Numerical Integration of the Primitive Equations on the Hemisphere

SEPTEMBER1959 333

NUMERICAL INTEGRATION OF THE PRIMITIVE EQUATIONS ON THE HEMISPHERE

NORMAN A. PHILLIPS

Massachusetts Institute of Technology, Cambrldge, Mass. [Manuscript received July 6, 1959 remsed August 31, 19591

ABSTRACT

A 48-hr. forecast for the entire Korthern Hemisphere of a barotropic hydrostatic atmosphere is made with the "primitive equations." Overlapping Mercator and stereographic grids are used, together with the finite-difference schemeproposed by Eliassen. Initial data corresponded to a Haurwitz-type pattern of wavenumber 4. The initial wind field wasnondivergent and the initial geopotential field satisfiedthe balance equation. The computations seem to be stable and well behaved, except for two small temporary irregularities. The amplitude of the gravity-inertia waves present in the forecast geopotential field is about 1/30 that of the large-scale field. It can be shown that this is due to the neglect, in the initial data, of the quasi-geostrophically conditioned divergence field. The computational technique itself therefore does not give any unreal prominence to the "meteorological noise." The computational charact,eristics and stability criterion of the Eliassen finite-difference system are investigated for a linearized version of the equations.

1. INTRODUCTION strophic forecasts. However, boundary conditions for the solut'ion of theprimitive equations by finite-differences The so-called "primitive equations" have not bcen used also require considerable care in their formulation, as has much in numerical forecasting because of two main diE- been pointed out, for example, b~-Smagorinsky [17]. This culties. First, if theinitial wind and pressure fields are problem is greatly simplified if the lateral boundary of the not known accurately, art'ificially large gravit'y waves will forecast region can be placed on the equator, where suit- appearin the forecast [3,8]. Secondly,t'he computa- able symmetry assumptions CRI~be imposed on the fore- tional stabilit'y crit'erionfor these equationsrequires a cast'variables. Although theequator as a boundary is time step of at most' 10 minutes compared t'o the 40- to readilJ- fitted int80 either a spherical coordinate system or 60-minute timestep allowed in the geost'ropbic system. intocoordinates on a Mercat'or map, both of these co- The development' of larger and faster computing machines ordinate systems have singu1arit)ies at the SorthPole. is rapidly eliminating the second difficulty. It also seems In an attemptto avoid thisproblem, the writer has probable that' a gradualimprovement of the rawin and suggested the simultaneous use of a Mercator map in low radiosonde net'work,combined with special ana1.vse.sof latitudes and a stereographic projection in high latitudes the initial data, may go far toward solving t'he first, diffi- [19]. However, thecomputational stability of this sys- culty. A stable and accurat,e computation scheme is t'hcn tem is then too complicat,ed a question to be examined by all that will be required to take advantage of the more mathematicalanalysis. A numericaltest of the scheme faithfulreproduction of atmospheric processes which is has therefore been made and t'he results are described in possible wit'h tjheprimitive equat'ions. (The geostrophic thispaper. The equations used were thoseappropriate system not, only fails at'short' wavelengths [2], but also to a homogeneousincompressible atmosphere moving loses its special prognostic value at' extremely long wave- h.vdrost'atically. The initialwind and pressure (geo- lengths [I]. In a,ddition,certain import'ant effect's such potential) fields were defined mathematically, rather than as the horizontal variat'iorl of stat'ic stabi1it.v cannot, be being obtainedfrom a weathermap. The computations incorporat'ed into the geost'rophic system [IO,] I].) are therefore a test only of this method of computation, There are t'wo aspects to the design of a good compu- anddo not purport to answer t'hequestion of whether tationscheme for the primitive equations: (a) t'he finit'e adequat'e initial data can be defined for real forecasts. difference equivalents of the partial differential equations themselves, and (b) the formulation of lateral boundar~- 2. THE EQUATIONS OF MOTION conditions.Eliassen (61 andPlatzmann [I51 have discussed the former and have arrived at, a finite-difference The horizontal equations of motion in spherical coordi- scheme for the primit'ive equat,ions which is more efficient' nates,assuming hydrostatic balance and neglecting t'han the type of finite-differences currently used in geo- friction, can be written

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.. 1 b4, u=-aX cos e sin x-ae cos X=m"k, aX COS 8=2~e(~+n) sin e-- - u COS e zn v=aX cos e cos X- ad sin X=m-ly, (9) aO=--aX cos O(X+2n) sin O--"-. 1 b4 a be (2) m=2(1+sin e)-l,

HereX=longitude, O=latitude, a=radius of theearth, so that u and v are the horizontal velocity components Q=angular velocity of the earth, and +=the geopotential along t.he x- and y-axes of the stereographic projection. of an isobaricsurface. Thedot (') is thesubstantial m is the scale factor for this projection. (77,V) and (u,v) derivative : are related by the expressions:

u= - U sin X- V cos X, (10) v= U cos X- V sin X. I ere p is the pressure, and itis clear that the independent variables which are being used are X, e, p, and t. Equa- Thc two horizontal equations of motion and the con- tions (1) and (2) do not contain the Coriolis term 2fki cos 0 tinuity equation (4) can now be written in the map co- orthe inertia term.s 2xa cos e and Zai. Theymust be ordinates: neglected for consistencywhen the hydrostaticappr0xim.a- tion is used, since their counterparts in the third equation bU " of motion have also been neglected. bt Applying these equations to them.otion of an incompress- ible homogeneous atmosphere with a free [surface we find that (1) and (2) carry over as written, if 6 is set equal to gz, where z is thevariable depth of theatmosphere. (3) becomes simpler bythe disappearance of the b/bp operator, since x and e may be taken as independentof the vertical coordinate p. The only other equation needed is the continuity equation, which for this atmosphere can be written bt

