diffusion scale. 1.10 Comparison with Higdon's Work In his 1984 paper , Higdon attributes small-scale density fluctuations to the passive mixing of t he ent ropy mode. In his 1986 paper , Higdon notes that if the entropy mode varies along t he m agnetic field , it is damped in a collision less medium . He t hen attri butes small-scale density flu ctuations to t he passive mixing of tangent ial pressure balances. Tangent ial pressure balances are structures 42 which are p arallel to the mean magnetic field. They are composed of both entropy waves and slow waves that have purely transverse wave vectors. Considering that Higdon 's papers preceded even a theory of incompressible MHD turbulence , they are a remarkable accomplishment. However , Higdon does not account for the parallel cascade. Entropy and slow waves with purely transverse wave vectors contain negligible power. They are cascaded along the magnetic field by Alfven waves. Consequently, they are damped when protons can diffuse across eddies in a cascade time, i. e., below the proton diffusion scale. 1.11 Compressible Turbulence When (3 < 1 In this section only, we consider compressible turbulence in plasmas that have fJ < 1, e.g., in strong oblique isothermal shocks and in the solar wind. Since Alfven waves are unaffected by the value of fJ, and since nearly transverse slow waves are only slightly affected (see Appendix) , the dynamics of the cascade is nearly independent of fJ. While the dispersion relation of the slow mode is changed from w = vA k z to w = cskz, the slow waves are still passively mixed by the Alfven waves. Therefore, the Alfven , slow, and entropy spectra in fJ < 1 MHD turbulence are the same as when fJ > 1. However , the damping of the slow and entropy waves is significantly changed . When fJ > 1, the proton thermal speed is faster than the Alfven speed. Therefore protons can stream across small eddies before they cascade. Conversely, when fJ < 1, the time for protons to cross an eddy is always shorter than the cascade time. Therefore slow and entropy waves cannot be damped by protons which cross eddies, and the density spectrum extends to smaller scales. We have ignored the effects of the electrons. However , in the following we show that the electrons' b ehaviour may be ignored for our purposes . Our discussion closely parallels that when fJ > 1 in §1.7, and uses similar notation. When fJ < 1, the equilibration scale (eq. [1.82]) is larger than the electron diffusion scale (eq. [1.80]). Thus the largest scale at which kinetic effects are significant is the scale at which the cascade time is comparable to the time for electrons and protons to equilibrate their temperatures. Below this scale, electrons and protons are thermally decoupled . Nonetheless, this thermal decoupling has no effect on the Alfven , slow, or entropy waves on scales larger than the electron diffusion scale. At the electron diffusion scale, the cascade time is comparable to the electron diffusion time. Below this scale, the electrons are isothermal. However , since the protons are thermally decoupled from the electrons , the slow waves and entropy waves are cascaded to smaller scales.13 Continuing to smaller scales , the next scale of importance is that at which the par allel size of an eddy is comparable to the proton mean free path. Below this scale, the cascade is collisionless. The 13There is a change in t he d ensity spectrum at this scale which is of order unity; it is due to the ch a nge in the elect ro ns' equation of state. 43 Alfven waves are undamped by collisionless effects. The entropy waves are undamped since protons cannot cross eddies within a cascade time. The slow waves are also undamped within a cascade time: although they damp within a waveperiod by Barnes damping, their waveperiod is longer than the cascade time. Consequently, the Kolmogorov density spectrum extends down to the proton gyro scale. 1.12 Future Work In a future pap er, we will examine in detail the density spectrum in the solar wind. In another future pap er , we will relate the theory of compressible plasma turbulence developed here to observationsprimarily those of diffractive scintillation. We will demonstrate that the observed amplitude of small-scale density fluctuations is surprisingly large, especially along certain lines of sight , such as the one toward the G alactic Center. Then we will attempt to determine which astrophysical sources contribute the bulk of the density fluctuations and why they do so. Acknow ledgments We thank Jason Maron for analyzing some of his simulations for us, for showing us how to use his numerical code, and for informative discussions. We thank the referee, Ben Chandran, for a very helpful report. Research reported in this chapter was supported by NSF gra nt 94-14232. 1.13 Appendix : Waves in Compressible MHD In this Appendix we derive the properties of the Alfven mode, the slow mode, and the entropy mode. The fast mode is not relevant to interstellar scintillation for reasons discussed in the body of the chapter. The Fourier-transformed, linearized equations of ideal MHD, with a/at -+ -iw and V -+ i k , read wn' (k·v)n , wv k[c2~ +bZVA ] -bVA kz , wb zVA(k . v) - v VA kz , (mass conservation) (momentum conservation) (Faraday's law) (l.89) (l.90) (l.91 ) where n is number density, p is mass density, B is the background magnetic field intensity, and VA == B / J 47r P is the Alfven speed. The z-axis is chosen to lie parallel to the background m agnetic field. Some other variables are the perturbed number density, n'; the fluid velocity, v; and the perturbed magnetic intensity, B ' , or in velocity units, b == B ' / J47rp. The sound speed, c, is defined 44 less conventionally here to be the square root of the ratio of perturbed pressure to perturbed mass density: c == (p' I p')1 /2. We can express the equations of motion in terms of v as follows: (1.92) For most of the purposes of this chapter, the energy equation may be left unspecified, and c may be assumed constant. However, we use the energy equation when we discuss slow mode damping at the cooling scale in §1.5.1. We assume that temperature fluctuations decay in a cooling time, teool; more precisely, T' iwTs' = - - (energy conservation) , (1.93) t eoo l where s' is the perturbed entropy per particle, and T is the temperature. Combining this equation with the following monatomic ideal gas relations p' p' T' 2, 5 p' P = P + T = 3s + 3p , (1.94) yields -2 _ p' 2 2 - i5wt eoo l c = - =cr p' 2 - i3wt eoo l ' (1.95) where c~ == pi p is the square of the isothermal sound speed. When wteool « I, c is the isothermal sound speed; when wteoo l » I , c is the adiabatic sound speed. 1.13.1 Alfven Mode The Alfven mode is incompressible; k . v = O. Thus the term involving the sound speed in the momentum equation vanishes, and the properties of the Alfven mode are independent of the equation of energy conservation. We obtain the dispersion relation (1.96) by forming the cross product of equation (1.92) with k. The eigenfunction satisfies n' = k . v = Z . v = 0, b = -sign(kz) v . (1.97) Note that both v and b are perpendicular to z: the Alfven wave is polarized transverse to the unp erturbed magnetic field . 45 1.13.2 Slow Mode We summarize the properties of the slow mode to lowest order in kz / k « 1, the limit appropriate to the MHD cascade. To obtain the dispersion relation we assume, subject to verification, that the perturbation in total pressure- i. e., thermal plus magnetic pressure- vanishes to second order in kz / k. Note that the perturbation in total pressure is proportional to the terms in square brackets in both equation (1.90) and equation (1.92). From the vanishing of this pressure term in equation (1.92), we then have V~ [ k . v = c2 + v~ 1 + 0 (k )2]k zv z · z (1.98) The z-component of equation (1.92), with k . v given by equation (1.98), then yields the dispersion relation: (1.99) Next, we solve for the eigenfunction to lowest order in kz/k: (1.100) where the x-axis has been chosen to lie in the plane containing k and z. We now see that our assumption of negligible perturbed pressure is self-consistent to lowest order in k z/ k . While we have not used the x-component of equation (1.92), both the left-hand side and the right-hand side of this equation are of the same order in k z /k . The vanishing of the slow mode's perturbed pressure may be understood as follows: the period of the slow mode is comparable to the time for a disturbance to cross a parallel wavelength at the slower of c and VA. Total pressure balance is set up within the time that the fast mode- which travels at the faster of c and VA - crosses a wavelength. This may be seen by solving equation (1.92) exactly (treating c as a constant): the slow mode's perturbed pressure is negligible , and our approximate solution is valid, when 2(kz /k) cVA /( C2 +v~) « 1, which implies that k z x min{ c, VA} « k x max {c, VA}. Note that both v and b are nearly parallel to z: the slow wave polarization is nearly aligned with the unperturbed magnetic field. Moreover , in the limit that c » VA, the slow mode is nearly incompressible. 1.13.3 Entropy Mode The linearized MHD equations yield four modes ; in addition to the Alfven, slow , and fast modes , there is a lesser known mode: the entropy mode. For adiabatic fluid motions, the energy equation is ws' = O. The entropy mode has w = 0, and its eigenfunction is given by p' = v = b = 0, 46 T'IT = -n'ln = 2s'/ 5. In the following, we calculate the eigenvalue and eigenvector of the entropy mode in the presence of cooling, i.e., when the energy equation is given by equation (1.93). We shall see t hat the entropy mode decays in a time comparable to t co ol ' We consider only wavenumbers sufficiently large that t he fast mode's crossing time is much faster than the entropy mode's decay time, i.e., k » kfast == [c2 + V~ ] -1/2 t~O~I ' Therefore the total pressure perturbation vanishes. Note that k > kfast in the vicinity of the cooling scale (eq. [1.39]) , where the present calculation is most relevant . The dispersion relation may be obtained from equation (1.92) , after setting the perturbed total pressure- the term in square brackets- to zero. The result is equation (1.99), which has three roots: two counter-propagating slow modes and the entropy mode. To solve for the entropy mode, it is convenient to rewrite equation (1.99) as follows: . 