The Mechanical and Nomographic Solution of RAOB and APOB Data Citation Fleming, Robert Ernst (1941) The Mechanical and Nomographic Solution of RAOB and APOB Data. Master's thesis, California Institute of Technology. doi:10.7907/dt2n-vj73. https://resolver.caltech.edu/CaltechTHESIS:02122025-164242576 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): Stewart, Homer Joseph Thesis Committee: Unknown, Unknown Defense Date: 1 January 1941 Record Number: CaltechTHESIS:02122025-164242576 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:02122025-164242576 DOI: 10.7907/dt2n-vj73 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16997 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 13 Feb 2025 20:02 Last Modified: 13 Feb 2025 20:14 Thesis Files PDF - Final Version See Usage Policy. 20MB Repository Staff Only: item control page

THE MECHANICAL AND NOMOGRAPHIC SOLUTION OF RAOB AND APOB DATA a thesis by Robert E. Fleming In Partial Fulfillment of th.e Requirements for the Degree of Master of Science in Meteorology California Institute of Technology Pasadena, California 1941 TABLE OF CONTEN TS Part I Separate Nomographical Solutions Ch. I An Outline of General Nomograph y --------1 Ch. II The Calculation of w ___________________ q Ch. III The Calc u lation of e -··----------------17 Ch. IV The Calculation of~ ------------------19 Ch. V The Calculation of ee ------------------ 25 Ch. VI The Calculation of Altitude -- --------34 Ch. VII A Criterion of Stability --------------41 Part II Master Nomograms 44 Part III Slide Rule Solutions Ch. VIII The Calculation of wand e -----------51 Ch. IX The Calculation of ~-----------------54 Conclusions -------------------------------------59 IN'l'RODUC'llON Anyone having frequent recourse to the Stuve, Neuhoff or Rossby diagrams must soon realize the tediousness of the operations involved in calculating repeatedly such quantities as w ee,, and altitude. There are two dis- tinct objections to these types of diagrams: The first is the complexity of nets or grids (systems of lines .for arbitrary constant values of variables). Not only is time wasted in following a given line to its coordinate or value, but the process is physically detrimental to the eyes and is a frequent source of error. The second objection is the small number of soundings for which one sheet may be used, if w or altitude must be found. The following treatise is an attempt to replace these diagrams within a limited scope by other means: and one mechanical. one nomographical , The scope covered by these means is confined to the ca-lculation of thermodynamical quantities, and cannot present a graphic picture of a RAOB or APOB sounding, nor is it ap plicable at present where the concept of work is involved, as in fog formation and dissipation, or in the probability and characteristic levels of a thunderstorm. Since at the present time RAOB and APOB teletype reports include altitude, pressure, temperature, relative humidity and mix ing ratio, the only es s ential thermodynamic quantity each office must calculate itself is eE", the equivalent potential temperature. The computation of this quantity is easily accomplished either by a norno grarn or a slide rule. However, both these instruments may be employed further, particularly the nomogram. With the latter will be developed methods for obtaining w, 8, ed , z and necessary associated quantites, of use both at the sounding station and at the transmitting center, where these quantities must at present be computed. An examination of the following charts will indicate their simplicity of construction and of operation as compared to the dia gr ams now in use. The slide rule has one added advantage, namely its compact size, but ha s not been consider.ed as thoroughly as the nomogram, because of the possibility of adding numerous other diagr ams upon the latter without being cramped for s pace. 1 Chapter I An Outline of General Nomogra phy. Before any charts are to be developed it would be well to consider briefly the construction of nomograms in general, and the properties of the types herein employed. Unfortunately the nomogram is known too little in proportion to its usefulness in the repeated solution of a formula or equation containing three or more variables. In any Cartesian graph this involves a set of lines in addition to, and intersecting, the lines parallel to the value s of ordinates and abscissae. Thus the general equation f(x) + f(y) + f(z)=o when plotted upon Cartesian principles would ap pear as in Fig I. If now a fourth variable ap pears in our equation, still another net, forming a grid with the former net, is added, as shown in Fi g II: ·:.:Z_-'.. ' -,.,.,,-- ~Z =Z3 -;.- -" ', ."--> < Z=-2y _,...---X.:. ~ L---~__,__ _.____~ f ( -K-} f(x) f(y) f(z) = O Fig J: f(x) f(y) f(z) f(w) Fig II =O 2 There is but a single nomogram~-termed a third order nomogram from the number of its variables--corresponding to Fig I, but there are a number of alignments or nomograms -c-orresponding to Fig II, as we shall see later. The general third order nomogram will consist merely of three lines, as shown in Fig III. Each of these lines is graduated according to definite functions of the type X0 = g(x) Yo x 0 = g(y) =, h(x) Yo= h(y) etc., where x 0 and Yo are the corresponding Cartesian grid supports, and are never shown upon the nomogram in practice, but only considered in choosing suitable scale factors and graduating the necessary lines. These supports may be of the simple type indicated in Fig IV, or of the more general type shown in Fig V, involving oblique Cartesian coordinates, so that· the range of values to be considered will be .included upon a reasonably sized sheet. '-(C,,) f (~ ] [z) y. re~) f ~ f:~ )', ~(,;) f( ) ,/fCq_) : q Xo z'f '1~ 2.3 <,\ z2- A X2- B x, Fig III x, Fig IV ' ~ 11 3 Xi. j1 )11 Y,, Fig V The procedure of solving for a dependent third variable can most readily be seen from Fig III. A straight line or -edge--sometimes referred to as the index--will be 3 laid from A to Bin such a manner as to intersect the given variables at Y~. 1 and z.,.8 for instance. Then the value of x uniquely corres ponding to the index would be x~.'7 • For a rigorous proof of the validity of these operations one should refer to a standard treatise on Nomography, several of which are listed in the Bibliography. Here but a general idea of the development of this chart will be sketched. From Cartesian geometry it will be recalled that the general condi tion that three points of P1 (x 1 y1 P2 (xi Yi ) , P:s (x:s y3 a straight line is given by (1) ), X2 Y,z. 1 ) lie upon =O 1 X3 A relation must then be found between P1 P 2 , and P3 , , and the position of three corresponding variable points so that for every given set of values of P1 , there will correspond three collinear points. Pi , and P3 This is accomplished by considering the parametric form of the equations for each P. The par ametric forms corresponding to equation (1) above may be 1ivritten X () = X,-- f(x) Yo= y, = g(x) :xo= xi= f' (y) Yo= Y2 = g(y) :xo=x,=f(z) Yo= Y,3 - g (z), and if maintained in this order Xo and Yo may be considered the Cartesian grid support upon which the loci of the curves for x, y, and z may be constructed. 4 Quite f r equently t he set of value s interes ted in, result in sc ale s of widely differing size . In some cases it is possible to introduce arbitrary constants, termed scale factors, so distorting the scales a s to even up t heir extreme values. Thus if we have the equation f(t)+ f( w) + f(z) = O and arbitrarily let X and =/',f (t) y =...f'3f (w) we will have three equations in x and y: 1•x+ O•y - _,µ,f ( t) = 0 O•x+l•y -,,,UJf ( w) = 0 .Lx+...Ly+ 01:5 ""4, f(z) = 0 The condition that any three points as loci correspond to given values oft, w, and z gives us the determinant (2) - f(t) f(w) = o. f(z) The latter will be termed the primary cons tructional for m, as contrasted to the primary basic form f(t) 0 1 f (w) 1 0 - O. -f(z ) 1 1 There are no set rules for t he method of fillin g in the various rows and columns of the latter; and ex per ience and trial and error are of prime neces si ty. Nevertheless, in the forms to be developed her eaft er, as close to an 5 inductive rea soning as possible will be presented, rather than merely the fina l or reduced form. To obt ain the reduced form, from whic h the nomograph ical functions may be constructed , one must realize first of all the form towards which he is striving. For a third order determinant an examination of equation I will show that no row may contain more than one variable. Hence even in the primary form the terms must be so arranged that no t wo variables a ppear in a single row. one column must consist of unit values only. Further, Hence we shall manipulate the determinant by columns, never by rows, so as to obtain unit values in one column. It would be a ppropriate to mention here the properties of determinants which we shall use: 1) If all the terms in a row or column are multiplied or divided by the same number, the value of the determinant will be multiplied or divided by that number. It follows a zero-valued det e rminant may have a column multiplied or divided by a number without change in value. 