+=-m[g+u-+u-dt bubx duld?J +v 2Qsine- (xv-yu~l2a2 (Here g, the acceleration of gravity,has been assumed constant, and variations in a have also been neglected.) We now define the map coordinates for the Mercator and stereographic projections as follows: ?=-m[$+u--bt +v- --u 2nsin 8- bvbx by [ (xv-2a2 yu)l X=aX, Mercator: Y= -u In h,

a=2ah cos X, Stereographic : (6) y=2ah sin X, The Mercator equations (11)-(13) are equivalent to the h=cos 8(1+sin e)-l. (7) stereographic equations (14)-(16) and also to the spherical equations (1)-(4). Following the procedure outlined in [13], we define Equations (11)-(13) have a singularity at the pole where M=se,c e becomes infinite. Equations (14)-(16) U=ax cos e=M-lX, have a singularity at the south pole,where m=2 (1 + V=ae=M-'Y, sin e)-l becomes infinite. As described in [13], the finite- difference solution of (1 1)-(16) is to be cmried out over M=sec e, one hemisphere by applying (1 1)-(13) on a Mercator grid in low latitudes and (14)-(16) on a stereographic grid in so that U and V are the horizontal velocity components highlatitudes. The Mercator gridshould extend from along the X- and Y-axes of the Mercator projection. M the equator to about 43O latitude, with the stereographic is the scale factor for this projection. grid being responsible for the area poleward of this lati- For the stereographic projection we define tude (see fig. 1). The singularities arethereby avoided

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To solve (1 1)-(16) we use the finite-difference scheme proposed by Eliassen [6j for the primitive equations. To describe this scheme in the present context we must define four lattices on the Mercator map and.four on the stereographic. First we introduce the constant space incrcment A to be used on both maps:

27ra A=- , (P=integer). I"

(It will be convenient to take 1 as a11 even integer.) Thc Mercator grid is t'herefore rectangular (P+2) x (Q+2) and thestereographic grid is square (1+2) x (1+2). The num.bers (po,qo) and (io,jo) determine the location on t'he map of the origin of eachgrid. We define thefour Mercatorand the four stereographic grids(denoted by A, B, C, and D)by the four sets of values of (po,qo) and (io, j,) shownin table 1. Figurc 2 shows therelative orientation on the map of thepoints on the fourgrids are defined at each point of a single grid at all time steps. which have the same subscripts (p,q) or (i,j). From the (Notethe discussion of fig. 5 insection 7.) For con- definition of X in (5) and A in (1 7), it can be seen that the venience in notation we will hereafter use a prime super- points p=o and p= 1 on the Mercator grid arc idcntical script (U', V', +', u', v', +') toindicate the quantities w-ith the points p= P and p= P+ 1, respectively. These defined at the intermediate times t= l/2At, . . ., (n+1/2)At. extra pointsare included in the gridsmcrely for con- For example, +,qn will indicate +[X=(p-l/2)A, venience in solving the equations. Y=(q-l/2)A,t=nAt] (st,oredon grid A, where po=1/2

The Eliassen computation scheme is obtained by de- and qo= 1/2). +',,%, however, will indicate +[X=pA, fining thc variables (U, V,+) or (u,n, +) on the grids as Y= (a- 1)A: t= (n+1/2)At] (stored on grid D, where follows. At time t=At, 2At,3At, . . ., nAt, U (or u) is po=O and qo=l.) represented at the latticepoints of grid U; V (or u) is For convenience in writing the finite-difference equiva- represented at the lattice points of grid P;and + (or +) is lents of (11)-(16) we also introduce the notation 6, and represented atthe latticepoints of grid A. Atthc up for the following operators: intermediate times, t=l/2At, 3/2At, 5/2At, . . ., (n+1/2)At, U (or u) is represented on grid C, V (or u) on grid B, ~,S,,=S,,"s,-1 9) and + (or +) ongrid D. Such anarrangement is much more efficient than when all quantities (e.g. U, V,and 6) ~,S,,=~~S,p+Sp-l ,). Similar definitions hold for 6,, u,, lit, 6j, ui,and uj. Finally, to eliminate unnecessary repetition of letter subscripts in TABLE1.-Origin coordinates of the four grids used in the Eliassen the formulae,a quantity suchas S,, will be written simply Jinite-diflerence scheme. as Soo,S, q+l as Sol,etc. Grid In the Eliassenscheme, the six equations (11)-(16) appliedto the Mercator and stereographic projections, result in 12 finite-differenceequations. Quantities ap- pearing on the right side of these equatiotls are understood to have the time subscript n in all cases. The symbol f 525468-59-3

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t 112 A .1

Y (y) B D

FIGURE2.-Relative positions on a Mercator (or stereographic) map of thepoints on the four grids A, B, C, and D whichhave the same {8i'#'l~+~j[(~iU11)(~iU11)I+Z)008j~iU11}+v00At finite-difference coordinates p, p (or i, j). A fooc-= [(i--oboo- (j-jo)~i~j~lIl is used for the Coriolis parameter 2Q sin e. j arid the scale factors M and m are identified by additional subscripts v;on-v& n-l= -rnOOB A, B, C, or D denot'ing the grid on which they are locatcd. ~~j'#'oo+*"o~i~j11'Oo+'Ti~~~~uoo~~6jZ)oo~l}"OOAt

