numerical methods used in atmospheric models

GLOBAL ATMOSPHERIC RESEARCH PROGRAMME (GARP) WMO-ICSU Joint Organizing Committee

NUMERICAL METHODS USED IN ATMOSPHERIC MODELS

VOLUME li

GARP PUBLICATIONS SERIES No. 17 September 1979

-...

CONTENTS Foreword . · · · ·

CHAPTER l:

· · · · · · · .

Page

.......

- iV

-

VERTICAL COORDINATES AND RELATED DISCRETIZATION Hilding Sundqvist

1.

2. 3. 4. 5. 6. 7.

Introducti on . . . . . . . . . . . . . . . . . . . Basic system of equations . . . . . . . . . . . . . Basic equations expressed in a generalized vertical coordinate system . . . . . . ..... Energy equati ons . . . . . . . . . . . . . . .. Various vertical coordinates and incorporation of orography Difference analogues and truncation Summa ry . . . . . . . . . . . . . . . CHAPTER 2:

3

4 5

,,.

9

20 39

DIFFERENCE APPROXIMATIONS FOR FLUID FLOW ON A SPHERE David L. Williamson

1.

2. 3. 4.

5. 6. 7.

Introduction . . . . Shal lo\'1-water equations Conformal projections . . Non-conformal projections Spherical geodesic grids . Latitude-longitude grids . Further remarks . . . . .

53 55 61

72 79 89 103

CHAPTER 3:

THE SPECTRAL METHOD

Bennert Maahenhauer

1.

2. 3. 4.

Historical introduction Basic principles . . . . Basic properties . . . . . . Models in spherical geometry CHAPTER 4:

,-~~-~-·

124 127 136 173

THE PSEUDOSPECTRAL METHOD

P. E. Merilees and S. A. Orszag 1.

2. 3. 4. 5.

Basic concepts . . . . . . . . . . . . . . . . . Application to passive advection in a periodic domain Application to the vorticity equation on a plane . . Application to modelling of large-scale atmospheric flow on a s phere . . . . . . . . . . . . . . . . . . The question of remote influence . . . . . . • . . .

278 278 284

286 290

j

- 121 -

Chapter 3 The Spectral Method

Bennert Machenhauer* European Centre for Medium Range Weather Forecasts Reading, Berkshire RG2 9AX, United Kingdom ABSTRACT The author reviews the spectral method for the numerical integrations of atmospheric prediction models. In contrast to the finite difference method considered in the previous chapters, the spatial dependence of the variables is represented by a finite series of smooth and, preferably, orthogonal functions. With this representation, the prediction eouations are expressed by ordinary differential equations for the expansion coefficients, which depend only on time. Since expansion coefficients are referred to as spectra, the approach is called the spectral method. After a historical introduction, the basic principles of the spectral methodare presented. The basic properties of the method are illustrated by application to the one-dimensional advection equation. Then the spectral method applied to the non-divergent barotropic vorticity equation model and the primitive equation model over a sphere is discussed. In particular, the interaction coefficient method and the transform method are described and their performance is compared. Finally, the spherical harmonic analysis of meteorological data is treated and same comments are made on the incorporation of orography and the efficiency of the spectral method in comparison with the finite-difference method.

*On leave from the Institute for Theoretical Meteorology, University of Copenhagen.

~

- 122 -

CONTENTS l.

Historical introduction

124

2.

Basic principles

127

3.

Basic properties

136

3.1

136

Introduction

3.2 The linear advection equation 3.2.l 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7

The analytical solution . • . . . . . . . . Choice of expansion functions . . . . . . . The solution to the space truncated system Convergence and consistency. . . The equivalent grid point method Discretization in time Concluding remarks . . . . . • •

3.3 The nonlinear advection equation 3.3.l 3.3.2 3.3.3 3.3.4 3.3.5

4.

Properties of the true solution The space and time truncated system Consistency and convergence . . . . Relation to grid point method . . . Comparison of a particular exact solution and corresponding numerical solutions by the spectral method.