2) ] f the te rms of t wo columns are identical or any single column elements all consist of zeros, the value of t he determinant is zero. 3) The value of any determinant remains unchanged if to the elements of one column are added a cons t ant times t h e corresponding elements of any other column . 4) The sum of t wo determinants, t wo of whose columns are identical, is a determinant consisting of t hese columns and a column whos e elements 6 are the respective swns of corresponding elements of t he dissimilar columns. In the light of the above we would add t he second and third columns of equation 3 and divide the third row by 2 obtaining (4) f(t) 0 1 f(t) 0 f(w) 1 O- f(w) 1 1 f( w) -f(z) 1 1 -f(z) 1 2 if(z) ½ 1 11 f(t) 0 1 l 1 =0 The nomogram corresponding to the reduced form is t hus seen to consist of three straight lines, wi th f(z) equidistant between f(t) and f( w). A similar manipulation of the constructional determinant gives us: 1 (5) 0 1 0 - 0 l - f(w) = 1 1 ' 0 __J_ I /4;s - f(t) f(z) J.-+.1... f (t) \ f( w) = ...L .,fa, _,,µ3 .,,,,U3 -f(z) 1 0 f(t) 1 1 f( w) 1 ~ ,)'yl,f ( ,A,+.ft:s A +/1:1 Comparison of (5) with (4) shows that a certain latitude of scale scale factors has been introduced into the constructional determinant, which may permit of a better distribution of curves. There are a number of methods of forming nomograms of the fourth order and higher. to grids. One is to resort Since this is what we are se eking to avoid, we will merely mention it in passing by indicating the general solution of (6) f(w)+f(z)+f(t)+f(v) = O is possible with a net as shown in Fig VI: z ) 7 f( ..,) f(w) flv) f(w) Q f{t) ..- / ..... F(-.i.) t =t3 t~ t4 A A.,... ...... ---c ...- Fig VII Fig VI Another method is often possible ·,l\!hen the functions are capable of being separated into t wo separate parts, of t wo variables each, neith er of \Vhich appears in the other. Thus if in (6) we let f(u) f( w) = Q = f(t) f(z), we may form two separate nomograms, as shown in Fig VII, with the Q function identical for both alignments, and hence unnecessary to .scale or graduate. This t ype will be used frequently, and will be termed a double-set nomogram. A third method, termed the set-square, will also be frequently employed. Essentially it consists of t wo nomograms, as does the double-set, but with the supporting Cartesian grid of one rotated through 90°, so that the two indices are at right angles. By t his device a formula involving four variables may be solved by a single setting. Without going through t he projective tran sf ormations and rotations necessary to develop this type, it will merely be stated here t hat if a determinant 8 of the form f(u) g (u) 1 1 f(v) g (v) . 1 1 f(w) g(w) 1 0 f(t) g(t) 1 0 =0 may be found, the four necessary functions will be constructed as shown in Fig VIII for ordinary rectilinear Cartesian supriorting grids or ax es, and as in Fig IX for oblique axes. Y. y, t /3 Fig VIII Fig IX The most complex type we shall wish to solve will be of the form f(u)+g(u)f(v)+f(w)+f(t) = o, or of such a nature as to be reducible to the above form. The solution of such forms as f(u)+g(u, w)f(v)+f(w)= O will not be attempted because of the excessive num ber of settings required. 9 Chapter II THE CALCULATION OF w The mixing ratio, w, for air is given by the expression -n 0 er (1) w :. where e is the saturation vapor pressure • 6 c,c,p ,* r is the relative humidity, in % and pis the total pressure of the air. To conform to general practice, we shall express p and e in millibars. The factor .622 is merely the ratio of the gas constants for dry air and water vapor, and is used in the definition of w to simplify other thermodynamical expressions. For the purposes of nomography it is best that each variable be in a separate term; hence we shall take the logarithm of both side s of (1), obtaining: (2) log w - log .622 - log r+log (p-er) = o. If now we assume the product er small in comparison top, we obtain the approximation (3) log w - log .6 22e - log r+log p = o. With this relationship between our variables, we shall construct a double-set and a set-square nomogram. *-------------------Brunt, pp. 30 , Equation 7 10 A: DOUBLE-SET NOMOGRAM Equation ( 3 ) may be re arrang ed so that (4) log w - lo g r = log . 622 e - log p. The reason for this particular grouping will become apparent later in our choice of scale factors. To arrive at a constructional determinant, let x =_/-1, log w y = _){elog r (4). We may then form in the left hand part of equation the three equations: (5) O•x.+ l•y -- \I lo g r l J A 1 X - fie Y =0 = 0. -Q The condition for consistency of this set in x and y is - log w log r = o. (6) Q Upon changing the signs of the third column, adding the first and the second columns, dividing the third row by ft ____!:_2!: I , and rearranging the order of the columns, we obtain ,,,L-<-02.. (7) l l log w. log w 1 I 0 1 log r -- log r 0 - o. 1 - 11 In the same manner we may construct from the right hand part of ( 4)' the follo wing de terminants: (8) ~ 3 - 0 -::--/id.og .622e 1 -__,«.;l o g _j_ -Q ,A ~ log . 622 e 1 1 J-o g p 0 1 :: o. 3 p ~ " 5 _/-<q G/ _/:_:_ 4 -=-- ,A~ - _A :f _.-A '/ - _/-( 3 1 If r is to vary from . 10 to 1.00 , w from 0.5 X 1 0 to 20 X 10, p from 1 00 to 1000 mb., and e as a function of T, the absolute temperature of the air, is to vary from 2 to ap proximately 150 mb., the a pparent s preads in the various functions will be given roughly by: , or log .10 to log 1.00 log . 0005 to log log 100 to log 1000 log .622 X 2 to log .622xl 50 , or .. 02 -1. 00 to o.oo (for r) , or - 3 .3 to -1. 7 (for w) , or 2 .0 to :s .o (for p) .15 to 2 .0 (for e) Appropriate scale factors would then be: As in many cases to follow, t he absolut e limit of one or mone variables will n ot be considered , becaus e of t h e tendency for the function to go rapidly towards i nfini t y, a s would the functions lo g p and lo g r above. Nevertheles s , we may choose as end values such limi ts as a re not usually exce eded in practice. As for the lower limit one, this will de pend upon whether t he vapor pre s sure i s taken over super-cooled water or over ice, and t he dilemma is avoided here because of the infrequent use of w a t low tempe rat ure s . our constructional determinants may now be ivr i t ten as 12 (9) 0. 6 log w 1 1 log r 0 1 Q 2 1 t:~: lo g . 6?? e l l p 0 l 2 1 = o. It must be borne in mind that Q will necessarily have the same value (be coincident) in both of the third order nornograms, whose scales, grid supports, and values are sketched below in Fig I and Fig II: X I I Iti' y I ------- .,/1 ___ .,.x e,,, tt-Y 1/.cO ,i ;I I I I ,...z I I 2. ,0 I I ~ -.:C q I I I I I / II Fig I Fig II The nomogram, in Chart I, is then constructed from the values tabulated in Table I. A 10 scale (i.e. 12 units per foot) was chosen, with each unit considered one tenth its assigned size in they direction; and because the second column may be multiplied by any satisfactory scale factor, the scales are merely separated horizontally by equal distances of 3.5 on the ten scale. It will be noted that the scales of p and r are constructed to the same logarithmic scale factor, and hence may serve as the A and B or C and D scales of a slide rule if necessary. 13 Since these scales coincide, one of them must be reproduced upon a different line, preferably that of Q. Values are then multiplyable provided t hat the index line remains parallel for the t wo settings. If values of e are graduated upon the T scale, then values of e r , the actual vapor pressure may readily be computed if desired. 14 B: SET-SQUARE NOMOGRAM Equation (3) may again be regrouped as: (10) =O ~ (log e-log_,f2)-log w] + ~ (log e-log }2.~+ log rJ It will be noted that two of t he variables are split and hence appear identically in the two groups. The reason for this will become apparent below when it will be desired to combine two- third-order determinants into a single fourth-order determinant. We may novv write each bracketed group, expressed as a determinant, so that (11) ½ log e 0 l log w 1 0 -1 1 1. 2 log .{2 2. 2 log e + -log r j 2 p log~ 0 1 1 0 = o. -1 1 Equation (11) may now be expressed as a single determinant, because we have two rows identical. (12) We obtain: -log r l 0 1 log w 1 0 -1 1z log e 0 1 0 -1 1 0 p 2· log ;T'2.. = 0 Upon shifting the minus sign of the fourth column into the first row and changing the sign of the latter, and then adding the fourth column to t h e third column, we obtain: (13) log r -1 0 1 log r -1 l l log w 1 0 1 log w 1 l 1 2 log e 0 1 0 0 1 0 i log -1 1 0 log .