The Use of the Primitive Equations of Motion in Numerical Prediction

By J. CHARNEY,The Institute for Advanced Stady, Princeton1

(Manuscript received September I 5, 1954)

Abstract

An obstacle to the use of the primitive hydrodynamical equations for numerical prediction is that the initial wind and pressure fields determined by conventional means give rise to spurious large-amplitude inertio-gravitational oscillations which obscure the nieteoro- logically Eignificant large-scale motions. It is shown how this difficulty may be overcome by the use of a relationship between wind and pressure which enables one to determine thesc fields in such a manner that the noise motions do not arise. The method is illustrated by a numerically computed example. The wind-pressure relationship is in a sense a generalization of the geostrophic approximation and may be used where the latter approximation is inapplicable, either to determine initial conditions or to derive a set of filtering equations for numerical prediction analogous to the quasi-geostrophic equations.

I. Introduction these equations have already yielded forecasts The recent widespread revival of interest comparable in accuracy to those produced by in numerical weather prediction has been conventional methods, and probably superior brought about partly by the development of in accuracy in the case of new developments. high-speed electronic computing apparatus In purely theoretical investigations the use of but also, to a large extent, by the very en- the geostrophic approximation, by automa- couraging early results obtained through tically filtering out high-frequency “noise” systematic use of the geostrophic approxima- disturbances, has aided greatly in the under- tion in what has come to be known as the standing of the physical properties of large- quasi-geostrophic equations of motion. In scale flow systems where the high-frequency these equations it is assumed that the wind is inertial effects are unimportant. and remains nearly geostrophic, at least Yet those who have participated actively enough so that the horizontal acceleration in this work are aware that this is barely the terms in the equations of motion may be beginning of a new era, an era in which approximated by the accelerations of the numerical methods will find increasing applica- geostrophic wind. Numerical integrations of tion in practical and theoretical meteorology, and that the quasi-geostrophic equations are This articlc is based on work done under contract but a first approximation to the equations N6-ori Task Order I with the Office of Naval Re- search, U. S. Navy, and The Geophysics Research Di- which will eventually be used to describe the rectorate, Air Force Cambridge Research Center. atmospheric motion. Experience has already Tellus VII (1955). 1 PRIMITIVE EQUATIONS OF MOTION IN NUMERICAL FORECASTING 23 revealed systematic errors attributable to the V!fS. The motion is said to be quasi-geo- geostrophic a proximation. The approxima- strophic if this number is small compared to tion tends to Bisplace depressions too tar north- unity. There are, however, cases where either ward over the United States, and does not, because I/ is large or S is small the Rossby apparently, permit the development of frontal number is not small. Nevertheless the large- discontinuities. It is of course possible to scale meteorologically significant motions are arrive at higher approximations by successive distinguishable from the inertio-gravitational iteration of the geostrophic approximation, oscillations, the other class of possible quasi- but this process, besides leading to severe hydrostatic motions. We shall arrive at the mathematical complications, is valid only required method for determining the horizontal when the geostrophic deviations are small in wind field by asking for their distinguishing the first place. Moreover, the time order of characteristics. the equations is increased by one with each There are essentially two types of small iteration, and since the time order cannot amplitude wave perturbation of an atmosphere exceed three, it is evident that the sequence moving with constant angular velocity. They of approximations would not be convergent. are distinguished by their frequencies and speeds In any case it is likely that in the process the of propagation. The meteorologically impor- simplicity and elegance of the geostrophic tant type is characterized by small frequencies approximation would be destroyed, and and by velocities of propagation of the same thereby also the reason for its introduction. order of magnitude as the speed of the im- One is therefore led to reexamine the possibility bedding current. The second type has frequen- of using the primitive Eulerian equations, as cies in excess off; periods less than a half pendu- Richardson1 first proposed. lum day, and velocities of propagation of the The difficulty here, as I have pointed out in order of (gHV In H/3z)x,where His the verti- previous papers, is that the initial values of cal scale height of the disturbance and H is the wind and pressure cannot as a rule be prescribed potential temperature. For the large scale dis- independently with sufficient accuracy. One turbances of the atmosphere H - 104 m and is apparently forced again to introduce for 2 In H/2z - IO-~m-l. Hence (gH23In 0/2z)% - the wind values either the geostrophic approximation or something very little better. As will - ($gH)’- IOO m/sec-1, which is much be shown later this not only introduces an gre‘ater than the speed of the wind in the initial inaccuracy but gives rise to spurious troposphere