Thunderstorm Mechanics Citation Levin, Gerald Balfour (1941) Thunderstorm Mechanics. Master's thesis, California Institute of Technology. doi:10.7907/wz0f-zp56. https://resolver.caltech.edu/CaltechTHESIS:03042025-183312396 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Meteorology) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Meteorology Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1941 Record Number: CaltechTHESIS:03042025-183312396 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:03042025-183312396 DOI: 10.7907/wz0f-zp56 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17043 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 06 Mar 2025 18:54 Last Modified: 06 Mar 2025 18:54 Thesis Files PDF - Final Version See Usage Policy. 15MB Repository Staff Only: item control page

THUNDERSTORM MECHANICS by Gerald Levin Meteorolo gy Cadet United States Army Air Corps In partial fulfillment of the requirements for the de gree of Master of Science in Meteorolo gy California Institute of Technolo gy Pasadena, California 1941 ACKNOWLEDGEMENT The author would like to acknowledge the kind assistance and cooperation of Mr. Herbert Shapiro, Assistant Engineer, United States Army Engineering Corps, in the preparation of this thesis. TABLE OF COl'TTENTS PART I Theromdynamical Structure of the Thunderstorm Page The convective condensation level ( 1) The critical temperature (2) Ma.ximmn acceleration level (2) Maximum vertical velocity level (2) The cloud top (3) Effectiveness of areas in accelerating particles (3) Maximum downward velocity (3) The Ice Crystal Level (4) Calculation of maximum velocity in a thunderstorm (5) Height that cloud particle rises above maximum velocity level (6) PART II Vertical Velocities in Thunderstorms as Found from the Motions of Hailstones and Raindrops The size of various particles occurring in the atmosphere (10) Schmidt's observations on raindrops (10) Lenard's observations on raindrops (11) Deformation of a raindrop (11) Development of the premise that "D" equals ·w11 11 ( 13) Formula. for a falling hailstone (15) Formula for a fallin g hailstone in terms of "Standard Atmosphere 11 ( 15) Table showing velocity and diameter of a falling hailstone at sea level (16) Table showing velocity and diameter of a falling hailstone at 10,000 feet ( 17) TABLE OF' CONTENTS Page Table showing velocity and diameter of a falling hailstone at 20,000 feet (18) Maximum values for falling hailstones (19) Velocity and size of hailstones with their frequence of occurrence (19) Some general conclusions concerning maximum hailstones and velocities (20) PART III Physical Structure of Thunderstorms Similarity of thundercloud to the cumulus cloud (22) Precipitation and streamline velocities in a stationary thunderstorm (22) Precipitation and streamline velocities in a moving thunderstorm (22) Origin of the cold air current preceding a thunderstorm (23) The squall or curtain cloud (23) The thunderstorm circulation (24) The Simpson rupture theory of electrical charl!,e accumulation (24) PART IV Analysis of St. Louis Thunderstorm Description of maximum rainfalls at St. Louis ( 27) Radio Sonde Data ( 28) Elevations of Important levels ( 29) Calculations of accelerations (30) Calculation of the maximum velocity (30) Calculation of the maximum velocity using heirht of cloud top (30) Sources of error in Fulk's diagram (31) Calculation of velocities by Fulk's diagram (32) Summary (34) List of Diagrams Die.gram (1) Thermodynamical Structure of the Thunderstorm Die.gram (2) Relations Between Velocities and Diameters of Falling Raindrops Diagram (3) Curve of Reynold's Number "R" Plotted Against the 11 Coefficient of Drag 11 Diagram (4) Terminal Velocities of Hailstones and Their Diameters for Elevations of Sea Level 10,000 and 20,000 feet Diagram (5) Air Streamlines and Vertical Velocities in a Stationary Thunderstorm Diagram (6) Precipitation Streamlines in a Stationary Thunderstorm Diagram (7) Air Streamlines and Vertical Velocities in a Moving Thunderstorm Die.gram (8) Precipitation Streamlines in a Moving Thunderstorm Diagram (9) Normal Distribution of Electric Charge in a Thundercloud Diagram (10) Sounding for St. Louis Storm Diagram ( 11) Fulk' s Dia.gram PART I THERMODYNAMICAL STRUCTURE OF THE TmJNDERSTOR.M l Thermodynamical Structure of the Convective Thunderstorm General) The pure air mass or convective thunderstorm is a direct result of insolational heating of the earth's surface. Its formation is most common in warm, moist air with a steep lapse rateJ i.e., Tropical Gulf, Tropical Atlantic, or Polar Pacific air in spring. Although the convective thunderstorm occurs most often in summer due to the strong heating at this time of the year it may be found, but less frequently, at other times of the year. If one refers to the diagram (1) on the thermodynamical structure of the thunderstorm, its formation can be easily followed. As the ground becomes heated the air layer near the ground becomes warmer partly due to conduction and partly to the absorption of the long terrestrial waves given off. This layer is then lighter than its surroundings and rises allowing a new and heavier layer to sink, which in turn is heated. If the ground continues to be heated the process goes on until the convectively mixed air reaches its saturation level. (eel). This level is called the convective condensation level It is the base of the cloud and can be found as follows: Assumine; that the specific humidity (w) at the ground remains constant and is the same at the eel. we have merely to draw a constant (w) line till it intersects the original sounding curve to find the eel. If (w) does not remain crnstant we must take an average value below the eel. Convective mixing below the eel. tends to establish a dry adi~batic lapse rate and therefore a particle arrivin~ at the eel. is in neutral equilibrium, that is, it has no te ndency to either r i se or fall. C: ~ - - - - , ~ - - - / C /ovd To~) ~ Sov,,J,;t --r----- Mo t.,,;,u>m Ve/4c; f( .