[2.z -1 1 0 .:2. 2. = 2 .t. 2 log e = o. 15 It will be seen that the above for m is that of a fourth-order nomogram employing a set-square index , as mentioned in Chapter I, with all four variables being straight line functions, due to the absence of functions involving va riables in the s econd column. Alth ough a constructional determinant could be developed, the consideration of the end values of t h e various functions precludes the necessity of proce eding beyond the basic, for the apparent spreads will be given roughly by: -1. 00 to 9.00 (for log r) to -1.8 (for log w) -3.3 0 .20 to 1.1 (for 21 log e) 1.10 to 1. 6 0 (for 2 logib) The scales, values, and grid supports may then be diagrammatically sketched as in Fig I II: ff, I ~x, A~ / ./ I ) / e. =f{r) I' o.,I \ \ \ \ ....:, 3.3 Fig III r w The values of major points are calculated in Table II, and plotted upon a 20 scale, whose unit values may be considered to be divided by 1 0 . The r a tio of • it y of th e axes, ~st , wa s t a k en as :z3 or .2.re .'f , an d it obliqu will be noted that this ratio must be a pplied to both 16 grid supports. As in the case of Chart I, two of the scales constitute an ordinary slide rule; these scales in this case are the rand w scales. Hence the quantity er may be calculated -by a single operation upon these scales. If greater accuracy is desired, a constructional determinant may be used, but the basic is deemed satisfactory, the maximum scale-spread ratio being only 2. 17 Cha pter III THE CALCULATION OF 8 The potential temperature,&, is defined as (1) e ...,- T ( - Pop ).288 , ,, where Tis the absolute temperature y pis the pressure of t he parcel of air p0 is the standard-level pressure of the air and Since p 0 is usually talrnn as 1000 mb, p must also be expressed Lri mb. o-1 of T The quantity .288 is the numerical value AR ore; for air. If instead of p we use pd , the partial pres sure of dry air, we obtain ed , the partial potential temperature. If now we arrange (1) in loga- rithmic form, we obtain (2) = 0 -log e+log T + .288 logJ3!!;' p or log T 1 0 • 288 lo g 1;e, 1 0 = log e 1 1 o. Upon adding the second and third columns of the latter, and dividing the third row by 2, this become s log T 0 1 .288 1 ogp 1 1 ::: o. 2log e l /f>Oe> J. 2 If we allow Tande to vary from 24 0 to 340 C, and p from 1000 to 200, the ratio of spreads does not *------------------------Brunt, pp. 38 , Equation (34 ) 18 exceed 2 , so that a basic determinant will suffice for illustrative purposes. An oblique system of grid supports must be intr oduced, such that the ratio~)(='~.! =19.67. The diagrammatic sketch of the chart thus obtained, reproduced · as Chart II, is shovm below in Fig I. p 0 , I ,_,I ,,- Fig I All values of y have been arbitrarily multiplied by a scale factor of 100, and the chart is laid off on a 20 scale. In practice the quantity e is not usually desired; instead eJ is sought. Rossby* has prepared a table of corrections to be applied toe to obtain ed. However the computer in practice is more bothered than helped by the necessity of using another table , so we should develop either a chart giving used directly, or else obtain it as a special case from a eE chart, considering w=O. Both these methods will now be developed. ------------------------- 19 Chapter IV THE CALCULATION OF 6J A: DOUBLE-~ET NOMOGhAM While the quantity 9J is not employed sufficiently in practice in an accurate fashion to justify its inclusion as a separate series of scales upon a master nomogram, when e or 9J may be obtained as a special case of er, it will nevertheless be instructive to construct such a chart. The exact definition of 9J occurs in: e'J =T(.l!.!..!:-)•288 f- er ( 1) While in the form above any nomogram is difficult to construct, if we substitute the variable w for r, where (2) w = .622p~;r= .622 , we have approximately: e = T[~) 288 d r(! -fJJ. 288 = T( ~ ) PW' where wt = 1- ,r,::i , , The latter is separable into two distinct equations, such that ·- (4) (~)3.47 T =Q =rw1, / (JO O I has been raised to the exponent ,m . where (3) J Upon taking logs of (4), we obtain (5) If we let log Q log+ log w'=log Q-t--3.47log T-3.4?log ed =-0 x =/', log Q y =-,;'1.3'log w' , 20 the three equations obtained are: l•x+O•y -__,,,q,log Q (6) =0 O•x + l•y -..,J.l_;og w'= 0 _/ X _,,u, + y + log....f-= o. /~t>() The consistency of (6) requires that (7) 1 0 ..Prlog Q 0 1 .,1-<3 log _j_ _,,A,(3 I w' - o. - -log-/4-;; Upon adding columns one and two, and dividing the third row by __,,-4,-1-.As, we obtain / t"t__,,,.,,«3 (8) 1 0 .-,,'<rlog Q l l ~log /f, ,,,~ log _,,,/4, -1-A ,,,4iV3 o. F Similarly, for the second part of (5), if we let X =~t1-og Q y = 3.4Y'fog T, we find that (9) 1 0 log Q 1 1 6.47 log T - O. 1 c: :'.'i!:'.".~log 8J ~a. "J87 .A< +/y. ,,/4~ +..fa '-/ A consideration of the spreads desired in our variables leads to the choices: _,,,t(I = • 667 /.3 = 20.0 so that the scale factor for Q is the same in both nomograms, and there is a great magnification in the w scale. A glance at the diagrammatic sketch Fig I and Fig II will 21 indicate that this magnification is accomplished at the expense of a tendency for coincidence in the Q and p scales; since the p scale is used in conjunction with thew scale, no loss in accuracy results, as long as w is not the variable sought. tY ' X. / q p T L,. I c.5' I l}d I I Q I 1/-30 I I I 'fl& -~ 30 w Fig I IJ.~:JS' ~- ....!3..:...?~ - --- --· -- ,r;I.S- I I --- ---... --- -----+ --- Fig II Chart IV has been constructed upon a 60 scale, from values tabulated in Table 22 Chapter IV B: SET-SQUARE NOMOGRAM While a satisfactory double-set chart may be constructed, it has not been possible to construct a set-square diagram with scales more evenly distributed than that developed by Millar. The reason for this will become apparent when we find that the constructional determinant employs the same scale factor for the two variables with ranges most dissimilar. Equation (3) may be written: (10) [i (3-log p-log w'µ- 3.47log ~+~( 3-log p-logW')-::; . 47log8JJO• If we let (11) Q : ~ (3-logp-logw' )+ 3 .47logT = -i (3-logp-logw' )+ 3 . 47log0cJ, two separate determinants may be formed. To obtain the constructional forms, let x =,,/4l.(3 - log p) y =~'t (log w'); if we include the Q term in the T and O functions, we have: (12) 1 1 7",_( 3-log p) 0 l ?"'~og w' I ,Aa_ -'t 6.94 log T -2Q I ,,,/4 1 0 -A,,( 3-log p) = 0 1 -fl'ilog w t I I -1.1z ,Ai -6. 94: log e.1+2 Q Each of (12) may be considered as composed of two separate determinants. = 0 23 (13 ) 1 0 7'12 (3-log 0 1 -.,!'+log w' _L .,,/4'! 1 0 _L 2. p) 1 0 0 0 1 0 6.94 log T I Ai, ..L __,/4.., 2Q 0 7 e_( 3-log p) 1 0 0 1 -_.11,il.og w' 0 1 0 I -6.94 log 9,J _j_ _,,A'r - 2Q - = o, and (14) 2 .,,/4"-t _, 2. = o. Expansion of the right-hand members of (13 ) and (14 ) shows us that these determinants have t he values 2Q and - 2Q res pectively. We may t hen add the left-hand members: (15) 1 0 - (3-iog p) 1 0 -A(3-log p) 0 l log w' +0 1 ~o g w' .J_ /" i. ..J_ fi'{ 6. 94 log T -:z;-~ -6. 94 log 8d .1-- /12 = o, or, upon combining into a fourth-order determinant: 1 0 -,,Ai_( 3-log p) 0 0 1 ~ ).og w ' 0 .J_ __J.... .,A't 6 . 47 log T -1 - 3 . 47 log 9d 1 (16 ) /r,z_ J__ _j_.. ..Az. /''f = o. We shall clear t he first t wo columns of fractions and subtract one from t he other , chang e t he sign of t he third row, add timesthe final column to the first, and divide t he rows so t hat the first column contain s only l's: /12. 0 - (3-log p 0 -;,,A'f _,,-'('f - log w' 0 0 -1 6 . 94 log T 0 l - 6 . 94 log ed 1 l /12. 0 = ( 3-log p) 0 - .,,)''I - log w r 0 ,}f2. -1 6 . 94 log T 1 .,Al 1 6 . 94 log e, 1 = o, 24 or, 1 0 (3-log p) 0 l -;)42. log w' 0 1 ...1 6.94 log T 1 1 1 6.94 log 9J 1 (17) ' = o. We see that the same factor multiplies both the p function, and w. Since the spread in log w' is very much less than that in log p, no satisfactory chart has been found. It is possible that other solutions exist, but, there being no inductive method of obtaining such, we must be satisfied with (17) or its equivalent, developed by Millar. 25 Chapter V THE CALCULATION OF eE: A: DOUBLE-SET NOMOGRAM The defining equation :for ee:, the equivalent potential temperature, will be taken as Lw (1) ee: = eci e "t', * or upon taking logs: lnee: = ln8.,1+/J. -,, Substituting the value previously given for eel , we obtain (2 ) c/::>l~g T-ARlnP'Po +~rcpln0i - 0 We shall regroup (2) so t hat I ; (3) AR log p + (c ,,1neE -ARlnp 0 )-Q =0 l w Cp 1n T +c=r- -Q = o, f where the arbitrary function Q ha s been introduced so that each equation contain only three variables. The second of (3 ) sug gests the following constructional substitutions: X or ~)Rlnp y 5MfQ , x+O•Y -:,,,U~ARlnp = 0 O•x+y _?U,t Q + 0 where =0 /4x ~,;Y + Cp ln9t: -p = O, P; = ARlnp 1 0 • The consistency o:f this set of equations requires that: --------------~~--------- * Brunt, pp. 97, equation 55 26 (4) 1 0 ~ARlnp 0 1 ~ Q -' _,A,('2. .::..L __.A'+ = o. -CP lnG/Pi Upon adding the first and second columns, and dividing the third row by , and rearranging, we obtain 0 1 1 1 - /2 1 (5) _,,,/4a._,,/4'f (P-C lne ) _,,,A __,,,,q2. f E ,r _.A'i- -:A2. =o. The first of equations (3) may most easily be put into a basic determinant: (6) Lw -1 0 CflnT __J_ T 1 Q 0 1 - o. In the above form the variation of L with T has been neglected; in some forms this would not be advisable, but we shall see that the scale of T chosen will be too small to be affected by this simplification. the variation ma~ be easily included. If desired, Upon adding the second and third columns of (6), dividing the second row byi..