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Produced by the NASA Center for Aerospace Information (CASI) ° i .R , .. a ,... y..., , a ..,tea.. ,w, ..^._...a ....p-'^,.., , r ,.^y +..+a ,^-R-".'.'^ 7 ►
NASA Technical Memorandum 80608
Documentation of the Fourth Order Band Model
Eugenia Kalnay-Rivas David H o itsma
, (N ASA-TH-606 06) DOCUMENTATION O.F THE FOURTH N80-17638 ORDER SAND MODEL (NASA) 117 p HC A06/MP A01 CSCL 048 a. Unclas G3/47 12902 I'.
DECEMBER 1979 -3 r
National Aeronautics and a ` Space Administration ^^^ x> ° °' '^ Goddard Space Flight Center q © Greenbeit, Maryland 20771 a. 6 J ^ - tti 4^ .a J rrl ^^.
9
a
p^ `3
f^xua....^.^.•+-+.^-^n - rn...w.....h+ +Frv'mr M^!^4nNaNY - ^a r^.+gtK#d^'.v+.ry+'^.H.k^as>dMY.HiN. y..:. i.^YS •.. Yr#+aYLi'kCA"^^NYbWUStRWY=^.fidYSiM.S#bk'u i^Yr.Nl.eu+3n:.n-.. x..L.1 +,wY^.:^rn^.I Y Y 6i1.1. Documentation of the
Fourth Order Band Model
Eugenia Kalnay-Rivas
David Hoitsma Sigma Data Services Corp.
November 1979
Modeling and Simulation Facility Laboratory for Atmospheric Sciences NASA/Goddard Space Flight Center Greenbelt, MD 20771 r e ^.
f TABLE OF CONTENTS Page Preface i I. Introduction I_1
II. Primitive Equations of Motion II-1 Derivation of the Equations at the Poles II-3 Computation of ZI.e Horizontal Pressure Gradient II-7 as suggested by N. A. Phillips
' III. Finite Difference Variables and Grid IIt-1
^i Periodic filtering of Short Waves II1-3 M v IV. Boundary Conditions IV-1 t V. Finite Difference Equations V-1 The Zonal (U) Momentum Equation V-1 The Meridional ( v) Momentum Equation V-2 .: The Thermodynamic Energy Equation V-3 The Moisture Equation V-4 The Pressure - Tendency Equation V-5 Vi. The Forecast Equations at the Poles VI-1 Zonal (U) Momentum Equation (Poles) VI-3 Meridional ( V) Momentum Equation ( Poles) VI-3 The Thermodynamics Energy Equation ( Poles) VI-4 The Moisture Balance Equation ( Poles) VI-4 ~ The Pressure Tendency Equation ( Poles) VI-5 The Vertical Velocity ,Equation ( Poles) VI-5 xr . VII. Diagnostic Equations (^,a,p) VII-1 VIII. The Time Differencing Scheme VIII-1
IX. Documentation of the Code ( Preliminary) IX-1 X. Flow Charts X-1
Appendix A Reprints of Three Papers A-;I. A Fourth-Order Forecasting Model IN-1 The 4th Order GISS Model of the Global A-3 Atmosphere The Effect of Accuracy, Conservation and A-16 Filtering on Numerical Weather Forecasting Appendix B Program Listing B-],
x;
it V__ Preface
We have decided to compile a preliminary documentatio: of the new GLAS Fourth Order General Circulation Model. The present documenta- tion has not been subjected to a careful editing process; we hope that its possible usefulness will compensate for some of its defects. The model dynamics (COMPO, COMP1 and COMP2) is still undergoing minor improvements, especially in the time differencing scheme which . we hope will improve its efficiency. The "physics" routine (COMPS) has not been documented because it is being thoroughly revised. The present version of COMPS, similar to the one used in the GLAS/GISS
models (see the documentation Iby Tsang and Karn), with modifications introduced by Y. Sud (;1979) is included in the code. Criticisms and suggestions for improvement will be reatly appreciated, since a final documentation will be prepared in 1980.
We are very grateful to all the people that have helped us generously. In particular, Dr. N. Rushfield had a major impact in the process of making the model operational. W. Connelly, D. Edelmann, D. Han, S. Breining, P. Anolick and M. Almeida were very helpful, in the devel- opment of the model. The documentation was expertly and cheerfully typed by S. Mathis; D. Edelmann and D. Rosen have also cooperated in its compilation. We want to express our special gratitude toward Dr. Y. Sud, who offered us generously both his advice and his time in the development of the "physics 1° routine, and most especially to Dr, M., Halem: without whose many useful suggestions, constant encour- agement, and long patience, this work would not have been finished.
Eugenia Kalnay-Rivas November 1979
i mR
I. Introduction
The band fourth order model is a G04 which uses quadratically oonservative, fourth order horizontal space differences on an unstaggered grid and second order vertical space differences with a Matsuno (forward-backward) or a smooth leapfrog time scheme to solve the primitive equations e of motion.
This program numerically solves these equations one latitude hand
I at a tim which greatly reduces the amount of computer core storage needed to run the program. It also uses the same variable names, order of computations, I/O, post-processing as the standard second order GCM. Appropriate modifications have been made for she fourth order differences and leapfrog scheme. (See the 1978 Goddard Model- ing and Simulation Research Review for an overview of the fourth order band model.)
The main feature of this model is that fourth order approximations are used for all the horizontal derivatives. The derivative aX is approximated by
4 ( q (x+ Ax) -q (x-Ax) , - 1 ( q (x+2Ax) -q (x-2Ax) ) 3 Ax 3 4Ax
and the derivative as (qT) by x
4 (T(x)+T (x+Ax) ) (q ( x) + (x+Ax)) - (T (x) +T (x-Ax))( (x) +, (x-Ax)) 3 [ 4 x
1. (T (x) +T (x+2Ax))(q (x) +q (x+2Ax)) - (T (x) +T (x-2Ax))(q (x) +q (x-'6Ax)) , 3 ` 8Ax
I-1 The second approximation is derived by averaging the flux qT to yield a conservative form of the dynamic equations. Note that if T is equal to 1 the second equation reduces to the first.
The primary variables are the horizontal components of the wind velocity, %V=(u,v), the temperature, T, the specific humidity, q, and the shifted surface pressure, 7r, ('r=ps-ptop' ptop^10 mb) .
The secondary variables are the geopotential,o, the vertical wind velocity,a, and the pressure,p.
The following ,pages give the differential equations of motion for the GCM model with the initial and boundary conditions. This is foAllowed by the equations with the corresponding fourth order approx- imations which use the same notation as the current second order model. A complete description of the primitive equations with the a coordinate system is found in the Arakawa UCLA notes (1976).
P
I-2 II. Primitive Eguat, ons of Motion 1
1.&2. Horizontal momentum equations
9
a
-Trvo - TT RT VW- (fiu t )kx P a f +
3. Continuity equation
+ V• (7AV) + aQ (TTc) = 0 , or (3.1) a 1 1 (3.2) V - (7i\V)da _ - V • 7^Vda at - - 0
4. Equation of state RT a = z P
S. First law of thermodynamics
at + V - (7AVT) + aQ (7r6T) _ fCa + CQ, (WAR) a P P k a IT From 6 =T/p , p=pT+Q7r, w=a7r+;G, W'ItW V7r, k=R/CP we get
D 7raT = pk a 7T;O Q(xT Replacing in 5, aQ Do + C P
a TrT + a 7T66 _ Tr ak T a Tr + ^V • V Tr) 7TQ V (7^VT) + pk + at as p at Cp
t 6. Humidity equation
a q + V • (AVq) = 0
7. Iiydrostatic equation
= -Cpa (from a _ -P = -^11) a A
II-1
q. s^' a`'S Y"E t'».t`GAOY'v"J^NaYiWIk:".^'_'=._- Ia:oan.. 1- 3 Of the varia bl e.^. Wr ,u,v0T , q 7 0'A WW upd ate th e 5Primary var4a es gr,u,v,T, and q using equations 1 ,2,3.2,5 and 6. From equations 3.1, 4, and 7 we can obtain ^,a, and; which are our secondary variables. Note that p=af'+Atop'
Sea level pressure (used only in the smoothing routine SMSHAP)
Hydrostatic eq. _ -p = A = -^ log( SLP ) _ RT cp s SLP - p(a=1) exp( a) RT
11-2
DERIVATION OF THE EQUATIONS AT THE POLES
Consider the continuity equation
Tr as = 0 t + V . (^►) + a^O
coupled with a conservation equation ^S which can be expanded into
P a t a ^r ct at + v . ( ^AVT) + a^ 71S
If we .integrate this equation over a polar cap of radius AO
2 r; -ao 27r 7T p0 ' I 'r a 2 cosOdOdA = - V. (7AYT) a2cosod^& a 0 7
2fr 7 /2-A o 2n 7r/2-Ao
a a^T a 2 cosododA a2cosododA 7r/2 T/2 and we assume the value of atT to be approximately constant over the polar cap
27 7T/2 —Qo 2n 7r /2 -Ao
T a2 cosododA = (a ffT) a2cosodo& f12 0 7r/ 2
27ra2 U-cosAy } (a tT) NP The first term in the rhs is, using Gauss' theorem
2fr ff/2-A^ 27r C V. (PVT) a 2 cos^d^dA - f 7vT a sinAoda ; ^r/2 0
II-3 This can be approximated as IM 2 7r -IM 71ivi Ti a sinA1 i=1
The third term on the rhs is
27r-A^ a7^T f a 2 cos¢ d^d A = 27a 2 (1-cos6fl (Dn;T ff
and similarly with the source term.
From these eauations we obtain
IM 8 ^rT m cotA^ /2 97TaT (at ) Pole - -(-1) aIM ^ T. (^iviT 7T - O Pole + S =1 I-A^
/2cos©cp/2 = since 2Tra sin A 2na 2 (1-cosQf ) = 2s nAO cotes IM / IMa 2sin ^ — a TM
This formulation is used in the continuity, momentum temperature and moisture equations. Note that the first term changes sign in the South Pole (m=1). In the momentum equations we make use of the transformation
U = -sinAu i - (-1) m cosAvi
Vi = (-l) m cosAu i sinAvi where m=1 for the South Pole and m=2 for the North Pole. U V Vi are the "cartesian" velocities at longitude Ja i , and u i , v i the corres- ponding spherical velocities.
The pressure gradient terms are computed making use of Green's
Theorem : a Dif Q dxdy = Qdy; fs a gP dxdy = - ^Pdx
II-4
For example
271 7wT/2-nm
a 2 cosOdOda _ —0 OaAO(-cosXda)
o T/2 Tr AO
+ aAo2zr IM OicosAi IM i=1
In the U-momentum equation we have then the following pressure terms IM AO (0i 7T ( atU) P — $ -cosh M +(2L
and similarly for 7V .
In the model we have approx ;i.rtiat,ed A^—'Qsin^
Then A^ _ 1 aTl=cos IM a sin A2 IM
Based on this formulation we construct the fourth order scheme at the Poles by taking 4 of the differences evaluated at A^ from the Poles (as expanded here), minus 3 of the differences at 2A^ from the Poles.
This formulation is not conservative at the Poles. However we have found that this has had no noticeable effect in the conservation of mass or energy in the model. In our shallow water experiments we studied a set of equations that were quadratically conservative, but inconsistent at the Pole,and another scheme analagous to the GFDL scheme, which is both quadratically conservative and consistent
II-5
44 .^ ,.,^ , ^ at the Poles, but suffers from a serious truncation errcr near the Poles in the pressure gradient term. The scheme that we chose gave better results than !';., le other two (Kalnay-Rivas, 1976).
11
11-6 ., r
Computation of the horizontal pressure gradient as suggested by N. A. Phillips (1) Let 0- T + 4 1 , ® 280 ° K/1000 i.e. constant
(2) 0 -T + Q', k R/Cp = .286
DO M -Cps 00 - CpOpk with 00 = Cp81000k apk
(3) T = T(p) + T' ► T(p) _ Wpk
• Thus our new dependent variables are V = 0 + Cp3 (pk-1000k) T ° _ T - gpk
VRT ' In this way w(nm + ©7), the pressure gradient in the momentum equations gets transformed into _ _ k , (4) ^r (9 + V + aR02 Cllr + IRA VTr) k apT 7r (off' + © Rt + 7r ( V + t3R^3^ O pt )
But t'A i ,,.,w i,)ond parenthesis is zero: _ k _ k Vy + aR6 QTT = - Cp80pk + ROp Vp = 0 P
In regions of steep orography, the second parenthesis in (4) is much larger than the first. When the horizontal pressure gradient terms are computed in their original form, the rear cancellation of the two terms introduces large truncation errors. The procedure sugges- ted by Phillips greatly reduces this truncation error. We have chosen a simpler definition of 0 o than the one suggested by Phillips.