Models in spherical geometry 4.1

Choice of dependent variables and expansion functions • . . . .

136 137 139 141 145 148 150 152 152 154 160 162 167 173 173

4.2 Types of truncation

182

4.3

187

Consistency and convergence

4.4 The non-divergent barotropic vorticity equation 4.4. l Introduction . . . . . . . . . . . . . 4.4.2 Properties of the true solution . . . . 4.4.3 The truncated spectral equations 4.4.4 Integral constraints for the truncated equati ons . . . . . . . . . . . . . . . 4.4.5 Exact solutions to the nonlinear equation 4.4.6 The interaction coefficient method 4.4.7 The transfonn method . . . . . . . . . .

a

136

'188

188 189

191 196 199 202 204

. _______.j

- 123 -

4.5

Extensions to more general models

213

4.5.1 Introduction . . . . . . . 4.5.2 Primitive equations models . . . . . 4.5.3 Vorticity and divergence equations models 4.5.4 The structure of present sigma-coordinate models . . . . . . .

213 216 226

4.6

Spherical harmonic analyses

241

4.7

Inclusion of orography

249

4.8

Performance and efficiency

252

References . . . . . . . . . . .

231

259

- 124 -

1.

Historical introduction

Since the pioneering experiments in numerical weather prediction in the late 1940's the grid point method has been used almost universally in numerical modelling of the large scale atmospheric flow.

During the last few years,

however, alternative methods have been made use of to an increasing extent.

One of these nethods is the spectral

method, as mentioned in the introduction to Volume I. The spectral method was introduced into meteorological modelling as early as 1954 by Silberman (1954), who considered the non-divergent barotropic vorticity equation in spherical geometry.

During the following years studies of the method

were performed (notably by Kubota (1959), Lorenz (1960), Platzman

(1960), Kubota et

al. (1961), Baer and Platzman

(1961), Baer (1964) and Ellsaesser (1966)). These studies demonstrated several desirable properties of the spectral method, and the study of Ellsaesser (1966) even indicated that fora balanced barotropic model the spectral method could compete with the grid point method used at the U.S. National Meteorological Center at that time with respect to performance and efficiency.

The spectral method seemed

feasible at a low resolution for this simple model.

It was,

however, not considered to be a realistic alternative to the grid point method for high resolution integrations of complex non-adiabatic models.

The reason for this was that the

method used in the computation of the nonlinear terms in the equations involved storing of a large number of so-called interaction coefficients, the number of which increases very fast with increasing resolution.

As this method

involves a number of arithmetic operations per time step

I

- 125 -

approximately proportional to the number of interaction coefficients it could be foreseen that the cornputing time and storing space requirements would exceed all practical limits at high resolutions.

Furtherrnore, it was

not easy to see any practical solution to the ~roblem of incorporating locally dependent physical processes, such as release of precipitation or a convective adjustment. The interaction coefficient rnethod survived unchallenged until Robert (1966) suggested the use of low-order nonorthogonal spectral functions based on elements of spherical harmonics.

This approach eliminated the complexity of interaction

coefficients at the expense of an orthogonalization procedure at each time step. Thus, the storage problem was eliminated.

The bias of a very rapid increase in arithmetic

operations with increasing resolution and the inclusion of locally dependent physical ,rocesses rernained, however, unsolved problems. The situation was cornpletely changed with the introduction of the transform method developed independently by Eliasen et al. (1970) and Orszag(l970).

In this method no interaction

coefficients are involved and the required storage as well as the nurnber of arithmetic operations is reduced substantially.

Furthermore, the method involves a stage in each,

time step where point values of the dependent variables are computed in an auxiliary grid in the physical space. As pointed out by Eliasen et

al. (1970) this made a

direct inclusion of locally dependent non-adiabatic effects possible, in a way sirnilar to that used in grid point models. A ane-dimensional version of the transform method was tested by Eliasen et

al.