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i(x,e, to)= i(x,- e, to),

e(x, e, to)=-e(x,- e, to), (3 1)

+(x, 0, to)=+(x,--, to), equations (1)-(4) will preservethis symmetry for all future t. It is clear that under these conditions computations need be made over only one hemisphere, and the appropriate boundary conditions at the equator may be inferreddirectly from (31). Forthe Mercator variables U, V, and (6, equation (31) may then be written a(x,Y, t) = U(X,- Y, t), V(X,Y, t) = - V(X,- Y,t), (32) The first six of th.esc refer to the IIercator grid, the second six to the stereographic. grid. [The nunl.bers io and johave 4(X, Y,t) =(6(X,", t). tbc value (I/2+1/4) in equatio~~s(25) and (29), and the As applied to the six hfercator grids of data, we find that value (1/2+3/4) in equations (26) and (28).] this implies The order of solvingt'he c~quations is (19)-(21) and U~Ob7i1 (po= 1/2) (25)-(27), follom-cd bJ- (22)-(24) arid (28)-(30). Fro111 the form of the equations it is clear that they can all be vio= - v-;, (pa= I) solved on the interior points of the grids (p= 1,2, 3, . . . , (6:o =(6:2 (qo= 1) P; q=l,2,. . . ~ 0) a.nd (i=1,2,3,. . . , I;j=I,Z,. . . , (33) I). A iurt'ber inspection will&ow that the left (i=O or up,= UP' (qo= 1) p=O) andtop (j=I+I or y=Q+l) boundaries all be cor~~putcdwith equations (21) and (27);the left and bot,- 1' PO -= -rP,(qo=l/2) tom (.j=O or q=O) boundaries CRI~ be computed in (19), (6PO S(6Pl (!lo= 1/21 (23), (25), and(29); the bott'om and right (i=I+l or p=P+ 1) boundaries in (24) and (30) ; and right and top (It can be shown from equations (20) that ViLwill always boundaries can be computed in (20), (22), (26),and (28). be zero.) The remaining boundary values (in general, two adjacent The left and right boundaries of the Mercator grid are boundaries for each grid of data) n>.ust'be computed by easily handled by applying the cyclic boundary condition othermeans than (19)-(30) in orderto regenerate t'hc that, the, point (p=O, q=q) is identical to the point (p=P, complete grids of data at each time step. The proccdure q=p) and the point (p=P+I, q=q) is identzic,alwith the that was used here is described in the next' section. point @=I, q=q). The spt,em (19)-(30), although written in a form sug- On the top boundary (q=Q+l) of the Mercator grids, gestive of uncent,ered differences, actuallv uses ce1ltcrc.d I-', 4', and 71 can be forecast by (ZO), (21), and (22). differences. (Thetruncation crror can be expressed as a The variables C:', V, and 4 at p=Q+l c,annot be forecast series in Az and (At)'.) A special starting procedure must by (19), (23), and (24), however. (Note that these latter be used to get the prirned variables at t,ime t= I /2At. In quant'it'ies are stored on grids A and C, which have the the tmestcom.putat,ion described in this paper, an uncen- smallervalue of po intable 1, and therefore the most t,ered step was used to get the initial values of C', V',ct'c. northerly position of the four Nerc,ator grids, as shown at n=O(t=1/2At). U', V', etc. were initiallyknown at by fig. 2.) In addition, boundary values of the six stereo- t=O. Theirvalue at t=1/2At(n=O) was obta,irled from graphic variables u',v', +', u,v, and + must be obtained (19)-(21) alld (25)-(27) by temporarilyreplacing At by on the boundasies listed in the third column of table 2. 1/2At on the right side of t'hosc equatiovs, a,tld the sccor~d (These are the only points on those grids which cannot be tern1 on the left side of thosc cquntions by I,;'(t=O), ctc. forecastfrom the firlitmedifference equations (25)-(30) .) In [1:3] a method was outlined for obtaining these boundary 4. BOUNDARY CONDITIONS values byinterpolation from the associated grid. For example, a Mercator boundary value of 4 is interpolated Weexamine first theequatorial boundary condition, from the corresponding stereographic grid of 4 values, and which is used to specify the variables on the bottom row viceversa. Inthe case of the velocitycomponents of the Mercator grid (0~0). If the motion at one instant (C, V9 and (u, v) the relations (10) must of course be used t=to over the entire sphere satisfies the symmetry condi- to supplement the interpolation process, as indicated in tions the last column of table 2.

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TABLE 2.-Interpolation of boundary values between the Mercator and correspondingdistances on the earth varied from 500.4 stereographic grids km. at thepole (m=1) and equator (M=l) toa minimum Variable 1 Grid Boundariesneeded Interpolated from grid- of about 350 km. (m-"1.4) inmiddle latitudes. It " I shouldbe pointed outthat horizontal space differences U' ...... 1 Mere. (C) q=Q+l Ster: u' (C)and u' (P) -1 -1 in the system (19) - (30) are taken over the distance A, 4 ...... Merc. $4) q=Q+l Ster: 4 (-4) V-...... Merc. (C) n=Q+l Ster: ZL (B)and u (C) rather than 2A as is customary at present in numerical u'...... Ster. (C) i=I+l andj=1+1 Mere: U' (C)and 1" (B) 71' ...... Ster. (B) i=O and j=O Mere: U' (C)and 1" (B) weather prediction.