(),wn ula. .,,.d} \. ~ I f "\ "\ \J ..... "'- o·~ /e .... f'e-r<:> W:: J•S. t1 I-u,,.e f 1/.10 Yo/'01/ /~ -· /l,r;J' /1~ Is ~ Salv~• /e / -4ct'J~Jo..l, c Y-4 "' tJ ;J /ld1,q·1, ~/, ·c A The temperature that the ground must reach in order that the ~;loi,l.d formation may take place is called the Critical Temperature (Tc). Its position can be found by running a dry adaibetic from the eel. to the ground. .An answer as to whether the Tc. will be reached at a particular station under specified conditions can be found by referring to previously plotted diurnal temperature curves. As soon as a cloud particle rises above the eel. it follows a saturation ada~bv;ic curve and is everywhere warmer than the surrounding air; this mee.ns that it is continually accelerated until the saturation adai be tic crosses the orie;inal sounding curve or the two temperatures, of the cloud particle and the surroundine air, are equal. The point of maximum acceleration is where the terr,.perature difference between the cloud particle and the surroundings is greatest. The lar gest vertical velocity is reached at the place where the saturation adaibetic along which the particle is ascending crosses the sounding curve, for the particle is continually accelerated to this point. If the particle's path lies to the right of the sounding curve, we call the area between the two curves the positive area. The greater this positive area is, the greater ~Qll be the maximum velocity reached by the ascending particle. Above the point of maximum vertical velocity the ascent curve lies to the left of the sounding curve. is then called negative area. The area between the two curves The rising particle is everywhere colder than its surroundings and is therefore negatively accelerated. Our cloud particle's velocity thus decreases above the point of ~[mum vertical velocity. 3 At the cloud top vertical velocities are zero. We may roughly estimate the position of this top by saying that when the ne gative area equals the positive area the vertical velocity will be zero. The ree.son that this expression is not very accurate is that a small area at 1000 millibars (mbs.) is tw:i.ce as effective in producing an acceleration as the same size area at 500 mbs. by the follo~~ng: .From the equation of state p -= pressure "fl.= gas constant p,,,. density T :=. absolute temperature ,At the 1000 mb. level let (1) denote the cl oud particle and (2) the atmosphere. P = -P f_ - F ~ :E. { -1.. - T.1 ) -=:.density of particle RT 1 , ~ At the 500 mb. level p' J 2. T"i. 1 ' - ;, I { F; - T density of air } _j__ - _I .... I~ ,. Since the lifting power is di~ectly proportional to the density difference which is in turn directly proportional to the pressure we see that the rising particle will be accelerated twice as much at the l00Omb. level as at the 500 mb. level for the same positive area. We can find the elevation of the maximum downward velocity of a cloud particle if we know how much liquid water is contained. If at the top of the cloud (w) the grams of water vapor per kilogram of air is equal to 0.5 and there is 0.5 grams of liquid water in a kilogram of air, the descending particle will follow a saturation a.dliAbfiv;ic until the line of 1'1/\' 11 equals 1 gm. intersects it. From this point on the descending particle follows a dry adaibetic, as it is no 4 longer saturated, until the original sounding curve is intersected. This intersection is the point of maximum downward velocity. Of especial interest is the position of the "Ice crystal level." This is the intersection of the line of 0 1 l tempeF.ature and the ascent curve of the particle. 1. If the 11 ICL" lies well up near the top of the positive area and the maximum vertical velocity is greater than 8 meters per second the situation. is well suited for a thunderstorm formation and no hail. This is true mainly because the rain drops are shattered before they reach the freezing level and if they do freeze there is not enough additional acceleration above the freezing level left to carry the frozen drops up and form hail. 2. If the "ICL" lies well down near the base of the positive area and there is a good deal of posjtive area. above it, the conditions a.re favorable for hail and no thunderstorm, provided that the vertical velocity at the "ICL" is less than 8 meters per second. The rain drops are allowed to reach the freezing level without shattering and there is then plenty of area to carry the frozen drops a.loft and form hail. 3. A combination of the two above cases is possible if the "ICC" lies near the center of a larGe positive area and the vertical velocity of the "ICL" is greater than 8 meters per second. In this situation it is possible to get a violent thunderstorm accompanied by heavy hail. It ha.s been found that an updraft of 8 meters per second is needed to support the large rain drops found in thunderstorms, and even greater 5 velocities are needed to form hail. Also, rain drops of any size can not fall through air of normal density whose vertical velocity is 8 meters per second or more. Calculation of Maximum Velocity in~ Thunderstorm For a rising cloud particle: c:i: acceleration .P =: density g = gravity p = pressure T = absolute temperature ff.