±:.I, and changing the sign of the first row, t here T results: -~ 2. (7) CeThT ~(I +T) 1 1 - 1 i+T -- o. 1 0 9/2 As we shall see, the scale of w and T are so dissimilar that a constructional determinant is required. This may be derived if the variables of (3) are rechosen as separable teTms involving non-s eparable functions of 27 several variables. Without developing this form in detail, we may immediately for m the constructional determinant corr esponding to (7) by analogy to one of the basic type s listed in Allcock and Jones*, as 1 1 ---- 1 0 1 - 61,.:{:£. /< f (8) 2. T _,/i3 Cp Tfa! ./41 ( rf--r) ~(1 +T)~ ( /~T)-t,A'.J ~l( (~)-11/'3 9 /2.... where 4ft = o, --0, --,/1.J • If we clear the denominator of the second ro w, we have: 1 /1 (9) 1 - 1 0 l r:;12,_ =o, 1 where T = A.,.A-+.,,A3 (l+-T) =A ¼_,.T Equation (9 ) will be reversed vvith respect to it s grid supports by subtracting the third column from the second: -~ 2. ~ I (10) ~Cpl fa_T_ 1 0 2. , ' I - ,,fl/ T' 9/z 1 1 = o. 1 Close comparison of (10) and (5 ) lead s us to the conclusion that the most practical position of scales is such that t he Q scale is ap proximately in t~e center, with the T and vv scales slightly smaller than those of p or 01; . The value of Q will t hen be centered somewhere near the c enter of the diagr am, and no solution of eE will t hen occur beyond the chart, even vvi t h maximum magnification of the sc a l es. ------------------------- *Allcock and Jones: This may be accomplisDed 28 if we impose the conditions that/4, and__,,/'12 are negative, while_.,.,/43 and/fare positive. Consideration must now be given the s preads of the basic functions: , or cPlngG -P from .85 to .94 e£ from 250 to 360 from 100 to 1000 , or T from 250 to 300 , or erT'TAT and 1- ;,variable 0 to 20 Lw from o.oo to 5.9 or z:-- p w from ARlnp . from ,, r, .0 ,::, to . 48 Whatever oblique system we adopt, changing Q and the spread of T with reference to the grid supports, but not in mathematical value, we must still impose the condition .,/1/>03 , in order that the spread upon the w scale be reduced to comparable size with other scales. We may now construct diagrammatic sketches of the two nomograms sought: q w p T ~ I I I I I ..... ...... I ...... I "'-'...., X ---- .....____ - I I I I I I I I --- --- -l.. ~I II Fig II ---- The nomograms of Fig I and Fig II will be employed later in the master nomogram. For the purposes of simplified induction we shall here employ a set of positive scale factors, whose values may be determined more easily. In this case the Q scale will be an end scale in the w-Tnomogram, leading to a smaller spread 29 upon the T scale. The sketches for this case are shown below in Figs III and IV: +Y p w 7o - 5 - - - 'S" - S I I - / / ,;- ,,, F/9 I!Z / X 1tC / I / j/ The values chosen were,,,,,A, = 200,.,-,'<,= 5,/<2 = 1,/~= 'i• It vvill be noted that the latter values of/t',e and.A--1lare those reducing (5) to a basic form. The values have been calculated and listed in Table V , and plotted upon Chart V , using a 20 scale. It will be noted that the extreme obliquity of, coordinates in Fig III further contracts that T scale spread, decreasing accuracy somewhat. 30 B: SET-SQUARE NOMOGRAM From equation (2 ) we saw that (2) This may be separated in such a way that t wo of the variables are repeated: ff1nT-ARlnp+ ~$) - (-~lnT +~p lnef -ARlog p0 -~ = O. (11) For the sake of brevity let p (12) 2 _-4/?lnP, cf i'.) The constant pi has been included with the term involving eE merely to maintain positive logarithmic values, and could be employed with any of the other functions. If, now, we as s ign the value Q to each of the quantities in parentheses, we find: w (13) ~lnT ARlnp 0 w ~ ~T 1 -- Q = (lnT)(-~) 0 1 -1 1 t f' (lne!: -P) 2 0 L, 2:T l 0 1 Since these determinants are not zero-valued, we must carry along upon both sides any operation we perform; however, it will be found t hat th ese operational factors finally cancel out. Upon changing t he sign of t he first row of the first of (13 ), adding its latter columns, and dividing its middle row to obtain unity t here results: -w (14 ) 1 CpTfa T 1-, L + 2 --r L -tz.--r ARlnp 0 1 31 Once again t he derivation of a constructional form corresponding to (14) involves the formation of a new set of variables; rather than develop this, we shall take the general analogous form listed in Allcock and Jones, obtaining: (15) - 1 1 ~3w ,,.., ,(, 3 CpT/ nT ,' 1 ~//., 7 ARlnp 1 (- = Q. l, +-i~ 2T 1 0 Similar operations upon the second of (13) gives us _:::!r w / '< / /:s cp T/4r (16) y, 1 ,/1,k T' 1 0 1 c f (lnB;;;. E ~) vvhere 1 ~-t2T) 2T = Q, -- Ly,-.,4):0, L-t-2 T) T' = Y, +- 2/,-3T • Subtracting the last t wo ·cteterminants, we have w (17) r. 1 ::r <!p Tkt .,,/4,l ,1 T' ARlnp 0 ~'W 1 T _.,,,µ, i.., 1 + .Jc T' T' 1 cf (lnG;-Pt 0 1 .,,Mi 1 - 1 - o, 1 or upon combining into a single determinant: (18) c F' (lneE.:.p~ 0 1 1 ARlnp 0 1 -1 /..J 1 0 1 0 ::1CT/4T T' -/I_~ ,.,-,q, ~ ~ 1 - o. If we change the sign of the second row, add the fourth column to the second, and divide the third column by4f, we find 32 (19) c r(lne;-~) l l l -ARlnp l -1 1 1 _f-. ,,,u,l 0 1 1 0 3 Cp1k _,.,,M. , I.. - ~3 ,,,a I 1 IN' _, = o. A consideration of the desired s pread of variables leads to the choice fa 3 = 2.5 ~ ,= l; actually these scale factors are not separate; rather the ratio,h is the single arbitra ry quantity , for / ' - / _ I ~,l-,-~ ~ 4J: T. _,,,a,L, - ~ , !... - 1-'-, 1 I-,.:;::::, The diagrammatic sketch resulting from (19) is shown below in Fig V and Fig VI, with the value s taken from Table VI • \ I\ lo(\ \ I 31 \ \ p ,/ ' / / 8,f- 97 I / I v:'.r<, I I I 1 w I I I I .5 - - - - - _I Fi g VI Fig V The angle,::,(.,. has the assigned value - I?, '? - /r:;3 . ,2__ - - • 109 • From these figures Chart VI has been constructed, using a 20 scale. Approximately the greatest ma gnification possible has been utilized, but t he w sc ale is still smaller than desirable; while t his scale may be increased 33 very simply by increasing the ratiob , it will be found .,,,a, that the distance between the T and w scales increases rapidly. The possibility of a negative ratio has also been investigated; it is found that the spread in the T scale decreases too rapidly because of the extremely small o< necessary. 34 Chapter VI THE CALCULATION OF AL'rITUDE A: DOUBLE-SET NO~OGR.AM From a consideration of the hydrastatic pressure of a column of air, Equation (1) may be deduced: e-~ RT, (1) p -- where p is the pressure at the altitude z , D ,.o P0 is the pressure at altitude z = 0 T is the temperature of the column of air between z = 0 and z =- z. g is the value of gravity and R is the gas constant for air Equation (1) is strictly valid only for an isothermal column, but little error is introduced if the mean temperature of the column is employed, as is the case with the Stuve diagram. Also variations in g with height and locality will be ne glected, and the standa r d value of the acceleration of gravity employed. Extracting the natural logs of (1) we have: (2 ) ln ¾+- ~ ~z= O*, where ~z is the difference in elevation between the pressure levels p and P., , and T the mean temperature between p and p 0 If now we let (3 ) x =~ 1 ln ~ ------------------------Brunt, pp. 34 , Equation 14 35 the following set of equations is obtained: J;,x (4 ) -+-#r.),] + O - 0 x + O• y- ,,,,.,,,/4/4,= 0 O•x + y = o. - .P·2j ,t,z The condition that these be consistent in x and y is: (5) 1 0 0 1 Upon changing the sign of the third column, adding the first to the second, and dividing t h e first row by we have: /2 RT-r/-<'1 :.,..::---/11/'vR,T (6) I 1 0 /---i zRT 1 0 /'1hn 1 0 f 0~ ,./2,fa"l P, 1 1 /<e3 "z 1 1 _/-< 2 'a [J p --""1+,.,,U2RT _,A, o. The conditions we shall impose upon end values are: 8 , C. rn2.. .S" - 2. g t1 z, 0 to 5 X 10 RT , 3 X 1 0 X 2 X 10 6 C 2. to 3 X 3 X 10 8 ~Po , 0.10 to 1.00. Since the diagrammatic sketch of Fig I shows us that and z are graduated in opposite directions, necessitating an oblique system of grid supports, appropriate values for the scale factors would be: p, (7) = 5.0 /2..= 5.0 X 10 The constructional determinant then becomes: 11. 5 lo g l;p0 0 1 4 . 