(1970) fora hernispheric shallow water

- 126 -

model (i.e. a single level primitive equations model). Subsequently Bourke (1972) and Uachenhauer and Rasmussen (1D72) tested the full two-dimensional transform method

for the same model. The results were very encouraging. Even at relatively low resolution an order-of-nagnitude improvement in efficiency was obtained compared to the interaction

coefficient method, and as this factor increases rapidly with resolution, spectral models with much higher resolution seemed now feasible.

Several groups started the development

of complex global or hemispheric baroclinic models utilizing the transform method. Descriptions of these models and reports on the first experiments were published during the following years (Machenhauer and Daley (1972 and 1974), Bourke (1974 ), Hoskins and Simnons (1975), Daley et (1976), Bourke et al.

al.

(1977) and in the report from the Sym-

posium on Spectral .Methods held in Copenhagen in 197~: et al.

Bourke

(1974) , Eliasen and Hachenhauer (1974), Daley et

( 1974), Gordon and Stern (1974) and Hoskins

al.

and Simmons (1974)).

At the present time spectral models are used for routine numerical weather prediction in Australia (Bourke et al. (1977)) and Canada (Daley et

al. (1976 )) and in several countries

the rneteorological services are in the stage of developing and/or testing spectral models. In addition spectral models are being employed at various institutions for studies of the general circulation and atmospheric dynamics. The presentation ..g..f---1;ae •PQQtrnl ros±bQQ ip - ~ o s is organised as follows. ., 'Il J ;. it