II-7 III. Finite Difference Variables and Grid
The notation used in fourth band model is the same as the standard GLAS(GISS) second order GCM except that we use a non-staggered hori- zontal grid. A complete description of the variables can be found in the TSA.NG-KARN documentation of the GISS 9 level model.
Let u ijk , v ijk , Tijk , g ijk , Br ij be the
finite difference approximations to the primary variables u,v,T,q,
and v at the mesh point (ipa^ jp^, (k,. ^pQ) , Also the scaled variables fru,frv, ...are approximated by ffu ijk' nv ijk , ... .
The finite difference equations also use the followinq geometric arrays:
DXP(j) = mj - a cos h jA X , DYU(j) = n j = aQ^ We use a factor of 12 in DXYP to make our scaled fourth odder differences simpler. DXYP(j) = 12.:,, j • n j , fl ij = DXYP(j)'7Tij ' U ijk = nj'ijuijk
S ij k = 11 i j . ijk ADLDP = 12.a*AXAc Vijk mj ' j jVijk
F 1jk DXYP(j)f j + AD1'DP • SIN^ UiQj
V Horizontal Grid
The fourth order bknd model uses an unstaggered grid in the hori- zontal direction. j=JM1(N.Pole
Q$ JM intervals between south pole and north pole j=l(S.Pole)
i=l i=2 ^X i=IM i=IM+I IM intervals in the longitudinal direction and periodic,
i=IM+l is identical to i=1.
III-1
r 0 I^h9 JI
A i = (i-1)G^-7T = '(i-1)51-160° 2 - (j-1) 9°-90° 7 JSP = 1 JNP = JM1 = JM+1
Vertical Grid
The vertical grid is staggered; the values of all.the variables u,v,T,q,ir, ... except 5 are computed at the center of each layer.
The values of a are computed at the edges of the layers.
d=0 __ — Cfz0 2
k u,v,T, qk k'+ 1 ^rk ' Pk °+1
V
N L A Y.^_
In the case of uniform vertical resole tion and NLAY=9 vertical layers, }'9 G = SIG ;:n) = 1 (0,1/9 . ...1)
a k , = SIGE(Y) = 2k18-1 = (1^,...^18) and Aa = 1/9 .
U ijk and its scaled version S i j k are defined at the eight interior edges, i.e. for ak with k=1 to 8 since 6(0)=6(1)=0 from the
boundary conditions. The pressure pk1=:,Trakv+ATOP is defined at the same level as &k°-1' Note for level k we need S ijk and Sijk-1 to form the second order vertical differences:
S ijk jijk;.-Vijk+l) - S ijk-1 (Vijk+Vijk-1)
(t1AGNAL PASk US, R III-2 I' I- II1-2 Periodic Filtering of Short Waves
An integral part of the numerical scheme is tht periodic applica-
tion (every ISMTH steps, generally 2 hours) of a 16th order Shapiro filter. This has the effect of removing waves shorter than 4Ax, which are not resolved in the model, while waves longer than 4nx, which are accurately computed by the difference scheme, are not affected by the filter (Kalnay-Rivas and Hoitsma, 1979).
The filter is applied to an array q j in the following way: Let ^'.. (q j ) = qj+ q 3' d - (q j ) yj qj-1
Then a Shapiro kilter of order 2N is given by
q j qj-(-l) (d+ d - ) qj
The response of the f.iltea, applied to a wave of the form,
qj =Q exPUL2^n' /1xj) is
sin 2N 'n'nx) qj L
The 2-dimensional filter is applied as a product of 1-di:mensior;al filters (first in longitude, then in latitude). In latitude we filter the fields on great circles ;formed by meridians of longitude
A and A+ ,n', where .0< A ..< ,rr. We are presently filtering only potential temperature and sea level pressure. These .fields were chosen because they are not very affected by orography. Winds are not currently filtered, because the adjustment between mass and velocity fields does .lot allow the development of short waves in the winds alone. However, in the tropics, where the adjustment of winds to the mass field ik. minimal, the winds are somewhat noisy, and we may opt to filter them too.
III-3 V. L'r-
IV. Boundary Conditions
Periodicity in the Zonal (East-West) direction
IT IM+mj 11mj m=1,2,3,..., j=2,...,JM
QIM+mjk-Qmjk m=1,2,3,..., j=21...,JM, k=1,...,NLAY
for Q=u,v,T,q,0,6,7ru,wv,...
Boundary conditions at the north and south poles.Def ne the array INDEX as follows:
k2 F>. INDEX(i) = i+ IM i=1,2,...,22 N ole P INDEX U } 2) i i=11, ... rim
i.e., INDEX: 2+1,I I-2+2....,IM,1,2,...,I2
Then we can define Tr 7r ts needed "beyond" the North Pole, iJM+2 INDEX (i) JM f or the poi.,►
and 7T for the points "beyoni, " the South Pole 11 i 0 INDEX(i)2 and similarly for T,q 4 .
For the horizontal velocity V=(u,v) we have
V iJM+2k ` -ViJMk
Vi0k = Vi2k
IV-1
V. Finite Difference Equations
THE ZONAL (U) MOMENTUM EQUATION
a7ru _ 1 a (7ru • u) + a (7rvco^^ I ax s •u.) a (7rvu) at - - acos ^ a^ ) - a o
7r [ a 0 + aRT a7r ) ^ acos^ -DT p + (f + U tana ) Iry + ^tFx
COMP1: HA: PU Ii PUli all U. _1/ 2j) (U +1/2 1 k = 4.*[ ( *. + )-(U*. a ijk U*i-ljk ijk +U i -ljk ijk +U i+ljk ) (U ijk +U..i+ljk )l PU2i PU2it1 -.5[ (U 1. +U* ) (U +U. )-( U * +'1 ) Jk i-2jk ijk 1-2jk ijk i+2jk ) (t)..ilk +U i +2jk PV1 +4* [(V# (U j-lk) +Vij+lk)(Uijk+Uij+lk)Hu PV2 . ijk+U 5*[(Uijk+Vij-2k ) (U ij-2k )-(Vim k Vij+20 (Uijk+Uij+2k) VA: +.5*[S ijk-1 (U ijk +Uijk-1)-S ijk (Uijk+Uijk+l)l /Aoyk COMP2: PC +(7r i .*n.* { 8*[^1-1'k-^'i+1.k+akRTijk (Tr i-ij-7i+ij)l + 7 7 J J Pk
dkRTia k I _ 1+2jk - i-2jk+ It ( ^i+2j ?ji-2j)) C: COMP3 + Fijlc*lij*Vijk)+(IIFX)
Note: COMP1, COMP2, COMP3 are the names of the three subroutines where the different terms are computed. HA: Horizontal ad vection terms. VA: Vertical advection terms. PG: Pressure gradient terms,C: Coriolis term. Also note that PV2 is set equal to zero for j=0 and j=JM, i.e., there is no transport of mass acres3 the poles
t V-1
THE MERIDIONAL (V) MOMENTUM EQUATION
a7rV 1 a ('nu °v) + a ('►rt"COS^ V afrav a t - aco s ^ [ ate-- a m 3)- 0
7r a^ pRT a utan ^ + P a) — (f+ a ) ^ru + rrFy a [ a m
COMP1: HA: artv.. ilk = {4°*I (Uijk+U -ljk) (V . +V )-(U*. +U* ) (V. +V. ) at i Jk i-ljk i.Jk 1+ljk i.jk .i +ljk
-.5[(Uijk+U*i-2jk)*(v^jk+Vi-2jk)-(Uijk+Ui+2jk)(Vijk+Vi+2jk)
+4.*[(Vijk+Vij-lk)(Vijk+Vij-lk)-(Vijk}v1^+lk)(V^jk+Vijtlk)
—.5*{(Vijk+Vij+2k)(Vijk+Vij-2k)-(Vijk+Vij+2k)(Vijk+Vij+2k)
VA: +. 5* — S ijk ( ijk ijk ) [S ijk-1 (Vijk+Vijk-1) V +V + l l /Auk3
COMP2: P: +(7r. *m, —^_k (7ri.-1-^ri +1)l 1 1 J *{8. *[`^i'J -lk '$i'+lk+J ^k Pk J J
akRTjk ( ^r ij .j-2) l } + [ ij+2k-ij-2k+ a°k +2-' C: COMP3: F.ijk *fij *Uijk)+(gFy)
V-2
s
THE THERMODYNAMIC ENERGY EQUATION
a 7rT 1 [ a 7TUT + a nycosm • T k a Tr&T k at - " acoso 3 a^ ] - p 3
+ ll(,KT a ^ U a ^ v a 7r 'IQ p (aat + acoso + a d + Cp
COMP1: HA: 3nT1 k
.[( )(T. +T. ilk 1-ljk i.jk i-ljk)"(Uiik+Ui+ljk)(Tijk+Ti+ljk)
-.5*[(Uijk+Ui-2jk)(T jk+Ti-2jk) - (Ulik+Ui+2jk)(Tijk+Ti+2jk)
(Vjk +4:' [(Vijk+Vij-lk)(Tijk+lij-lk) " +Vij+lk)(Tijk+Tij+lk)
- .5*[(V*. +V* ) (T.. +T. ) - (V*. +V* )(T. i3k ij-2k Jk j+2k ilk ij+2k ijk +T.ij+2k)] VA: ( Tick +. 5p^' [S .. -T-1-^-k=1 - S (T-^k + Tom ijk ijk-1 k + k ) ijk k k ) ] /L^^k } p ijk p ijk-1 pijk pijk+l COM.P2 :
QkKTijk a 7r + U {8. _lj (+ [^r k * [^i+lj-^i ]+^ i-2j ^i+2j } i 7 at pij ?c
+ Vl jk{ 8 .*[^r ij+l - Br ij-1 +7r ] ij-2 -Br ij+2 }] ) COMP 3:
+(^Qijk Cp
V-3 THE MOISTURE EQUATION
a 7Tq = _ . 1 [ a^ ,^, a ffycosc^ • q ) — a^rc^q t acos^ a s a 7T + r 11 E—C) COMP1: HA a Rqi k at (4.*[(Uijk+U-ljk)(gijk+qi-ljk)-(U*ijk+Ui+ljk)(gijk+qi+ljk)I
-.5*[(Uijk+U*i+2jk )(g ijk+q i-2jk )-(U ijk+U +2jk)(gijk+qi+2jk)l
+4.*[(Vijk+Vij-lk)(gijk+qij-lk)-(Vijk+Vij+lk)(gijk+qij+llc)j
Y 4 -.5*[(Vijk+Vij-2k) (gijk+qij-2k)-(Vijk+Vij+2k)(gijk+qij+2k)]
+.5* S .. ( q. +q. )—S. (q. +q, [ i3k-1 ijk ijk -1 ijk ijk ijk+l )l/^Q k} COMPS: +n(Eijk-Cijk)
Note: The current transport scheme for the moisture field is being modified.
r
V-4 THE PRESSURE-TENDENCY EQUATION
a7T _ _ `° Q) k ^a ,►ru + a7r vcos at k=lE acos^ am 7 COMP1: CONVi^j =©v&(8.*(ULljk- Ui+1jR) +Ui+2jk-Ui-2j2 +V -Vi 8.*(Vij-1R -Vij +1R^ ij+2k j-22} COMP1: a II .. L 1 ^ = E CONV at 2=1 izj
THE VERTICAL VELOCITY EQUATION
a 7r I ( a TrV + a 7rVcas l _ a (76) at - - acos^ aX a^ aQ Thus
A°k all = CONVikj - ASijk giving: COMPI : - Dok a T1 S ijk {S ijk-1 + CONV kj TT}
V-5 P V._. T The Forecast Equations at the Poles
The fourth order band model uses a spherical cap at the poles, and the finite difference approximations to the equations of motion must be derived for this spherical region. Stereographic projection is used to give us a well-defined velocity vector at the poles.
From trigonometry, (Fig.Vla) if the vector ( U,V) in X-Y coordinates is represented by (U',V') in X'-Y' coordinates where the prime co- ordinate axes are rotated by an angle A then, Y - (U',v') Y' v'co a - --^%\ 1 (11 U = U I cosa - V°sinA i x' 7 (2) V = ul sina - V' cosa U'sina -_ I Al We want to interpret "polar stereographic" velocities ( UNP , VNp ) also as "spherical" ^i'^ x U'cos7. velocities (U,V) ,, as we approach the Fig. VIa North .Pole along a meridian of longitude A. From Fig. VIb, which shows a unit vector in both coordinate: systems, we see that, at the North Pole, spherical coordinates are rotated by an angle ( 7+X) with respect to the polar stereographic coordinates. Therefore, Ucos(a +2) - Vsin(a UNP = +2)
V_ VNP = Usin(X+2) + Vcos(a +2)
or UNP = -13sinX - Vcosa (3) VNP = Ucosa - Vsina
Similarly, for the South Pole: J ^^UA USP = -Usina + VCOsA (4) VSP _ -UcosA - Vs ina VNP r also,
UNP U = -UNP sinA + VNPcosa (5) V = -UNPCOSA - VNPsina
U = -USP sina - VSPcoSX Fig. VIb (6) V = USPcosX - VSPsinA VI-1
^f
a:
*#•w y "1.'?" !^r....),'6^'isl;J^i1!'^t:rb've'+:eftciiGn.:..,. The initial values for UNPk ' VNPk ' USPk ' VSPk are obtained from equations 3 and 4 by averaging in the zonal direction on the line of latitude j=45 for the north pole and j=2 for the south pole. In the forecast stereographic velocities are advected and 'transformed back into spherical velocities after each time step by equations 5 and 6.