9xl0 -Az 1 1 0 ~l ~+.0l'/3T = o. 36 I '1f X / to/ro ;t X { 1, / //,!, / / Q / / / ~ ·/ / / t? L\z; / / § -~ Po I f / / I I I ,j. .l, y I I / :l. 5' / ,{ ' p 'j Fig I Fig II The values according to which the sc a le s of Chart VII have been graduated are listed in Table VII, which also contains the scalings for Fig II. A lar ger scale for T andAz could be used, but was not deemed jus t ified by experimental data. Since in practice t he r a tio P/~ is not known , while p and p., are, a nomogram solving this ratio must be included, with the same P/Po scale. Since we want (8 ) or Q : P/R, log Q - log p + log p0 = 0, where the Q scale is the same as the P/P,, scale in t he alt i tude nomogram, t he determinant resulting will be: (9) log Q 0 l log Q 0 1 log Po 1 0 - log Po 1 1 - o. log p 1 1 i log p 2 1 - The second determinant result s from adding the second and third columns of the first, and then dividing t he 37 third row by 2. A diagrammatic sketch of the resulting nomogram is shown in Fig II. It will be noted that the direction of they grid support has been chosen positive dovmwards; this leads to a pressure scale with decreasing pressures upwards, to which the computer is accustomed upon adiabatic. charts, and which will. enable this scale to be used with but a change in scale factor upon the composite master nomogram later to be developed. There is one undesirable feature of this choice of variables: Az is calculated rather than z, and any mistake in a previous calculation will enter into all suc.ceeding calculations of that particular sounding, and is not readily detected . Personal errors in setting, of course, will tend to cancel out, however. 38 B: SET-SQUAR~ NOMOGRAM The differential altitude may also be found in a single setting by a set-square nomogram. We may split·(2) into two parts such that (10) lnp+j~;= Q = lnp -o2W. 0 If now we let we obtain the sets of equations: x+O•y -,,,kf, (lnp - Q) = O =0 (11) =0 and x + 0 • Y -,,,k(.;:(lnp0 - Q) = 0 (12) Ox +Y -_,,1,11g.6Z = 0 ;;zt_ x - _;;q;I Yo2_____l.-R-r -+ 0 -- o. The conditions that these be consistent in X and. y is that (13) l 0 ::;ti,(lnp-Q) 0 1 ::::,l{gL\Z _!.. _L o2 Rp13 I .3 0 1 0 7/ ln Po -Q = 0 1 :::,/4'f'gL'.l z L ~ ;;.. :2 R;/14 0 Equations (13) may be expanded in the third columns so that o. 39 1 (15) 0 _, 0 _,kl}n pb 1 0 7'-1:1._Q 1 /'1f'g~z t 0 1 0 -I 0 -I I 0 - = o. R]A + An expansion of the right hand members of (14 ) and (15) .Qf(Jµ3 ~ ,;:;._ ,2 will show that these determinants have the value -Q. Vle can then subtract the t wo left hand components, obtaining (16) 1 0 _,.,u, lnp -1 0 1 ~ 3 g.dz 0 _L o<f?½ 0 If we now impose the condition and change the signs of the third column and first row of the right hand determinant, we may combine the two determinants into the single fourth-order determinant -1 0 ___A,lnp0 1 1 0 ,,,,u, lnp 1 0 1 _,,A3 g6Z 0 ____L 0 0 (18) I .,z RV13 -- o. Upon multiplying the fourth row by 2Ry.H and adding the second and fourth columns, we obtain 1 -1 1 ,,M, lnp 1 1 1 ,,A,lnp 1 1 ~3g.D-Z 0 0 0 1 _,,;A,lnp 0 _,,,t11 lnp 0 1 ,)'{.3 g ..62, 0 1 0 -1 ,;) RJu3 __,k(, 0 0 0 0 ,A, 1 0 1 = o. A suitable scale factor for both z and T is not immediately apparent, but if we multiply the second column byA and add it to k times the first column, we 40 vdll have a suitable form: .,,u,log P.,, A+k 1 1 _,,a,log p A-k 1 1 ~ g,c,z \_ 1 0 0 A-2R~Jk 1 0 (20) :::. o. Assuming the same end values of our functions as for the double-set nomogram, appropriate values of our arbitrary constants will be: k = 4. 50 , A: 10. 00 A= 8. 0 0 ,_),(3= 2 X 10 -Ir The diagrammatic sketch according to which Chart VIII wi ll then be constructed is shown in Fig III: Yi t I I I I I 5 -t X i A:Z. 3.o I :L(J.O p p. T --+-J I :2. .0 )''l,~ -- - :Z,ll .i:f.5'-'- - t.:?..o )( I Fig III The second grid support is so placed that the T scale is placed upon the same line as p 0 , eliminating the construction of an extra line. It will be found advisable upon the master nomogram to reverse the coordinate systems, so that p and p decrease upwards; Chart VIII is 0 reversed because of the easy transition from the grids of the e.: chart to those of this one . The values accord- ing to which it has be en graduated are shown in Table1Zlll. W'Dile a setting is eliminated in the set-square diagram, as compared t o t he double-set, the same difficulty remains, in that any initial error in z carries over into succeeding calculations. 41 Chapter VII A CRITERION OF STABILITY One of t he main disadvantages of a nomographic substitute for such charts as t he Stuve is the ina bility to determine the degree of stability of a given sounding directly. Of course, we could proceed to calculate ~ , leading to a simple three scale nomogram scaled logarithmically; however, it is sometimes desirable to have an idea of stability wi thout first calculating ee. In practice, once ee is known, t he computer can estimate for the purposes he requires without computation. A method of determining stability, without knowing 8€, will now be developed. (1) :: where e is t he density of t he air; while t h e adiabatic decrea s e of temperature with height is given by: (2) li clz - Aq -Cp • We may divide these equations, obtaining (3) ~ _ ARr:. ;J p - pep - S 'd • The quantity 6d is, then, t h e adiabatic increase of temperature with respect to hydrostatic pressure, and hence is analogous in use to the quantity ?[J, t he dry adiabatic lapse rate . .Any 6 we should obtain by estimation from the given increments of temperature and pressure, if less than t he 6dfor the mean temperatur e and pressure of the 42 incremental colurrm , will i ndicate a stable layer, while a 6 longer than 6c1 will indicate an unstable layer. (3), we have Taking t he lo gs of log p + log ~"-log T = O. (4) AR A constructional determinant may be formed by letting x ~ ,log p y =:Alog T or l•x + O•y -;_µ, log p = 0 =0 -1-- x- J y -t- log CeARDJ = 0 _,,a, O•x + l •Y ~ 3 log T (5) ,,P-3 Equations (5) require t hat (6) 1 0 log p 0 1 log T = O, -I - I ~ 3 -loge AR or, upon add ing t he first and second columns, and dividing t he third row by,){3~ : ~.3 (7) 1 0 lo g p 1 1 log T = o. 1 ~ ~0.Jlog cebJ fa 3-;,,;,<. 1 A f? _,M-3_;«, From previous knowledge of t he s pread in t he p and T functions, we will choose fa ,= 1 / 3 = -5 , obtaining the constructional dete r minant (8) -log p 0 1 5 log T 1 1 - %log ~ Ye, AR 1 o. 43 The scale factors were chosen of opposite sign in order that p would be upon an outer scale, to fit into the master nomogram. The diagrammatic sketch resulting from (8) is shown below in Fig I. A 10 scale has been used to graduate the values, upon Chart IX , listed in Table IX • \ .3.0 T \ p \ - - _j_ - /f.S\ : \I ~ Fig I. 44 lfiASTSR NOMOGRAMS A: DOUBLE-SET NOMOGRAlVi The double-set charts previously developed will now be combined into a single chart, with as much duplication of lines as practical, so that the fe west possible number of lines will remain. A compa rison of the various determinants shows us that all the p functions may be made to coincide, but t ha t the T scales can not. If the chart is to be approximately 11" x 16", the ge chart, previously plotted upon a 20 scale, may readily be plotted upon a 10 scale. The values used in plotting this . /~re indicated in Fig I and Fig II . •7t r r -,1 r'( " ½1 ~1/ I °'1 ,A )' I w / / I I : I , II [ I I II I Q I I // I I I ~ y I ~ Fig I 31.t. 7i, ' 'f7 I y♦ Fig II It will be noted that t hey axes have been inverted, and the x axes then reversed, in order that the p function be graduated with lowest pressures at the top, corresponding to the direction of pressures upon other charts. We will not be able to graduate all T f unc tions in the same direction, however, and the computer will 45 have become accustomed to each scale individually. It is recommended, of course, t hat t he T sc a les be gr ad ua ted in degrees Centi grade, while 0£ be in degrees ab s olute. The pressure-altitude chart is no w constructed , with its p sc ale superimposed upon that already graduated. This necessitates multiplying they coordinates of all frmctions by 1.373 , orJ,-:;,, effectively multi plying the p scale by 1.58, the s pread on the former p scale, and dividing by 1.15, the scale factor formerly em ployed upon the p function. As s hoV'm in Fig III and IV, t he x axes have been reversed, allowing t he max imum hori zontal distance between scale s. X" \ Q r- - - - - ; I\ I I \ J , \ I 3 /-' ~, llol \ l \1 \ \ f p' l{-7. 