4'"" ...... ~ter.(D) i=I+1 andj=O Mere: 4' (D) It might be thought that all boundary values on the u ...... Ster. (B) i=O and j=O Mere: U (B)and 1' (C) u ...... Ster. (C) i=I+l andj=l+l Mere: U (B)and V(C) stereographic grids, including those which can be forecast +...... Ster. (A) i=O and j=I+l 1 Mere: 4 (A) by- (19)-(24), could be obtainedby interpolation. This procedure was in fact tried in a preliminary computation. (It is logically easier from the pointof view of the machine In order to perform the boundary interpolations in a program to interpolate all four boundaries on the stereo- "neat" manner it is necessary that the individual Mercator graphic grids than it is to do only those listed in table 2.) and stereographic grids overlap one another to a certain However, the resultsof this preliminary computation were extent. Tomake this statement more precise, let us quiteunsatisfactory compared to the results described define the sub-boundary points of a grid as those points in thispaper. When interpolation wasdone on all located next to aboundary (i.e.where q=Q onthe boundaries, the flow patterns tended to move at different Mercator grids or the points with i= 1 or I, j=1 or I speedson thetwo grids, and discontinuities developed on thestereographic grids). We requirethat the near the grid boundaries. points q=Q+l onMercator grids A and C (for which Thedetails of theinterpolation process were as de- Y=(Q+1/2)A) lie north of themost northerly sub- scribed in [13]. Computation of one interpolated boundary boundarypoint of thestereographic grids. This latter value took less machine time than did a computation of point is located at distancea r=Jx2+y2=(A/4)[(21 one of the equations (19)-(30) at one pont. Thus, only -3)2+ 1]1/2from the pole on the stereographic projection. about 5 percent of the total computation time was spent The boundary values of U', 4, and V required in table 2 on the boundary computations. can thenbe obtained by interpolation on the stereo- graphicgrids without reference to the boundary values 5. INITIAL DATA on those grids.Referring to (5), (6), and (7) we find that Y and r arerelated by the equation exp(- Yja) The initial velocity and geopotential fields for this test =r/2a. We musttherefore have computat'ion were defined by a flow pattern of the type hated by Haurwitz [7]. The initial velocity field v was e~p[-(2&+l)a/P]<(?r/4P)[(21-3)~+1]'/~.(34) non-divergent, and given by the stream function $:

Another constraint which should be satisfied is that the +=-a2w sin B+a2 K cos sin e cos RA. (36) northernmostboundary point on a stereographic grid (at r=(A/4)[(21+1)2+ 1]1/2) belocated south of the w, K, II, and a (radius of theearth) are constants. As southernmost sub-boundary point on the Mercator grids shown byHaurwitz, a flow patternlike this will, in a (located at p=Q on grids B and Dl with Y= ((3- 1)A). non-divergent barotropic atmosphere, move from west to The, boundary values of u',v', tp', u,v, and tp required in east without change of shape with the angular velocity v: table 2 on stereographic boundaries can then be obtained by interpolation on the Mercator grids without reference R(3+R)w"2n to any Mercator boundary points. This leads to a second '=(1+R)(2+R) ' (37) inequality : Theequations used int'he computations, however, are notthose for a non-divergentatmosphere, butfor one with a free surface.(37) will thereforeonly be satisfied If Q and I are not large enough to satisfy both (34) and approximately. The presence of divergence in the baro- (35), the interpolation process becomes more complicated tropic atmosphere, will, as is well known, slow up the rate and will undoubtedly lead to mathematical instabilities. of progression of the flow pattern, especially forsmall The values of P,(3, and I used in this test computa.tion values of the wave number R [16]. were In the non-divergent barotropic atmosphere treated by P=80, Q=12, 1=22. Haurwitz,the pressure field (pip) associatedwith the initial flow pat'tern (36) canbe readily determined by This gives a grid increment A=2aa/P of 500.4 km. The integration of the equations of motion (1)-(2), using the

Unauthenticated | Downloaded 10/03/21 06:49 PM UTC SEPTEMBER1959 REVIEWWEATIIER MONTHLY 339 angularphase velocity I toevaluate d+/bt. Wereplace pip by +. The distribution of + obtained in this way is given in the following formulae:

+=~,+a2A(e)+a2B(e)cos RX+a2C(e) cos 2RX, (38)

A(e)=~w(23+w)~2+~K2~2R[(R+1)~2

+ (2R'- R-2) -~R'c-~],

B(ej = 2(3+w)K c"[(R2+2R+2)-(R+1)*e2], (R+l) (R+2)

C(e)=%K2C2R[(R+l)C2-(R+2)],

C=COS e.

Both (36) and (38) satisfythe symmetry conditions (31). I'

+Q in (38) is an arbitrary constant which will determine the average height of the free surface in the atmospheric model beingused here. This in turn will determinethe "J- - - - speed of propagation of gravity-inertia waves and also the order of magnitude of the divergence in the model (div V= - +"d+/d t). The initial distribution of + and + used for the compu- FIGURE :<.-Initial distribution of theheight of thefree surface, tations was that given by (36) and (38), with the following shown 011 a stereographic projection. Only one octant is shown, values for the constants: the pattern repeating in the other three octantsof the hemisphere. The outer circle is the equator. Isolines are labeled in km.