4 + 2,8[1 - (vAk z l w)2 ] ~wtcool = 6 + 5,8[1 - (vAk z lw) 2] , (1.10 1) where,8 == 24I v~, and where equation (1.95) has been used for c. Presupposing that w is of order t~o~ l' we see that the right-hand side of this equation has two limiting behaviours, depending on the value of k z . On small scales, where min{vA,cr } x tcoolkz» 1, .2 1 w = -~--- 5 t co ol (1.102) In this limit the slow mode crossing time is faster than cooling, so the thermal pressure perturbation vanishes, as m ay be seen directly from equation (1.95). Thus, to lowest order in (k zvA t cool) -l, the eigenfunction is t he same as for the adiabatic case, p' = v = b = 0, T'IT = -n' ln = 2s'/5. On large scales, where kzVAtcool « 1, equation (1. 101) implies that .4 + 2,8 1 w = -~----6 + 5,8 t coo l , vA tcoolkz« 1 . (1.103) Substituting this into equation (1.95), and using total pressure balance, yields the eigenfunction , n 'l n = bzl vA = _(pI Ip)( ,8 12) . The remaining non-zero components of the eigenfunction are T ' IT = - (n' In) (1 + 2/,8) and s' = - (n' In )(5/2 + 31,8) . 47 B ibliography [1] Armstrong, J . W ., Rickett, B. 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L. 1972, A First Course in Turbulence (Cambridge: MIT Press) 49 Chapter 2 Imbalanced Weak MHD Turbulence Written with P et er Goldreich ; to be submitted to the Astrophysical Journal ABSTRACT MHD turbulence consists of waves that propagate along magnetic fieldlines, in both directions. When two oppositely directed waves collide, they distort each other, without changing their respective energies. In weak MHD turbulence , a given wave suffers many collisions before cascading. Imbalance means that more energy is going in one direction than the other. In general, MHD turbulence is imbalanced. A number of complications arise for the imbalanced cascade that are unimportant for the balanced one. We solve weak MHD turbulence that is imbalanced . Of crucial importance is that the energies going in both directions are forced to equalize at the dissipation scale. We call this the pinning of the energy spectra. It affects the entire inertial range. Weak MHD turbulence is particularly interesting because perturbation theory is applicable. Hence it can be described with a simple kinetic equation. Galtier et al. derived this kinetic equation . We present a simpler , more physical derivation , based on the picture of colliding wavepackets. In the process, we explain why Goldreich and Sridhar claimed that perturbation theory is inapplicable, and why this claim is wrong. (Our "weak" is equivalent to Goldreich and Sridhar's "intermediate." ) We perform numerical simulations of the kinetic equation to verify our claims. 2 .1 Introduction MHD turbulence is ubiquitous in astrophysics . For example, it is present in the sun , the solar wind, the interstellar medium , molecular clouds, accretion disks, and galaxy clusters. Theoretical understanding of incompressible MHD turbulence has grown explosively in the last decade . Nonetheless, it remains underdeveloped. Iroshnikov (1963) and Kraichnan (1965) developed a theory for MHD turbulence. They realized t hat the magnetic field at the largest lengthscale in a cascade directly affects all of t he smaller lengthscales . Small-scale fluctuations can be treated as small-amplitude waves in t he presence of a large mean magnetic field. By contrast, the large-scale velocity is unimportant for sm all-scale dynamics ; it can be eliminated by a change of variables, since the equations of MHD are invariant under Galilean transformations. Despite their realization of the importance of the mean magnetic field , Iroshnikov and Kraichnan assumed that small-scale flu ctuations are isotropic. Numerical simulations later showed t hat this assumption is wrong. Even with isotropic excitation at large scales , fluctu ations on smaller scales are elongated along the mean magnetic field (e.g., Montgomery and Turner 1981 , Shebalin , Matthaeus , 50 and Montgomery 1983). In retrospect, it is not very surprising to find elongated fluctuations. Arbitrary disturbances in incompressible MHD can be decomposed into shear-Alfven and pseudo-Alfven waves. Each wave travels either up or down the mean field at the Alfven speed, VA, which is the magnitude of the mean field in velocity units. Consider stirring a magneto-fluid with a spoon that is moving at speed V « VA, for a time comparable to the spoon 's width divided by v. (In a turbulent cascade, one would expect V « VA on small scales, since V decreases towards small scales, whereas VA is unchanged.) Alfven waves are radiated away from the spoon, parallel to the mean field with speed ± VA. After the disturbance is finished, there are two wavepackets travelling away from each other. Each wavepacket is elongated along the mean field , with parallel-to-transverse aspect ratio ~ VA / V » l. The characteristics of MHD turbulence depend critically on the amount of elongation. When parallel-to-transverse aspect ratios are smaller than VA/V, waves collide many t imes before cascading. Hence the turbulence is weak, and perturbation theory can be used to derive a kinetic equation and a spectrum (Sridhar and Goldreich 1994; see Zakharov , L'vov, and Falkovich 1992 for a general review of weak turbulence). Goldreich and Sridhar (1997) and Ng and Bhattacharj ee (1997) deduced the spectrum of the balanced weak cascade from scaling arguments. 1 However , similar scaling arguments are inadequate for the imbalanced cascade (see §2.3.2 of the present chapter). Galtier et al. (2000) derived the kinetic equation for the weak imbalanced cascade. Their balanced spectrum agreed with that of Goldreich and Sridhar (1997) and Ng and Bhattacharjee (1997). They also presented a partial solution for the general imbalanced case. In §2.3.2, we explain why their solution is incomplete; in §2.4 we give the complet e solution. Even if aspect ratios are smaller than vA/von large scales, at a small enough scale they become comparable to VA/V. Below this scale perturbation theory breaks down , and weak turbulence becomes strong. Goldreich and Sridhar (1995) worked out the scalings for the balanced strong cascade. They argued that aspect ratios are comparable to vA/vat all scales in the strong regime. Strong turbulence is difficult , largely because it is non-perturbative. Although strong and weak turbulence differ in a number of ways, they also share many similarities. One of our main motivations for studying weak turbulence is to gain insight into strong turbulence. In particular , turbulence in the 1 We relegate some of the history to a footno te because it can be confus ing. Sridhar a nd Goldreich (1994) d eveloped the first theory of MHD turbulence that accou nted for the a nisot ropy of flu ctu at ions . They cla imed that three-wave processes vanish in weak MHD turbu lence , a nd four-wave processes must be co nsid ered (i .e., perturbation theory is trivial to first order , so second order terms a re import ant ) . As a res ult , they used four-wave co uplin gs to d erive a kinetic equa tion and a spectrum for weak MHD turbulence . Montgomery a nd Matt hae us (1995) cla imed, and Ng and Bhattacharjee (1996) showed , that Sridhar and Gold reich (1994) are wrong, and three-wave pro cesses do not vanish. Goldreich and Sridhar (1997) expl a in ed the co ntradi ction: Sridhar a nd Goldreich (1994) h ad unknow in gly ass um ed t h at the fi eld line wa nder is small ; in th is case, three-wave co uplings a re negli gibl e a nd t he kin et ic equat ion based on four-wave couplin gs is correct. In the more realist ic case t hat fi eldlin es wand er s ubstantially, three-wave pro cesses are important. Goldreich and Sridhar (1997) went on to argue t hat , in the la tter case, perturbation theory is in a ppropriate, a nd couplings of a ll order a re of comparable m agnitude ; so they call ed t hi s interm ed iate turbulence . Galtier et a l. (2000) argued t hat perturbat ion theory is appropriate , even when three-wave processes are important. In the Appendix of the present chapter, we use Goldreich and Sridhar's pi ct ure of wavepackets following wand ering fi eldlines to clarify the controversy, and to ex plain why perturba tion th eo ry works. Because it does work, we call the cascade weak instead of intermediate. 51 solar wind is observed to be imbalanced ; it cannot be understood wit hout a t heory for imbalanced strong MHD turbulence. Yet this theory is unknown . In a future paper, we will work it out by extending the results of the present chapter. 2.2 Basic Equations Ideal incompressible MHD2 is described by the following equations of motion: V'v -VP+B·VB , (2 .1) B · Vv , (2.2) V · B=O. (2. 3) The density is set to unity; the fluid velocity is v ; the magnetic field in velocity units is B == (magnetic field)/(47r) 1/2; the total pressure is P == p + B 2/2 , which is the sum of t he thermal and magnetic pressures . Viscous and resistive terms are neglect ed in the above equations; they are important on small scales, and will be included where required. We decompose the magnetic field into its mean , VA Z , where VA is t he Alfven speed and Z is a unit vector , and into its fluctuating part b == B - VA Z . With this decomposition , the equations of motion may be written in terms of the Elsasser variables, w t + VAoz Wt = -w.J..·Vw t == v - b and w.J.. == v + b, as follows: VP , (2.4) Ot w.J.. - VA OzW.J.. = - w t ·Vw.J.. - VP , (2.5) V·w t (2. 6) Ot wt - =V·w.J..=O. Note that P is not an independent degree of freedom . Taking the divergence of either equation (2.4) or (2. 5) yields (2.7) where \7 -2 is the inverse Laplacian. When w.J.. = 0, w t propagat es un distorted upwards along the mean magnetic field wit h speed VA. Similarly, when w t = 0, w.J.. propagates downwards at VA- Nonlinear interactions occur only between oppositely directed wavepackets . It is these interactions t hat are responsible for t urbulence. There are three conserved quantities in incompressible MHD. Two of these are immediately apparent from equations (2.4)-(2 .6) : the energies of the upgoing and of the downgoing waves, i.e., (w t )2 and (w.!