In the following equation we will denote the polar velocities by VPk'm , VPk,m and the temperature and other variables in a similar
way, TPk,m' ffPk,m' ••• where m=1 for the south pole and m=2 for the north pole. The following constants are used,
COEF1=(-1) m COEF2=-COEF1 RADIM=3•a•IM CONI=4COT(.5A^)/RADIM CON2=-COT(AO)/RADIM CON3=4DT/(RADIM*SIN(.50O) CON4=DT/(RADIM*SIN(AO))
V- JPOL(K,M)=(2- JM } 3 JM-1
r=JPOL (1,m) =first interior value of j ( 2 for the S. Pole, JM for tae N. Pole) s=JPOL (2,m) =second interior value of j(3 for the S. Pole, JM-1 for the N. Pole) .
VI-2
ZONAL (U) MOMENTUM EQUATION (POLES)
COMP1 : a TrUP IM k,m {COEF1*[CON1*lEl-7TirVirk irk S nXi+COEF1*VirkCOSai) att - (U
IM +CON2* i=1E -'r isVisk(UiskSinxi+COEFI*ViskCOSAi))
+.5[SPk-1 (UP k +UPk-1 )-APk (UP k+UPk+l))/AQk}
COMP2: IM o RTP^ 0 + TrPm[ CDT3 *iE1 ($irk+ kPPkk Trir),COSai 1,
4 kRPk 7is)COSXi) +CDT *IE (disk+G PP i=1 k COMPS: + fm7TPm*VPk.sn) + (7TPm.FPk,m
MERIDIONAL (V) MOMENTUM EQUATION (POLES)
COMP1: IM a TrVP k,m = {COEF1*[CON1* E —ff V irk. *COSa. SinA ) at i=] it irk*rk irkrk i IM +CON2* E -'nisVisk(COEF2*Uisk*COSai+ViskSinai)) i=1
+.5[SP k _1 (VP k+VPk-1 )-SPk (VP k+VPk+l)]/Aak}
COMP2: CO IM Cl RTPk NS +TrPm.[ DT * E _._ i=1 (do irk P?k ^ir)Sinhi IM a RTP +CDT4 * E (disk+ k P kTris)Sinail =1 P k i COMPS: - f mTrPm'UPk,m + (TrPm•SPk,m)
VI-3
s
THE THERMODYNAMICS ENERGY EQUATION ( POLES)
9 COMPl: aTrTP IM IM k m_ _ (COEF1 *( CON1* E +CON2* E ^irVirkTirk nisViskTisk) 8t i=1 iml k * • TPk,m _TPk-1,m k ._ ) +.SPPk , m [Spk-l,m ( c + pPk,m nPk-1,m
TP - APk,m ( TP= m + --Y—. ))/Aak} PPk ,,^ PPk+ l , m COMP2:
+('n mP ak KTP a IM k,mf_+C:ON5* E [(COEF1*UPkmCOSxi+VPkmSINAi ci=1^tnl ° ° ppk,m
COMP3: (87tir-'is)) } ) Tr p QPk) +(-Cpm-
THE MOISTURE BALANCE EQUATION (POLES)
COMPl: a 7rgPk , IM m iM ={COEF1*(CON1 * E 7T irk irk +CON2*E 7TisViskgisk) at i=1 irV g i=1
+.5[6Pk-1,m(gPk,m+qPk-1m) SPk,m(gPk,m+qPk+l,m))/Aak}
COMP3: +(ffPm *EP k)
VI-4
THE PRESSURE TENDENCY EQUATION ( POLES)
IM IM CONVPLk=COEF1 *( CON1* E irVirk+CON2* E 7ri^Visk) Aok ,m i=1 i=i
a TrP NLAY E CONVP'Lk m at m k=1
THE VERTICAL VELOCITY EQUATION (POLES)
D 7r Pm 91., SP SPk—1,m + CONVPLk Q^k at
VI-5 VII. Diagnostic Equations (^,v,p)
Once the updated values of ffU, nV, wT, ffq are found we unscale:
Ui jk = fui +j k,/ir3jl for all i,j,k.
Similarly for V,T,q. We also filter the fields near the poles to prevent linear instability (see subroutine AVRX). a is obtained from A by un.^caling also.
We determine from nr 4," and ak r Pijk
PTOP=constant. F Ijk ck^ij l + PTOP
^s is the surface geopotential (a function only of latitude acid longitude) For the Phillips ge^pvtential we define (p) k At the center of the layer in the following way: k+1k+l_ k+l) ( ,^ ) k = pijk+l° Pijk° = 1 ap k (pk-1 k k21 a 1 j (k+l) (pijk ,+lRpi jk,',) p , ... , NL:AY where +1 2 pig , = (SIGE(k)*^ij +pTOP)k (SIGE(k)=cV, i.e.
pk+l is obtained by exponentiation and ( p }k by differences.
The following equations represent the geopote.ntial calculations used in the old fourth.order model.
VII-1
j
Let k k C 10 'k I p ijk + i k-1 Pijk 1 k' k # -1 ' L Pijk Pi jk for k=l,...,NLAY
with
k k j , and p k Pijo = Pi ijNLAY+l - PijNLAY
An optimized version of C, IS:
it R& ij k k .5c k k l3k A o +0 (P k k p ' ij k kt(Pijk+l-pijk ) k-I Pijk ijk+l-pijk-I
Rather than compute ^ and then subtract T' wv do everything at once.- NLAY (1) ^!IjNLAY CPTH*(PSKAPA - ) + E C ^s - P^IjN-'r,AY ijz Tijk
(2) k T— T I j ^^t ljk+l ( + :-iA+l ) - 2r)) 2 ljt+l -Pijk) k — P ijk Pijk+l where
CPTII = C • C) PSKAPA = 1000k = p k p s
VII-2 VIII. The Time Differencing
The fourth order band model ha time scheme or the smooth leapfrog scheme (see MWR-Vol 100 (487- 490) R. Asselin!.
Let Q represent a typical variable that is to be updated to time n+l, and let D(Qn ) represent the nonlinear space differences. The Matsuno (Euler-backward) scheme is as follows: d = Qn+©tD(Qn) Qn+l = Qn+4tD(4)
The standard leapfrog scheme is given by
For the smooth leapfrog scheme we replace Qn-1 by Qn-1
n+1=Qn'-1 + 2 OtD (Q (2) 111 n ) with
( 1+Qn+1) (3) Qn 1 -v)Qn + .5v(5n-
Equation (3) represent: a simple time filter except V1_1 is used instead of Qn-1 in order to save core storage. The above equa- tions (2) and (3) represent the order in which the smooth leapfrog scheme is evaluated. For n=1 we define Q O=Q0 then we update in equation (2) followed by the filtering in equation (3) which is needed for the next time step.
VIII-1 i
The smoothing step introduces dissipation with respect to time, controlled explicitly by the parameter v, as compared to the im- plicit dissipation in the Matsuno scheme. The amplification factor can be found in the paper by Asselin.
A further modification must be made to the smooth leapfrog scheme when source terms are included. Essentially the idea is that we must include the source term effect over two steps rather than one.
If we do not do this, then the source effects ( COMP3) are included only in every other step which will introduce large discretization errors. ( For details see the attached report, Appendix B.)
If the source terms are called every NCOMP3 steps, then for step n=NCOMP3
Q* = Qn-2 + 2AtD(Qn-1)
Qn-l = (1-v) Qn-1 + .5v(Qn- 2 + Qn)
Then compute the ..source terms $ n and include in both steps n and n-1,
Q = Q* + NCOMP3'AtSn
Qn-1_ Qn* -1 + NCOMP3•^tSn
The actual code is complicated by the fact that we actually use scaled and unsealed variables, but the generalization is straight- forward.
VIII-2 IX. Documentation of the Code (Preliminary)
The band fourth model uses special equivalences and nonstandard dimen- sions in order to have the variables P,U,V,T,SH stored in contiguous locations for each line of latitude. Thus, we desire to have the variables stored as follows.
IM NLAY*IM IM NLAY*IM IM NLAY*IM IN NLAY*IM Zr9 1 IM+l (NLAY+1) *IM+l ~ i PHIS TS V sfis T P ,- - SH , PHISy U J=1 t- J The scaled variables PT, UT, VT, TT, SHT are to be stored as ,follows:
IM NLWIM IM NLAVIM IM NLAY*IM IM NLAY*IM IM
1 1M (NLAY,F1) *D1+l PU UT GT VT TT FT SHT FU UT --- - GW
The above storage designation is accomplished by dimensioning PHIS(4*(NLAY+1)*IM,JNP) instead of PHIS(IM,JNP) (for a fine grid we have PHIS(2880,46), for ultrafine we have PHIS(4800,72)). Then we equivalence NLAY*IM locations of the first line of latitude of U with PHIS(IM+1,1) to PHIS((NLAY+1)*IM,1) (the first IM locations of
PHIS are used for the quantity PHIS itself).
Similarly, we equivalence TS(1,1) to TS(IM,1) with PHIS((NLAY+1)* IM+1,1) to PHIS((NLAY+2)*IM,I) and so forth. (See the enclosed com- puter codc3 for the exact values in the fine and ultrafine versions.)
IX-1 1
In order to have the successive lines of latitude arranged properly in storage we dimension ou)r variables to achieve this purpose. Thus, we have U(IM,4*(NLAY+1),1) instead of U(IM,NLAY,JM), similarly for V,T,SH,UT,VT,TT, and SHT. For P we have P(4*(NLAY+1)*IM,1) instead of P(IM,JNP) and similarly for TS, SHS,GT,GW, and PT. Note that the variables U,UT,... are only computed at IM*NLAY*JNP points, the special dimensions are needed to properly align the variables in storage. It is important to recall that we are using two proper- ties of the FORTRAN Compiler:
First, by equivalencing two elements of two different arrays we im- plicitly equivalence the other elements of the arrays. Second, that last array dimension can be left as 1 as long as it is dimensioned properly in the calling routine or it is equivalenced to a properly dimensioned array. Note we could have dimensioned U as U(IM,4*(NLAY+1),JNP)we would use the same amount of storage. It is crucial to have the IM*(NLAY+1) dimension because it would cause the computations on i) for the successive lines of latitude to by shifted IM*(NLAY+1) locations where the next line of latitude of U are stor- ed. For clarity and simplicity in programming we use the standard equivalence of U,V,T,SH with Q(I,L,N,J) with U equivalent to Q(I,L,1,J) and so forth. Since we want P,U,V,T, and SH in contiguous storage locations and the equivalence of Q with U,V,T, and SH, then we must dimension Q as Q(IM,NLAY+1,4,1) instead of Q(IM,NLAY,4,1). We fill in the extra locations by including PHIS,TS,SH, and P.
IX-2 COMPO Description
This subroutine contains the time schemes and controls the calling sequences of the routines COMP1, COMP2, COMPS, and the polar filter- ing routine AVRX.
The logic of this subroutine involves three considerations. First, it permits one to choose either the Matsuno or the smooth leapfrog time scheme. Second, the latter scheme involves the use of storage arrays PSM and QSM. Third, the unscaling, smoothing of the updated variables QT and the call of subroutine COMP3 occur at the value JS2=J-2 when J is the value of the current line of latitude which is being computed. The routine COMP1 and COMP2 require the values of U,V,... at JS2 in order to compute the updated values at J because fourth order differences are used in the meridional (J) direction. Only after the COMP1 and COMP2 are called for value J can we unsca.le, smooth, and finish processing the variables at JS2.
The code for the poles is identical in format with the code for the other J values except the variables are scaled by the pressure PPOL only.
The main program contains two calls to COMP which cannot be treated
independently because of the calling sequence: COMP0(Q,QT), COPIPP (QT,Q). If the first call is a leapfrog step (LF), the second one must be also LF. If the first call is Matsuno predictor (MP), the second one must be Matsuno corrector (MC). This is represented
symbolically by LF -;P LF, or MP-+MC. The second call can be followed by either MP or LF. Each of these combinations requires different transfers.
IX-3
F Description of the Time Step Sequence Parameters
NSTEP: Counts the time steps. Starts and restarts both begin with NSTEP=0.
NSEQ: The number of steps (combined matsuno and leapfrog) in each (repeated) sequence of time steps.