~=K=7.848XlO-~see. -l (-O.lQ),

R=4 * (V is thehorizontal gradient operator on the sphere.) The advantage of using +- and +-fields which satisfy (39) +o=9.8 (8X lo3) m.2set.-'. is that b(div v)/bt is initially zero. According to Charney [3], this will resultin much smaller amplitudes of the These values for w and K give rise to large velocities, the gravity-inertia waves than would appear if only the maximum values of ai cos e (=-a-*b+/de) and ab[= geostrophic relation were used to relat>e the initial +- and (a cos e)-'b+/bx]being about 99 and 65 m. sec.". Figure v-fields to one another. 3 shows the distribution of the initial height of t'he free It is of some interest to examine thevorticity field surface z=+/g. The total variation of 3.5 km. in z is sev- corresponding to (36) : eral times as large as the typical variation in the height of the 500-mb. surface in winter. f+(=2(w+~)sine-((1+R)(2+R)KcosResinecosRX It is clear from the way in which this initial +-field was -f[l.l-1.5 cos4 e COS 4x1. (40) determined, that + and + togethersatisfy the so-called "balance equation" [3] : (We have here introduced the valueR=4 and theapproxi- V*E'VJ.+V-A=V'+, matevalues w=K-O.lQ). Because of thelarge value of K there are four regions of negative absolute vorticity where in low latitudes.These regions extend poleward toa latitude of about 22' wherecos4 0=(1.1)/(1.5).The minimum value off+( (reached at 8-7.25") is only about "0.093, however. Ax=ap3 sec eJ !@!!!@ =-v.V(ai cos e), (39) ( x,e ) 6. COMPUTATIONAL STABILITY Thecomputational stability of the finite-difference equations derived in section 3, with the boundary condi-

Unauthenticated | Downloaded 10/03/21 06:49 PM UTC SEPTEMBERla59 tions discussed in section 4, is difficult to investigate because the forecast equationsarenonlinear withvariable F=):fAt, coefficients, and the boundary conditions are not simple.

Considerableinformation can be obtained,however, by y'@" sin -1 "=(A!) 2 examining the computat'ional stabilit'y of linearizedver- sions of the forecast equations. To further simplify the _V= - 3 $ sin -. analysis we make the coefficients constant' and consider (3" ; only solutions only which are periodic anin x andon y, as (45) infinite plane. As an anahW to (11)-(13) Or (14)-(16), we thentreat U,, V,, . . ., Hh are now the non-dimensional ampli- the following simplesystjem:tudes ofperturbations the functions are and of time (n'At). We also define the following matrices: z,= [I]v, , Zh= [:I v:, , (43)

Here .f, ug,ug, and @ are constants, while u, v, and + are perturbationquantities. (43) couldalso include a term - (u&P/dx+vb@/dy) = --f(Ut+-vUg). It also is omit'ted for simplicity. Fordisturbances of theform exp i (pt +as+by), (41)-(43) are sat'isfied by tthree solutions for The Elinssen-grid finite-diffrrcnce equivalents of (41)- the frequency p: (42) can then btl written(nfttr some algebra)in the following compact, form:

An rquntion for Z (or 2') nlonc is rasi1.v obtainedfrom p1 corresponds t,o a geostrophic,non-divergent wa.vr, t,hrse : while p2 and p3 are gravity-inert'ia waves. We now write (41)-(43) in finit'e-differences,usingt'he z,,,-2Lz,+zn~,=o. (48) same basicscheme that was used toget (19)-(24) and (25)-(30). We introducesolutions of the form qpq= exp i (ap+Pq), where a=aA,P=bA, and p and (I are the Here L is the matrix I+gG2, where I is t,he unit matrix. finite-difference space coordinat,es. F~~ conveniencein L is Hermitian (lij is t'he complex conjugate of Zji) and the analysis we introduce following new quantities:thrreforc has t,hrcereal eigenvalues Xj andthree orthog- onaleigenvectors ej. Further calculationshows that the X's are equal tjo cos .$,where .$ can assume any one of the I" upqn=lJn\ three values

II~~~=Vn\l r@ qpqe 1/22 (01+8) 7 sin ($&) = m', 4pqn=H,$ qpqe'/2ip, sin (+&J = 11'- [F2+M2+N2]1/2, (49)

~b~~=l~h-J.q~~e~/~i(a+~), sin (it3)= 14's [E'2+M2+AT2]1/2.

Expanding Z, now as a series in the ort)hogonal eigenvec- tors ej,

ffP W=- ugsin - cos -+?lo cos At(A 2 2

Unauthenticated | Downloaded 10/03/21 06:49 PM UTC we find that (48) leads to the following scalar equations only the three frequencies given in (44). It can be shown for the 6,: that three of the sixfrequencies in the finite-difference system are similarin form to the three continuousfre- 6j (2 cos tj)a,,+ ai n”l=O, j = 1,2,3. (50) quencies in(44), and that the remainingthree finite- difference frequencies differ only by a reversal in the sign of theadvection term W. Theseextra solutions are The solutions for aj, are therefore very similar to the “computational wave”which is present inthe conventional way of solving the geostrophic vorticity equation [14].