-)2. (Technically, these are twice t he energy per unit mass. We refer to t hem as 2I n t his cha pter , we consid er o nl y in compressible M HD turbulence ; com pressibili ty does not a lter the dy na mi cs very much (Li t hw ick and Gold reich 2001 ) . 52 simply energies throughout the chapter.) These energies are directly related to t he total (kinetic plus magnetic) energy C( (wt)2 + (w.!-)2 and to the cross-helicity C( (wt)2 - (w.!-)2. The focus of this chapter is turbulence where the energies in the up and down waves differ, or , equivalently, where the cross-helicity is non-zero . The third conserved quantity is magnetic helicity; however , for reasons explained below, its conservation does not playa role in this chapter. In MHD turbulence, on lengthscales much smaller than t he outer scale, there is effectively a strong mean magnetic field that is due to fluctuations on the largest lengthscales. Gradients transverse to this mean field are much larger than gradients along it (e.g., Shebalin, Matthaeus, and Montgomery 1983, Goldreich and Sridhar 1995, 1997, Ng and Bhattacharj ee 1996) . This allows the MHD equations to be slightly simplified . The transverse components of equation (2.4), assuming that w;8z wt is much smaller than w+·V ..l..wt , are (2.8) where transverse components are denoted with the symbol ..1 == (x, y) . We assume t hat t he parallel component of w+ is either comparable to, or less than, its perpendicular component. We will see below that this is typically the case in the inertial range of a turbulent cascade. From equation (2.7), P:::::: _V :.L2( V ..l..wt:V ..l..w+) ; therefore (2.9) Similarly, (2. 10) (2. 11) If we change:::::: to = , equations (2.8)-(2 .11) form a closed set. They are called the equations of reduced MHD. They apply also in compressible MHD whenever transverse gradients are larger than parallel ones (e.g., Biskamp 1993). The main goal of this chapter is to solve these equations. Although the complete equations are not much more complicated, it simplifies our discussions to neglect parallel gradients relative to perpendicular ones at t he outset. There are two conserved energies in reduced MHD : (wl)2 and (wiY. The magnetic helicity associated with wI and wi vanishes, and is therefore unimport ant for their evolution. 3 We also consider the parallel components of equations (2.4) and (2.5). Neglecting parallel gradi3In 3-D MHD , th e m agneti c helicity is fA · B d3 x, where A is th e magnetic potential (V X A = B ). If the m agnet ic field is bro ken up into its mean vAi, and into flu ct uations b , t hen the on ly non-vanishing part of the heli city is fa· bd 3 x, where V X a = b. In reduced MHD , b is t ra nsverse to i and a is nearly parall el to i , so t he he li city nearly vanishes. 53 ents relative to transverse ones yields (2.12) (2.13) Clearly, (wI)2 and (w*)2 are conserved quantities. The transverse equations describing reduced MHD are unaffected by these parallel equations because the former are independent of wI and w*. Nonetheless, the parallel equations have observable consequences. It is conventional to decompose the normal modes of linearized incompressible MHD, w t and w.j., into shear-Alfv'en and pseudo-Alfven waves. These correspond to the Alfven and slow waves of compressible MHD . When perpendicular gradients are much larger than parallel ones, w 1 and wi are nearly equivalent to shear-Alfven waves; wI and w* are nearly equivalent to pseudo-Alfven waves. Also observationally relevant is the evolution of a passive scalar s, which satisfies 8 t s + v, V s = O. In terms of Elsasser variables, and after neglecting parallel gradients, t he passive scalar satisfies 8t s ~ -(1/2)(wl + wi)·V.LS . (2.14) To avoid a proliferation of subscripts, in the remainder of this chapter we drop the -L from w 1 and wi . To denote the parallel components, we use wI and w*. 2.3 Weak Turbulence: Heuristic Discussion 2.3.1 Scaling Relation MHD turbulence can be understood from the dynamics of w t and w.j. (eqs. [2 .8-2.11] for reduced MHD, dropping -L subscripts). To linear order, w t is a wave that propagates up the mean magnetic fieldlines at the Alfven speed, VA; w.j. propagates down at VA. Each wave perturbs the mean magnetic fieldlines. Nonlinear terms describe the interaction between oppositely directed waves: each wave nearly follows the fieldlines perturbed by its collision partner. 4 Consider an upgoing wavepacket that encounters a train of downgoing wavepackets. As the upgoing wave travels up the length of the down going train, it is gradually distorted . It tries to follow the perturbed fieldlines in the downgoing train, but these fieldlines wander, i.e., the transverse 4The equation for a quantity f that travels upwa rds at speed VA, whi le fo llowing the magnet ic fieldlin es of the down-going w.j. is (at + VAa z + w.j.· V jJf = O. Equation (2 .8) for w t differs from thi s because of the pressure term, which is required to keep w t incompressible, while conserving the energy (wV. Thus w t do es not exactly follow th e fieldlin es of w.j.. Nonetheless , we show in the Appendix that this deviation does not great ly affect the b eh av iour of the turbulence. Dissipation is a second effect that prevents the fo llowing of field lines. In the present discussion , we co nsider lengt hscales that are sufficiently large that dissipat ion can be neglected . 54 sep ar ation between any two field lines changes. After each downgoing wavepacket, t he amount of wander is random ; after many wavepacket s, the wander tends t o increase, as in a random walk. When the up-wave has travelled sufficiently far along t he down-waves that the typical amount of wander is comparable to its own t ransverse size, the up-wave cascades . To be quantitative, let each downgoing wave in the train have a typical amplitude wi, a transverse size A, and a parallel size A, where "transverse" and "parallel" refer to t he orientation relative to the mean magnetic field. The most important collisions are between wavepackets of comparable t ransverse size (see §2.3.4). So let t he upgoing wave have t ransverse size A as well. Since each downgoing wavepacket has a typical perturbed magnetic field of magni tude ~ wi (neglecting the factor of 1/2), it bends the fieldlines by the angle Wi / VA; t he transverse displacement of a fieldline through this wavepacket is (Wi / VA )A ; and the wander of two typical fieldlines t hrough t he wavepacket , if they are initially separat ed by A, is also (Wi / VA )A. In weak turbulence, the wander through a single wavepacket is smaller t han t he wavepacket's transverse size, (2. 15) i.e., the parallel-to-t ransverse aspect ratios are smaller t han VA/wi and VA/wI. (When these inequalities are not satisfied , strong turbulence is applicable; see §2.3.4.) An upgoing wavepacket must travel t hrough many downgoing ones before cascading. After N downgoing wavepackets, fieldlines have wandered a distance ~ Nl/ 2 (w±lvA )A , assuming tha t wavepacket s are st atistically independent. The up going wavepacket is full y distorted- and hence cascaded- when t he fieldlines it is following wander a dist ance A, i.e., when N ~ (AVA / Awi? Since each downgoing wavepacket is crossed in t he time A/VA, t he cascade t ime of t he upgoing wavepacket is (2. 16) In this time, the up going wavepacket t ravels a dist ance vAtJas, which is much larger t han A (see eq. [2.15]). Consequently, if t he upgoing wavepacket also has a parallel size A, t hen as its energy is cascaded to smaller transverse lengthscales, it does not cascade t o smaller parallellengthscales: A = scale independent . (2. 17) We assume t hroughout this chapter that the upgoing waves' parallellengthscale is the same as that of t he downgoing waves, A; t he extension to t he case when t hey differ is t rivial (as long as t he inequalities (2 .15) are both satisfied , with the appropriat e A's) . We calculat e the steady state energy spectra by using Kolmogorov 's picture of energy flowing from large t o sm all lengthscales. The energy in up-waves fl ows from lengthscales larger t han A to 55 those sm aller than>. at t he rate (2.18) We call this the flux. In steady st ate, it must be independent of >. , so (2.19) 2.3.2 Insufficiency of Scaling Arguments for the Imbalanced Cascade When the cascade is balanced, the steady st ate solution is simple: wI = wi ex >.1 / 2 and tOt = to.!. ,...., (WI )4A/>.2vA , (Goldreich and Sridhar 1997, Ng and Bhattacharjee 1997). When it is imbalanced , a number of complications arise. By the symmetry between up- and down-going waves, equation (2.18) is also equal to t he flux of down-going waves: wtw.!. ] 2 A >. VA to'!' ,...., [~- . (2 .20) Const ancy of to.!. is forced by the constancy of tOt, and does not yield new information . One implication is t hat scaling arguments are insufficient to determine the flux ra tio tOt / to.!.. Physically, any flux ratio should be possible. But without the dimensionless coefficients of equations (2.18) and (2.20) , tOt / to.!. cannot be det ermined. The coefficients depend on t he spectral slope of wI (or equivalent ly of wi ex >'/wt, eq. [2.19]) , and cannot be obtained from scaling arguments . Galt ier et al. (2000) calculated the coefficients using kinetic equations. (We explain how below.) Therefore t hese authors were able to relat e the flux ratio to the spectral slopes. However , the arguments presented thus far are still insufficiently constraining. Equations (2.18) wI wi. There are seemingly an infinite number of solutions wit h given values of tOt and to'!', since wI can be multiplied by any const ant as long as wi is divided by this same constant . Furt hermore, we expect on physical grounds that if the values of wI and wi and (2.20) constrain only t he product at a given lengthscale are fixed (inst ead of the values of tOt and to'!'), the cascade should be completely constrained ; however , in t his case the constancy of equations (2.18) and (2 .20) leaves t he sp ectral slope of wI completely undetermined- even given the coefficients derived by Galtier et al. (2000). Do t he two spectra cross? Are t hey cut off by dissipation at the same scale? All of t hese problems for the imbalanced cascade can be resolved once the dynamics at t he dissipation scale is understood . 