= gas constant The upward force on the cloud particle~~ P Buoyant force = P, g - ?~where?, surrounding air density These two forces are equal so: From ( for par tic le) f: ~?, /fT, ( for surrounding air) ,, T- -r;_ = ' At -r, The acceleration of the particle is therefore equal to the acceleration of gravity times the difference in temperature between the cloud and the surrounding environment divided by the temperature of the surroundings~ The above formula does not give an accurate value for the accelere.tion because it assumes the cloud to be a particle in a stationary atmosphere. In reality the surrounding atmosphere must be 6 •oving dcwnard 'b7 the ve17 nat\lM ot the u,-I"d mo.,._nta ln the thund:erat()ffl• ?he dC!Wffllal"d motion along • dey ad&Aba,,tio tend.a ~o 1-&t ~• atmoaphen and bring ita temptti•ature olcffJ" -to th.e oloud'•• ~ aoulel"at-1.on ca.loulated tor the aloud particle 11 tc:io laJ-g• beeau" AT tor e. 111oving envil"oment is ~U•r than the Ll T \teed• W. oau f'1tld an a.ppfl'ft:inate ftlua to't tt~ 116ttlmum wlcoity in e. thund•rttol"II by th• f'ollowing .f'ermule.1 V = (:z. 21h) ~ h a Mlght h a 'ba•• ot olcud to level of mui1LMD vel<JOl ty {S.. Diagram( l)) a at: • • pobi.t • • tcttlld to be g A -r , T, It T, la tM uaa tapel"ature of the eurroundillg alt" t r . the bate of the olwd t.o the altlmtm ftlcoii:7 level, and 4 T' 1e the aean t~re.turre diffet-H .nM bet.en th• oloud and it-• tt'rroundinge in thie tnttH'"'ml ct' height., the Jllnn e.no--el.,..tlon 1t ~ = I AT' T'I The axtaa wlcoit:, 1a thJa, V = ( .2 f h 4•'1-;::r• .) I t'hla le an &,P Pf'OX1•te method tor finding the helcht tu.~ th• •loud -.,..t't!ol•• will rl•• ab-ow the level of ma.rd!IMll ve:J;'Uoal veloe.H:iye I-I-= belpt olou.d piarid.ol•a r1• above mu1m.um wlooity level ,J, := denat t, ot alt that le nliQg ~ = density- of ,-,n-rKM-lng alr T,: topel"Ct.ture ~r rising air ,;_ = tempoi-a-tmte or euf"'roul'ld:lllt air 7 R.,. gas constant p = pressure V = maximum velocity .d r~ uniform change in temperature difference between risin g air and surrounding air per cer.timeter of elevation. f = acceleration of gravity '-'I= mass of air considered The kinetic of mass (m) at t he maximum velocity level is 1/'.2 h1 Va. This kinetic ener gy must equal the work done a ~ainst gravity in lifting the mass (m) a hei ght of /-I The wei Eht of the mass (m) is effectively zero at t he maximum velocity level as its density is the same as the surroundine; air density. At some other level its wei ght is greater due to the f 0 .ct that the density of its environment is less. Weight of mass (m)-= ""s(F.-Ji) From r.;as law?~_!_ (e.t any level) = 7_;-T; ..P, ~' -~ la_-T, Wei ght of mass (m) at same level is then: Weight of mass (m) at level"H" centimeters above ~imum velocity level - T is an average absolute temperature i n the interval Avera ge wei ght of mass (m) ~,·- = ho'\ { 4 1 J 1-1 ~ Work done in liftinb(J' average wei ght ,.'H ., _ ·- II+. Equating above work to t he kinetic ener gy Y2 l"Yl v '2. ~ yY"\ g A T )( 1-1 ~ :a. r /-4 .. u ,. r, 8 For our limits of error T and 4r can be found accurately enou~h from the sounding on the adta,bQ,tic chart. centimeters per second. second. V is expressed in g is expressed in centimeters per second per 4 Tis expressed in degrees absolute per centimeter of elevation. The value of greater than the " H ,, . H ,. found from the above equation will be actually attained because of the effect of viscosity which was ne glected. 9 PART II VERTICAL VELOCITIES IN THUNDERSTORMS AS FOUND FROM THE MOTIONS OF HAILSTONES AND RAINDROPS 10 Raindrops The following table shows the size of different particles in the atmosphere. Size or Diameter Phase (nnn) Rain 6.1 6.6 Clouds and Fog 0.004 0.1 Snow-crystals .}'raction mm. 0.5 1.5 Dust and Smoke 0.0003 - 0.0017 Various formulas have been developed to find the descending velocities of the above particles. Among the most practical are those of Stokes, Oseen, Goldstein, and Zahm. The difficulty is that these formulas are applicable to the smaller drops while in thunderstorms the lareer drops can easily be supported. Zahm's formula ~mich can be applied to larger drops then any of the other formulas is not accurate for drops larger than 3 mm., but raindrops do not break up until they are from 5 to 7 mm. in diameter. The observations of Schmidt and Lenard furnish the best information as to the size and velocity of falling raindrops. Their observa ':; ions are included here. Schmidt's Observations Diameter (cm.) Velocity (cm/sec.) 0.04 0.06 180 270 0.08 340 393 577 0.10 0.20 0.30 0.35 692 740 l] Lenard's Observations Dia.meter (cm.) Velocity ( cm./sec.