where the three values of ti are still given by (49). Corn- 7. RESULTS OF THETEST COMPUTATION putational stability is achieved by demanding that all ti be real.Referring to (49) we see that thisrequires the A 48-hr. forecast was made from the initial wind and following inequality to be satisfied: pressurefields given by (36) and (38). Since thisfore- cast cannot be compared with either a real atmospheric flow patternor a mathematically knownsolution, the results will beexamined only from the following view- points: a. Smoothness of the fieldsin space. In particular, It should be noted that a weakening of this to permit the theagreement between the stereographicand Mercator equality will adlow cos e= - 1, whereupon (50) will contain an unstable solution of theform 6n=n,-1)n. representations in the areas of overlap (see fig. 1). b.Smoothness of t’hefields in t,ime-thequestion of Introducingthe definitions of W, F, A{, ant1 N from “meteorological noise.” (45),and taking the worst possible oricntltltiorl for uo Figure 4 shows the forecast field of z=+/g at 48 hours, and 210, the simplified computationalstability criterion in thearea covered bythe stereographic grid.The for this Eliassen grid system can finally be written: A waves have moved about 18O to the east in approximate agreement with (37). Of special interest is the agreement between the Stereographicisolines andthe Mercator isolines (heavydashed lines) in thearea of overlap of the two grids. In general, the two sets of lines are both In this formula At and A are the time and space incre- smooth.They agree with one another extremely well ments over which the partial derivat,ives with respect t,o exceptfor one areanear the upper right corner and t and z (or y) are expressed asfinite differences. IvO/ is anothersmaller area near the lower left corner. The equal to ~’u~+v~.As is clearfrom thepreceding maximum value of the difference between the two grids analysis, the satisfact’ionof (53) will not’ necessarily insure of z-values in these areas is about 90 meters-about 1/20 the stability of a c,omputation where the lat,eral boundary of themaximum differencein z betweena trough and conditionsare morecomplicated than t’hesimple ones ridge at the same latitude. The field of z on that portion implied by(45). In suchcases (53) is bestthought of of theMercator grid not shown in figure4 was very as a necessary, but not sufficient’, condition for st,ability. smooth,even in the low latitude regions where f+{ Criterion (53) allowsa maximum time st,ep of At= was negative. 12.5 min. to be used in forecasting the flow pattern de- A severe t’est of the smoothness of the forecast z-field is scribedin section 5. Thetest computat’ions were made shown in part A of figure 5. Here the quantity -4xii+ with a time step of (1/7) hr.kg.5 min. It took approxi- zi+l,+ri +zi-r j+zi j-l at t= 36 hr. is plotted for an mat,ely 30 sec. on an IBM 704 to comput’e one time step; areacentered near the North Pole.(Only thestereo- that is, to solve the 12 equations (19)-(30) at all points graphic grid covers this region.) There is some tendency concerned, and to do the necessary boundarycomputa- for a “checkerboard”pat,tern to appcar, but) it is not tionslisted in table 2. A 24-hr.forecast therefore re- very pronounced. quired about 84 minutes of computertime. (Checking Smagorinsky [17] andHinkelmann [9] havemade ex- of the results is not included in this figure.) Any further perimental forecasts with the primitive equations which increase in camputer speeds, say by a factor of 10, will were notbased 011 theEliassen type of finite-difference certainly make it possible to use the prin~itive equations grid. In theirscheme, only one grid is used(instead of over an entirehemisphere for even a mult’i-level baro- the four grids described in section 3), and the geopotential clinic atmosphere. and both velocity components are stored at all points of It is clear from (51) and (49) that the linearized finite- this single grid at all t’ime steps. Time and space deriva- differencesystem (47) possesses six frequencies. The tives are expressed as centered finite differences over the continuous system (41)-(43), on the other hand, possesses intervals 2 A t and 2 A, muchas is done in the usual

Unauthenticated | Downloaded 10/03/21 06:49 PM UTC 342 MONTHLYREVIEW WEATHER SEPTEMBER1959 way of makingnumerical weather predictions with the geostrophic model [4]. In order to compare this method with the Eliassen method of solving the primitive equations, a special 36-hr. forecast was made with the appropriate difference equationsfrom the same initial flow pattern. Figure 5 B shows the resulting field of “4zij+ zi+l j+zi j+l+zi-l i+zi j-l at 36 hr. from this special forecast. The aueragr! value of theplotted numbers is almost the same, 7.1 infigure 5A and 8.2 in figure 5B, but the range in the plotted values is 25 in figure 5A and 4i in figure 5R. The tendency to a checkerboard pattern isvery marked in figure 5B. It is clear then that fore- castsmade this way will be muchmore irregular than t’hosc obtained with the Eliassen grid system. The main purpose of the computation described in this paper was to test the computational stability of the overlapping stereographic-Rilercator grids,since this feature of the computation was not amenable to the type of computationalstability analysis carried out insection 6. Although the results shown in figure 4 certainly indicate that the scheme is at least reasonably stable, two small t,cnlporary “wiggles” did appear during the course of the forecast. They did not appear until after 24 hours, and FJGURE4.-The forecast field of z at 48 hours, drawn on a stereo- as shown by figure 4, had practically disappeared again graphic map, showing only the area covered by the stereographic by 48 hours. They appeared only near the top boundary grid A. Isolines are drawn at intervals of 500 m. Thin continuous isolines are drawn from the stereographic grid-point. values (grid of the Mercator grid, the stereographic grid point values A). Heavydashed isolines are drawn from theMercator grid- being quite smooth at all times. Figure 6 shows the de- point values (grid A). The Mercator grid A extended only in to t-ailed st>ructure at 36 hours of the wiggle locatednear the circle. Thesmall dotted areas show the spacing of the X=56’. Theother irregularity was verysimilar and lattice points on the Mercator and stereographic grids. locatedexactly on tho other side of thehemisphere, in the same part of the wavelike flow pattern. Both of the irregularities seemed tobe quasi-geostrophic in character, with the 0’ and V components following the geopotential field shown in figure 6. The writer has not been able to isolate the cause of thesetwo “errors,” which, although small, disfigure what otherwise seem to be an excellent computation. Since they appear only on the Mercator grids, and in only two of the four waves, it is safe to conclude that they do not represent anything 4 14 6 12 5 -I 12 222 -4 22 -4 19 2 real, but represent rather some peculiar type of trunca- 5 0 II 5 5 15 IO I9 -7 25 -14 22 -7 22 tion error. In this connection itmay be important to recall from section 6 that the Eliassen grid system does 4 . 2 8 2 21 -4 10 27-4 22 -4 -4 25 -4 vontain three false computational frequencies in addition