and very much greater than the inertio-gravitational oscillations which obscure speed of propagation of the observed large- the meteorologically significant motions. The scale systems. Even when the motions are of purpose of the present note is to show that large amplitude and therefore not linearly accurate initial winds can be determined from superposable solutions of perturbation equa- the pressure field alone, and that if this is done tions, a distinction can be drawn between the inertio-gravitational oscillations will not arise. two types, at least in orders of magnitude of frequency and velocity of propagation. II. Characterization of large-scale atmospheric From what can be inferred from available motions observations the bulk of the energy in the We shall deal with statically stable motions troposphere is confined to the first class of in which the horizontal scale far exceeds motion. This partition of energy may be the vertical scale. It is therefore permissible attributed to the circumstances that the to assume that the motions are in quasi- principle energy sources in the atmosphere hydrostatic equilibrium. If f is the coriolis have characteristic periods lar e compared to parameter 2 SZ sin (latitude), V a characteristic the pendulum day and there sore excite little horizontal velocity, and S a characteristic of the second type of motion and that the horizontal scale distance, the ratio of the inertial first type is stable with respect to whatever force to the coriolis force is in order of mag- perturbations of the second type are excited, nitude the non-dimensional Rossby number either direct1 by the external sources of Weather Prediction by Numerical Process, Cam- energy or inirectly by nonlinear interaction bridge, 1922. with the first type. Tellus VII (1955). 1 24 J. CHARNEY 111. The balance equations by integration. In this sense the balance equa- Let us now examine the order of magnitude tion is a generalization of the geostrophic of the horizontal divergence in the meteoro- approximation. Both determine the velocity logically important systems-systems that are from the geopotential. characterized by particle velocities and speeds Now it is characteristic of the inertio- of propagation small compared to the gravity- gravitational motion that the horizontal diver- inertia speed (gH2alncY/az)%. If a flow is gence is not relatively small. Hence if the u quasi-geostrophic it can be shown by the use and v, whch must be prescribed independently in the primitive quasi-static equations of of dimensional arguments'v that the ratio of au av motion, are determined from the balance the horizontal divergence -+- to one of ax av equations, the inertio-gravitational motions are its component terms, say au/2xx, is just the automatically excluded initially, and since Rossby number and therefore this is small. In they are not generated to any appreciable the case where the Rossby number is not small extent during the motion, they will never the ratio is found to be V2/gH2a1nH/az appear. and is therefore again small. Thus the sig- The balance equation has been derived by nificant motions of the free atmosphere are Fjplrtoft (unpublished work) in an independent characterized by small horizontal divergences, investigation as a necessary condition for the stability of the non-divergent flow with re- and we may therefore express 14 and v in terms ot a stream function- spect to perturbations with horizontal divergence. The neglect of the divergence terms (% in the divergence equation has been justified It= -- , v =-av oy ax, empirically by S. PETTERSSON~who finds them to be one to two orders of magnitude smaller It may furthcr be shown that the convective than the remaining terms. acceleration terms involving w in the horizontal equations of motion are smaller in order of Iv. Illustrations of the use of the balance magnitude than the remaining convective equation acceleration terms. If we omit these small terms, substitute the stream function ex- I had thought at one time that the use of pressions for u and v, and take the horizontal geostrophic 11's and v's as part of the initial divergence of the equations of motion, we specification of the flow would lead at most obtain, when pressure is used as the vertical to small amplitude high-frequency inertio- coordinate, gravitational fluctuations superposed on an essentially correct low-frequency trend motion. However a recent numerical integration of the non-linear Eulerian equations for a barotropic atmosphere carried out in collaboration where cp is the geopotential. The above equa- with B. Bolin led to the suspicion that large tion, which shall be called the halance equation, amplitude inertio-gravitational perturbations suffices to determine the horizontal velocity can and do arise. In retrospect this is not field if the field of geopotential is known. surprising, for although the initial divergence This equation is of the Monge-Ampere type of the geostrophic wind is zero or nearly and can be solved for y as a boundary value zero, the first time derivative will not be zero problem providing since the balance equation will not be satisfied. The experiment was not conclusive because !!Y+f>O the fluctuations could have been due in part f2 to computational instabilities arising from in- v2p. correct boundary conditions. (The boundaries i.e., if the geostrophic relative vorticity ~ is f were geometrical surfaces immersed in the greater than -f/2. If the non-linear terms are fluid.) To investigate the matter further and small we obtain the geostrophic approximation to verify the filtering properties of the balance