0 ' \ \ I I I Fig III ~y Fi g IV The p' or p scale has be en gradua ted upon t he Q sc a le of Ri g I and Fig II, and t he z fu..nction upon t he w line, t h us elimina ting the addition of t wo lines. The w chart must also have its y coord'inates multiplied before the p sc a le coincides with t hat of e~ '• and z. In this case the factor is 1.58. A further simplification has been effected by bringing the rand w scales into coincidence. The value s with which t h is 46 has been accomplished are indi ca ted in Figs V and VI. r w .3 . r I\ I \ I - ·- - -- I.I I I I y i Fig V The use of another Q scale has been eliminated by using the temperature scale of the ee chart. Fi~ally, a stability chart has be en addede They coordinate s of this chart must also be multiplied by 1.58 to achieve coincidence upon the p scales. Other- wise no alteration has been found necessary from t he type of Chart IX. It was found useful to gradua te the T sc ale upon the dummy line of t he w chart. Because a number of lines are found upon the mast er chart, some of them for the same quantity, t h e cases where these may be confused have been distinguished by adding a subscript to the variable, indicating the calculation with which the scale is to be used. A brief summary is also printed upon t he chart, indic a ting the combinations one uses in solving t he various problems. A subsc :cipt n placed before the variable indicates;that this variable is t o be used as a dummy index only, about which the alignment index rotates, and the value graduated is not the true value. 47 B: SET-SQUARE NOMOGRAl~ The various separate set-square charts are combined in the most advantageous manner feasible in Chart XI. Unlike Chart X, several scales have been used to graduate its functions. This chart is best begun by considering the best possible axes and scale factors for the e.e diagram. In Chart VI the spreads in variables require a nearly square chart; hence a new set of oblique axes have been developed for a 16 11 x 11" sheet. The new angle between axes has a tangent equal to 0.0763, considerably less than that of Chart VI. This increased obliquity does not decrease the spread of the temperature scale noticeably, as a check thru Figs I and II will indicate: Fig II With the values indicated a 10 scale fills the chart very well. In superimposing the altitude diagram a reversal of axes is first required, in order that p may 48 decrease upwards. Because the cycle of' the p scale differs from that in the ee diagram, an adjustment must be made. Two possibilities exist; one is merely to multiply the column of the constructional determinant by the suitable quantity, and the other is to change the scale factor in Equation (17) of Chapter VI. Closer scrutiny revetals that any decrease in__,µ,--as would be necessary--increases the spread of the T scale. Since the latter is already calibrated for points beyond the spread in p, a decrease in the spread of the former is more desirable. This is effected by a decrease ink; it is still possible to decrease?,, but a larger decrease in the spread ink is then necessitated, with consequent rapid convergence in the p and p scales. Hence the former alternative is elected, with a multiplying factor .,/.5'i -- O o f ,Ii¥ . • 859 , and a new k • a 1 so se 1 ec t e d • = 3 • 60 1.s The spreads in the variables thus resulting is indicated in Figs III and IV: -5.S T I t------.Y AZ. Fig III .j, X Fig IV A consideration of the variables occuring in thew diagram might l ead one to believe that its pressure 49 scale could also be made to coincide with the other pressure scales; however, t he remaining sc al e s ha ve larger spreads than that of p, and we have made the pressure sca le the largest one in t he other diagr ams. It is possible to develop a constructional dete r minant and satisfactorily warp the scales; howev er, t nis has not been done in Chart XI because it would either lead to confusing double gradua tions upon a single side of a line or to additional lines. The lines already re- sulting from the 8e- and A z d iagrams may be employed in limited ranges t hat coincide with t he spreads of Chart VI, so t hat no alteration from the lat t er other than a small compression of the x axes ha s been made. The actual spreads are shown below in Fi gs V and VI, which, unlike the other diagrams, is laid off still upon a 20 scale. yt I ).,,., .,,... .,,...'f.,,. .,,. .,,.~x I .,,. 17 T / ,. e, 111\ \\I 11 l. 'i 10.0 I I p 33 r w Fig V Fi g VI While accuracy is lessened by this arrangement, i t 1s justified not only by the elimination of ex tra lines, but beca use of t he less fr equent use of t his d iagram, compared to that of 0€ . 50 Finally a stability chart has been added, with its p scale superim posed upon that of the 8e: and ~z diagrams, and its temperatures graduated upon t he same line as w (labeled wt) upon the w diagram. The same scale factors are employed in this as in the double-set chart, but the slo pe of the axes has been altered and the hori zontal distances between lines changed to allow T upon w1 • Th e diagrammatic sket.c.h of its construction is indicated below in Fig VII: Fig VII A short note has been added t o the chart to indicate which scales are used in conjunction. 51 Cha pter VIII THE CALCULATION OF w AND 8 Millar, in the Bulletin of the American Meteorological Society*, outlines a satisfactory method of obtaining w and e upon an ordinary slide rule. A new scale -must be graduated upon the slide rule for each quantity. We have se en that w is defined from (1) (~22 - er W -- .v PJ Hence if e,.. , the existing vapor pressure, and p./, the partial pressure of the dry air, are known, and set upon the A and B or C and D scales, w is readily obtained. Ordinarily, however, eris not known, but T and r, the relative humidity are. The relationship between t hese quantities is (2) er= re= rf(T), where e is a function of T only, and may be obtained from the Smithsonian Tables. We may us e the B scale for r, and the A scale for er and e; since, however, e is not given, we must graduate corresponding values of T above the values graduated upon t he A scale . The procedure in calculating w is t hen to set the B scale index line (1 or 100) opposite t he given value of T, and read t he value of e opposite r upon B. upon A Scale Bis t hen reset with t he value of Pcc1 under t he -value of e,,., and w obtained upon ------------------------- -;:-Mi ll a r, F. G. Bull A. M. S . Oct 05 Page 229 52 A opposite 62 .2 upon B. Since PJ is obtained by mentally deducting e from p, which is given, a slight mental effort is required. However, p may be used in place of pd i f accuracy is not paramount; sinc e , however, no aduitional calculations are necessary to obtain er, it will not be found difficult to employ pcl • • Another scale must be constructed to obtain e or gJ. We have seen that , (3 ) or upon taking logs (4) T. log e = log T + . 288 log 1 Inspection of (4) will show t wo alternatives are possible; one is to construct a scale for Tande, and t h e other, to construct one for p. Since the exponent . 288 compresses the scale, accuracy will be greater if the entire scale is raised exponentially: (5) 3 . 47 log e - 3 . LJ,7 log T + log 1~ , f> and the p function is graduated along the D scale. Now · Upon most slide rules the lenght of the cycle for the C and D scales is 29. 50 upon a 3 0 engineering scale. Hence we must multiply (5) by this quantity in order t hat p may be used upon the D scale: (6) 102 .4 log e =L02 . 4 log T+ 29 . 5 log ~ . The same. scale will be used for T and e, t hen, and graduated according to the equation (7) y = 1 02.4 l og T = 1 02 . 4 log e. The values .to be used in this graduation are 53 tabulated in Table • The position of t h e pointy= 0 is quite arbitrary, but should be sufficiently far off the left end of the slide rule that the usual range of Tande be inc1:uded within the cycle of C and D. The quantity ed may be obtained if instead of p we use pJ; in general, however, this results in ex tra computation and is not done. An empirical method of correction will later be given, or else Rossby's correction table may be employed. It will be noted that the above method of obtaining e requires but a single setting: the index point of the C scale is set opposite T upon the new scale, and e found opposite p upon the C scale. All other slide rule calculations to be outlined will require at least t wo settings. This fact is intimatety correlated with the necessity of using fourth-order nomo grams in all cases except fore, sinc e a slide rule is but a special type of second order nomogram: one in whic t. t h e aligning index shrinks to a point, along with the third scale. 54 Chapter IX THE CALCULATION OF 8£ FROM 0c1 A: From the previous chapter we saw that an ordinary slide rule could be used to calculate e4 directly, providing a special scale was constructed for 9o1• We shall now show that this same scale may be used as a scale for 9£; however, a linear scale will have to be added. We have defined 9i:: as (1) where /...W I w=z;-,- The above may be written as (2) Since the C and ,D- scales a:re constructed upon a 10 log base, we shall rewrite (2) as (3) The constructive scale of a4 was taken previously as y = 102.4 log e4 ; hence we will multiply (3) by a factor of 102. 