6 17 0 II I 17 -3 22 -14 -2027 27 -14 22 to thethree physical frequencies.Experience withthe geostrophic vorticity equation has shown that such false 5 I 14 3 13 IO I3 -4 25 -4 27 22 -4 -4 frequencies frequently become important near boundaries. In order to give an idea of the amount of “meteoro- 5 9 5 IO 8 4 7 22 -7 22 -14 25 -7 I9 logical noise” present in the computations, a record of the 6 3 II 8 6 6 8 2 19-422-422 2 height a at 2-hour intervals is shownin figure 7 for 3 A B selected points. (Unfortunately, a record of the forecast fields was printed out only every 2 hours=14 time steps.) FIGURE5,“Crid point valrlcs of the quantity (-4~ii+zi+~, 3- Point I is located near t’he equatorin one of the regions of zi-1 j+ z: itl+ zi i-1) at 36 hours for an area centered near the pole;negative Point I1 is located initially in the trough (A) from the forecast made with the Eliassen grid system, (B) f+{. from a special forecast made with thefinite-difference system used at 46O N. near the top of the Mercator grid. (It is one of by Smagorinsky and Hinkelmann. Units are in tens of meters. the points in figure, 6, where it is marked 11.) Point I11

Unauthenticated | Downloaded 10/03/21 06:49 PM UTC SEPTEMRER1959 REVIEWWEATHER MONTIILY 343 is located 175 km. from the North Pole. All three points have been purposely chosen in regionsof small net changes in z (except for the lasthalf of curve TI), so that anysmall short-periodoscillations will standout clearly.Such oscillations areindeed present and evidently have an amplitudecorresponding to aheight change of f50 meters. For comparison, the maximum net 48-hr. change in z at anypoint was equal to f 1450 meters- --a value 29 times as large as the amplitudeof t,he meteorologicalnoise. Heighttendencies measured over intervals of less than about 4 hours would therefore represent primarily “noise,” and not the slower quasi-geostrophic changes. This result.--the presence of a small but noticeable inertia-gravity oscillation- -is at first sight contrary to the results obtained from “balanced” initial data by Charney in [3]. Sn Charney’s t’est computation with the primitive equations, no meteorological noise appeared at all when the initial wind and pressure fields satisfied the balance equation (39). The explanation for this difference is that Charney’s initial flow pattern was a stat’ionary wave,while theflow pattern usedhere is not stationary but moves slowly to the east. In a barotropicat’mosphere a quasi- geostrophic wave has, as is well known, a small, but sig- nificant divergence field associated with it if the wave is not stationary.This divergenceassociated withthe geostrophic wave disappears only if the wavelengt’h hap- pens to be such that the wave is stationary. Therefore, unless the initial wind field also has this small amount of’ divergence, the forecast must contain some high-frequency gravity-inertiaoscillations (“noise”) in additionto the low-frequency geostrophic motions. From the linearized treatment of the noise problem by Hinkelmann in [8],it is possible to estimate t’hemagnitude of the noise which is introduced byneglecting in the initial data the (small)divergence associated with a moving FIGURE6.-Detailed st,ructure of one of the two temporary irregularities which appeared on the Mercator grid. The isolines of z geostrophic wave. Sf c1 and c2 are the phase velocities of (for t=36 hours) are drawn on a Mercator projection, the small the geostrophic and gravity-inertiawaves respectively, the crosses being the-points of Mercator grid A. The top of the figure fictitious gravity-inertia wave will have an amplitude in is at q= Q+ I. The heavy dashed isolines represent the Mercator the geopotential 4 approximately equal to (c1/c2) times the analysis and the thin continuous lines an independentanalysis of amplitudein 4 of the quasi-geostrophicwave. Forthe thecorresponding stereographic grid. Both sets of isolinesare example treated in this paper, (cl/c2) is about 1/30, giving drawn at intervals of 100 meters, the minimum value isolines in the upper right portion of the area being in both cases that for good agreementwith the numerically computed ampli- 8300 m. The grid point marked I1 is the point corresponding to tudes in + of the two types of motion. There can be no curve I1 in figure 7. doubt then that thenumerically computed noise shown in figure 7 is due to thechoice of initial data andis not caused by the numerical technique. The importance of including this geostrophically-conditioneddivergence in the initial data for the primitiveequations has also beendemon- strated recently by Hinkelmann [9]. and the vorticity of v2 is given by an equation similar to According to the t’heory of the geostrophic approxima- the balance equation(39), but with thenon-linear terms in tion as developed by Monin [12] (cont’ained to some ext,ent + evaluatedgeostrophically. Evidently Charney [3] and also in [5]), the second geostrophic approximation to the Hinkelmann [9] have each tested separately the value of true wind is given by v2, say, where the divergence of v2 is adding to the geostrophic wind a correction either for the precisely that divergence which appears (multiplied byf) vorticity or for the divergence. The elimination of noise in the usual geostrophic form of the vorticity equation, in both of their results is due to thespecial choice of initial