2 .3.3 Dynamics at the Dissipation Scale: Pinned Spectra The m ain result of the present chapter is t hat t he energies of t he up- and down-going waves are forced to equalize-they are "pinned"- at the dissipation scale. This completely constrains t he cascade. 56 It is very surprising that the dynamics at the dissipation scale has such an important influence. In t his subsection we explain why pinning occurs. In §2.4, we give the resulting solut ion of t he steady state cascade. From equation (2.16) , the cascade time of the up-going waves is inversely prop ortional to the energy of t he down-going ones: t ras = (A/ wi)2 (VA / A), and similarly for t he downgoing waves. We consider how the spectra evolve if initially, on lengthscales comparable to t he dissipation scale, waves going in one direction are more energetic t han the oppositely direct ed ones . To facilitate the discussion , we refer to Figure 2.2 , which presents t he results from a numerical simulation that we discuss in detail below. In the middle panels of Figures 2.2a-d , we plot et(k) ~ (AW1) 2 and e.J..(k) ~ (Awi) 2 as functions of wavenumber k = I/A. Note t hat wi > wL so t~as > tras ' We consider a lengthscale-dependent dissipation time, tdiss, that is t he same for both up- and downgoing waves. 5 On large lengthscales, the dissipation timescale is much longer than t he cascade times. Towards smaller lengthscales, tdiss decreases faster than both tras and t~as ' The effects of dissipation are felt on lengthscales where tdiss is smaller than , or comparable to, eit her tras or tLs' Since t~as > tras in the vicinity of t he dissipation scale, the largest lengthscale at which dissipation effects are felt is where t~as ~ tdiss. In Figure 2.2a, t his is at k ~ 1, 000 . We now let the spectra evolve (Figs. 2.2b-c) . Since wi feels t he dissipative effects at k ~ 1, 000 , its spectrum is exponent ially cut off at smaller scales . This implies t hat the cascade time of t he up waves, tLs, increases exponent ially towards smaller scales. As a result , up-wave energy t hat is being cascaded from large t o sma ll scales cannot be cascaded fast enough at k 2: 1, 000. Therefore the up-waves' energy flux is backed up , and the wI spectrum rises. Furthermore, as wI rises, t~as falls, so t he cascade t ime of t he down waves on sm all scales decreases, and t he down-wave spectrum falls. The final result is that t he two spectra are pinned at the dissipation scale. This pinning occurs very quickly: on t he dissipation t imescale. 2.3.4 Two Peripheral Issues This subsection may be skipped on a first reading, wit hout affecting our main line of argument . Locality In §2. 3.1 , it is assumed that t he dominant interaction is between t hose wavepackets th at have comparable t ransverse lengthscale, i.e., interactions are local in lengthscale. In this section , we justify this assumpt ion. We focus on t he cascading of an upgoing wavepacket by downgoing ones. Let the u pgoing wavepacket have transverse size A, and let the downgoing wavepackets each have t ransverse size l , parallel size A, and amplitude wt. The upgoing wavepacket cascades when the fieldlines it is 5For example, in t he simul ation presented in Fi gure 2.2, t d iss ~ ).. 2 lv, where v is bot h t he vi scosity a nd t he resistivity. 57 following wander a distance comparable to its transverse size, A. We consider first the case that l < A. Two field lines that are separated by A at the head of t he downgoing wavepackets wander independently of each other. Their transverse separation after N downgoing wavepackets increases by (2.21) Conversely, if l > A, then the magnetic field at two points separated by A differs by ~ wt AI l ; consequently, the fieldlines separate by (2.22) as long as this separation is not much larger that a few A. For interactions to be local, i.e., for the amount of fieldline wander seen by an up-wave of transverse size A to be maximized by those down-waves that have l ~ A, the following two conditions must hold: (i) wt is an increasing function of l (eq. [2.21]); and (ii) wt Il is a decreasing function of l (eq. [2.22]). So the cascade is local if (2 .23) The same condition clearly holds for wI. In terms of the steady-state scalings, wI ex A{l +a)/2 and wi ex A{l -a)/2, the condition -1 < Ct < 1 (2.24) is required for the cascade to be local; otherwise, nonlocal effects are important. Galt ier et al. (2000) derived t he inequalities in (2.24) from the kinetic equation. There is a second reason why the condition d In wI/d In A < 1 is necessary for our heuristic arguments to be valid. When this condition is violated , the energy in up-waves that have lengthscale A, i.e., (w1)2, is smaller than the contribution to the energy at lengthscale A by upgoing waves with l > A, i. e., (wi AII) 2. This is a different kind of non-locality than considered previously: energy cascades directly from large to small lengthscales, skipping over intermediate lengthscales. Transition to Strong Turbulence Weak turbulence is applicable when wi AlVA « A and wIAlvA « A (eq. [2.15]). Since A decreases faster t han both wi and wI (eq. [2 .23]) , even if these inequalit ies are satisfied at large lengthscales, they are violated at small ones. Thus weak turbulence has a limited inertial range. Of greater relevance than weak turbulence for describing astrophysical sites such as the solar 58 wind is strong turbulence, which is applicable when the above inequalities are violated. In this case, the fieldline separation within a single wavepacket is not smaller than the transverse size of the wavepacket. This has two implications. First, equation (2 .16) for the cascade t ime is no longer valid; and second, the parallel size of wavepackets A decreases towards smaller scales because the head and tail of a wavepacket are independently cascaded. The balanced strong cascade is worked out by Goldreich and Sridhar (1995); we discuss the imbalanced strong cascade in a future paper. Strong turbulence is more difficult than weak turbulence because perturbation theory is not valid. Nonetheless, a number of the features of weak turbulence that we develop in the present chapter are applicable to strong turbulence. This is our main motivation for studying the weak cascade: to gain insight into the strong cascade. Since this chapter is concerned with weak turbulence, we choose the dissipation scale to be sufficiently large that the inequalities (2.15) are never violated, and the entire cascade is weak. By choosing a sufficiently small A, the lengthscale at which weak turbulence transitions to strong can be made as small as desired. 2.4 Steady State Energy Spectra wI wi ex A (eq. [2.19]) , and We denote the value of wI and The steady state energy spectra are simple to calculate given the scaling the fact that the spectra are pinned at the dissipation scale, Adiss. wi at Adi ss by Wdiss. We can then express the energy spectra as follows A) (1+,,)/2 Wdiss (- , Adiss A) (1-,,)/2 Wdiss ( Adiss ~ (2.25) (2.26) These spectra are valid in the inertial range, i.e., on lengthscales larger than the dissipation scale, and smaller than the outer scale, at which the turbulence is stirred. There are three parameters that must be calculated to constrain the spectra: Wdiss, Adi ss, and Ct. For definiteness, we assume that dissipation is caused by a diffusive process, with a viscous term of the form vV 2 v in equation (2 .1) and a resistive term of the form vV2 B in equation (2.2). This implies that the magnetic Prandtl number is equal to one. 6 The dissipative timescale is (2.27) At the dissipation scale, the cascade time is equal to tdiss, i. e., t cas( Adiss) = tdiss(Adiss ), which implies 6It is straightforward to consider other forms for the dissipation. We consider hype rviscos ity in §2.8. 59 that (2.28) after using equations (2.16) and (2.27) ; so V VA )1 /2 Wdiss:= ( A (2 .29) If the dissipation is caused by a diffusive process, then Wdiss is independent of the energies and the fluxes. This is not true for Adi ss or a. To calculate Adi ss and a, we consider two alternative scenarios: first , the energies at the outer scale are specified, and second the fluxes are specified . 2.4.1 Fixed Energy at the Outer Scale We assume that the energies are specified at the outer scale, Aout, where wI and wi are denoted by w tA o ut and w+I\ o u t . Since w t w+ I A = wdiss 2 I Ad iss , 1'\ 1'\ Adi ss = Aout t .j. VVA (2.30) A ' w"\out w>" out after using equation (2 .29). Dividing equation (2.25) by equation (2 .26) yields wL"lwL" = (Aoutl Adiss)"', so (2.31) With wL,,IwL,, fixed , the cascade is balanced (a --+ 0) in the limit that the inertial range is infinitely large (Aout I Ad iss --+ (0). Inserting equations (2.29) , (2.30), and (2.31) into the spectrum (eqs . [2.25) and [2.26)) gives the solution to the steady state imbalanced weak cascade, assuming that 2.4.2 wtu, and wLu, are specified. Fixed Flux Heuristic arguments are insufficient for calculating the ratio of the fluxes, lOt IE.j. (see §2 .3.2) . We would like to determine both lOt and E.j. given the spectra of wI and wi. Conversely, we would like to determine the spectra when lOt and E.j. are specified . To accomplish this, the dimensionless coefficients of equations (2.18) and (2.20) are required. For a given power-law solut ion , and wI ex A(1 +<»/2 wi ex A(1 -<»/2, these coeffi cients depend on a: t .j. 2 A f(a) [WAWA ] _ , A VA (2.32) 60 t .\. 2 A f(_a)[WAW A] - , A (2 .33) VA where f( a ) is a dimensionless function of a that must be calculated . By the symmetry between up and down waves, f t and f.\. are both proportional to the same function f , evaluated at ±a. Since heuristic arguments are insufficient to calculate the fun ction f , it is fortunate that weak t urbulence can b e analyzed with perturbation theory. We compute f in the Appendix (eq. [2 .85]) ; Galtier et al. (2000) also computed it . The ratio of the fluxes is related to a by (2.34) The limit la l « 1 is particularly interesting. For given outer-scale energies, if the inertial range is very large then the steady state cascade is nearly balanced (see the discussion below eq. [2 .31]) , and la l « 1. In this limit, we show in the Appendix that f(a) :::: f(O) . (1 + 0.5a) (see eq. [2 .86]). Thus (2.35) To linear order in a, the product of the fluxes is independent of a : (2 .36) neglecting the unimportant order-unity multiplicative factor [J(0)F. In sum, if ft and f.