MLF(I): MLF(I)=0 or 1 according to whether the Ith step in the sequence is Leapfrog or Matsuno, respectively. First step is always Matsuno (MLF(1)=1). ISMTH, Smoothing routine (SMSHAP) is called MOD(NSTEP-NSM1, NSM1: ISMTH); if ISMTH=O, there is no smoothing.
NCOMP3, Physics routine (COMPS) is called MOD (NSTEP-NCM1, NCM1: NCOMP3).
Sample Runs
Matsuno only:
NSEQ%z , MLF(1) =1, MATSUN=1, DT=750., NSM1=0, NCM1=0 BCINO3=4, ISMTH=8
Leapfrog only:
NSEQ=1, MLF (1)=1, MATSUN=O, DT=600., NSM1=0, NCMl=O NCOMP3=5, ISMTH=10
1 Matsuno, 4 Leap- Frog:
NSEQ=5, MLF= (1,0,0,0,0), MATSUN=not needed, NSM1=0, NCM1=0, DT=600., NCOMP3=5, ISMTH=10
IX-4
h COMP1 Description
The COMP1 subroutine contains the horizontal and vertical advection
differences. The DO loops over I are arranged to make use of the
periodicity of the variables in the zonal (I) direction. For example,
suppose we are to compute for I=1,...,IM. Then the corresponding code is
I=IM DO 10 IP1=1,IM D(I)=Q(IP1)-Q(I) I=IP1 10 CONTINUE
Where we used the periodicity Q(IM+1)=Q(1). r
In the meridional (J) direction we compute our difference approxi.ma-
tions in stages in -^ ler to make maximum use of each line of latitude
of a typical var4lable when it is in core. The fourth order differ- ence approximation to (ikj0^) =D(i,k,j) aQa
(1) 3(Q j+l -QiRj-1 ) - 3(QiYj++22-Qikj-2)
Thus we see that Qi.,j will be needed in the difference approximations
to aQ for j-2, j-1, j+l, and j+2. The corresponding code is
DO 20 I = 1,IM QFLUXI= 4*(Q(I,L,JP1)+Q(I,L,J)) QFLUX2= .5*(Q(I,L,JP2)+Q(I,L,J)) D(I,L,JP2)= D(I,L,Jp2)-QFLUX2 D(I,L,JP1)= D(I,L,JP1)-QFLUXI D(I,L,J) D(I,L,J)+QFLUXI+QFLUX2 20 CONTINUE
For simplicity the array D is inititalized to zero and Q is scaled
so that (1) contains no divisions.
At the poles (j-1 or J=JNP=46) we use the values given in jection IV
on boundary conditions. We have special code for these cases denoted
J=2 or J=JM corrections. IX-5 COMP2 Description
This routine contains the Coriolis force term, the geopotential calculation (which should be made into a separate subroutine), and the pressure gradient and energy term calculations.
The geopotential PHI is dimensioned PHI(72,9,5) since we only need at most five storage locations for any computation. We use modular arithmetic (MOD5) to compute; the indices MOD, JPIMOD, JP2MOD which correspond to the standard index values of J,J+1, and J+2. Thus PHI(I,L,6) is stored in PHI (I,L,1), and we avoid shifting array values by using the MOD index as a pointer.
For the south pole calculation we need geopotential values at J=2 and 3, thus, for J=1 we compute PHI for J=1,2, and 3. For successive values of J we need only compute PHI at JP2 which is needed in the pressure gradient calcualtion at J. Therefore, the calculation of PHI and the associated array PK are coded for calculation at JP2. Except for the first J value we are only computing the geopotential at one latitude value for each pressure gradient calculation.
IX-6
T . ry ^^e . ,...^' ...u^t'^+^.:..,-. ,o.^^..u..^..uarl. .f, .^.M^.^f^'u.Yxl.'ia.3LFa.`ud.! ... _.1,.,_., . ....,.. .. X: Flow Charts
CCMPO (0, or)
JS1=1 JS2=1
((MAIN LOW)) DO 10 J=1,JM 1 ( (COMPEY'E ALL J-PAR*WrERS) ) IF(JP2>JM) GO 70 25 IF (J>1) CO TO 18 JP2=JP2MOD=2 ( (SAVE OTPOL (M) IN QM1POL (M) ,Wl, 2) ) 18 ((SAVE Q7(JP2) IN OSM(JP2MOD))) IF(JP2>2) GO TO 25 ((INCRM4MT JP2 , JP2WD)) GO :b 18 25 CALL, COMP1(O, QT, J) { CALL COMP2(Q, QT, J) ((ELIMINATE NBGATIVE HUMIDITIES) IF(PT4(400,1100.)) STOP IF(J<3) GO TO 200 9
IF (J=3) GO TO 70 i 29 ((UNSCALE QT(JS2))) CALL AVRX(QT,JS2) ((LEAP FROG, MATSUNO AS RELINED TO THE SEQUENCE OF 2 CALLS TO COMPO: 1st CALL 4 LF # LF or MP 4 MC, 2nd CALL, 4 IF 4 LF or LF 4 MP or MC 4 DIP) IF (LF+LF) GO TO 45
IF (LF-►MP. OR. MCMP) GO TO 58 ((STATUS: MP4W OR MC+LF)) ((P(I,JS2)'-PSM(I,JS2MOD),Q+QSM)) IF(MC+LF) GO TO 63 (STATUS: MP+MC) )
(contin.) a X-1 a
(Q• QT) j
S8 ((P(I,JS2)+-PT(I,JS2) oC)*.WDXYP*PT)) GO TO 64 45 ((P(I,JS2)4 *P(I,JS2)+a*(PSM+PT), 01-$*DXYP*P*QMt* (QSM+QT*DXYP*PT)) ) a 63 CON'1nLE ((SOURCE TER4 CORK MOM FOR LEAP FROG) ) A IF (NOT 00HP3 CALL) GO TO 67
((Qa4-M*DXYP*PT) )
64 IN NOT COMP3 CALL) GO TO 67
IF (MAT`SUNO PREDICTOR STEP) GO TO 67
CALL COMP3(QT,JS2) ((OOMPUTE SOURCE TERI CORRECTION, Q° Q+WDXYP*PT) )
67 IF (J, ((1NCREMT JS2,JS2MOD)) IF (JS2, 70 ((POLES)) 200 JS2=JS1 J 'I-? 10 CONTINUE ((END OF J LOOP)) F:F:ITJRN CW1 (QIQTIJ) {(COMPUTE JP2-JS2MOD)) IF(J>2) GO TO 2150 JS1=JS2-JSIMOD=JS2MOD=1 (contin.) X-2 COMPl (Q,QT,J) 2150 IF (J--JM) GO TO 2158 ((Cawm PV 1 ,Pv2 =V^ + v^+2}} IF(J=1) GO TO 2225 2158 ((COMPtTI'E PUl , PU2= U* + Ui+2 )) a ((COMPUTE HORIZ.ADVEC.-IN TANG. DIREC.)) IF (J JM} GO TO 2237 2225 ((COMPUTE HORIZ.ADV.-IN LAT. DIREC.)) IF (J>l) GO TO 2290 (0=2 CORKED'. TO HORIZ. ADVEC & COW.)) ((CONY. CALC. FOR CONT. EQ.)) 2290 IF (J IF (J=1) GO TO 2600 ((COMPUTE SIGDOT,PT)) ({COMPUTE VERTICAL, ADVEC.)) IF (J 2600 ((POLES, M=1 or 2)) ((HORIZ. ADVEC.)) ((CONT. E.Q.)) ((SIGDOT PL, PTPOL)) ( WERT. .AD,., EC.) ) RETURN END OF COMP1 Alternative code from to -e-----^ IF(J.EQ.1) GO TO 2222 ((COPIPUTE .PU, PU ) ((COMPUTE HAiI) )) GO TO 2235 2222 ((J=2 CORREC )) 2235 IF (J 2225 ((HORIZ ADVEC(J) )) ((CONV.CALC.FORCONT.EQ.)) 2405 ((CONV CONV*DSIG(L))) ----•---- X-3 COMP2 (Q,QT,J) ((COMPUTE JP2=... )) IF (J>1) GO TO 3001 ((JP2=JP2MOD=JPKP2=1)) :3005 3001 ((CORIOLIS )) I F (J >JM) GO TO 3032 ((FIRST MAIN LOOP IN L)) 4 3005 ((DO 3030 LX=I,NLAY )) ((COMPUTE ENERGY TERM IN THERMODYNAMIC EQN )) ((L=NLAY+1-LX )) IF(L 3007 ((COMPUTE PK(I,LL,JPKP2)FORLL= I,NLAY)) ' ((COMPUTE PHI'(I,NLAY,JP2MOD)) :3065 3055 ((COMPUTE PHI' AT JP2MOD FOR L (( 2nd MAIN LOOP IN L) ) 3032 ((DO 3031 IX--1, NLAY)) IF(J=1) GO TO 3111 ( POLES) IF (J>2) Go TO 3085 ((J=2: VT,TT COPMC.)) 3085 MP) Frig is m-)) IF (J--2orJM) GO TO 3135 (((P) FOR V EQ.)) 3135 IF (J < JM) GO TO 3031 ( ( CORREC. TO (P) J=JM) ) 3111 ((POLES)) 3031 ((CONTINUE)) END X-4 A FOURTH-ORDER FORECASTINV MODEL* r (E. Kalnay- Rivas, D. Hoitsma, and P. Anolick) The GISS fourth-order model, (Kalnay-Rivas, et al., 1977), which is a fourth-order, energy-conserving C M on an unstaggered grid, had shown promising capabilities. It produced forecasts that showed an 1;141provement over the second-order GTSS forecasts with the same fine grid (4 1 x 5 0 ) resolution, but that were somewhat inferior to the "ultrafi.ne" forecasts. , , , However, the first version of the model required excessive amounts ,of computer memory and time for execution,. The model has been reprogrammed into the "fourth-order band model." The, new program solves the primitive equations one latitude band at a time The arrays are stored in an interlaced way, with all arrays being u pdated at the same latitude stored contiguously, and similarly for all arrays used in the computation of the time derivatives. This design of the program makes effective use of the virtual memory capability of the Modeling and Simulation Facility's IBM 370/165 or Amdahl computers. The virtual memory facility permits the execution of programs whose c,)re size is larger than the one available, by placing the excess on disk and reading in those pages of information needed in the current calculations. The band fourth-order data structure and computations were constructed to optimize this virtual 1/0 process; a possible improvement may be to interlace also the arrays being updated with those used to compute the time derivatives. The use of the virtual memory facility avoids the explicit 1/0 used in the current Kern model, and yields a simpler program. (The band structure was suggested by G. Russell.) The band Fourth-order model computations have been optimized so that each time step is computed in the Amdahl in half the time requi,od by the old model. The array structure has also been designed to reduce the amount of overall storage and high- speed memory by a factor of two (see Table 1). Table 1. Comparison of Fourth-Order Model Computing Requirements. Original 6,nd 4th-Order Model 4th-Order Model Core (bytes) 3500K 1500K CPU time per step (COMPI, COMP2) 34 sec 17 sec *Reprint from Atmospheric and Oceanographic Research Review--1978, C'AS, NASA Technical Memorandum 80253, A-1 TM^ The model has been programmed so that it can use both the Matsuno and the leapfrog time schemes, the latter with the Robert time smoother. I f Q and D ( Qn ) denote the fields at time nAt and the corresponding time derivative computed from thEk space differences, the smoothed leapfrog ^%,J ,L eme is Qn+l = Qn - 1 + 2At D(Qn) Qn = Qn + .5v ( Qn-1 + Qn+l -2Qn) with Qo = too . The use of a smoothing coefficient requires a slightly smaller time step. For example, with v = .1, the model is marginally stable with At = 288 sec., compared to = 300 sec. for the Matsuno and leapfrog schemes. The smooth-d leapfrog scheme is further modified to include the sub grid '°,.,Ahysics" terms, and scaling and spatial smoothing proc ,,I ures. Since the "physics" is called every few time steps, unless the leapfrog scheme is restarted after every call to the "physics," only one of the two consecutive fields will be affected. The restarting procedure is time- consuming, so it has been replaced by the following algorithm Qn-1 = Qn-1 Qn Qn = Qn + Sn Q n-1 = Qn-1 + Qn where S n corresponds to the "physics" terms. This procedure, which ensures that the physics is applied to two consecutive time steps, has been tested with good results. The model can be extended into an "ultrafine" version in a straightforward way. Preliminary Results. The first numerical integrations performed with the new band fourth-order model show dramatic improvements over forecasts made with the GLAS model with the same resolution ("fine grid" or 4 1 latitude by 5 1 longitude). The quality of the forecasts is now comparable with those produced with the "ultrafine°' (2.5 0 latitude by 3 0 longitude) version of tie model. Results from a 3-day numerical integration are present:e.,r,.',I in Figure 1. During an extended 8-day integration of the new fourth-order model, the atmospheric systems remained remarkably smooth, exhibiting a realistic behavior both with respect to position and intensities. References Kalnay-Rivas, E., R Bayliss, and J. Storch, 1977: The fourth- order GISS moc.-. of the global atmosphere. Contributions to Atmos. Phys., 20, pp. 299-311. A-2 G. The 4th Order GIBS Model of the Global Atmosphere* E, K61nay•Rivas t ) and '), A, Bayliss ` ) t) and J. Storch') t ) Massachusetts Intitute of Teelmology Dept, of Meteorology, Cambrid ge, MA 02139, USA a) Goddard Institute for Space Studies NASA, New York, NY 10025, USA °) Present affiliation: (CASE, NASA Langley Research Center Hamptc.1, Vir ginia 23665, U,S,A, Abstract: The new (HISS 4th order model of the global atmosphere is described. It is based oil order quadnttically conserving differences with the periodic application of a l6th order filter oil sea level pressure and potential temperature equations, a combination which is approximately enstrophy conserving, Several short range forecasts indicate a significant improvement over 2nd order forecasts with file same resolution (- 400 kilt), Ilowever the 4111 order forecasts are somewhat inferior to 2nd order forecasts with double resolution, This is probably (tile to the presence of short waxes in the range between 1000 kilt and 2000 kilt, which are computed more accurately by the 2n(i order high resolution model. An operation count of the schemes indicates that with similar code optimization, the 4th order model will require approximately the same amount of computer tints as the 2nd order model with the same resolution. It is estimated that tie 4th order model with a grid size of 200 kill provides enough accuracy to make horizontal truncation errors negligible over it period of a week for all synoptic scales (waves longer than 1000 kiln), 1. Introduction It is generally accepted that the use of 4th order finite differences is more efficient in reducing space truncation errors than the use of higher resolution on a 2nd order model (KREISS and 01JUER, 1972), Linear analyses and shallow water type of experinents give an upper limit of the inll rovenient that can be expected from the reduction of errors in the horizontal differences, For example, Table I corresponds to it linear wive e(Itiation with phase speed c = I I ins - t , typical of atmospl-.°ric motions, It provides a measure of the "computational predictability period" after which the errors introduced by horizontal truncation alone become very serious. The table ,suggests that for waves longer than 2000 km ,I Order - 400 kin grid model is preferable to a 2 nd order , 200 grid model. Waves shorter than III 2000 kill are forecast more accurately by it 2nd order - 200 kin grid model. to insure that horizoni'al truncation errors are sinall in the 1 •3 week period of atmospheric predictability for all synoptic scale waves it is necessary to use either 4th order differences with a grid resolution of the .)rder of 200 kilt or 2nd order differences with a grid of the order of 100 km. Numerical experiments with simple nonlinear models (WI LI.IAMSON and BROWNING. 