Unauthenticated | Downloaded 10/03/21 06:49 PM UTC 344 MONTHLY WJLI‘I’H&;R RICVIEW SEPTEMBER 1959

7700-

a 0 - 0 em 0 - ;= I2 7600,; 0* 0 - e 0 0 1’ = //

7500 - d

0700 - . - 0. 0 8600 - 0 -

8500- - p =I/ e 84QO 0 0 0 9 =I2 0 *.. OO. *.*: 83001~ -

0 I 0 0 *e* p=3 II 100 o.o..* .e* - ..e- 4=2 I I I I

I2 24 , 36 48 FIGURE7.”Variation of the height of the free surface at three selected grid points, st lo^^ at %hour intervals. I!nits are mctters.

data used by each of them; in Charney’s special test case REFERENCES the correctionfor divergence was unnecessary, as men- I. A. P. Burger, “Scale Considerations of Planetary Motions of the tioned above, because the wave was stationary, while in Atmosphere,” Tellus, vol. 10, No. 2, May 1958, pp. 195-205. Hinkelmann’sspecial case the linearterms in (39) were 2. J. Charney, “On the Scale of Atmospheric Motions,” Geofysiske not only rnwh larger than theneglected non-linear terms, Publikasjoner, vol. 17, No. 2, 1948, 17 pp. but the correction for divergence was quite important be- 3. J. Charney, “Thc Use of the Primitive Equations of Motion in cause of the very strong baroclinicity in the zonal flow. NumericalPrediction,” Tellus, vol. 7, No. 1, Feb. 1955, pp. 22-26. 4. .J. Charney, R. FjGrtoft,and J. yon Neumann,“Numerical ACKNOWLEDGMENTS 1ntegrat)ion of theBarotropic Vorticity Equation,” Tellus, VOI. 2, No. 4, NOV.1950, pp. 237-254. The research described in this paper was sponsored by 5. A. Eliassen,“The Quasi-Static Equations of Motionwith the Office of Naval Research and the Geophysical Research Pressure as Independent Variable,” Geojysiske Publikasjoner, Directorate under contract Nonr 1841 (18). The compu- vol. 17, No. 3, 1949, 44 pp. tations were performed at the Computation Center of the 6. A. Eliassen, A Procedurefor Numerical Integration of the PrimitiveEquations of the Two-ParameterModel of theAt- Massachusetts Institute of Technology. Miss K. Kava- mosphere, Scientific Report No. 4, on Contract AF 19(604)- nagh and Mr. A. Katz assisted in the programming of the 1286, Dept. of hleteorology,Univ. of California at Los computations. Angeles, 1956, 53 pp.

Unauthenticated | Downloaded 10/03/21 06:49 PM UTC 7. B. Haunvitz,“Thc Xfotion of AtnlF1spheric Disturbances or1 13. X. A. Phillips, “A 1Iap Projection System Suitable for Large- theSpherical Earth,” Journal of AlarineResearch, vol. 3, Scale Numerical Weat,her Prediction,” Journal of the Meteor- 1940, pp. 254-267. ologicalSociety of .Japan, 75thAnniversary Volume, 1957, 8. K. Hinkelmann,“Der Rlechanismus des met,eoroloaischcll pp. 262-267. Larmes,” Tellus, vol. 3, No. 4, Nov. 1951, pp. 285-296. 14. G. Platzman,“The C9mputational Stability of Boundary 9. I<. Hinkelmann,“Ein Nurnerisches Experiment mit der Condit,ions in Numerical Integration of the Vorticity Equa- primitivenGleichungen,” C.-G. Rosshg Memorial C’olume, tion,” Archiu ftir Meteorologie, Geophysik, und Bioklimatologie, Esselte A.B., Stockholm, 1959. Ser. A, vol. 7, 1954, pp. 29-40. 10. G. Hollman,“uher prinzipielle Mangel der geostrophischen 15. G. Platzman, “The Lattice Structure of the Finite-Difference Approximation und die Einfiihrung ageostrophischen Wind- Primitive and Vorticity Equations,” Monthly Weather Review, komponenten,” MeteorologischeRundschau, vol. 9, Nos. 516, vol. 86, No. 8, Aug. 1958, pp. 285-292. A‘lay/June 1956, pp. 73-78. 11. E. N. Lorenz,“Stat,ic Stability and Atmospheric Energy,” 16. C.-G. Rossby and collaborators, “Relation Between Variations ScientiJicReport, No. 9, on ContractAF 19(604)-1000, in the Intensity of t,he Zonal Circulation of the Atmosphere (GeneralCirculation Project,), nept. of Meteorology, RIas- andt,he Displacements of theSemi-permanent Centers sachusetts Institute of Technology, 1957, 41 pp. of Action,” Journal of Marine Research, vol. 2, 1939, pp. 38-55. 12.A. S. AIonin, ‘Tzmeneni13. DavleniG v Baroklinnoi htrnosferr,” 17. J. Smagorinsky, “On the Numerical 1nt)egrationof the Primitive [PressureChanges in a Baroclinic Atmosphere] Izvesfiz^a Equations of Motion for Baroclinic Flow in a Closed Region,” AkademizZ Nuuk SSSIZ, Ser. Geojiz., SO. 4, A\pr. 1958, MonthlyWeather Review, vol. 86, NO. 12, Dec. 1958, pp. pp. 497-514. 457-466.

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