\. are specified, then the spectra of wI and wi are given by equations (2 .25) and (2.26), with Wdiss, a, and Ad iss given by equations (2.29) , (2.35), and (2.36) , in t he limit of small la l. Note that, to first order in a, the only relation that depends on the kinetic equation (i. e., on the form of f( a)) is equation (2.35). 2.5 Kinetic Equations in Weak Thrbulence We show in the Appendix that perturbation theory can be used to describe weak turbulence. As a consequence, the evolution of the energy spectra of the up- and down-waves is described by a closed set of two equations; in other words, the two-point correlation functions evolve independently of all higher-order correlation functions. This is a great simplification. Evolution equations for the energy spectra- the kinetic equations- were obtained in Galtier et al. (2000, 2001). We present an alternate, more physical, derivation in the Appendix. Such a derivation is useful because it clears up a number of erroneous claims that have been m ade in the literature- in particular, the claim of Goldreich and Sridhar (1997) that perturbation theory is inapplicable. 61 In the following, we summarize the result derived in the Appendix. The kinetic equations are given in Fourier-space. We Fourier transform w t (x, y , z, t) and w+(x , y, z, t) in x and y (but not z ), and denote the transforms by wl( z, t) and wk( z,t), where k is purely transverse (kz == 0) . We define the energy spectra e t and e-l- such that (wl( z, t) "Wl,( z, t)) et(k, t)J(k + k') (2.37) (wk(z, t) "Wk'( z , t)) e-l-(k, t)J(k + k') (2.38) where J(k) is a two-dimensional Dirac delta-function , and angled brackets denote both an ensemble average and an average over z . (We assume that the turbulence is homogeneous in z .) Bot h e t and e-l- are real. The turbulence is isotropic in the transverse plane, so e t and e-l- are functions of the magnitude of k. Within an order-unity factor, k 2e t '" (W!)2 when). = 11k . Recall from §2 .3 that wI is the typical value of w t on lengthscale ).; more precisely, it can be defined as the square root of the structure function of w t . Therefore the steady-state scaling to e t ex: k-(3+o); similarly, wI ex: ).( 1+0)/2 (eq. [2.25]) corresponds wi ex: ).(1- 0)/2 corresponds to e-l- ex: k-( 3-0). In comparing our results with those of Galtier et al. (2000), note that these authors use the one-dimensional spectrum, which they denote E± , while we use the two-dimensional spectrum. So, E+ '" ke t and E- '" ke-l-, within a multiplicative constant. 7 From equation (2.80) in the Appendix, the kinetic equation for the up-waves is (2.39) where (2.40) There must also be a term describing dissipation on smalliengthscaies. If the dissipation is diffusive and the viscosity (v) is equal to the resistivity, then -vk 2 et is to be added to the right-hand side of the above equation . Because of the symmetry between up- and down-waves, the equation for e-l- is the same as for et, but with e t and e-l- everywhere switched. The steady-state relation between the energy and the flux that we use above (eq. [2.32]) is obtained in the Appendix by setting the right-hand side of equation (2.39) zero. 7The one-dimensional spectrum is defined such that its integral over dk is proportional to the energy. Simil arly, the integral of the two-dimensional spectrum over d 2 k is proportional to the energy. 62 2.6 Numerical Simulations of Kinetic Equations In this section we present the results of numerical simulations which verify our previous heuristic discussions: in pa rticular , t h e pinning of spectra and the scaling of the sp ectra in steady state. It is much faster to simulate kinetic equations t ha n the full equations of motion for w t and w"!'. There are a number of reasons for this. First, the kinetic equations are only one dimensional, since homogeneity has b een assumed parallel to the mean magnetic field , a nd isotropy has been assumed in the plane tran sverse to the mean m agnetic field . Second , a nd more importantly, the averaged energies e t and e.j. a re much smoother functions of k than w t and w"!'. Thus, a logarithmically- sp aced grid can be used, which logarithmically reduces t he number of variables t h at need to be evolved. The result is an enormous reduction in computational time. A typical kinetic simulation takes a few hours on a PC to reach steady state. A comparable fully t hree-dimen sional MHD simulation , would require many months , if not years, on the fastest sup ercomputers. Galtier et al. (2000) also perform numerical simulations of the kinetic equation. However, their investigation of the imbala nced cascade is very incomplete. In pa rticular, t hey only plot spectra of the product et e.j.. They do not discuss the pinning of the spectra at the dissipation scale, which is crucial to the evolution of the cascade. For our numerical simulations of the kinetic equations (eq. [2 .39] a nd the a nalogous e.j. equ ation) , we set A / VA = 1. This corresp onds to absorbing A/VA into e t a nd e.j. . We evaluate all functions of k on a fixed , logarithmically-spaced grid with k = 2i / 8 , i = 1, ... , 100. kout = 21 / 8 = 1.09 a nd t he m aximum k is k m ax = 2100 / 8 At t he outer scale, = 5793. The double integral in t he kinetic equ ation is performed by summing the values of the integrand evaluated on the k-sp ace grid . T he factor that depends on B is averaged in t he vicinity of each grid point, i.e. , at each (kt,k.j.) . Since B is a function of k t / k and k.j./ k, and since the grid is logarithmic, the averaged angular factor can be precomputed and stored in a two-dimensional matrix, each element of which is t he average in the vicinity of (kt/k, k.j. / k).8 We integrate in time with second-order Runge-Kutta. For the discussion s of the simulations t hat follow , recall that the energies of the waves at a given lengthscale >. = l/k are, within unimporta nt multiplicative constants, (2.41 ) 8In evalu at ing t he integral near kout, it is necessary to use t he values of ei( k i ) and e.!.(k.JJ for ki,k.!. < kout. Since t hese values are off of t he grid , some method of ext rapo lation must be used. In th e run s that we present in this chapter, we extrapolate w it h ei,e.!. 0:: k- 2 We have a lso experimented with a fl at spect rum for k < k out : e i = ei( kout) , e.!. = e.l.(k out ). While this cha nges t he b eh aviou r near k ~ kout- in particular, it leads to a shar p drop in e i a nd e.!. between k = k out and k = 21/Bkout by a factor of around 3- the remainder of the spectrum for k > 21/8kout is nearly unaffected. With the k- 2 extrapolation t here is a lso a drop at k = kou t, as can b e seen in t he top pan el of F igure 2.1d, for exam pl e; but t hi s drop is m uch less drastic t han with the fl at spectr um extrapolat ion . 63 The cascade times of the up- and down-going waves are, respectively, (2.42) (see eq. [2.16]). The dissipation time is (2.43) (see eq. [2.27]). For the simulations in t his section , the viscosity is set to v = 3.10- 5 ; (2.44) this implies that tdiss(k max ) = 0.001. 2.6.1 Fixed Energy at the Outer Scale For our first simulation, we fix et(kout ) = 1 and e-l-( kou t ) = 0.1 throughout t he simulation. See Figure 2.1. For the initial condition , we use the spectra e t ex k- 3 and e-l- ex k - 3 (Fig. 2.1a). These initial spectra are seemingly valid solutions of the steady-state flux relations (eqs. [2 .32] and [2 .33]) , with a = 0 and Et = E-l-. But the spectra are not pinned . When they are evolved in t ime, it seen that they become pinned to each other at the dissipation scale. This pinning happens very quickly- at the dissipation timescale (Fig 2.1b). The reason for this pinning was discussed in §2.3.3 , and can be traced in Figures 2.1a-c. The entire e-l- spectrum adjusts to et on the timescale t~as(kout) ~ 1. The e t spectrum takes longer to adjust, since t!as(k out ) ~ 10. By t = 50, steady state is reached (Fig 2.1d) . We can compare t he behaviour in steady state with our calculations in §2.4. From equation (2.30), the dissipation wavenumber is (2.45) which is approximately three times larger than the value seen in Figure 2.1d. The reason for this discrepancy can be seen in the bottom panel of this figure: dissipation is imp ortant where the cascade times are ~ tdiss/3, rather than ~ tdiss as we previously assumed . To account for t his difference, we can use an effective viscosity, VerT == 3v, instead of V in our steady state formulae. Equation (2 .31) gives (2.46) 64 a) t=O b) t=0.005 c) t = 0 .5 d) t = 50 3 .5 3 2.5 100 10- 5 10- 10 100 10- 2 100 10 2 k 10 4 10 0 10 2 k 10 4 10 0 10 2 k 10 4 10 0 10 2 10 4 k Figure 2.1 : Simulation of Kinetic Equations with Fixed Energies at the Outer Scale 65 so dlne t - -din k d ln e-l. --dink 3 + a = 3.14 (2.47) 3 - a = 2.86 , (2.48) in good agreement with the top panel of Figure 2.1d . From equation (2.36), the product of the steady state fluxes is approximately (2.49) In the numerical simulation, the values of (0 t and (0-1. in steady state are obtained by outputting t he instantaneous decay rates of et(kout ) and e-l.( kout ) when the forcing is turned off. We find that (d/dt)et(k out ) = -0.136 = -(Ot and (d/dt)e-l.(k out ) = - 0.114 = -(0-1., so (Ot(O-I. = 0.016 , in reasonable agreement with the predict ed value. Recall that heuristic arguments suffice to derive all of the steady-state formulae t hat we have used thus far in this section, and the kinetic equation is unnecessary. But the kinetic equation is necessary to derive the relative value of the steady state fluxes. From equation (2.35) , we should have (2.50) if we use the value of a from equation (2.46). The actual value is 0.136/0.114 - 1 = 0.19, once again in reasonable agreement. 2.6.2 Fixed Flux For our second simulation , we inject energy at a fixed rate at k = k out . We use the same inj ection rate as we found in steady state in the first simulation: (Ot = 0.136 , (0-1. = 0.114 (2.51) Thus et(koud and e-l. (k ou t ) are free to evolve. For our initial condition , we use the spectra obtained in steady st at e in the first simulation , but multiplied by a constant; specifically, et(k) ~ e t (k)/ lO and e-l.