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Numerical Integration of the Primitive Equations on the Hemisphere

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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 compu- tations 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 attempt to 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 obtained from a weather map. 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 tation scheme 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 dis- cussed 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 Unauthenticated | Downloaded 10/03/21 06:49 PM UTC 334 MONTHLY 'CVEA4THERREVIEW SEPTEMBER1959 .. 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 Unauthenticated | Downloaded 10/03/21 06:49 PM UTC SEPTEMBER1959 MONTHLY WEATHER REVIEW 335 3. FINITE-DIFFERENCEEQUATIONS 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 stereo- graphic. 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.
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