( ) 0.10 0.20 0.30 0.35 0.45 0.546 0.636 440 590 690 737 805 798 780 A. curve showing Lenard's and Schmidt's observations is included in die.gram (2). In the same die.gram is a plot of Zahm's calculations. It must be remembered that these results are not accurate for rain drops over 3 l11!ll. in diameter. Looking at Lenard's table it is seen that the velocity decreases ~fter a raindrop reaches a diameter of 0.45 cm. This decrease in velocity is probably due to deformation of the rain drop and the presentation of a larger cross sectional area to the direction of motion. The deformation takes place in the range of "Reynold's Number" / from 740 to 3,950 or in rain drops whose diameters very from 1.7 mm. to 6.6 mm. The following formula which determines whether a drop descending through a viscous fluid is a prolate or oblate spheroid has been developed by Saito. ~ o .S ( I + m) - ~ o <> - 6 1 f n-. - · '-/ ~ .y n-i ~ _ 3 ,.,, ~ ..)(t~internal dynamic viscosity of the drop Jt = internal density of the drop ...,CC = dynamic viscosity of the flu.id y: density of the fluid ) I\ ,- e- ~ . ~- - ,- - ... - )./ n r, I - I ~I ~ , ,. \ ' I t::~ 1 \ -~lI \ ~, \ \~ \ lJ ~ ~ ")I ' \ I ~ ~ - I -..... ~ I ,- .J Op- 4 = 1/ t..~ ..I) ~ ~ ~ c::. -- tc ~ ~l,jJ ' \.) r\ > \ v') -.J <) ')(. .\ "'\._" - ~ ~\ -~ , "\ ' K,- ' "' ' ' ~' ' -s.. ' 4'--c:?--' r- S'd nl - 1,.1.no1:1/\ ' ' ' ..... ... 12 If the above expression is positive after substitution of correct values, the drop will assume the shape of a prolate spheroid. If the result comes out to be negative, the drop will have the form of an oblate spheroid. A rain drop falling through the atmosphere has been shown to take the form of an oblate spheroid, that is, a sphere flattened at both poles. From diagram (2) the conclusions can be drawn that (1) raindrops can not fall through still air with a velocity greater than 8 meters per second, (2) rain drops can not fall through air whose upward velocity is 8 meters per second or more, (3) rain drops shatter when they reach a diameter of from 5 to 7 nun. From an analysis of rain drops only it is known that velocities in thunderstorms exceed 8 meters per second. An idea of the larger velocities autained in thunderstorms can be found from the study of vertical currents needed to s upport hail stones. up in the next section. This will be taken 13 Hailstones Development of formula to find the velocity of falling hailstones: 1. It is assumed for this development that the falling body is a smooth sphere. 2. It is also assumed that the resistance to motion is proportional to the relative velocity of the falling body. v = relative velocity vo initial velocity Q Vt=- terminal velocity m =- mass w l:!. weight g = acceleration of gravity c = a resistance coefficient y = vertical velocity t ;a time According to the laws of motion by Newton, and if a downward direction is positive the equations of equilibrium give J" 11'\ ,__ dt = "rt'\ e - C:.. y Integrating:s:-..!!L ::Z. C ,2.. • o t' l"t"I 'I ~ -::. 'n1 ds- 2'\'\(va.-~)...,. A C:. .. i - C~ - (dv :v~) dt- dS A is a constant of integration to be determined by the boundary condition This gives an A= ~ ;;z. C. 14 s ~ a-, e For all practical purposes as Therefore '4 ~ = 7'Yl g <:... Vt- • terminal velocity or final velocity attained when S is lar ge. C V-t ~ =D where D is the drag offered by the surround in~ medium. Since ,.,,.g -=-W we get JJ..::_ W'.'.,.. The premise that D =W is a fundamental used in several formulas; for the velocities of particles in a viscous medium. Dynamic resistance is introduced by t he equation D =- CJ. 'r A C J>::zabsolute eoefficient of drag (non-dimensione.1) <f = --!£" p V ~ ( 7 I, s. -f t. - .i.) A-=. cross sectional area of sphere normal to direction of motion -= 'IJ.fI J 2... ( f "t ;i.J r'-=density of the air (slugs ft.- 3 ) V = velocity relative to t he air ( ft. sec. - ' ) d ~ di ameter (ft.) W-= ~/"frJs(ArJ where A-:.wei E;ht of wat er per unit volume ( lbs ft. - 3 ) j":. specific gravity of hailstone From D = \v = CD r_ A V..1. p V .2 CI> '7r • l"f • 1. y ing : S imp tettin"': y ~-- •A l !L 3 K~ 4 ( -1 Y) I:, we get:\/ :i== -g (.P} K,":.!L Co Reynold's Numb er is: t ~ /4 'I/J (A y) 3 :t (A o) _, 4 (?J Co V= K o ,,.- R::. ; J ( J CO )k:~ -V- ~~inematic viscosity of air 1,-. -- :;s ..Ac wh e r e ~ - dy namic vis c osity 15 If we In the above expression if values of R are assumed we can find C by its relation to R (diagram 3). ~), ·I -"f N can be found if__r'is known as are constants once a value for hailstone) has been chosen. be calculated. ~ (specific gravity of The diameter of the hailstone can then Its velocity is equal to: _~?r=-]LA ~ d F V - Another form of the equ(a~:)v= /((~~)½:is Q = ~ where a.. ; 0 V ~(~: ):i --1 v . By using constants for the "Standard Atmosphere" set up by the National Advisory Co:rnmi ttee for Aerona 11tics in 1925 the development can be extended. The following relations are good below the isothermal The O subscript refers to sea-level. level (35,332 feet). .?= Yo a ( I- 4f h} i. :z.s-6, = temperature gradient 3 • 566 ){ 10 - 3 4,., .L ft.-' -r;= ground temperature (absolute)= 518.40° . / ; === 2.378.Xl0- 3 lbs. sec.2. ft.-i' ....{: 62.4 A=- 3.575 ,00- 1 V= L3L/. /2(,,tr'J~ -1--:::5 1 CD p,. A ( I - ~ h) 1. s- I~ To Co can be found from dia gram (3). t' (specific gravity of hailstone) varies from pure ice Y = O. 7 / r--- The relation between Rand r= by Reynold's Number O.l, o is representative. to The diameter can be found \ 0. 4 \ \ O.l I \ \ ' \ ""-... 