(k) ~ e-l.(k) . 10 (see Fig. 2.2a) . With these fluxes and initial spectra, the flux relations (eqs. [2.32] and [2.33]) are satisfied- since they were satisfied before the multiplication and division by 10. But , once again , this is not a valid steady state solution because the spectra are not pinned at the dissipation scale. When the spectra are evolved in time, t he sp ectra are first pinned (Figs. 2.2a-c). In this simulation, the spectra are initially pinned where t±as ~ tdiss, which is at k ~ 1, 000. Therefore the pinning timescale is ~ tdiss(k = 1, 000) ~ 0.03. 66 a) l = O b) l = O.Ol c) l = O.l d) l = 25 0 3 .5 3 2 .5 10 0 10- 5 10- 10 10 0 10 0 10 2 k 10 4 10° 10 2 k 10 4 10 0 10 2 10 4 10° k Figure 2.2 : Simulation of Kinetic Equations with Fixed Fluxes 10 2 k 10 4 67 As exp ect ed , the same st eady state is reached as in the first simulation (Fig. 2.2d). In reaching steady state, the two spectra must cross. This happens at t ,...., 40, when both spectra are ident ical, with slopes equal to -3, and with outer scale energy equal to ,...., 0.3. The time to reach steady state is considerably longer in the constant flux simulation than in the constant energy one. Steady state is reached at t ,...., 250, as opposed to t ,...., 50 in the previous simulation. This may be attributed to the fact that most of the flux that is input at large scales is continuously being lost at small scales. Therefore, the constant flux simulation is less efficient at growing the spectra than is the constant energy spectra. In sum , these two simulations show that the pinning of the spectra occurs very quickly. In addition , both the evolution of the spectra towards steady state, and the behaviour in steady state, can be understood with simple heuristic arguments . However , there is one exception: the relation between the ratio of the fluxes and the spectral slope, Et IE:" - 1 ::::: Ct, the coefficient of which (i. e., 1) can only be obtained by considering the steady state kinetic equation. 2.6.3 Decaying Turbulence For our third simulation, we allow the spectra to decay without inj ecting any energy. The initial condition is the steady state spectra from the previous simulation , see Figure 2.3 . At the outer scale, the cascade time of the down-waves is ,...., l/ et (k out ) ,...., 1, so e:"(koud decays on this timescale. More precisely, from the values plotted in Figure 2.3, e:" ex exp( -0.5t) at fixed k. The cascade time of the up-waves is much longer. Initially, tras ,...., 10 at the outer scale; as e:" decays, t ras increases exponentially. So et (k out ) does not evolve. Nonetheless, tras is smaller at small scales, and there is some evolution of e t at large k. The spectra appear to evolve in a self-similar manner, with the spectra remaining pinned at their dissipation scale. The end result is that the energy in the down-waves disappears , while the energy in the up-waves is nearly unchanged. Decaying weak turbulence is unstable: an initial imbalance between up and down waves is magnified exponentially. A similar instability occurs in strong turbulence, as suggested by Dobrowolny et al. (1980) in the context of the solar wind , and as seen in numerical simulations of strong turbulence (Maron and Goldreich 2001, and Cho et al. 2002) . 2.7 A Model of the Kinetic Equations: Coupled Diffusion Equations It is instructive to consider a very simple model of the kinetic equations. In this section, we develop a model in the form of two coupled diffusion equations. Investigation of these model equations 68 10 4 k Figure 2.3: Decaying Simulation of Kinetic Equations Spectra of e t (dotted lines) and e.J. (solid lines) at the following times: t = 0,25,50,75, 100, 125 (from top to bottom). 69 illustrat es that the pinning of spectra is quite general; the only requirement is that the cascade time of one type of wave be inversely relat ed to t he energy of the other type. A second reason for considering model equations is that they can be simulated much fast er and more easily t han t he kinetic equations. Finally, by contrasting the kinetic equations with the model equations, we can gain insight into the behaviour of the kinetic equations. 2.7.1 Derivation of Coupled Diffusion Equations Our model equation is (2.52) where €f is the flux . Since the energy of the up-waves is proportional to the integral of e t over d 2 k , ke t is proportional t o the energy per unit k . For €f, we choose a form that depends on t he following quantities evaluat ed at k: e.!., e t , and Bk et . By contrast , the kinetic equation depends on e t and e.!. evaluat ed at a range of wavenumbers, so our model equation is much more local in k (in fact, it is exactly local). For t he cascade time of the energy in the up waves to have the correct form (eq. [2 .16]) , i. e., tJas = ()../W;) 2(VA /A ) ~ (k 4 e.!.) - 1(vA /A) , we choose (2. 53) where /3 is a constant that we specify below. Of course t he above relation for the flu x gives the correct st eady-st ate scaling: e t e.!. ex: k - 6 . The evolution equations for e.!. are t he same as for e t (eqs. [2. 52] and [2.53]) , with t 's and .j..'s interchanged. With e t ex: k -(3+a) , e.!. ex: k -(3 - a), we can relate the steady-stat e fluxes to the energies: j(a)[k6et (k) e.!. (k)]~ , (2.54) j( - a)[k6et(k)e.!.(k)]~ , VA (2.55) j(a ) = 2 + a - /3 . (2 .56) VA where These relations are analogous t o equations (2 .32) and (2.33). To make the analogy closer , we choose /3 so t hat j has the same dependence on a as does f in the limit that t he cascade is nearly balanced (i.e., lal « 1). Since €f / f'!' = j(a )/ j( -a) ::: 1 + a /(l - /3 /2) in this limit, we see by comparison with equation (2 .35) that we should choose /3 = O. Collecting results, our model equations are two coupled diffusion equations: (2.57) 70 a at -e-l- (2.58) after including dissipative terms. These equations, while much simpler than the kinetic equations, share many of the same features. Since the cascade time of the up-going waves is inversely proportional to the energy of the downgoing waves (and vice versa), the argument for the pinning of the spectra in weak turbulence (see §2.3.3) applies here as well. Moreover, in steady state, these diffusion equations suffer from the same degeneracy as does the kinetic equation: the constancy of {t and E-l- is insufficient to determine the scaling of e t or e-l- separately. This degeneracy is partially broken by the dependence of the fluxes on the slopes (eqs. [2 .54] and [2.55]); we have chosen this dependence so that it is the same as for the kinetic equations in the limit that the cascade is nearly balanced (Ial « 1) . In the following section, we present numerical simulations of these equations. 2.7.2 Numerical Simulations of Coupled Diffusion Equations We run two simulations of the coupled diffusion equations. As with the kinetic simulations described in §2.6, the first simulation has fixed energy at the outer scale, and the second simulation has fixed flux. And, as before, A/VA is set to unity, functions of k are evaluated at k = 2i / 8 , i = 1, ... , 100, and the viscosity is // = 3 . 10- 5 . Fixed Energy at the Outer Scale For our first simulation, we fix et(kout ) = 1 and e-l-(kout ) = 0.1. The evolution is shown in Figure 2.4. It is very similar to the evolution of the kinetic simulation (Fig. 2.1) . As before, the spectra are pinned quickly. And since, by design , the predicted steady state relations for the diffusion equations are the same as those for the kinetic equations (eqs. [2.45]- [2 .50]), the steady state behaviour of the two simulations are nearly identical; compare Figures 2.4d and 2.1d. Nonetheless, there are two differences worthy of note. First, the slopes of e t and e-l- do not have a spike near k = kont · The presence of such a spike in the kinetic simulation is due to the extrapolation of the spectra to k < kon t (see footnote 8). Since the diffusion equations are exactly local in k , such an extrapolation is unnecessary, and so the behaviour is much smoother near k = kont . Second, dissipation is important where the cascade times are ~ tdiss/15 (see bottom panel of Fig. 2.4d), instead of ~ tdiss/3 as for the kinetic simulation. Therefore, we should use an effective viscosity //eff = 15// in the predicted steady state relations. In particular, the prediction for a (see eq. [2.46]) becomes a = 0.17 with this From the output of the code in steady state, the fluxes are Et = 0.308 and E-l- = 0.260. Thus Et /E-l- - 1 = 0.18, while the predicted value is a, i.e. , 0.17. 71 t = 0 .005 t = 0.5 d) t = 50 3.5 3 2.5 10° 10- 5 10- 10 10° 10° 10 2 k 10 4 10° 10 2 k 10 4 10° 10 2 k 10 4 10° 10 2 10 4 k Figure 2.4: Simulation of Diffusion Equations with Fixed Energies at the Outer Scale 72 b) t=O.Ol d) t=1000 3.5 3 2 .5 100 10- 5 10- 10 10° 10° 10 2 10 4 10° k 10 2 k 10 4 10° 10 2 10 4 10° k 10 2 10 4 k Figure 2.5 : Simulation of Diffusion Equations with Fixed Fluxes Fixed Flux at the Outer Scale For the second simulation, we inj ect energy at k = kout at a fixed rate, equal to that found in steady state in the previous simulation; i.e., Et = 0.308 and E.(. = 0.260. For the initial condition, we divide the previously found st eady state value of e t by 10, and we multiply e.(. by 10. The evolution is shown in Figure 2.5. It is very similar to that found in the corresponding kinetic simulation (Figure 2.2) . 