* Charney, J. G., Gcof. Publ. 17, No. 2, 1948. Tellus, 5. No. 3, 1953. Tellus VII (19i>), 1 PRIMITIVE EQUATIONS OF MOTION IN NUMERICAL FORECASTING 25 equation the following numerical experiment yielded the side condition was recently performed in Princeton : Initial conditions were prescribed for the flow of an atmosphere consisting of two homogeneous incompressible layers of different at y = o and y = D. The solution of the density bounded above and below by rigid balance equation subject to this condition was surfaces. The lower layer was assumed to be found to be so deep that its motion could be ignored. The 2nx ny upper layer would then move as though it H+Afsin-sin- + were a single layer with a free surface.. but with LD gravity reduced in the ratio (I - e’ and c),n \ CI e being the densities of the upper and lower layers res ectively. This was done to simulate the actuaP static stability of the atmosphere. It is eas to show that if the divergence had The mean thickness of the upper lay& was vanisheB exactly the motion would have been exactly stationary. Since the use of the balance taken to be Hand I - -- ) was chosen to equal equation was supposed to prevent the occur- rence of large divergences it was to be expected Haln tj/az for thec actual atmosphere, i.e., that also in the present case the motion would about 10-1. In this way the buoyancy forces, as well as the speed of the internal gravity remain nearly stationary. This was found waves, became comparable to those in the to be so for a forty-eight hour period to actual atmosphere. The coriolis parameter f within the limits of a quite small computa- was taken to be constant and was set equal to tional error. A numerical integration was next performed 10-4 sec-1 corresponding to a latitude of about 43’. The motion was assumed to take using the geostrophic initial conditions place in the x, y plane between the rigid lateral walls y = 0, y = D. The initial velocity field was prescribed by the periodic stream function

2nx 7dy y = A sin -sin with L = 4,000 km and D = 2,000 km and um,=v,,=2nA/L = 50 m sec-l. To check - KINETIC ENERGY KINETIC ENERGY first that the balance equation did indeed ( lnitiol velocities ( lniliol vebcitier exclude inertio-gravitational motions the initial rotirfy balance geostrophic 1 values of the thickness field k were determined - condition) from the balance equation, which in this case took the form

POTENTIAL ENERGY ( lnitiol velocities geostrophic) where p=g(~-$)k.

Since v, the y-component of velocity, is t zero at y = o and y = D, the second equation of motion 0 2 4 6 8 10 dv-+fU= --Jv t (hrs) df aY Fig. I. Variations in time of kinetic and potential energy. Tellur VII (195S), 1 26 J. CHARNEY which of course also satisfy the boundary The foregoing example indicates that the conditions fu = - 2y/Jy at y = o and y = D. balance condition is far more suitable for In this case large divergences ocurred which determining the initial wind pressure fields in turn produced large potential energy than the geostrophic condition. changes according to the formula It should be remarked finally that the preser- vation of the velocity-pressure adjustment LD demanded by the balance equation necessarily dP- =-“Is-.(I -,>Q‘ h2 (zJu +$) ddy entails horizontal divergences. I have merely at 2 attempted to show that these divergences 00 must remain so small that the motion is approximately non-divergent at all times. in which P is the potential energy per wave Because the balance condition always holds, length. The kinetic energy was accordingly it is possible in principle to employ it in the highly variable. Figure I shows the kinetic same manner as the geostrophic approxima- and potential energies plotted as functions tion. It may be taken together with the of time. The upper curve represents the kinetic hydrostatic equation, the vorticity equation, energy and the lower the potential energy. the physical equation and the continuity The horizontal line represents the constant equation as determining the motion. The kinetic energy in the non-divergent and difficulty then is that it is not possible to com- balanced cases. bine these equations into a single equation in We note that there is a fluctuation of some as was the case with the geostrophic ap- 14 % in kinetic energy. The local fluctuations proximation, and I have succeeded in solving were considerably larger than this and must them numerically only in the barotropic case be regarded as errors. (unpublished result).

Tellur VII (1955). 1