1913, KALNAY-RfVAS 1976a, from now oil also indicate dramatic reduction of errors by the use of 4th order differences, All these studies indicate that a considerable improvement in forecasting skill into be expected from the use of 4th order differences if a substantial portion of the f m-caving errors is clue to horizontal tnu►ra- tion errors. III numerical forecasts there are several are several ether important sources of errors: vertical truncation errors, errors in the initial conditions, and poor "physics" (i.e. parameterization of physical processes like radiation, dissipation, subgrid transports, cumulus convection, boundary layers, etc.) (Paper presented during; the DMG-AMS Meeting, Hamburg 1970; .see Preface to issue I 211977) 9eltnige zur P hysik der Atmosphlire, 50, Bard, 1977, Seite 299-311 A-3 ^y ^g1 • Wable 1, Fiapsed lime `I' after which a wave of wuvelrngill 1, moving with it speed c I 1 ms' ► lags by more utalt 1 00 kill due to space irwication errors. Ax r 400 km Ax ; 200 kn1 2nd order 1. 2000 ku► _ f , 0,4 ch►ys L - 2000 kin, I' - 1.6 days differences 1. 4000 kilt, T" 1.6 days h ., 4000 kin, T" 6,4 days 41h order 1. 2000 lint, T;^ 1.5 days L 2000 knt, 1'_- 21 days dit'feronces I, = 4000 kilt, '1' _= 21 Buys I. - 4000 knt, Tr ;328 days At the Goddard Institute for Space Studies (( SS), New York, we have developed it 4111 order general circulation model ((;CM) with the expectation that it will yield more accurate short range forecasts, and probably more realistic climate simulations, but with the inlprovetlicnl limited by the other sources of errors, The model is described fit Section 2, The results of several experimental short rouge forecasts are discussed in Section 3. A sample 30-hour forecast is presented and compared with forecasts by the standard 2nd order GISS model, which has the same resolution as the 41h order nu►del, and by the "ultrafine" 2nd order CISS model, which has twice the horizontal resolution, Section 4 contains the conclusions and a discussion of future work. 2. Description of the model The 4th ord,.r (HISS global atnu►spheric tilt►del is a primitive equation model using longitude (X), latitude (0) and sigma (o) coordinates, The hasic equations and the paranteterixttfious of physical processes are the same its those of the standard 211d order GISS global model (Somerville of al, 1974). fit The finite difference scheme is quite different and is described following subsections, 2.1. Finite-difference scheme The horizontal grid is u0fform (constant AX and J(p) and non-staggered, There are two ways to define such it depending oil variables are defined of the poles., as do-ne by wl I IANISt1N and n ROW NING (1073), or half a grief size aw.ly from the poles (I101,1,OWAY et al, 1 1)73), We chose the lornler method because in the absence of smoothing near the poles it allows a lime step twice Its large its the lattcr. The singularity that shcrical coordinates have at the poles is e yplicitly avoided by the use of it polar cell where "stereographic" velocities are used, III vertical direction we use it grid with !r, the vertical velocity, dctini=d ill tlu_ boundaries of each layer, atld all other variables in file center of the layer. The vertical grid can he non uniform although experiments have been made so, far with constant 8n. Experiments reported in I indicated that no significant improvement ill was obtained when a 4 0l Order staggered Conservative schcane ill was itself vertical direction. `I'llerefore the uuxlel,has 211d order vertical differences, The eompu- tation of* the geopotclitial (1) is performed as indicated by .ARAKAwA (1072), with iI modification suggested by h111 i,i,i vs (1974). a) Forecast equation away from the poles Several systems of horizontal difference werl tested as reported in I, 'I'lle scheme chosen for ► he h asis of these experiments consists of the Simlllcst possible quadrahcally conservative 4th order difl'erenc:es. a A^4 We define the finite difference horizontal divergence operator at die vertical level k as ^ 4 6,v (Ihi^`9 A ) 1 6 2 ,\ (Ilu2^g 2^) Dk(b) = atscs + 5 0 (IIV o is^ 0 9 ^52,,(llv COS 0 2` 920)J k and the finite difference gradient operator as d14 ^ ._ 3fi2^g 2A 6 (2.2) Vg ucosc^ ^ 3fiAf3 + a [ -- 9) ^ .. 62^,t3 20i here II = pS p,r, is the difference between the pressure at the surface and the constant pressure at the top of the model, a is the radius of the earth, w = u6 + v is fhe horizontal velocity vector ill coordina(es, and we use the finite difference notation 8,\ g = jg(X + n AX/2, 0, o, t) f? 1 _, nAN/2., t)II(n AN), (2.3) 9 11h = . ig(X +nAX/2, 0, o, 1)+ g (N nAX/2, 0, o,t)1 12, and similar formulas for the other independent variables, With this notation, the continuity equation away from the poles is 0113t _=. R( )... _ l) 1 11fi ti h ( . 4a ) or, if we integrate it in the vertical and slake use of (lie boundary condition n =r 0at0 = 0, 1, l< 11 = V 1)k (1) Aok (2.4b) It ^ l where K is the mimber of vertical layers. Equation (2.4b) is the forecast equation for 11. The momentum equation is Ilk tarlo _ lmo r 11S a (ow ")k + ^t + a IK X (Ihv)k c)t rc II ( V ,I" + our/ V 11] + IFir, k The first law 6f thermodynamics is 11r:^^k=,,.Dk(T)^-(p"11fizT^o l'n (1'/p k ) ^}k IIox T r31f II + 8t + V.VlI } } + Q. (2.6) K 1 k r> A•, 5 I lore '1' is absolute temperature, IF is the 1'rictiunai force, )i = R/C1„ Q is the diabetic heating per unit fit mass, p is the pressure and p" is computed as indicated :, Lct The moisture equation for the water vapor mixing ratio q is i) Ilrl 1 I) b) Forecast equations at the Poles We have followed the n►ethord used by W1 ► HANISON and BROWNINC (1973)and define it polar Call of' radius a ny, on which we use "stereographic" (or rather "cartesian") velocity components defined by tilt lransl'ormatioll L usin\ t vcos\ V. I,a coO v sinx (2.8) with Inverse it sin\ 4 V.,pcas\ v FII, CON \ V j pSlllN ( ,i)) where the top and bottom signs correspond to the north and south poles respcctivvly. The positive x-axis of the carrtesia n coordinates coincides with lho meridian of longitude N 0, The difference in signsl at (lie South pale is due to the choice of I right hallded system ol'coordinaies with tile verti- ca unit vector pointing outwards, formulas (2,8) :n d t,Z. o)) are used for define "spherical" or "carlesian„ velocity components wherever !!Icy are required in thi s finite difference equations. The finile dit'terenco horizontal divergence operator :11 file pales is defined by all average aver all longitudes \a: 11 1, k(3) : + a'r 1Aa (ilvi;) ;, I i A (Ilvg),\;, ^ ('; z as") j (^.11t) where \I w (t 3) AN, AN M 2n/l, and A,, -- JNasin(n 10)/1 2na = (1 vos(nAo))j, The unite difference gradient operator at the poles is also defined by an average over all longitudes: f V1 p9 ^.+ gs 1 , r ads) t~^ i (; ^ao)1' leosXa ,' sinNi j, t ,I I) I:► ^¢ 3 l`; 6 A- 6 71ren the continuity equation at the poles is 3 ' t^ = 4 ^ D, 11,W) 11ak, (2,12) k-1 the momentum equation is d II V) =T DtPk(V) - 11 ,P h o(a tP V„P) +fipkx(11V) at^, k k 1111 oR7'' I II tP ^O,P ^It +_4'_.. Dil'llI k +IFtP,k (2.13) where Vt1, = U tt, i + V tt,j. The thermodynamic equation is J 1 1T = +D,,P,k (T) — {PtP I I tP b a Oil' (TsP/PsP k (2.14) cat .tP,k ill t1, a K Ttp + ._. _ ptt^ #t' lip _ C^c7l^ + Vi VtI,flk+ 'Qtt, and the moisture equation is similar, C) Diagnostic equations In a-coordinates, the pressure is defined by p = all + p ,t, . hollowing a suggestion by Pill UlPS (1974) we define at the center of a layer pk = I /(N + 1) S„pk ” 1 /hepk, instead of pk = ( )” as in ARAKAWA (1972). This formula was derived under the assumption that ti►e potential temperature varies in a a-layer much less than either the temperature of the pressure. Ptn I_I.IPS indicated that a more accurate relationship between temperature and geopotential can be expected from this formula. The geopotential The coefficients C k are determined From Iiak R ook C1^ Ck _._.. pk , _,.._ p.k (a (2.17) k The vertical velocity in o-coordinate is determined from (2.4a) and (2.41)), and the boundary condition b = Data = 0, 1: K k (l) 6, bk I1 I N' D Oak Dk ( l )^ (2.18) k 1 A-7 dl Boundary conditions In the cast-west direction we use periodicity: g(N + 2 rr, 0) = g(N, 0) for all variables, When the value of a variable is needed "beyond" the poles, we define it by continuation along the same meridian: ^v^N,±^+^cp!}- Iv^rr+Nt^1~d^?}'g^N,+(2+`1$)I^g(tt+,*\^^o/!> where g represents all variables other than the two horizontal velocity components, In the vertical the top and bottom are material surfaces through which there is no flux (o= 0 at o = 0, 1) except for subgrid boundary layer fluxes of moment,mr and heat included in F and Q, 2.2. Filtering near the poles and high order filtering The CFL computational stability condition requires a very small time step unless linearly unstable short waves are filtered out near the poles. For this purpose several alternative procedures liave been tried but so far the method found to give most satisfactory results is the fourier filtering of the prognostic variables u, v, T and the indirect smoothing of` Il through the filtering ol'the sea level pressure (SLP) field, The fourier filtering is performed polewards of 66 0 latitude. The amplitudes of the fourier components of zonal wavenumber n are multiplied by a transfer function which is I for n ` N and decreases linearly to zero between n = N and n = N + S. The number of retained modes is defined by N(0) = integer part of (90 cos 0). We have tried filtering the stereographic velocities U, V instead of u, v but no improvement. was obtained. Since both the surface geopolential and the surface pressure fields contain large amplitude short wave components due to the presence of orography, an artifical smoothing of these fields represents a distortion of the real geometry of the boundary, Oil other hand, the SLP is an instrinsically smooth field, and its high wavenumber Components more closely represent atmospheric waves. Therefore in our model the SLP is filtered near the poles and 11 is recovered by solving the transcendental equation used to relate them It was found that the 4th order model forecasts were less smooth than those of the 2nd order GISS model, especially in regions with steep orography. Based oil same considerations we have introduced a periodic application of a high (I 6th) order filter (SHAPiRO, 1970) oil SLP and potential tem- perature fields, which are not very affected by orography. .h. The filter is of the form g = { i — (F')s){ 1 (F')s } g, where F'(gij) = (gi+ij"' gij +gi -ij)/4 and has a response F' Iexp(ikX)} = sin' (kAN/?) exp(ikN). This filter eliminates waves shorter than 40N and even after hundreds of applications has negligible damping effect oil longer than 4AX. In the meridional direction there are three simple alternative forms for F2: F,Pi (gij) = (gij + t ._ ?gii + gii - j )/4 v. g F q511(gij) = i(gij + 1 ij) COS Oj + 1/ 2 — (gij — gii - 1 ) cos Oj -- 1/ 2 1/(4 Cosoj) and F^,rri (gii) i F.'i (gij cosO)I/cosoi FRANCIS (1975) used Pali , which is the only one that conserves the area weighted average of g. However,we have found that the three meridional filters produced virtually identical results, This could A-8 be expected because the waves affected by the filter are too short to be strongly influenced by the convergence of the meridians. Since F^ and Foi l, can be programmed as efficiently as F' we find them preferable. In the model we use Fp l , with g continued "beyond" the poles as indicated in Subsection 11 d. At the poles g t p is defined as the average over all longitudes N I of the filtered values gt , f p. At the present we are applying the high order filter to the SLP and potential temperature fields once every two hours. 2.3. Enstrophy constraint BAYLISS and ISAAC'SON (1975) have developed a simple method to make any finite difference scheme conservative with respect to any quantity. The method has been tested in our model by forcing conservation of enstrophy on the dry adiabatic version of the model, although in such a model it is potential enstrophy that is conserved. The procedure is the following: Let the vectors U and V represent the values of the velocity components discretized over the grid and let the functional G(U, V) denote the consistent 4th order approximation of the mean square vorticity. At each time step a correction U, V is added to the predicted values U, V such that G(U + U, V + V) = G(Ue, VQ ) where Uo, Ve de , iote the velocities at time t = 0. Since this equation cannot be solved explicitly, it is linearized about the predicted values U, V: U' + f ^G V = G ( U n, Vii) — G(U, V:) (2.19) g'ia, V WV U, V If we assume / aG aG) 20 au av then o can be determined from (2.19): ..G(U, G ( U 0i, Vo•) V) ^ 8(122 + 1 2 ( '?1) r »^ ail aV) It may be easily shown that the choice (2.20) minimizes II U' II + I) V' II ,the norm of the correction vector.. It has been found that this procedure improves the forecasting skill of the model. However, when it is used in combination with the periodic application of the high order filler the improvement is marginal. Therefore this option is not included in the standard version of our model. 