73 2.8 The Bottleneck Effect The bottleneck effect appears in numerical simulations of both hydrodynamic and MHD turbulence (e.g. , Borue and Orszag 1995, Biskamp 2000) . A trick that is commonly used in simulations is that the viscous term is modified from the diffusive value vk 2 v to vnknv, where n is typically 4 or 8, and Vn is t he hyperviscosity; the resistivity is modified in a similar manner. This trick makes the dissipation lengthscale slightly smaller. Thus a smaller part of the spectrum is subjected to dissipation , and more of the true undissipated spectrum can be observed for a given dynamic range . Although this trick does work, and the dissipation lengthscale is made smaller , there is a problem: the spectrum on scales slightly larger than the dissipation scales is made fl atter. The energy, in effect, is backed up . This is the bottleneck effect. It can be particularly problematic in simulations of strong MHD turbulence, where the energy backup can affect lengthscales considerably larger than the dissipative scales (Biskamp and Muller 2000, Maron and Goldreich 2001 , alt hough Cho and Vishniac 2000 show a less extended backup) . This motivates us to investigate the bottleneck effect in weak turbulence. If the bottleneck effect appears, its interpretation will be simpler t han in strong turbulence. We perform two simulations of the kinetic equation, one with hyperviscosity of the form v4k4, and the other with vsk s. In each simulation, we fix t he energies at the outer scale, et(k out ) = 1, e.!.(k out ) = 0.1 , and allow the spectra to reach steady state. The steady state spectra of k 6 et e.!. are shown in Figure 2.6, offset for clarity. Also shown is the simulation with ordinary viscosity described in §2.6. 1. From this figure it is apparent that weak turbulence suffers from the bottleneck effect; t he effect becomes larger with increasing hyperviscous exponent . The bottleneck effect can be understood as follows. Consider an up-wave on a lengthscale slightly larger than dissipative scales. It is cascaded by down-waves that have slightly different lengthscales than its own . Hyperviscosity gives a sharper dissipative cutoff to the down-wave spectrum than ordinary viscosity. Therefore a hyperviscous simulation has fewer down-waves in the vicinity of t he dissipation scale, and the cascade time of t he up-wave is longer. For the up-wave energy flux to be independent of lengthscale, a longer cascade time implies a larger energy. As a result , the spectrum is flatter on scales slight ly larger than dissipative ones. Falkovich (1994) has a similar explanation of the bottleneck effect in hydro turbulence. 2.9 Discussion In this chapter we described imbalanced weak turbulence and solved the steady state cascade. In a future paper, we will extend the result to the strong cascade. One of our ultimate goals is to use the theory of imbalanced strong turbulence to explain turbulence in the solar wind, where imbalance is 74 10 1 ................. ;... v: ~ ........ .. ...... ... ...;...................... . 10- 1 10 0 . ... -.- -. --.- .". -- -- --_. ........ . <.· _-- ----- _. _-- --_ ........... :-.- ....... _-_ . .. _.- -. -.-----;.-·· .. ·· .. ····· ····· · · .... .. .... 10 1 10 2 10 3 10 4 k Figure 2.6; The Bottleneck Effect Three spectra, offset for clarity; t he top spectrum has viscous and resistive terms vsks, the middle has v4k4, and the bot t om has vk 2 . 75 observed. Although strong turbulence is more generally applicable than weak, the latter is a simple and useful model. There are a number of issues in strong turbulence that are not understood . Weak turbulence can be used as the first step in explaining them. For example, in this chapter we found that the bottleneck effect appears in weak turbulence, where its interpretation is straightforward. There are a number of other issues that we intend to investigate in weak turbulence as a prelude to understanding them in strong turbulence; for example, turbulence in the presence of a high magnetic Prandtl number and reconnection. 2.10 Appendix: Kinetic Equation In Weak 'IUrbulence 2.10.1 Preliminaries The kinetic equations describing weak turbulence were derived in Galtier et al. (2000) using the theory of weak turbulence. In this appendix, we present a more physical derivation. We consider the evolution of w t in a plane that is transverse to i and moving with velocity vAi, i.e. , with fixed zt == z - VAt. (Recall that i is in the direction of the mean magnetic field.) Changing variables from z to zt in equation (2.8) gives (2.59) In weak turbulence, the parallel length over which disturbances are correlated, A, is small; in particular, in the time that an up-wave crosses a down-going wavepacket, Ilt ~ A/VA, its distortion is less than unity: Ilw1/wI ~ (wi/vA)A/>"« 1 (eq. [2.15]).9 Therefore, w t undergoes small uncorrelated changes each Ilt ~ A/VA, and its evolution is analogous to a random walk, with step-size IlwI ~ wI (wi/vA)A/>.. « l. If, at fixed z t, w+ is a known function of time, then the evolution equation for w t is linear in w t . Nonlinearity arises because w t modifies w+ (through eq. [2.10]) ; this modification backreacts on w t through equation (2.59). Nonetheless, in weak turbulence the backreaction is unimportant for the random walk of w t at a fixed zt. Every Ilt ~ A/VA, the backreaction changes wI by ~ wI [(wi/vA)A/>..][(w1/vA)A/>"], which is smaller than IlwI by the small factor (w1/vA)A/>"« l. Furthermore, since this backreaction is uncorrelated on timescales larger than A/VA, it only effects a small change in the step-size of the random walk. Therefore, this backreaction can be neglected, and it suffices to consider equation (2.59) as though it were linear in w t . Our goal in this appendix is to quantify the random walk; the resulting equation is the kinetic equation . Before doing so, we solve a simpler problem: a one-dimensional linear oscillator, de9We neglect the factor of 2 associated with the fact that the relative velocity of up and down waves is 2VA· 76 scribed by a linear equation with random forcing. The extension to weak turbulence will then be straightforward. 2.10.2 A Toy Problem: the Linear Random Oscillator We consider the evolution of a simple random oscillator 'IjJ : :t 'IjJ(t) = iA(t)'IjJ(t) , (2.60) where 'IjJ is a complex scalar and A is a real random variable; the factor i ensures that the energy 1'IjJ(t) 12 is conserved. Our goal is to calculate the evolution of ('IjJ(t)) , where angled brackets denote an average over an ensemble of A's, not over time.1° We assume that the values of A at two different times are statistically independent of each other whenever the two times are separated by more than the correlation time, Teorr . (We make this more precise in footnote 11 , below.) For simplicity, we take A to have zero mean ((A(t)) = 0); the extension to A with non-zero mean is trivial. Finally, we assume that the statistical properties of A- such as Teorr and Arms == ((A2))1 /2 - vary only on times cales much larger than Teorr . Note that 'IjJ corresponds to to wI in equation (2.59); A corresponds wi, or, more specifically, to wi/A; and Teorr corresponds to A/VA . There are two limiting regimes for the random oscillator, depending on whether ArmsTeorr is less than or greater than unity. The former case corresponds to weak turbulence , the latter to strong turbulence. We consider ArmsTeorr « 1. The change in 'IjJ within the time Teorr is of order ArmsTeorr'IjJ, which is much smaller than 'IjJ. Since A is uncorrelated on timescales larger than Teorr , 'IjJ undergoes small un correlated changes each Teorr , and thus its long-time evolution is a random walk. Intuitively, we expect that the time evolution of the statistical properties of 'IjJ- in particular, ('IjJ) - can be represented by a differential equation. Our goal in this section is to derive the differential equation , and to understand the approximations that are made in deriving it. The solution of equation (2.60) is simply 'IjJ(t) = 'IjJ(0) . exp(iAt) , (2 .61) where At == f~ A(t')dt' . We expand as follows , 'IjJ(t) = 'IjJ(0) . (1 + iAt - (AAt)t + ... ) , t (2.62) t' where (AAt)t == fo A(t') fo A(t")dt" dt'; note that (1/2)(At)2 = (AAt)t is an identity for any A(t). lOFor a textbook discussion of the linear random oscillator, see van Kampen (1992). His derivation of the equat ion governing ('IjJ) is more rigorous t han ours. 77 Equation (2.62) may be thought of as an expansion III powers of A; as we discuss below, the dimensionless expansion parameter is really the ratio of the correlation t ime to t he cascade time. We evaluate ('lj; (t)) as follows. Since A(t) is unaffect ed by 'lj; (0) , A (t) and 'lj; (0) must be uncorrelated: (2. 63) Since (A ) = 0, (At) = O. Next, we consider (( A A t )t) . The values of A at two different t imes "st a tistically overlap" only when the two times are separat ed by less than the correlation time; i.e., (A( t') A (t' + Ilt)) is non-zero only if Ilt < Teorr . Thus (AAt) :::: A ;msTeorr when t ~ Teom and (( AAt )t ) :::: A ;msTeo rrt Y Therefore (2.64) Equivalently, d 2 dt ('lj;(t») = - A rm sTeorr ('lj; (t )) . (2. 65) This equation is the main result of this section. Its interpretation is as follows: since 'lj; changes by Il 'lj; ~ A rms Teorr 'lj; in the time Teo rr , then after N steps, each Teo rr long, the change in 'lj; is VN Il'lj;. Thus for order unity changes in 'lj;, ('lj; / 1l'lj;)2 steps are required. The result ing time is the cascade time: (2.66) as in equation (2.65) , and Teorr/Teas ~ (Arms Teorr ) 2 « 1. Alt hough ('lj;(t )) decays to zero on the timescale Teas, the energy 1'lj; 12 remains const ant ; the cascade of ('lj; ) is analogous t o phase mixing. We may now proceed to derive the kinetic equation in weak turbulence. Before doing so, we consider the linear random oscillat or in more detail. If the reader is satisfied with the above derivation of equation (2.65) , the following subsection may be skipped . The Validity of Perturbation Theory and Goldreich and Sridhar (1997) Goldreich and Sridhar (1997) incorrectly claim that perturbation t heory fails in weak turbulence, i.e., in their intermediate turbulence.12 We explain their claim and its resolut ion in t he context of the linear random oscillator. In the process, we clarify the validity of the pert urbation expansion. We consider the terms neglect ed in equation (2.64). For example, the fourth-order term is 11 For t he expressio n ( AAt)t ) ~ Arms2 Teorrt to be a pproxim ately valid , we see how fast A m ust decor relate: (A(t' )A(t' + Ll. t)) must go to zero faste r t h a n 1/ Ll.t for large Ll. t > Teorr ; ot herwise, ( AAt)t ) rises faster t h a n t he first p ower of t, a nd ou r derivation is invalid . In add it io n , any order uni ty co nst a nt that mul t ip li es th e right -h an d side of t hi s eq uat ion can be a bsorbed into t he defi ni t ion of Teorr, so in t he followin g we set ~ to = . 