0- I ,__. - -- -- ----- 0 0 z. 8 A I O,,. /{}S' 16 Three cases giving values of R,,De.nd V for different values of the constants are given below. tables. The formula used was The values used are shovm above the ,, ~ '-- (p.2..Co)¼' N G Case I a= O. 'o Elevation-aaSea Level ;V = ft l~t> d (inches)= R j/Os-' 2. ) ,t.- CR Ca. ...:Lb 3 :., 0.50 1.00 1.50 1.75 2.00 2.25 2.50 3.00 3.50 4.00 5.00 6.00 7.00 8.00 9.00 10.00 1.01 1.55 1.89 1.97 1.88 1.76 1.81 1.97 2.14 2.37 2.83 3.29 3.76 4.18 4.63 5.10 For sma ll diRmeters: 't/ = K 3 /:J. N G _R o J:r I 2c. 7 ;{./D 3 1.25 1.91 2.34 2.42 2.32 2.16 2,23 2.43 2.64 2.92 3.49 4.06 4.64 5.15 5.71 6.29 0.25 0.50 0.75 1.00 > 72.3 94.3 116 130 156 188 203 223 240 247 259 267 272 280 285 287 (c5!:.o ) ~ == ~ I f d ~;.,,, CD == O• 'i 'Y ,- CI> is co nsidered constant d (inches) v(rt: s~c.-'l= N. {-e- d J-' . v{+t. 5' ec -1 31.4 44.4 54.4 62.8 17 Case 2 Elevation 10,000 \I =- o. '" ~ ::. V= ~-"~Y/" -'I y 'P-=- 1? v d. -=- ¥. 9~ X Jo 3 3 { 82..CD~X: I{) 3 I J. A'( R: 3 C0 }~10 V ( f- t- sec - 'J = 'If'(~ <I}-' a.so 1.00 1.50 1.75 2.00 2.25 2.50 3.00 3.50 4.00 5.00 6.00 7.00 8.00 9.00 10.00 1.01 1.55 1.89 1.97 1.88 1.76 1.81 1.97 2.14 2.37 2.83 3.29 3.76 4.18 4.63 5.10 1.38 116 2'. 57 2.68 2.56 2.39 2.46 2.69 2.92 3.23 3.85 4.49 5.12 5.69 6.31 6.95 143 160 191 230 248 273 293 303 317 326 334 344 347 352 For small diameters V :=. K (d- ) .!:2 = :i..s- 3 r <1. k~ Co d (inches) 0.25 0.50 0.75 1.00 1.25 88.4 2.ll V {+--r-. sec.-') 36.5 51.6 63.2 73.0 81.9 18 Case 3 Elevation 20,000 y::: o. 6 o I{= '11/ '710 ¢ = --;, ~ i, = ~- 9 ~ 'i Io - ~ K= I 9 'J I\' =- 3. S' ~ /Iv / () 3 Vd. d (inehee) == .• . 3 R ·x1 o _ v{f-t . ')'ee.-'hK{¢J)-' ·1.00 1.01 1.•.66 2.s5 1.89 2.87 110 144 177 1.15 1.97 2.99 198 1,88 1.16 2.85 238 2.74 2.99 3.25 308 0.50 .· 1.so a.oo ,.25 2.50 1..54 2.sa 3.50 1,81 1.97 2.14 4.00 s.oo s.oo 1.00 s.oo s.oo 2 .. 37 :s.so 3,29 3.76 4.18 4.63 4.99 s.7t~ 10.00 s.10 6.34 7,02 7,74 0~25 43.l 0.,75 14.5 86 .l !hOO 2.as 340 366 377 394 407 4.29 o.so 416 427 434 438 60.8 1~00 1.2s 1.so 296 96,4 106 The plot o:f' hailstone diam9ter a gain.st velocity is shown for the three oases tabulated in diagre..in ( 4) • 5 In the region of Reynold's Number eq1-,a.ls o.silO 5 to 2 )( 10 there a.re three terminal velocities that a smooth hailstone may attain. This is shovm by the interval on +:he three curves plottod :i.n diagram ( 4) 32.0 '-') 240 ~ . \.,.__ I '' 2 O(J \.... ~ '-.{ ~ 160 ~ ~ ~ t:t:: ~ /20 Bo '/0 0 0 2. 4 3 OIAM£TcR - IN 6 19 where the slope is infinite. This transitional zone is caused by a change from laminar flow to turbulent flow. It is possible to pick the most probable maximum size for hailstones by choosing the point where the slope of the curves becomes infinite; this is the most stable condition. Below are ta.bu lated the results for the three c urves plotted. r Eleva.ti on \I (j!:s-ec·j 0 >b +;,. m,, m(JJ 0.60 0.60 0.60 Sea. level 10,000 20,000 7b. s.. 0.154 0.220 0.310 2.42 2.70 3.00 133 160 204 W-4.ig h"t· From the above tabulation we can conclude that if hailstones are formed and sustained by vertical currents at an altitude of 20,000, the maximum probable size would be 3.00 inches and the vertical current _, about 200 ft. sec • The size of hailstones and their frequency of occurrence as classified by W.J. Humphreys is a.s follows: Diameter (inches) Occurrence very common often reported not extremely rare doubtful 1 2 3 4 We can classify vertical velocities in thunderstorms accompanied by hail in the following manner assuming that the hailstone is supported at 10,000 feet a nd O~ 0.60 Diameter (inches) 1 2 3 4 5 Velocity (ft.sec. 73 112 296 319 332 ) Occurrence very common often reported not extremely rare doubtful very rare 20 I included diameters of 4 inches and 5 inches because at DallasJ Texas, stones of 4 inches were found during a storm and a stone of record sizeJ measuring 5.4 inches in diameter and weighing one and one-half pounds was found at Po.tter, Nebraska, on July 6, 1928. The hi ghest conceivable vertical velocity in any hail storm should therefore be 332 ft. sec._, ::c 102 meters sec.-' = 227 miles per hour. Velocities of 200 miles per hour have been reported in some cases by pilots unfortunate enough to get caught in a thunderstorm and fortunate enough to get out. The above discussion of velocities does not take into account turbulence except of the 2 per cent that was present when the relation between ~ and CZ> was determined in the N.A.C .A. varia.b le dens i ty wind tunne 1 for a smooth sphere. The turbulent factors are too many and too variable to be included here. 21 PART III PHYSICAL STRUCTPRE OF THL1lJDERSTORMS 22 Physical Structure A thunderstorm is simila.r to a cumulus cloud with regard to the circulation of air but the motions and velocities are much more violent. In general there is an upward circulation of air inside the thunderstorm and a downward circulation in the air surrounding it. The cloud may flatten into an anvil shape on top if the tropopause is reached. a protrusion boils up in the center of the cloud. Often This is caused by a sudden release of latent heat when water condenses causing the air to expand and rise. The precipitation in a stationary thunderstorm is usually heavy in the forward and rear sections but moderate in the center. The reason for the lighter precipitation in the center is the higher velocities that are experienced there than near the edges. The rain drops and