2.4. Summary of the properties of the model The finite differences of the model have the following properties: a) Horizontal differences are performed on a nonstaggered grid and have 4th order truncation errors. b) Vertical differences are performed on a staggered grid and have 2 A- 9 a ,'; R^ 3 e) The model contains no horizontal diffusion expect for the possible use of it dissipative time scheme i and for the periodic application of the high order filter. It has been shown in 1 that the combination of a 4th order quadratically conservative scheme with the periodic application of a high order filter t replaces success/411), the use of ap scheme. This is because waves shorter than four times the grid size are the ones subject to aliasing and to large truncation errors, and they are removed by the filter before they at.rain finite amplitude. Waves longer than four tintes the grid size are accurately computed by the 4th order scheme and are not affected by the filter, 3. Numerical forecasts 3.1. Analytical initial conditions ne model was tested using 3 dimensional extensions of both the nontrivial steady state solution and the Rossby-Haurwitz wave initial conditions used in I. The forecasts remained smooth S during the several days of interation, Errors in the steady state case were an order of magnitude larger than in I because of the use of single precision arithmetic oil IBM/360.95 computer. However, it was found necessary to use double precision for the longitudinally averaged pressure gradient at the poles in order to avoid excessive local error growth. 3-1. Forecasts with real initial data As mentioned before, the parameterization of physical processes included in the terms F, Q, C and E of Equations (2,S) • (2.7) has been adapted front standard 2nd order GISS model (SOMER VI LLt. et al, 1974), The only change that has been introduced is the reduction by it of 2 of the surface all drag coefficient, TMG was found to be necessary in order to avoid damping of the pressure systems after one or two day forecasts. We believe that the fact that in the staggered grid the surface winds are obtained by a horizontal average reduces the relative effect of friction in the standard model, At the present time (November 1976), we have performed 5 experimental short range forecasts, Since they have been made with slightly different versions of the 4th order model, we don't yet have a reliable measure of the model's forecasting skill, The initial data so far has been available in the staggered grid used by the standard model so that winds have been linearly interpolated to the nonstaggered grid. This is a source of error whieli nlay have adversely affected the 4th order forecasts. We have performed a single 4-day forecast which indicated that the model remains very stable and that synoptic systems show no tendency to become unrealistically weak. All the numerical 4th order forecasts show a significant improvement over the 2nd order forecasts performed with the same resolution. This improvement appears as a better estimation of the changes in position and intensity of several pressure systems. This in an encouraging result, especially in view of the study made by BAUNINEI M and DOWNEY (1976) which indicates that the standard 2nd order GISS model forecasts compare favorably with those made by the NMC and NCAR models with similar resolution. On the other hand, and contrary to our expectations, the forecasts made with the 2nd order GISS model with double horizontal resolution were found to be either similar or superior to the 4th order forecasts. This is more true in the sea level pressure than in the 500 nib forecasts. We consider that there are several passible reasons for this result. One is the extra errors introduced in the 4th order model initial j A-,10 I e^ N 12 4 O 0 —4 H 28 24 L 6 8 4 8 M^ 12 12 8 .8 40 0 4 1 •1 60° 4 20 0 .1 8 •4 24 4 0 L 28-S, 28 40 8 32 H 0 8 32 H M 7.0 28 28 24 24 ae 20 24 — 24,, 20 20 16 I6 12 12 12 12^ 180• 1600 1400 1200 1000 BO• 600 40• 200 W O° OBSERVED SEA LEVEL PRESSURE ( FEB 22, 1976 122) • Figure I. Observed 36 hr, sea level pressure at 0000 GMT, 22 February 1976 data by the averaging of the winds. A second reason (and probably the most important one) is the existence of waves with significant amplitudes and wavelengths between 1000 and 2000 km. These short waves are better detected in the initial data and more accurately forecast by the 2nd order model with double resolution. The nonlinear interaction of these short waves with longer synoptic waves which according to linear theory are better forecasted by the 4th order model is clearly very important. A third important reason is the fact that errors introduced by the parameterization of subgrid processes become less important as the grid size is reduced. We present an example of a 36 hour forecast of the sea level pressure corresponding to February 22, 1976, 1 2Z. Figure 1 is the verification sea level pressure map. Figure 2 is the forecast computed with the .standard 2nd order GISS model, which has a resolution ON = 5°, A0 = 4°, Aa = 1/9. Figure 3 is the 4th order forecast computed with the same resolution, and Figure 4 is the forecast obtained using the "ultraf`me" 2nd order GISS model which has a resolution AN = 2.5 0 , A0 20, A a = 1/9. The most striking improvement showed by the 4th order forecast is in the position and intensity of the low in the eastern portion of North America, in fact, this low and the high over the Atlantic Ocean have been forecasted better by the 4th order model than by the 2nd order model with double resolution. On the other hand the developing cold high in central Canada and the intense low pressure center south of Greenland have been predicted better by the double resolution 2nd order model, As usual in this A.-11 f6 N 12 80' 4 1 20 .4 1 4. 0 H 8 8 4 1 60` 4 8 V 1 6 _ q 0 12 20 2 6 24 0 28 X82 2 24 40° 1 H ^I 32 20 28 40 24 H a 24 16 20' 20...:. I6 16 1 12 6 12 .. 12 12 12 180° 1600 1400 1200 1000 800 609 40 20° W 0• 36 HR. FORECAST —2nd ORDER MODEL • Figure 2. 36 hr. horerast using; the standard 2nd order GISS model _ type of comparison, the three forecasts share several important deficiencies, For example the intensity of the high over southwest USA has been overpredicted by the three models, probably because of difficulties introduced by the orography. The low south of Alaska has been erroneously split by the three models, This splitting, which is slightly less pronounced in the 4th order forecast, may be due to both errors in the initial data and orographic and coastal problems, 4. Summary and conclusions We have described the characteristics of the 4th order GISS model of the global atmosphere, It is based oil quadratically conservative scheme with the periodic application of a 16th order filter on the sea level pressure and potential temperature fields. As shown in I this combination is approximately enstrophy-conserving. An operation count of file numerical schemes indicates that with similar code optimization the 4th order model will require approximately the saute amount of computer time as the 2nd order enstrophy-conserving GISS model with the saute resolution. We also plat: to introduce a simplified serni-implicit scheme, and to study the possibility of using the combination of a nonstaggered vertical grid with the Kreiss 4th order method described in 1. A-12 1 3 --- •--•----^^ 12 N 20 4 0 . a 80' ^^8 20 1 -8 4 60' _p —.. 4 ^ 0 8 ^^ 4 4 20 12 H 12 12 232 8 40 3s H ,.^ H 0 m i ^0 16 16 2 24 \\^12 6 28 H 24 +r 2 a20 20 2 0, _u^ 16 16 12 ®^ 12 I O° 180° 1600 140° 120° 100° 800 600 40° 20° 00 36 HR FORECAST -4th ORDER MODEL • Figure 3. 30 hr, forecast using the 4th order GiSS model with the sane resolution as in Figure 2 The results of several short range forecasts indicate a significant improvement over the 2nd order forecast with the same resolution. This improvement is shown in the estimations of changes in position and intensity of several peessure rystems, We plan to study the impact that the greater accuracy of 4th order differences has on the forecasting skill of variables of more practical importance, such as tempe- rature and precipitation. It has been found that the 4th order forecasts are somewhat inferior to the forecasts made with a 2nd order model with double horizontal resolution. We consider that one of the most important reasons for this result are the presence of waves in the range between 1000 kin and 2000 kin which are corn- puted more accurately by the high resolution 2nd order model than by the 4th order model. Another important reason is that errors introduced by the parameterization of subgrid processes become smaller as the size is reduced. It is estimated that the 4th order model with a grid size of 200 kin provides enough accuracy to make horizontal truncation errors negligible over a period of a week for all synoptic scales (waves longer thr,n 1000 km). A-13 to 36 f1ti FORECAST -2nd ;?RDER MODEL WITH DOUBLE RESOLD PION • Figure 4, 36 hr, using titc "ultrafine" 2nd order GISS model with double horizontal resolutiml Acknowledgements We are grateful to Drs, k1, Baler, W, (quirk and Profs, J. Churney and P, Stone for their continuous support and encouragement, Prof, E, Isaacson provided generous advice and Prot', Dr, Randall made useful contitrcnts ott the manuscript, Prof, F, Sanders continents on the numerical forecasts are also gratefully acknowledge(l, References ARAKAWA, A. 1972: Design of the UCLA GCM, Teehn, Rept, No, 7, Dept, cat' Meteorology, UCLA, BALIMHEFNER, D„ and P, DOWNEY, 1976; Forecast intercomparisons between large-scale numerical weather prediction models, Annalen der Meteorologic, 11. Simulation of large scale atmospheric processes (extended abstracts) 205 208, BAYLISS, A, and E, ISAACSON, 1975; How to make your algorithm eonse) r'ative, Notices of the .Arn, Matti, Soc„ Augur; 975, FRANCIS, P, E„ 1975; The use of a multipoint filter as a dissipative mechanism in a numerical model of the general circulation of the atmosphere, Q, J, Roy, Meteor; Soc., 101, 567- 582, A-1.4 HOLLOWAY, J. L., M, Spclr.,an and S, Manabc, 1973; Latitude-longitude grid suitable for numerical time integration of a global atow;pheric model, Mon, Wea, Rev., 101, 69»78, KALNAY-RIVAS, L. 1976x; Numerical experimc+nts with 4th order conservative finite differences, Annalen der Meteorologic, I I , Simulation of large scale atmospheric processes (extended abstracts), Addendum, KALNAY•RIVAS, E„ 19766: High latitude truncatio,.; errors of box-type primitive equation models. Mori, Wea, Rev., 104, 1066- 1069. KREISS, H,-O„ and J. OLIGER, 1972: Comparison of accurate m, ibods for the. integration of hyperbolic equation. Tellus, 24, 199 -215, PHILLIPS, N. A., 1974: Application of Arakawa's Energy-Conserving Layer Model to Operaiional Numerical Weather Predictip,% NMC Off"w Note 104, SHAPIRO, R., 1970: Smoothing, filtering and boundary effects, Rev, Geophys. and Space Phys., 8, 359--387. SOMt RVILLE', R„ P, H, STONE,, M, HALEM, J.F. HANSEN, J.S. HOGAN, L,M, DRUYAN, G. RUSSELL, A, A. LAOIS, W. J, QUIRK and J, TENENBAUM, 1974: The GISS model of the global atmosphere, 1, of Atmos, Sci„ 31, 84--117, WILLIAMSON, D, L, and G. L. BROWNING, 1973: Comparison of grids and difference approximations for numerical weather prediction over a sphere, J, Applied Met„ 12, 264--274. A-15. THE EFFECT OF ACCURACY, CONSERVATION AND FILTERING ON NUMERICAL WEATHER FORECASTING i,% ja gugenia Relay-Rivas Department of Meteorology, Massachusetts Institute of Technology and NASA/Goddard Space Flight Center and David Nottema(a) Sip* Data Corporation. c/o NASA/Goddard S;,aca Flight Center 1, IK'[RODUCTION 2. DISCUSSION OF THE DESIGN OF THE MODEL The design of a numerical model for atmospheric Different numerical analysts and atmospheric 4 simulation is not a straight -forward procedure. modelers often take different approaches in I Both in the areas of mathematical and numerical the resian of a numerical model for weather ` analysis, and in the parameterization of predietion. Sometimes they even have diffor•int physical processes not explicitly resolved, basic philosophies. For example Arakaws (1966, the modeler faces several difficult choices 19721, has been a pitt-neer in the development between equally reasonable methods, and and use or numerical, 3chtmes that reproduce uometimes between similarly unsatisfactory as closely as possible the conservation pro- methods. parties of the eontinous equations of fluid dynamics that determine the motion of the In this paper we discuss the considerations atmosphere. On the other extreme, Krelsa leading to the numerical design of the GLAS and Oliger 1 1972, 19731 have advocated the fourth-Order Global Atmospheric Model. This use of more accurate schemes even when they model, which was briefly described in Kalnay- don't formally satisfy any conservation gives at al., 1 19771, has been restructured, properties. and several minor changes were introduced. The computation time and memory requirements The current GLAS model t o based on Arakawa°s for the 4th order model are now similar to 119721 sLjond order scheme wing a staggered those of the present aeoond order GLAS grid B. The scheme is energy conserving and model with,the same 4° latitude,, 5° longitude approximately enstrophy conserving for non- and 9 vertical - level resolution (Sommerville divergent flow. In this section we isolate at ,a^l..