12 A note o n t erminol ogy : the tu r bulence tha t we and th at Galti er et al. (2000 ) call weak , Goldreich and Sridh a r ( 1997) call intermediat e . T hey call it intermedi at e precisely because of t heir cla im that p erturb ati on th eory is invali d . 78 (-1/;<4)) / ('!f;(O)) = ({A[A(AAt)t]t}t). In this quadruple time integral , A is evaluated at four different times. Whenever two of these times are separated by less than Teorr , the values of A at these two times "statistically overlap," and hence can give a non-zero contribution to the total integral. When two of the times are separated by less than Teorr , and the other two times are also separated by less than Teorr , but the first two times are separated from the second two times by more than Teo", then this gives a contribution of ('!f;(4)) / ('!f;(O)) ~ (A;msTeorrt)2 = (t/Teas)2. Since ('!f;(2)) / ('!/J(O)) = -A;msTeorrt = -t/Teas , the contribution of the fourth-order term is as large as the second-order term after the time t = Teas. Similarly, if we consider the contribution to ('!f;(2n)) / ('!f;(O)) from correlating pairs of A, the result is ~ (t/Teas)n; so all terms are of comparable value when t = Teas, and it seems that the lowest order term is inadequate. This , in effect, is the claim that Goldreich and Sridhar (1997) make in the context of weak turbulence. It is incorrect; although equation (2.64) for ('!f;(t)) is only valid for t « Teas, equation (2.65) for (d/dt) ('!f; (t)) is approximately valid for all times , with corrections of order powers of Teorr/Teas « 1. Since '!f;(t) undergoes small uncorrelated changes every timestep of length Teorr , we expect on physical grounds that the evolution of its statistical properties should be governed by a differential equation that is invariant under time translations. 13 Equation (2.65) is the only such equation whose smalltime behaviour is given by equation (2.64). Its right-hand side may be interpreted as the lowest term in a perturbative expansion in Teorr/Teas . All of the contributions to ('!f;(t)) that are of order (t/Teas)n must be derivable from equation (2.65). For example, if A;msTeorr == I/Teas is constant, then equation (2.65) has the solution ('!f;(t )) / ('!f;(O)) = exp( -t/Teas) = 1 - t/Teas + (1/2)(t/Teas)2 + .... Terms of order (t/Teas)n in ('!f;) are generated from the lowest order term. We can solve the one-dimensional oscillator exactly when A(t) is Gaussian, and thereby illustrate the validity of equation (2 .65). If A(t) is a zero-mean Gaussian random variable, then so is At, and (exp(iAt)) = exp(- ((At)2) /2) .14 It follows from equation (2.61) that (2.67) Equivalently, (2.68) which agrees with equation (2.65) when (AAt) = A;msTeorr. Although we are mainly concerned with the limit ArmsTeorr « 1, we conclude this subsection by briefly discussing the opposite limit , ArmsTeorr » 1, which is particularly interesting because it corresponds to strong turbulence. Perturbation theory is inapplicable because '!f; does not undergo small un correlated changes every Teorr . Rather , its cascade time is ~ 1/ Arms, which is much shorter 13It should be inva riant on the timescale Teorr; Arms and Teorr are effectively constants on this tim escale. 14Proof: a zero-mean Gaussian x has probability distribution P(x) = (21rX~ms )-1/2 exp( _ x 2 /2x~ms); so (ex p(ix)) = J~oo P(x)exp(ix)dx = exp(-x~ms/2). 79 than the correlation time of A . Therefore A is nearly constant in the time that ('l/J) cascades. The inapplicability of perturbation theory makes strong turbulence difficult, if not impossible, to solve. Nonetheless, we can solve the one-dimensional oscillator in the strong limit. Equations (2.67) and (2.68) are still valid , but (AAt) ,...., A~mst, so ('l/J ) ex exp( -A~mst2), and equation (2.68) is not invariant under time translations. 2.10.3 Derivation of Kinetic Equation We take the Fourier transform of equation (2.59) by transforming in x and y (but not z). We denote the Fourier transform of wt by wk: (2 .69) where k is purely transverse (kz == 0) ; similarly, wk is the Fourier transform of w.J.. k is perpendicular to both k and Z, it only represents a single degree of freedom. Hence it is convenient to use a scalar potential 'l/Jk ' defined by wk = i(kxz)'l/Jk' with k == k/k. Similarly, wk = i(kxz)'l/Jk' The Fourier transform of equation (2.59) is then Since w 81 t at k ,zl 'l/J kt (t) -_ / d2 p Ak,p(t)'l/Jp(t) , (2.70) where (2.71) We suppress the functional dependences of 'l/J t and 'l/J t on z t because this equation is evaluated at fixed z t; in the following , we replace the partial time derivative by a total derivative, with the understanding that z t is fixed. 15 We use angled brackets to denote an ensemble average, in a plane with fixed z t. We define the spectral energy densities e t and e t as follows (Wk(t)·Wk,(t) ) (wk (t).wk,(t) ) ('l/Jk(t)'l/Jk , (t)) = et(k , t)J(k + k') (2.72) ('l/Jk (t) 'l/Jk , (t) ) = et( k, t)J(k + k') , (2.73) where J(k) is a two-dimensional Dirac delta-function that is required by homogeneity (i.e., by the assumption that (w t (X.L) . Wt (X .L + aX.L )) is independent of X.L); note that e t and e t are real. is real, wk = (w~k)*' where * denotes the compl ex conju gate, and so 'l/Jk = ('I/J~k)'; similarly, 'l/Jk = ('I/J~ k) ' · Note that Ak,p = A :' k, - p and ak,p = a:,k ,_ p ' as required by the reality of w t . The differential energy in up-waves within t he k-space area d 2 k is proportional to d 2 kJw t J2 = d 2 kJ'l/J t J2 So Ak ,p = -A*p , k a nd k k 15Sin ce wt ak ,p = -a;,k' as required by the conservation of up-wave energy. 80 From §2.10.1 , '!j;~ (and hence Ak ,p) can be viewed as evolving independently of '!j;t, since the backreaction is negligible at fixed z t. Therefore, equation (2.70) is similar to the simple random oscillator (eq. [2 .60]) , with A k ,p a random function that has correlation time A/VA. We solve it in the same manner , using perturbation theory. Since the bilinear quantity ('!j; k) = 0, we derive the evolution equation for ('!j; k'!j; k'); from equation (2.70), (2.74) where (k f-+ k') represents a second term t hat is the same as the first, but with k and k' interchanged. We expand in A k ,p ; to zeroth order, (2. 75) To first order, (2 .76) where Atk ,p == J~ A k ,p (t') dt'. To second order , ol,t ](2) - j d2Pd2q{ (A At )t[ol,t ol,t] (O) +(A At )t[ol,t ol,t ](O) } +(k f-+ k' ) [ol,t 'Pk 'P k' k ,p p ,q 'Pk ' 'Pq k ,p k' ,q 'Pp 'Pq . (2.77) The sum of these last three equations is directly analogous to equation (2.62) for the simple oscillator. Using the same reasoning here as we did for the oscillator in deriving equation (2.65), we take the t ime derivative of the expected value of equation (2 .77) ; we get, after setting (AAt) = (A/VA) (AA) (with the appropriate subscripts on A) , and after using equation (2.72) and integrating out the delta functions on the right-hand side: i5(k + k' ) :t et(k, t) = V~ j d 2p{ (Ak,pAp ,_ k') et(k' , t) + (A k ,p Ak, ,_ p) et(p , t) } + (k f-+ k') . (2. 78) We use equations (2.71) and (2 .73) and et(k , t) = et(- k , t) to re-express t his as (2.79) Since each upgoing plane with fixed z t interacts with many statistically independent downgoing waves before cascading, it is reasonable to assume isotropy in planes transverse to z; after inserting equation (2 .71 ) for ak ,p and changing variables, (2 .80) 81 where (2.81) We have redefined A to absorb the factor of 71", i.e., A/71" -+ A. Thus far , we have only considered a single plane with fixed zt. If we assume that the turbulence is homogeneous in z, the z-average of the above equation is trivial. Similarly, we re-interpret angular brackets to denote z-averages in addition to an ensemble average. 2.10.4 Steady State Fluxes In steady state, we set the right-hand side of equation (2.80) to zero. We try power law solutions, et(k) ex: k-(3+<>t), e.!.(k) ex: k-(3+<>~), in which case the right-hand becomes (2 .82) Note that the square-bracketed term is independent of ko. The above expression follows after breaking the kt integral into two pieces: one from 0 to k, and t he second from k to 00 . Then, the change of variables kt -+ k 2 / kt is made in the second piece (a Zakharov transformation), so that its limits are now from 0 to k . Finally, this second piece is combined with the first. In steady state, equation (2 .82) must vanish, so (2.83) Since wI ~ (k 2 et ) 1/2 ex: k-(1+<>t)/2, and similarly for wi, equation (2.83) is equivalent to wI wi ex: A (eq. [2 .19]). The vanishing of Gte.!. in steady state yields the same relation as equation (2.83), and so does not give new information. The flux associated with e t is given by integrating the right-hand side of equation (2.80) over d 2 k from 0 up to any particular k. The steady state flux is thus given by integrating equation (2.82), with a.!. = - at. The result of the flux integration is (2.84) where f (at) is a dimensionless function of at: (2.85) A technical note: although equation (2.82) vanishes in steady state, its integral over d2 k gives a 82 factor of at + a-/. in the denominator , which also va nishes in steady state. This 010 a mbiguity can b e resolved by considering the limit as at + a-/. -t 0 (L'Hopital's rule). Equation (2 .84) and the corresponding equation for €-/., a re equivalent to those used in the heuristic discussion (eqs. [2.32) and [2.33)). Galtier et al. (2000) plot the function f( a) after numerically integrating the steady-state flux integr al.1 6 They show t hat -1 < a < 1, t hat t he steeper spectrum always carries more flux (i.e., f(a)1 f( -a) is a monotonically increasing function of a ), and that in the limit that a -t 1, €t I €-/. -t 00. (Clearly, t his also implies t hat in t he limit a -t -1 , €t I€-/. -t 0.) In §2 .3.4, we explain the physical reason why -1 < a < 1. When this condition is violated, the cascade b ecom es nonlocal. Galtier et al. (2000) find infinite fluxes when these inequalit ies are saturated because they consider a n infinitely extended sp ectrum , which leads to unphysical results when the cascade is nonlocal. As discussed in §2.4.2, in steady state we are prima rily concerned with the case that the up- a nd down-going fluxes are comparable, so lal « 1. In t his limit, we linearize f about a = 0 , yielding a pproximately f( a) ::: f(O) . 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