s~aller hailstones are blown upwards and allowed to descend in a ring around the center in areas where the vertical velocities are less. As was stated before no rain drops can descend through a vertical current greater than 8 meters per second. Diagrams (5) and (6) show respectively the schematic representation of the air circulation and velocities in an average thunderstorm, and the precipitation stream lines. It will be noted that very little of the precipitation from the anvil edges reaches the ground. It either subliw~tes or evaporates before doing so. Diagrams (5) and (6) refer to a stationary thunderstorm. reality a thunderstorm moves with the prevailing air currents. In This has the effect of tilting the axis of the storm forward in the direction 0 A;,- 5£re'1-wr L;.,es A,./ Ve,-l,a,/ Vekcihes l11 II Ji(,j./"'••t'I"/ T ,v.. de r J fo Yntff. 23 of motion and concentrating the precipitation products in a comparatively small area in the forward edge of the disturbance. The tilting is ca.used because the wind velocity increases with altitude. In diagram (7) the air stream lines and velocities a.re shown for a. moving thunderstorm, while diagram (8) shows the precipitation stream lines in a. moving thunderstorm. In diagram (7) a component of air is shown descending from the thunderstorm and running out in front of it. current can be explained as follows: The origin of this air Convective cooling of ascending air in the storm by evaporation causes precipitation products to form. These products in falling to the ground chill a. column of air. The chilling is caused partly by the initial low temperatures of the precipitation products but ma.inly by their continued evaporation a.s they fall to the ground. The cool column of e.ir is set in motion by the frictional drag of falling precipitation and the fa.ct that the air being cooler and denser than its surroundings sinks. Just before a thunderstorm reaches a station, a cool current of air blowing from the storm is felt. The current owes its origin to the principles discussed above. The low clouds, often called squall or curtain clouds, which a.re sighted near the forward edge of a. thunderstcrm are a by product of the cold outflow of air in front of the storm. The cool air under runs the warmer air in front of it and the lifting thus produced coupled with the cooling caused by the admixture of cool air with warm causes clouds to form in the warm air. 0 ,4i.- Sl-,w,.,tn,5 An ✓ v,./,c. / y'.,/,,,-f,;J I n A l'loir,;t T ~IJn /e yJ foy ~ 24 The circulation of a. thunderstorm is not one typical of many encountered in meteorology; that is, a closed one whose energy is supplied by heating in one portion of the circuit and cooling in another portion; i.e., in land and sea breezes, mountain breezes, etc. The cool air that runs in front of the storm comes from high in the cloud and the warm air which forms the squall or curtain clouds from the surface in advance of the disturbance. These two air masses are not used over and over again but are continually replaced by new air as the storm advances. In other words a thunderstorm is an open convection, Of especial interest is the origin of the electrical charges which cause the li ghtning flashes so often observed in thunderstorms. Many theories have been advanced, but the Simpson Rupture Theory and the Wilson Capture Theory seem to be the only plausible ones which stand up under careful scrutiny. While Simpson investi~ated the charge on rain drops in Simla, India, experiments were conducted which showed that if distilled water has spray torn from it by an air blast the breaking of the water drops is accompanied by the production of positive and negative ions. About three times as ~~ny negative as positive ions are released and thus the water drops retain a net positive charge. In a thunderstorm it is not difficult for the large velocities present to break up innumerable water drops and release many negative ions. The water drops may coalesce and break up several times, but e 13_ch successive rupture is accompanied by the production of a greater positive charge on the wa.ter drop. Light negative ions are blown 25 up toward the cloud top where they unite with cloud particles. heavier rain drops fall out as positive rain. The The above explanation is the reason that heavy rain is usually positively char ged And li f ht rain negatively charged as observed by Simpson. A thunderstorm should have a preponderance of ne gative char ge above and a preponderance of positive charge below if Simpson's theory is correct. Actually it has been observed that a thunderstorm cloud is charged positively above and ne gatively below. This discrepancy is explained by Wilson in his "capture the ory" as follows: A rain drop is charged positively on its lower half and ne gatively on its upper half due to the normal electric field of the earth. In fallin g t hr ough space filled with positive and ne gative ions the drop accumulates negative ions leaving the positive ions to b e carried up to the cloud top. The ne gative charge predominates in the lower part of the cloud where it is carried by the r a in drops. Diagram (9) shows the normal distribution of electric cha r ge in a thundercloud (af ter Simpson). /D JO 8 8 (. ., t l. I, • ::- ::- iVe.1• , re ll11. 