^ Z9741, However, the fourth-order the areas in which the Fourth-Order model is nasal foreeaet skill is significantly better different from the GLAS model, discuss some than that of the current GLAS model, and of the alternatives, and the j ustification of after 3 days it ie e^c-ayable or better than our choice. that obtained with the 2.5° by 3 1 version of the GLAS model. 2.1 Accuracy A discussion of several of the basic charac- There is A consensus among modelers that for teristice of the model design is contained finite difference models with second-order In section 2. For each of them we present accuracy, horizontal resolution of about scat of the possible alternatives, their 400 ko. and about 10 vertical levels, hori- advantages and disadvantages, and the reason zontal truncation errors are the most important for our choice. In section 3. we discuss source of errors. The truncation errors the effect on numerical forecasts of changes can be reduced by any of the following methods: In the accuracy, resolutio4 A nd conservation peopertieg of the models . ( , Section 4 e. Increased horizontal resolution. contains some final remarks. retaining second-order differences.. Advantaitesi If the horizontal grid size s reduced by a factor of 2. cruncation error• are reduced by a factor of 4. A ^1) As of this writing, several of the coaeparlson of the effect of truncation errors numerical experiments are not complete. on the computational phase speed and group velocity of a limear wave to presented In c ,present affiliation: Department of Fig. i+ Another advantage is that Increasing Mathematics, Wesleyan University of Connecticut A-16 I^` the resolution allows smaller but possibly Advantages: These fourth-order differences Important scales to be explicitly Included are considerably more accurate than the In the model. fourth-order differences of the first kind Dleadv^ants est The reduction of error is (Fig. 1). The crossover with double-^eaolution slow., ind t he computation time is lncrused second-order differences in the phase speed by a factor of 8. error occurs At 3634 and In the group velocity error at about 44x. Disadvantages: in the simplest case, Kreles' COMPUTATIONAL PNASEGPM oVe t ourt h order differences require the solution 10 of tri -diagonal matrices. Finite element schemes require the solution of at least block os s.r; tri-diagonal matrices. Even though there are 0: 01, efficient methods to perform these inverefono, 0e, they are still computattonally expensive. sr The extra accuracy can be compensated with u 4th order differences of the first kind by increasing the resolution. which has othor 1 7K:WAVt ER07% in st a tsrr • s : a za advantages, as we mentioned earlier. These schemes, as well as In higher order finite difference schemes, waves In the range of 24x COMPUTATIONALIA GROUP VELOCITY op/o to 4 Ax ere still grossly misrepresented. d. Spectral schemes. 11A Advantages . .1J the baets of the apevtroal 0.0 as ORDER eSpan iLQn are the oigenfunetions of the wave MnORDERfa equation, spectral schemes have no phase a4 0h,0RD:^nn speed errors. Because of this they require am oa ono:nlnxat less resolution than finite difference schemes. encr:n► Disadvantages: Because of the large number 111) :wawuNaon:lnxt M :os : ra : of computations required for the nonlinear terms, spectral schemes are competitive with finite difference schemes only in combination 140 with the use of less resolution and semi- 44 Implicit time schemes. Our choice: Wr chose to use fourth order -Is finite differences of the first kind because they are computationally efficient and have small trWncation errors except in the range Fig. 1 Computational phase speed and group of waves with wavelengths between 2Ax -and 4Ax. velocity fot a linear wave of constant phase speed c, for different 2.2 Type of Grid numerical schemes as a function of wavelength L. Both staggered and unstaggered grids have been widely used by atmospheric modelers. a. Unstaggered grid: The advantage of this grid is its simplicity. Higher order schemes are easily developed with this grid. Its disadvantage is that all centered Fourth order schemes of the first b. differences have to be computed over a kind. We include in this group the common distance of 2Ax. 5-point explicit fourth order schemes )Kreles and Oliger, 1972). b. Stammered grids: Several staggered Advantages: Phase speed errors are such grid configurations are possible as reviewed "aller especially for long waves (Fig. 1). by Arakawa ( 1972). The one he called scheme C. Waves longer than 5 Ax have smaller errora in which Is the most commonly used, has the their phase speed with fourth order differences pressure defined at the center of a grid than with second order differences with twice cell. and the velocity components u and v the resolution. If we consider the computs.- defined at their corresponding normal walla. clonal group velocity. the crossover actors This grid has the advantage that the pressure at about 7Ax. The computational time is only gradient and velocity divergence terms are Increased by a factor between 2 and 3. computed over a distance of only 14x, so Disadvantages: Wdves in the range of 24x to that inertia gravity toavee are oamputed 4Ax are still grossly mlarerresented. with double resolution. Therefore, as pointed out by Arakawa, geostrophic adjust- Fourth order schemes of the second C. ment is best represented In this grid. On kind. In this group we Include linear finite the other hand. advection terms are com- elements, cubic splines and the "compact" or puted with no more accuracy than In the Fade type 3-point 4th order differences unstaggered grid, and in the Coriolis' 1 0rszag i Israeli, Introduced by Kreles acceleration term, it Is necessary to take 1972), all of which have the some fourth horizontal averages of the velocities. The order accuracy. A-17 higher resolution of inertia gravity waves and be eventually dissipated (Charney, 19721. reduces the msuimum time step for erplieit In a numerical modal, the finite resolution time scheme by a faotor of two. FulI fourth *Voses an artificial Utl" at the short and order schemes can be developed with staggered of the spectrum, inducing an excessive schemes but they are very involved (Nalnoy- accumulation of energy in the shortest waves. Rlvas, 1976). To date, modelers using This problem to worst in noncone*rvative a staggered grids have introduced fourth order schemes, but it appears even in allaa-free, differences only In the advection terms. energy- and enstrophy-conserving spectral models. This justifies using some par&- Our choicse During extended ranges fore- asterizatlon of the unresolved subgrid casts,,w seorder error* in the non-advective addle• to withdraw energy from the smallest toes* may become important. For this mason resolved scales. we chose to use an unstaggered grid that allows the use of a simple, full fourth-*,order For this purpose, Leith (1972) suggested the scheme. use of nonlinear horizontal diffusion in which the eddy diffusion coefficient is pro- 2.3 Conservation Properties and the Us* of portional to the local gradient of vorticity. Horizontal Diffusion This formulation to consistent with the trans- fer of energy to subgrid scales In two- With respect to conservation properties, there dimensional turbulence. In another widely are basically 3 types of finite difference used formulation, suggested by Smagorlasky schemes for the primitive equations: 11963' and based on a three-dimensional a) Nonconservative schemes, the simplest of turbulent cascade theory, the diffusion which is the one based on the advective form coefficient is proportional to the deformation of the equations; b) quadratically or energy tensor. These formulations are better than conserving schemes; and c) enstrophy conserving the use of linear diffusion, but they both schemes. Advective and quadratically conserve- share the following problems: a) the diffu- tive schemes can be easily developed using sion coefficient is computed inaccurately staggered or unstaggered grids (Lilly, 1965; for the shortest waves, and, more Importantly, Bryan, 19661. Enstrophy conserving schemes b) when the diffusion coefficient is large for the primitive equations have been developed) enough to avoid the spurious growth of the on a grid C by Grammeltvedt (19691 and Arakawa smallest scales, It produces excessive and Mintz 119741. Sadourny 11965a, b) con- damping of the larger scales ;Merlleea, structed a potential enstrophy conserving 1975; Williamson, 1978]. Furthermore, at the scheme on grid C, and Arakawa (19781 has short end of the spectrum (scales of the order recently developed a potential enstrophy and of 100 km), neither quasi-geoetrophic nor 3- energy conserving scheme also on grid C. dimensional isotropic turbulence theories are really justified. Williamson 119781 Honconservative schemes require a procedure genera7,ized a higher order diffusion of the to damp waves shorter than 44x, which other- form O KV2 , suggested by Kreiss and Oliger wise grow spuriously causing catastrophic 11972;, still using the deformation type of nonlinear Instability (Phillips, 1959). diffusion coefficient. This formulation This has usually been do,ie by means of linear has the advantage that, because it is more or nonlinear horizontal diffusion, or by scale dependent, there is less damping of using dissipative numerical schemes such as the longer waves. M_ the Lax-Wendroff, or schemes that contain In our model, we have taken an approach closer explicit horizontal averaging. to Fourier filtering the shortest waves, Quadratically conservative schemes avoid the as suggested by Phillips 119591. Our "subgrid unbounded growth of the solutions associated parameterization" is based on the following with catastrophic nonlinear instability. arguments The 4th order scheme Is adequate However, as Arakawa (19721 and Sadourny accurate for waves longer than 46Y. but and (1975a, b) have pointed out, in the course grossly 9naccurate for wh4es between 2 Ax 36& Since the shortest waves cannot provide of long integrations, there is still a spurious build up of energy in the shortest waves, any uaef6Z information in a finite difference we filter them out of the system which appears as an unbounded growth of the eoheme, total enstrophy. In the absence of hort- while their amplitude is still small. Even zontal diffusion, this type of though they are filtered out, the shortest Blow non• waves still play an important role in the linear instability will completely distort the solution. Enstrophy conserving schemes, model: they act as a buffer or "sponge layer" on the other hand, impose a stronger con- In the spectrum domain, allowing energy to straint on the growth of the smallest scales trickle down from longer waves and avoiding present in the model. For this reason, the the accumulation of energy that would other- UCLA and the CLAS models which use enstrophy wise occur at the short wave cutoff. conserving schemesedo not need to include horizontal diffusion. The elimination of the short waves to performed In the model with the periodic use of a 16th It should be emphasized that conservation of order Shapiro 119701 filter, first Introduced enstrophy does not necessarily Imply a more In GCN@ by Francis (1915). Figure 2 indi- accurate or realistic simulation. In the catoo the response of Shapiro filters of real atmosphere, the'constratnt of ®wsi- order 4, 8 and 16. It can be observed that ogeostrophic motion implies that very little In the case of a 16th order filter waves of the energy generated In the bdroclinically longer than 4 are scarcely affected at all unstable scales can reach the smallest scale* A-18 even after 128 applications, which in our model correspond to a 10 day integration. At: 4th order. non-conservative, noemoothin . Waves shorter than 4 are virtually elimi- A,1: to" ,linear diffusion.aa10^W/s of nated. Lower order filters introduce too A3: " .16th order filter/4 bra. to much decay at long scales. A4: " " /time step. at: is, quadrat„ conserv., no smoothing. to It 82: order filter/4 bra. 83: ifI " , 8th " of of 54: 4t1h to .6 to C: 2nd order, no smoothing 6 16th order ^` 1 application , .4 TABLE is Characteristic # of the different runs in Fig. 3. .2 128 opphcotiona,. O 16 8 6 4 3 2 L/ax 1.0' % .6 8th Order I .4 1 opp 128 opplic. .2 Ways) 0 16 8 6 4 3 2 L /ax Fig. 3 Variations of total enstrophy during 6.0 numerical integrations as indicated In Table 1. The scale corresponding .8 to experiments Al and A2 is multi- plied by 10- . 