4 : (4) 102.4 log et: = 102.4 loge~ + w / / where w // ,-, ,I , W = .z. /o.,_. w = 109.8 -::,: 7 3() Equation (4) shows that theed scale will also be the 0£ scale. However, we must construct a new scale, linear in~, according to the scale (5) y = 109.8 ~- 55 These values have been tabulated in Table • The zero value of y may be placed at any convenient point along the scale. Since the zero value of must be placed opposite the given value of ed , it is recommended that the pointy= 0 be chosen to coincide with the left end of the C scale, and that the scale be graduated above the C scale, or above the CI scale, if the slide rule has the latter. It will be noted that the cheaper slide rules, with but A, B, c, and D scales on the face are ideal for this purpose, but that space is lacking on better rules. The procedure, then, is to obtain the ratio? first. The computer will probably find it best to run through a whole sounding, jotting down these ratios before finding e~. With ·the scale factor chosen above w will be expressed as grams per kilogram, or 10 its assigned value. times The computer will also find that less sliding need be done if Tis set upon t he D scale and vv upon the c, and the ratio read upon t he C scale opposite either end of the D scale. Having obtained the ratio¥, they= O or index point for this scale is set opposite the given value of eJ, and t he value of 0~ found opposite the calculated value of 1• Since the station is not particularly interested in ed, and the latter must first be known, in order to obtain ee in this manner a direct method of obtaining e£ will now be developed. 56 B: FROM p, T, AND w DIRECTLY As before let eE;; But we saw from Equation 0 6' = e 4 e w' • (1), Cha pter rn ( !_!!!!!!. ). :l.'i"i J. PJ , t ha t • Hence we may write (6) Upon taking logs to the base 10 equation (6) becomes (7) p,,i + :i.~ 1 og e. E. -- 1 og T - • 288 1 og 7ooo . 30 • It will be noted that the pressure ratio has been inverted, in order to obtain a minus sign in front of the term. We are thus able to consider t he pressure scale as an inverted scale and add the magnitude of t he function to t h e value of log T by merely placing the value of p opposite the value of T, thus eliminating an or- dinary setting. There are now t wo main possibilities in the choice of scale for. T and et: • One is to construct another scale, similar to t he C or D scale, but ranging in value only from 200 to 360--the usual range of T and e£--, thus obtaining an accurate setting , or else it is possible to employ the D sca le (or the C scale, for tha t matter) for T and~, and merely construct one composite scale of p and';- • For the sake of simplicity we shall choose the latter al--ternative. We have seen that 57 (8) y::: 29 .5 log T would be the equation of graduating T upon the C scale, using the 3 0 engineering scale . Therefore we shall re- write (7) as (9) 29.5 log +31.6 -,w. e = 29.5 lo g T-8.50 log_& 1000 E. Here again w is expressed in grams per kilogram. The equations of graduation for p and w will then be: y = 8.50 log 7 ~;0 y - 31.6 --¥ • The pointy= 0 may be arbitrarily placed for graduating p4 , but the pointy= 0 must coincide with p = 1000 for the r scale. Since p does not greatly exceed 1000, the fe w graduations of P J above this value mi ght conveniently be placed above or below the line of' graduation, wh ich could then be used for f• Once a gain the procedure r equires t he calculation of the ratio * ' though we shall indicate a method of obtaining an approx imate solution below, t hat dispenses with this calculation. Having obtained f , the value of pq is set opposite the value of T upon the C scale, and e t value of obtained upon the C scale opposite t he *• i ~ere the calculation is wanted ra pidly, a rough value of ee may be obtained by cons idering T constant in the expression t";-. This assumption leads only to a second order er ror, because of t he small fluctuation in T compared to that in w, and because the total term is small compared to other terms . Such an 58 assumption has been used to plot the Rossby diagram. Here, however, it would be best to assume t wo mean One could be T = 250, and the other T = 295. values. Two scales could then be constructed for w, one in black and one in red, say, and the proper graduation for 1 estimated according to the temperature prevailing. Another simplification may be used, in that p may be substituted for P<A• The err-or arising from this approximation may be compensated for by adding increments according to the Rossby table.* For the sake of rapidity however, the author has formulated the empirical rule that there should be no addition until w =4, that 1 should be added until w = 10, that 2 should be added from w = 11 tow= 17, and that 3 should be added if w = 18 or more. This rule will give 8e upon the border of experimental and calculating error for the case of the a pproximate w graduations. ------------------------*Rossby s correction t able 1 see biblio gra ph;v 59 Conclusions Two similar charts have been developed, both of which will compute RAOB data with greater rapidity than the old type charts. Moreov,er, one of these, the set-square, actually affords greater accuracy than the old charts of similar size. For the solution of thermodynamical problems such as fog dissipation, however, the old type charts still must be used , a nomogram being unable to express a relation betwesn empirical data and a fi xed relationship. For the restrict ed purpose of merely computing Be , an even faster and more compact method of computation has been devised; since a slide rule is an instrument of almost univ ersal usage, the beginner will not find it difficult of use, and since every office has need of this quantity before all others, the slide rule solution is of prime importance. It is hoped that enough l a titude for revision has been indicated, that each individual office may alter the general types herein developed, so that maximum use may be obtained. Table I wxl0 3 log w r .oo 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 1 6 .0 17. 0 18.0 19. 0 20.0 -1.65050 -1.61090 -1.5774 5 -1.54845 -1.52290 -1.50000 -1. 4 5155 -1.41195 -1.37850 -1. 34950 -1.30105 -1.2614:5 -1.19895 -1.15050 -1.11090 -1.07745 -l.048L15 -1.022 90 -1.00000 -0.97930 -0.96040 -0.94305 -0.926 95 -0.91195 -0. 8 9795 -0.88480 -0.87235 -0.86060 -0.84 950 T e .5 .6 .7 .8 .9 1.0 1.25 1.50 1.75 2.00 ~2 .o 3.0 Ll. 0 260 265 270 275 280 285 290 295 3 00 30 5 310 315 3 20 .5 2.25 3.35 4.89 • 7.06 Ufb. 02 14.02 19.38 26.44 35.65 47.55 62.76 82.00 106.10 .10 .15 .20 .26 .30 .65 . 40 . 45 .50 .55 .60 .65 .70 .75 .80 . 85 .90 . 95 1.00 ,5log,622e .0730 .1594 .2416 .3213 .3973 .4702 .5405 .608 0 .6229 .7354 .7957 .8538 . 8 097 log r -l.0000 -0.8269 -0. 6 990 -0.6021 - 0 ...52 2 9 -0. 4 559 -0.3979 -0. 3468 -0.3010 -0. 2596 - 0 .2218 -0.18 71 - 0 .1549 -0.1249 -0.0969 -0.0706 -0.0458 -0.0223 -0.0000 ;p log p 100 150 200 250 300 650 400 450 500 550 600 650 700 750 2.0000 2.1761 2. 3 010 2. 3 979 2. 4771 2.5441 2.6021 2.6532 2.6990 2.7404 2.7782 2.8129 2.8451 2.8751 2.9031 2.9294 2.95 42 2.9777 3.0000 3.0128 800 8 50 900 950 1000 1030 Table II wxl0 3 .5 .7 .8 .9 1.0 1. 25 1.50 1.75 2.00 2.5 3.0 4. 0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 log w -3.3010 -3.2218 -3.1549 -3.0969 -3.0458 -3.0000 -2.9031 -2.8239 -2.7570 -2.6990 -2.6021 -2. 5229 -2. 3979 -2.3010 -2.2218 -2.1549 -2.0969 -2.0458 -2.0000 -1.9586 -1.9208 -1.8861 -1.8539 -1.8239 -1.7959 -1.7696 -1.7447 -1.7212 -1.6990 T e 260 265 270 275 280 285 290 295 30 0 3 05 310 315 320 2.25 3.35 4.89 7.06 10.02 14.02 19. 38 26.44 35.65 47.55 62.76 82.00 106.10 .6 r .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .so .85 .90 .95 1.00 ~ log e .1761 .2625 .3447 .4244 .5002 .57 34 .6437 .7111 .7760 .8386 .8988 .9569 1. 01 286 log r -1.0000 .8239 .6990 .6021 .5229 .4559 .3979 .3468 .3010 .2596 .2218 .1871 .154 9 .1249 .0969 .0706 .0458 .0 223 .0000 - -- - p 100 150 200 250 300 350 400 450 500 550 600 65 0 700 750 800 850 900 950 1 000 J_ 2 1 -r t e·~ 1.10014 1.19119 1. 25;:s5 9 1 .. 30206 1. 341 66 1. 3751 4 1.4,0410 1. 4:2969 1. 4 5257 1. 4 7 327 1. 49217 1.50956 1.52557 1.54067 1.55462 1.56772 1.58023 1.59196 1.60310 Table III T Log T p Log p 240 2.38 0 2 2.3892 .3979 .4065 .4150 . 4233 .431 4 .4393 .447 2 .4548 .4624 .4698 .4771 .4:843 .4.914 .4983 . 5052 .5119 .5185 .5250 .5315 .5378 .5441 .5502 .5563 . 5623 1030 1000 950 900 850 800 750 700 650 600 55 0 500 4 50 400 375 35 0 325 300 275 250 225 200 3 .0128 3. 0000 2.·9777 2.9542 2. 9294 2 .9031 2.8751 2.8451 2.81 29 2. 7782 2. 7 4 04 2.6990 2.6532 2.6021 2.5740 2.5441 2.5119 2 .4771 2.4393 2.3979 2.3522 2.3010 245 250 255 260 265 270 27 5 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 36 0 365 .8640 .2ss lQg:p 240 - 0 . 003 0.000 250 260 0.006 0.013 270 .020 28 0 .028 290 300 .036 .045 310 . 054 3 20 .064 33 0 . 075 340 350 . 087 .100 360 .114 .122 .131 .141 . 1 51 . 1 62 .173 . 