26 PART IV ANALYSIS OF' ST. LOUIS STORM (August 25, 1939) 27 Ana.lvsis of St. Louis Storm (August 25, 1939) On August 25, 1939, St. Louis experienced the severest thunderstorm on record. At the Garrison and Lucas Streets station 5.02 inches of rain fell between 4: 35 and 5: 35 a.m. This was 1.4 inches greater the previous record of 3.6 inches. 3.92 inches was recorded in 30 minutes which exceeded any 30-minute or 60-minute previous rainfall. For the total storm 5.79 inches of rain fell at Garrison and Lucas streets. I have chosen this storm to analyze, first, because it is a good example of a severe thunderstorm and, second, because a continuous record cl>f rainfall was kept .by automatic rain gauges during the entire storm. The Radio Sonde data is shown on the next page while the sounding is plotted in diagram (10). The most representative data that could be obtained as to the air that would be present st St.Louis during the storm wa.s a sounding at Oklahoma City taken at 3:00 a.m. The storm commenced at St. Louis at 1:43 a.m. and continued intermittently until 7:05 a.m. The synoptic chart for August 29, 1939, showed a warm front just at St. Louis and since the thunderstorm occurred in the ea.rly morning it was suspected that the trigger action to set it off was supplied by lifting along the frontal surface. However, when the lifting condensation level was found and the particle's ascention curve plotted next to the sounding very little positive area was found. Certainly not enough to result in a thunderstorm. A plot of the ascention curve from the convective condensation level resulted in good positive areas. Although ~onvective thunderstorms usually EL EV 228 1 ,,,,,, llO 69 119 1 .11.9 1 i:: r:; 1 SJ S AL• 3 30M 3•4 1 - fi l 'ZLI.() - 53 _c:; o 4 00 5 -8 -9 -18 - 40 90 A <"l ("\ Ft::C 1 '2. !"i 7 0. 2 1. 2 1. 8 ? .4 Q 7 .6 9. 7 46 90 49 59 46 46 a .~ 47 1 fl\ : .5_., 14 . 8 63 48 ·· q : • 8 ? 714 789 R70 OA_ (l Vt 8 8E GRs./KG , DEGS ABS , DEGS, ABS Rf:. t:: /"\ '2 $ 23 25 24 21 16 TEMP DE GS C 07 () PRESS MBS 7R 4 7 14 22 30 45 50 74 H NO S M RH WNO S t:,-1 LIPTS 5 , 000 FT, PRESS URE DATA 10 , 000 FT. p DESIG _ _ 15 , 000 FT , P. AIR MASS 20 , 000 FT , ~ R E MA R K S DATE- Aug . 25, J 0 30 STAT ION Qk)s bo~lil C L O U DS IS ENTROP I C DA TA Q ~ 0EGS ABs/KM AEROGRAPHIC DATA SHEET COMPUTED BY_ _ _ _ _ _ _ __ 03 CALIFORNIA INSTITUTE OF TECHNOLOGY TIME SOUNDING EST 1939 ST. LOUIS STORM Aug. 25. co N ,2 I r • f I tr} ') r r I i t t L t- t i 7 ~ t t· t t: :tf t- T +-+- +00 + i l- t- .:''iOI 20 t r t t + t t t t + ++ t 2JO ti -: f t 30 ti j f t t t t + r + t +- t t + l- r j(J +- +- -1-+ +- H -+ + t t t t + +- ' t ' + + + +- + + ,._t q.r O t .j.. : + + + +- H +f t t t: t t+' i+ + - : .j.. t tf I r t + + t: -1 30(, J t + t n +t 1 4 JC . (C:t,vo tJ11Jt) t Hl5Q[ 70 =°f' L·f 8 +t L POl!/) ,iil:hluii~:ti t 70 -~ E.O _ ~ 11_ ~"'f,- 4J.1q.- ~ L 200 210 50 l' ,_p 60 t ·40 3t a .2. \ lj \~\ 'y ' u '..J 29 occur during the warmest part of the day it was assumed that this storm was a convective type one especially since there was a temperature of 77° F. present at the time. The storm actually was probably due to a combination of high humidity, steep temperature lapse rate, convective mixing and cooling at the cloud top. The convective condensation level was found by averaging the lower four specific humidity points and running the constant value of specific humidity obtained straight up until the sounding curve was intersected. Very little change in the level of the convective condensation level was found if the lower three points were used instead of the four points in the calculation. The elevations of pertinent points in the cloud obtained from diagram (10) are listed below. Convective condensation level (cloud base) 4,500 meters Maximum acceleration level 10,900 meters Maximum vertical velocity level 13,600 meters Cloud top 17,000 meters Ice crystal level 6,500 meters In the following discussion I have tried to apply some of the formulas developed in Part I on "Thermodynamical Structure of the Thunderstorm. 11 It should be remembered that the results will be correct only to the right magnitude because the formulas do not take account of certain factors and because only average values are used in the calculations. 30 Maximum acceleration . a-=gg g-::;acceleration of gravity in centimeters per second T A11\,.Temperature difference between the particle and the surrounding air 'r~ Temperature of the surrounding air Average acceleration The primes denote mean values. I/. S' e ..., / 3 e c . ;i, Maximum velocity V = 'l~&'h. h:: hei ght that particle is accelerated Maximum Velocity using Hei ght cf Cloud~ The formula. developed in Part I for finding the hei ght that the cloud rises above the maximum velocity level is: H=li~~) ~ 'I' is the average absolute temperature in the - · ,.,, " interJ!al J-1 c1 'T' is uniform change in temperature difference between rising air and surroundine air per centimeter of elevation If II H" is taken off of the sounding in dia gram ( 10) as found by equating positive area and ne gative area., a. value of \I can be calculated from the above equation. V =- 1-1 , L)-i: { g4-,- = 31 This value differs from the previous value for the maximum velocity of 46 meters/sec partially because the previous formula considered the surrounding air as stationary partially because this formula does not consider viscosity of air, and partially because average values were