4th order .6 We found that all stable rune conserved total 4 1 opplicotion...V% energy with a high degree of accuracy. Fig. 3 shows the variation in time of the total ,2 128 oppircotions potential enstrophy, which is conserved exactly in the continuous equations. The results indicate that when the 16th order 0 filter is applied every time step ( 10 minutes) 16 8 6 4 3 2 L /ax even a formally nonconservative scheme con- serves both total energy and total potential enstrophy during a long integration ( Run A4). Fig. 2 Response of Shapiro filters after 1 The quadra tically conservative sche :de controls and after 128 applications. better the amount of energy going into the smallest scales, so that in Run B2 it was enough to apply the filter every 4 hours to conserve potential enstrophy within 0..052, even though such conservation is not formally In order to explore the effect of filtering guaranteed in the scheme. short waves we performed a series of 20 day forecasts with a simple shallow water equa- tion model IKalnay-Rivas, 1976) 0 using diffe- rent combinations of schemes, smoothing operators and frequency of application as Indicated on Table 1. A-19 .. v ,f It may bo questioned whether the application our choices we chose to use a quadrat• of a high order filter eliminates small leally (energy) conserving scheme because it scale features like frontal zones or the to only slightly lase efficient than an advec- effects of orographic or cumulus convection tive scheme and It requires the application forcing on small scales. Fig. 4 indicates of a high-order filter only every few hours that this is true for lower order filters. to avoid the effect of slow nonlinear Insta- Rovever a 16th order filter eliminate* only bility associated with a spurious growth those components which are not resolved any- of enstrophy. We apply the filter to the way, and still allows for the formation of and potential taTer+a- sharp gradient son@* and strong local maxima. seat teust pressure twe fields in order tt compensate for the affect of topography.(^l 10 PASSES ZONE DAY) OF SHAPIRO FILTER 3. EXPERIMENTAL FORECASTS: PRELIMINARY " RESUL 5 FRONT" 16th ORDER J We plan to perform an extensive series of forecasts to study the effect of using different schemes and varying resolution on the quality of actual wu.ther forecasts. In this section we present saame preliminary results. 8th ORDER Ji We tested the Fourth Order model by making 3- day forecasts from several initial conditions. In every case the model performed much better than the CLAS model with the same 4° by V resolution. After 3 days the forecasts were comparable to the 2.5 • by 3' version of the 4th ORDER CLAS model in the sea level pressure maps, ..J1 and had oll htly less phase errors in the 500 mb maps. Fig. So shows the verification sea level pressure map corresponding to a 3-day forecast with February 19, 1976, OZ (a case that has 2nd ORDER been studied in detail by Atlas et al., 1979). Fig. 5b is the 3-day forecast with the Fourth Order 4 1 by 5° model verifying on the same date. The excessive gradients, especially at 10 PASSES 40011E DAY) OF SHAPIRO FILTER high latitudes are due to a faulty computation of the ground temperature by the radiation routine used in that run. Figs. 5c and 5d `nth ORDER are the forecasts generated by the CLAS models with 4' by 5 • and 2.5° by V resolution res- J L pectively. Fig. 6 displays the corresponding 500 mb maps. In Fig. 7a, we present the verification sea level pressure map corresponding to a 3-day 8th ORDER forecast with February 1, 1976, OZ initial ^` conditions. The results of six different 3= day forecasts are shown in Figs. 7b to 7g. The forecast in Fig. 7b was computed with the new full Fourth-Order model, using the same resolution and parameteri_stion of physical processes (except for the long-wave radiation routine) an in the 4 • by 5 • ALAS model 4th ORDER (Fig. 70. in Fig. 7d, the model was the same as in 7b, but full Second-Order accuracy was used.. If we compare these three made with the same resolution, we see that the fourth order model did considerably better, especially In forecasting the development and motion of the cyclone southwest of Greenland. f 2nd ORDER The two second-order forecasts are close to each other. Fig. 4 Effect of 10 applications of a 16th order Shapiro filter un a) a step (3) The idea of filtering these fields to due function b) a spiko. to Dr. A. Bayliss. A-20 Fig. 7 shows the forecast made with the 2.5' We are performing integrations with a 2.5 • by by 3' CLAS model starting from the NASA 3' version of the Fourth-Order model. From Initial conditions with assimilation of linear theory, this should make horizontal satellite data (Atlas et &1., 19791. All truncation errors negligible for most synoptic other forecasts were made from NMC's Global scales for periods of a week or longer. Analysis initial conditions. The cyelo- Still, narrow atmospheric features like those genesis forecast was poorer than with the due to sharp orographic forcing on the ITCZ fourth order model, but over continental will remain poorly resolved. North America the forecast was better. This may be due to the higher resolution or, We have found the fourth order model to be possibly, improved initial conditions. extremely sensitive to the parameterization Fig. 7f presents the loreeast made with NMC's of physical processes. For example, it became 6-layer, 380 km resolution, which has resolution unstable through excessive cooling near the comparable to our 4' by S' grid. For these surface. apparently due to a flaw in the Initial conditions. NMC's forecast was better radiation routine. None of the second order than the CLAS second order forecast, although models was sensitive to this problem. It shared some of its errors (such as a spurious anticyclogeneels over the Great In this paper we have discussed the use of Lakes). formally nonconservative schemes coupled 'with periodic filtering of the shortest waves as Fig. 7g shown a forecast made with a slightly an alternative to the use of conservative different version of the fourth order model. schemes. It is clear from the experimen4al The differences were as follows: In the 7g results that formal enstrophy conservaalon forecast we used a Mateuno time step, a has had no beneficial impact on the quality surface drag that was .75 of that of the of the forecast (Figs. 7c and 7d). Conservation standard model, the Shapiro was applied every of potential enetrophy (Arakawa, 19781 might two hours, and a slightly different scheme have a more positive effect. for the vertical advection of moisture was used. In the 7b forecast we used a combined We think that the use of nonconservative Matsuno-leap-frog scheme, the surface drag schemes with high order periodic filtering Is wee the same as in the standard model and the also justified in climate simulations as long filter was applied every hour. The two as the rates of energy or enetrophy loss due forecasts are quite similar, but from other to filtering remain much smaller than the experiments it seems that the positions of observed rates of generation and dissipation, the oceanic lows west of Spain and south of Alaska were somewhat affected by the change In frequency of the Shapiro.filter. Acknowledgements: Dr. A. Bayliss' contribu- tion was crucial in the development of the model. The authors have benefitted from stimulating discussions with Profs. M. Cane, 4. FINAL REMARKS E. Isaacson and D. Randall. The authors are grateful to Dr. Me Halem for his patience, Although these are preliminary results.'the encouragement and many useful suggestions. new Fourth-Order model seems to have very Dr. N. Rushfield and Mr. We Connelly were good forecasting skill. Similar excellent extremely helpful in the development of the results olth fourth order-schemes were reported program. Dr. Me Wu provided us with her by Campawa 11978, 19791, and by Williamson accurate radiation routine. (1978). Campans found that most of the Improve- ment over the second order model was obtained dust by introducing fourth order accuracy on the horizontal averages performed on the Shuman-Hovermaile model. Campana also obtained that for 2-day forecasts, fourth order differences were important in the advection terms but not in the pressure and continuity terms. We made shallow water equation experiments that showed some deterioration of the solution after about 8 days when only the advection°terms were written with fourth order accuracy. Since baroclinic models are much more unstable than the SWE, rampana's results may not hold during extended forecasts. We are performing experiments to study this Fig. 5 Sea level pressure maps corres- possibility in our model. We are also ponding to 3-day forecasts with repeating some of the forecasts using a February 19, 1976, OZ initial vector invariant form of the momentum equa- conditions. tions. The NMC 6-layer PE model used in Fig. 7f is based on such a formulation, and Fig. 6 Same as Fig. 5 but 500 mb we want to determine if it affects the geopotential heights. solution. fig. 7 Same as Fig. 5 but for February 1. 1976, 02 initial conditions. A-21 i f ...^"^ r` i I t ei r 4!1 a ^i ),V-11 s_ p. r ,i P1 ml F. Y r. ' ^ G •R t l ^' l I ' t v/ I ^J I s 1^ a^ 1 I o ^; ^ f 1, 'f <1•, 1^ ^ \ ^ . . - `^ T- ^ ^^ ^^ ^ , ^,+^ p;^'>r I i N te r/, ;;^ 1 A-22 ORIGINAL pgol OF la PGGrt QUALI'Iy 1 (^ Ate.' . ,^^ _ ^ A-23 I II A-24 •_.._ lam' C' c i a . .M W. REFERENCES I Arakswa, A., 1966: Computational design for long term numerical integration of the equation of fluid motion: Two dimensional I Incompressible fluid. Part 1. J. i Compu • Phys . , 1, 119-143. CD Arakava, A., 1972: Design of the UCLA CCM, 0 Report No. 7, Dept. of Meteorology, UCLA. * Arakawa. A. and Y. Mintz, 1974: The UCLA atmospheric GCM. Workshop Notes, Dept. a At of Meteorology, UCLA. a Y Atlas, R., M. Halem and M. Chil, 1979: Subjective evaluation of the combined u ^ e Influence of satellite temperature sounding data and increased model re resolution on numerical weather fore- casting. To be prevented at the 4th u Conference on Numerical Weatherredic P - tion, Silver Spring, MD, October. ^ s Campana, K., 1978: Heal data experiments with a fourth-order version of the opera- tional seven-layer model. NMC Office Note 188. . Campans, K, 1979: Higher order finite diffe- rence experiments with a semi -implicit model at NMC. J. Atmos. Set.. 107, 363-376. A-25 ry Williamson, D. L., 1978: The relative turbulence, Charney, J. C., 1971: Ceostrophtc importance of resolution, accuracy and J. Ateas. US., 28, 1081-10950 diffusion In short-range forecasts with the NCAR global circulation model. Francis. P. L., 1975: The use of a multi- Mon. Wea. Rev., 106, 69-88. point filter as a dissipative mechanism in s numerical model of the general circulation of the atmosphere. 1. J. Roy. Mat. Soc.. 101. 567-582. Cram ltvedt, A., 1969: A survey of finite difference schemes for the primitive equations for s barotroptc fluid. Mon. lea. Fov.. 97. 384-404. Ka lnay-Rivas. E.. 1976: Numerical experiments with fourth-order conservative finite differences. Annalen der Mateorologle. 11. Simulation oilarge scale atmos- pFieric processes (extended abstracts). Addendum. Kalnay-Rivas, E., A. Bayliss and J. Storch, 1977: The 4th order CISS model of the global atmosphere. Cbnt. to Atmos. Phys.. 90, 306-311. Kreiss, H.-O. and J. Oliger, 1972: Comparison of accurate methods for the integration of hyperbolic equations. Tellus. 24, 199-215. Kraiss, H.-0. and J. Oliger. 1973? Methods for the approximate solution of time dependent problems. CARP Pub. Series N^o- 10, WMO. Leith, C. E., 1972: Internal turbulence and vertical momentum transfer. Parame[eri- zation of Sub-grid Processes. CARP Pub. SeHes No. 8 0 40-51. Lilly, D. K., 1965: On the computational stability of numerical solutions of time- dependent nonlinear geophysical fluid dynamic problems. Mon. Wes. Rev., 93. 1;-26. Marilee@, P. E., 1975: The effect of grid resolution on the instability of a simple baroclinic flow. Mon• Wes. Rev-, 103. 101-104. Orszag. S. A. and M. Israeli. 1974: Numerical simulation of viscous imcompressible flow. Ann. Rev. of Fluid Mech.. 6. 81-318. 2 Phillips, N. A., 1959: An example of nonlinear computational instability. The Atmosphere and Sea in Motion. Rockefeller Inst. Preis, New York. 501-504. Shapiro, R., 1970: Smoothing, filtering and boundary effects. Rev. Ceophya. and Space Phys., 8, 359. Smagorinsky, J.. 1963: Ceneral circulation experiments with the primitive equations. Mon. Wea. 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