187 .202 ½Log e 1 . 1 901 1 . 1 989 1 .2075 1. 2157 1. 2236 1. 2 31 2 1. 2 385 1 .2457 1. 2 526 1. 25 92 1. 2657 1. 2720 1. 2781 Table IV 20 l og(J-~ wxla3 T 6.94Jag T ] ,735] O~frl p 0.0000 .0140 .0278 .0418 .0560 .0700 .0842 .0984 .1124 .1266 .1408 .1550 .1692 .1834 .1976 .2118 .2264 .2552 .2696 .2838 .2982 .3128 .3272 .3418· 0 1 2 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 16.642 16.701 16.760 16.817 16. 874: 1 6.929 16.983 17.036 17.089 17.140 17.191 17. 2Lll 17. 290 17. 358 17. 386 17.432 17.478 17. 524 17. 568 17.612 17. 656 4.1604 4.1753 4.1900 4.2043 4.2184 4.2322 4.2458 4.2591 4.2722 4.2851 4.2978 4.3103 4.3225 4.3346 4.3464 4.3581 4.3696 4.3809 4.3921 4.4:031 4.4140 100 150 200 250 , 300 35 0 400 450 500 550 600 650 700 750 800 850 900 950 1000 1 050 -- -- --- -- - .3562' 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 /000 • 6451Q~ f'> .6450 .5314 .4508 .3884 .3372 .2940 .2566 .2236 .1941 .1674 .1431 .1206 .0998 .0805 .0625 . 0454 __.0295 .0142 .0000 -.0137 Table V ef: 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 360 • 552loga.: -.474 p .158 log p w, .295w 8496 8544 8591 8636 8681 8725 8768 8811 8852 8893 8934 8973 9012 9015 9088 9126 9162 9198 9234 9269 9303 9337 9371 100 125 150 · 175 200 250 300 350 400 4 50 500 550 600 650 700 750 8 00 850 900 950 1000 1050 .3160 .3313 .3438 .3544 .3636 .3789 .3914 .4020 .4111 .4192 .4264 .4330 .4390 .4444 .4495 .4543, .4587 .4628 .4668 .4705 .4740 .4773 o.o 0.00000 0.00369 0.00737 0.01106 0.01474 0.01843 0.02211 0.02948 0.03685 0.04422 0.05159 0.05896 0.06633 0 .07370 0.08107 0.08844 0.09581 0.10318 0.11055 0.11792 0.12529 0.13266 0.5 1.0 1.5 2.0 2.5 3.0 4.0 5. 0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 1 3 .0 14.0 15. 0 16.0 17.0 18.0 ..2,0c, T T' 5C~ -=r 2~0 1450 1500 1550 1600 1650 1700 .56970 .57685 .58365 .59015 .59640 .60234 .1380 .1333 .1290 .1250 .1230 .1176 260 . 270 280 290 300 Table VI e, 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 360 p ,158 log p wxlO 100 125 150 175 200 250 300 350 400 450 500 550 600 650 7 00 750 800 850 900 950 1000 1050 .316 0 .3313 .3438 .3544 .3636 .3789 .3914 .4020 04111 :4192 .4264 .4330 .4390 .4444 .4495 .4543 .4587 .4628 .4668 .4705 .4740 .4773 1 2 3 4 ,522logee-,474 .8496 .8544 .8591 .8636 .8681 .8725 .8768 .8811 .8852 .8893 .8934 .8973 .9012 .9051 .9088 .9126 .9162 .9198 .9234 .9269 .9303 .9337 .9371 t:: v 6. 7 8 9 10 11 12 13 14 15 16 17 18 19 T .06234 T lo~ T 'T' T' 250 255 260 265 270 275 280 285 290 295 300 1.402 1.435 1.469 1.502 1.535 1.569 1.603 1.636 1.670 1.704 1.738 250 255 260 265 270 275 280 285 290 295 300 1840 1865 1890 1915 1940 1965 1 990 2015 2040 2065 2090 3 2,5 w .0025 .0050 .0075 .0100 .0125 .0150 .0175 .0200 .0225 . 0250 .0275 .0300 .0325 .0350 .0375 .0400 .0425 .0450 .0475 '/ __.c- g,'--t.._ 3.118 3.161 3.203 3.245 3.288 3.330 3 . 372 3 .41 5 D.457 o.soo 3.542 Table VII '%Po =Q J.1,5 log f4>,, p ll,5x½log p 1.000 .900 .800 .700 .600 .500 .400 .300 .200 .100 0.0000 0.5273 1.1156 1.7833 2.5536 3.4654 4.5810 6.0201 8.0475 11.5129 100 200 3 00 400 500 600 700 800 900 1000 11.5128 13.2455 14.2592 14.9787 15.5365 15.9924 16.3775 16.7114 17.0056 17.2692 _L.DZ 4,905xlo T .5 1.0 1.5 2.0 3.0 4.0 5.0 2.4525 4.9050 7.3575 9.8100 14.7150 19.6200 24.5250 250 260 270 28 0 290 300 310 5-,.,oN3s-r .5822 .5727 .5634 .5544 .5458 .5373 .5292 Table VIII z I l2. ~~'.ZQ ~lQ~l .980 1.960 2.940 ":3'J 920 ' 5.880 7. 840 9 . l~0 0 200 210 22 0 230 240 250 260 270 280 290 300 310 12. 9150 1 3 .5608 1 4 .206 5 14 . 8 523 15. 4980 1 6. 1 438 16.7895 17. 43 53 1 8.0810 1 8. 7 268 1 9.3725 20.01 83 - 12=:g" 18,40 log p .1z(:xlOl 1. 96 ..:6 1 050 1 0 00 950 900 850 800 . 750 700 650 600 550 500· 45·0 1 400 350 300 250 200 175 • 1 50 :J.25 100 55.59oi 55.200 54.7897 54.3573 53.9010 53.4170 52.9018 52. 3498 51.7574 51 . 1189 50.4234 49.6616 48.8189 47.8786 46.8114 45.5786 44.1214 42.3384 41.2722 40.0040 38.5830 36.8000 .5 1 .0 1 .5 2.0 3.0 4.0 5.0 Table I X ~ Log P T log T ◊J 100 125 150 175 200 225 250 275 300 350 400 450 5 00 550 600 650 700 750 800 850 900 950 1000 105·0 2.0000 2.0969 2.1761 2.2430 2.3010 2.3522 2.3979 2.4393 2.4771 2.5441 2.6021 2.6532 2.6990 2.7404 2.7782 2.8129 2.8451 2.8751 2.9031 2.9294 2.9542 2.9777 3.0000 3. 0212 200 205 210 215 220 225 230 235 240 245 250 255 26 0 265 270 275 280 285 290 295 300 305 310 11.505 11.559 11.611 11.662 11.712 11.761 11.809 11.856 11.901 11.946 11. 990 12.033 12. 075 1 2. 117 1 2. 157 12.197 12.236 12. 274 12. 312 12. 349 12. 386 1 2.422 12. 4 57 .oo .05 .06 .07 .08 .09 .10 .125 .150 .175 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 /ilog3.4934 &4 -.63149 -.56551 -.50971 -.46139 -.41876 -.38062 -.29970 -.23388 -.17809 -.12977 -.04902 .01697 .07277 .12111 .16371 .20185 .23616 . 2678 2 .29680 .32362 Table XII p 1050 1000 950 900 850 800 750 700 650 600 550 5 00 450 400 350 300 275 250 225 200 175 150 125 100 . 8.50 log 11/2000 8e: 102.4 log 0& 1YI T . 18012 .00000 -.19448 -.38896 -.59993 -.82374 -1.06199 -1.31665 -1.59027 -1.8857 3 -2.20694 -2.55876 -2.94772 -3.38249 -3.87541 -4.44448 -4.7657,0 -5.11751 -5.50647 -5.94125 -6.43416 -7.00324 -7.67627 -8.50000 260 265 270 275 280 285 290 295 3 00 3 05 310 315 320 325 330 335 340 345 35 0 355 360 247.293 248.141 248.971 249.787 250.589 251.376 252.150 252.910 253;.657 254.392 255.115 255.827 256.527 257.217 257.895 258.564 259.224 2 59. 873 260.513 261.144 261.765 0 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050 .055 .060 YJL_ ,. ,. 1 6 T XO 9« 0 .158 . 316 .474 .632 .790 . 948 1.106 1.264 1.422 1. 580 1.738 1. 896 BIBLIOGRAPHY Brunt : l:'hysical and Dy namical Meteorology, Cambridge Universit y Press 1939 Rossby's Correction Table : Elements of Weather Prediction : Krick u mimeographed lab notes", Ch. VI Page 19 Allcock and Jones : The Nomogram:: the theor:v and practical construction of computation charts, London Pitman, 1932 Hewes and Seward Design of Diagrams for Engineering Formulas, McGraw Hill, 19 23 Millar, F. G. : Bull. A. M. S., Oct. 35, Page 2 29 Smithsonian Meteorolo g ical Tables : Wash., 1939 Fifth Revision The author wishes to express his sincere appreciation for the invaluable assistanc e of Jack Guerin i n preparing t h e tables and drafting t he charts and Charles Hight, and also to acknowledge the valuable sug ge s tions of Dr. H.J. Stewart, particularly in the development of t h e inverted ~\: slide rule . 100 CHART APPROXIMATE ··w 90 DouBLE 320 80 70 1000 310 60 900 800 16 50 300 700 600 12 40 zIw u ~ liJ c:O ~ 500 w a: 14 ::) v >- ~ -l 0 V, :) V) I Ul ~ a.. 400 <( a_ I- 30 10 ~ w w :) I<C a::: I- ~ uJ s C! u.J .J 8 6 0 > <t 290 a.. 280 ~ 4 3 ~ 20 300 270 2 1.5 1.0 200 260 10 0.5 0 I<t Q:: z'° -~ X I . SET CHAR. T APPROXIMATE SET "w" CHART SQUARE. CH AR, II i ~ ~ ~ ~ ~ ~ re ~ TEMPERATURE .. T ~ . DEG C.A. ~ I.rt ~ \\j 20 IS 18 0 ID t\j 17 I~ 15 14 13 12 II 10 9 8 7 5 4 100 95" 90 85 75 70 o5 bO 55 0 I- 2 LL) u It I{) PRESSURE,, p ,, MB 0 0 ~ S'. 3 0 - 2.5 I<! so w CY'. 45 z 1.75 Q.. 40 t. 35 > 30 25 I- Q X _ /.SO 2 1.25 ~ :J I Ill 20 2. <.!) ->I<! 1.00 .<:} .8 .7 .J LL.I 15 CY .6 .5 , 10 365 CHART 200 m 360 POTENTIAL TE.MPERATURE. (HART 355 225 350 345 25"0 340 335 275 330 325 300 320 325 315 310 350 305 37S 300 400 295 290 285" 280 <; u I.!) L&J 0 w Q'.' ::> I- 275 2.70 -C( et: w Q.. ~ 450 360 ~ 340 0 320 265 11.l w Q:'. 300 ~ ::> II) Ill uJ 5S0 oc Q. 600 :, I- 290 & LlJ 280 ¢! 260 255 " 0 310 500 LJ i3o <! 11.J I- ca '350 65"0 ~ ~ 700 ...J 250 270 245 260 240 250 <( I2 w l- 0 a.. 750 800 85"0 900 240 950 IOOO 1030 ed CHART DouBLE 5E.T : ~ CHART N p T 150 oOO 0 i: ~ a::: ~ z - )( 2$5 200 ~ w 0 290 2 3 250 285 4 5 G 7" 300 <l:• 280 0 350 w II 0 0d 450 350 345 IO <J) I- 270 <r lb 17 I 18 I- 19 u.l u V1 15 Cl. 315 <( 310 :J 0: :::> 0:: Lu 320 er 13 14 325 550 w 265 6: 700 290 750 285 CD ~ 22 23 260 - 85"0 - 275 270 900 - 265 950 260 /000 255 1050 250 24 2'5 280 800 20 21 305 \!) 300 w 0 2!35 w 12 U) 330 CL 650 I- 335 500, 600 275 340 ~ 9 10 I.!) 400 8 255 250 CHART JZ eE CHART DOUBLE SET 1050 1000 950 900 850 BOO 360 0 355 : uJ 750 350 <D 700 345: 340 335 2 330 - 3 325 320 4 3t5 3(0 305 5 ~ ~ 5 ~ ~ lo 7 : 28S £l.. 9 10 I/ ~ w _J I-- 270 Ct 265 ~ 260 LU <t z )( 2 255 250 450 J <( 295 290 275 500 l 6 0 550 0.. 1- 280 GOO <t 300 ~ 8 = 650 <{ 400 10 ~ Q.. 300 ~ > w cr V') -::> 350 ~ :> V'I 250 uJ a:: C.. 12 200 13 175 14 150 15 /6 '25 17 /8 19 20 100 eE CHART - CHART SET SQ.uARE. TI 0 r:? 150 350- \75 340 200 3'30 320 2.50 p 300 310 300 290 350 280 400 270 450 500 550 600 650 700 750 800 850 900 950 1000 1050 260 250 CHART AL Tl TUDE DouBLE YII (HART SET Po 200 /::;.Z o.o 0.5 1.0 p - .3 1.5 2.0 100 3.0 400 - 5.0 .4 200 ,.:,~ n_O ~ - .5 0 - f<I: ~,:,. 0 ~ >500 0 ,5' ..90 .,0 300 600 a: .6 400 700 500 .7 600 .8 4.0 800 700 900 800 900 1000 1000 1050 ALTITUDE. -1000 CHART 950 CHART 1050 1000 VIII 950 900 900 0 ln 850 q -800 -.::t" 750 850 0 0 M .6. z.. 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