used for temperature functions. = The hei ght taken from the sounding by equating positive and ne gative areas was found to be 3.4 kilometers. It is to be expected that the va lue of "H II from the sounding should be less than "11 11 calculated because of viscosity effects. The only explanation is that the positive and ne gative areas a.re not equated very well or that too low a. ve.lue was a.rri ved at for a maximum velocity in the thunderstorm. Calculat ions of Ve loci ties in Lower Levels of Thunderstorms -by Fulk' s Diagram. In the "Monthly Weather Review" of October, 1935 a discussion of 11 Fulk 1 s Dia. ~~rrun of Approxime.te Precipitation" is given. Fulk has constructed a diagram from thermodynamical considerations which shows the precipitation received in millimeters per hour from a column of saturated air 100 meters thick ascending at the rate of one meter per second. Some sources of error which make Fulk's dia gram only approximate a.re as follows: 1. A pseudo-~daibetic process is assumed. That is a.11 the products of precipitation are assumed to fall out as soon as they are formed, 2~ It is assumed that the column of air does not change its thickness ~Qth height. de 3. An approximate value of di\ ( The rate of change of the saturation vapor pressure with height) is used. This may cause an error as much as 5%. All that is required to find the rate of precipitation in millimeters per hour from the "Fulk's Diagram" is the temperature and pressure distribution, the column thickness, and its vertical velocity. If the precipite.tion rate is known and the vertical velocity is not known one can work backwards and find this velocity. "Fulk' s Diagram II is shown in diagram ( 11). for the The sounding curve St. Louis storm is given in diagram (10). Knowing the ratio of precipitation for this storm some average values of vertical velocities in the lower levels can be arrived at. The calculations are shown below. Maximum rainfall for one hour-=: 5.02" ::a 127 nun. Average rainfall per hour for entire storm ... 1. 74" = 44.1 nnn. Let x ~ vertical velocity in meters per second. MM per hour Elevation 4.5 to 5 KM 5 6 6 7 7 8 9 10 11 12 13 2.7 X 4.9 X 4.2 X 3.3 X 2.7 X 1.8 X 8 9 10 11 12 13 17 1.3 X Total 0.7 x (Extrapolated) o. 2 X (Extrapolated) 0.1 x (Extrapolated) 21. 9 x rri.m per hour ., I ' IC'\' V '\ / I ;f\• I - J V I 400 I/ I/ J I/ I V ) ·~ I I I L;- J ~ V I - /J ) / I/ I J / I I ~oo I ½ ~ tl,l /l I ' ' Ill,. rrE. V V I V /_ "',- /11 r M Pt:. ,iq I-II) v/t. I 700 I I ~ ,r I I/ ~ I '"" ~ t( I ~ I I 800 V I I V / I I I I f\ - I ooo - Jo "I -20 - /0 0 I ,....v ~ ~, '1 I., I IO V V I r •I 11 I):< '/ I I I I h/ I I I .o V J I I A ./l I/ I C/00 V ' I I / L / I ,. (V I/ ti 1/ I/ 20 33 For maximum rain fall: 21.9x :::rl27 X: 5.8 For a.vere.ge rain fall:: 21.9x = 44.1 X:. meters per second 2 •.0 meters per seCiond If a value for the velocity below 10 KIVi. (where most of the rain fall takes place) is found by the formula previously used, the following results: c3., = 4,' ~-:: T' #!f'lo x :l, 3 = V.~ c.t7t/sec-<~, 7 A velocity of 30 meters per second is a.t the 10 KM. level, so a.n average value below this level is about 15 meters per second. This value is a. good deal hie;her than that found by the method using "Fulk's Dia.gram." The fa.ct that the surrounding atmosphere is considered stationary for the solution probably contributes to the lar ge answer obtained. 34 Summary Although a substantial part of the material presented in this thesis has been garnered from various references, it is difficult to ·write a thorough discussion of "Thunderstorm Mechanics II from a purely original point of view. Upon commencing this paper the writer contemplated answering certain pertinent questions concerning thunderstorms. For example: What is the correct magnitude for minim1lJll and maximum velocities in a thunderstorm, and how can these values be computed? How are the stream- lines of precipitation and velocity concentrated in a thunderstorm when moving and when stationary? 'ifuat is the most plausible theory as to the production of the large electrical charge necessary in a thundercloud to produce the electrical phenomena observed? Is there a maximum size to the raindrops and hailstones that can be produced, and what shape does this precipitation take while falling? Can the height that a thundercloud rises above the maximum velocity level be computed by equating kinetic energy to potential energy, and can the maximum velocity be calculated if this height is found from the adaa,bA±ic cha.rt sounding? What are the main physical factors causing the cold outflow of air that precedes a thunderstorm? All of these questions as well as others have been answered to the writer's satisfaction in this manuscript, and if only a little knowledge concerning "Thunderstorm Mechanics" is imparted t o others reading the manuscript, then it has fulfilled its purpose. BIBLIOGRAPHY 1. Miscellaneous Papers J. Bjerkness 2. Studies of a Near Maximum Storm at St. Louis, (Civil Engineering, April, 1940) Horner 3. Fulk's Diagram of Approximate Precipitation Fulk (Monthly Weather Review, Oct., 1935) 4. Synoptic Meteorology Notes Irving Krick 5. Weather Analysis and Forecasting Sverre Petterssen 6. Thunderstorm Forecasting Lts. McGinnis and Barry 7. The Dynamics of Raindrops and Hailstones Albert A. Koch 8. Physics of the Air Humphreys