principles upon which

~

methodS

~~~g

In Section 2 the basic (tf'{,

i;;

based are .presented,

and in Section 3 the basic properties of the methodsare illustrated by application to the ane-dimensional advection

'Å.

. ..

. .• /,,

equation. The spectral ifu.;thed applied to models in spherical geometry is finally treated in Section 4 .

- 127 -

2.

Basic Principles The complete set of equations used in any atmospheric

model may quite generally be written in the form

cl /" • atd,i (w1)

=

F.(wl 1

2 ,w '

I

••• ~. 'w )

i= 1,2, ... ,I.

(2.1)

where the prognostic variables wi = wi(;,t), i=l,2, ... ,I,

....

are scalar functions of the space coordinates, given by r, and the timet.

Fi is a function of the prognostic variables

generally including linear as well as nonlinear terms which involves space derivative:; and in same model:3 eyen space integrals.

In Fi the diagnostic variables are supposed to

be eliminated by rneans of the diagnostic equations of the model. J,i is a space differential operator which in most cases becomes the identity operator, i.e. cli(wi)

= w,.

In

models in which the vorticity and the divergence equation is used the stream function and the velocity potential may be used as prognostic variables in which case for these equations. operator.

Here 'i/

2

cl.1 (w.) = 'i/ 1

2

(w.) 1

is the horizontal Laplacian

We shall refer to the system of equations (2.1)

as the partial differential model eguations. In the numerical solution of the partial differential model equations different numerical methods may be used. In the preceding

chapters we have considered almost exclusively

the grid point method, where the spatial dependence of the variables

is represented by values at discrete points in

physical space and where derivatives and integrals are approximated by finite difference quotients and quadrature forr.mlae.

- 128 -

An alternative approach is to approximate the field of any dependent variable at a certain time by a finite series expansion t'

. \'

~),->;

in terms of linearly independent analytical functions,wncn, which are defined over the whole continuous integration region S. Thus, any of the variables

wi is approximated by a series of

the form " 1-+ .

w (r,t)

=

N ,

.

1

-+

w (t)w (r), n=l n n l

(2.2)

wbere Nis a constant positive integer. Us:i.ng such a representation space derivatives and integrals can be evaluated analytically so that no finite space difference approximations or quadrature formulae are explicitly needed.

The representation (2.2) is of course

equivalent to a representation in terms of the values of :i in N grid points distributed over the region ahd the

use of (2.2)

as an interpolating function fitting exactly in all N grid points. Evaluation of space derivatives and integrals using (2.2) may therefore be considered as equivalent to the use of certain finite differences and quadratures on this grid.

With the

expansion functions used in practice these finite differences

and quadratures are of a higher degree of accuracy (defined in Volume I, Chapter.l) than those usually used in the grid point method. We shall call a method built upon a representation of the form (2.2) a series expansion method. The spectral method, to be considered in this chapter, as well as the finite element method and the pseudo-spectral method cbnsidered in the following chapters are all series expansion methods.

In such

methods the time dependence of the prognostic variables is determined by the expansion coefficients, that is by the values of w!(t), and consequently the methods involve the transformation of the partial differential model equations

____

- 129 -

into a system of ordinary differential equations which determine the time derivatives of the expansion coefficients in the finite series.

This transformation is analogous

to the transformation carried out in the grid point method when the system of equations determining the time derivatives of the grid point values is constructed.

(A simple

example is the transformation of equation (1.1) to equation (1.3) in Volurne 1, Chapter 3 .). Thus, when a series expansion method is used as well as when a grid point method is used a finite set of ordinary differential equations is obtained. We shall call these equations the space truncated model equations. In numerical models utilizing a series expansion method a discrete representation in time is used as in y,rid point models. Thus the time differencing schemes which were introduced and treated intensively in Volume I may be used also for the numerical integration of the space truncated equations for such models. A finite series representation of the form (2.2) can only be an exact solution to the partial differential model equations in very special cases, therefore the transformation of the partial differential model equations to the space truncated equations must in general involve some approximations. These approximations are minimized by determining the transformed system of equations subject to some "best fit" criterion. It is in the choice of "best fit" that the spectral method and the pseudo-spectral method

differ.

The spectral method is

based on a least square approximation, which as we shall see is equivalent to a so-called Galerkin approxirnation, whereas the pseudospectral rnethod

forces the mathematical equations

L

- 130 -

to be exactly satisfied in a number of grid points equal to the number of expansion coefficients (the collocation method). In the finite element method either of the approximations may be used, but quite different expansion functions are used. Fere we shall consider only the transformation procedure when based on a least square criterion. In order to simplify the presentation let us consider a model with only one variable, w. (The following presentation may easily be extended to the case with more prognostic variables). Equations (2.1) are then reduced to ~t /,(w)

=

(2.3)

F(w)

and the initial conditions becorne -+-

-+-

w(r,0) = f(r).

(2.4)

In order to avoid problems connected with complicated boundary conditions we shall furthermore assume that these are of a type which can be satisfied by a proper choice of the expansion functions for any truncation of the approximate solution "

-+-

w(r, t)

N

=

l

w

n=l

n

et)

1

n

er)

(2.5)

and for any set of the expansion coefficients. As we shall see this is the case in the applications of the spectral method to be considered in the following. The space truncated equations are now derived by minimizing the mean square integral of the residue R(w) =

h, d,(:) -

(2.3).

That is, at any time we choose those values of dtn which

F(:) , obtained by substituting (2.5) into (!W

- 131 -

minir.iize N

N

(I dwh, [ ( '!' n) - F ( I wn '!' n )) S n=l dt n=l

f

J ( w) =

2

d S.

These values are obtained by setting the derivatives of J(w) with respect to

dwn

equal to zero (the condition fora

dt minimum) and the resulting space truncated equations become

N

fs

'l'\,-711.'

N

( n I'-1 -

dwn,J('I' ,) - F dt n

0. Choice of expansion functions

3.2.2

We shall consider the numerical solution to (3.2) obtained by the spectral method. functions,'

n

A natura! choice of expansion

(A ), in this case is the trigonometric functions

- 138 -

'l'n 0..)

It is well

=

rl

c~s

rn 21 A)

for n

=

1, 3, ...

SJ.n

(~

for n

=

2, 4, •.•

A )

known that these functions form a complete

system of orthogonal functions and that a wide class of functions can be represented by infinite Fourier series, which converge rapidly for sufficientl!' smooth functions. Furthermore, each of the functions is periodic with the period 2 n, so that they satisfy the given boundary condition (3.3), and also they behave very simply under various

operations of analysis, notably differentiation.

Correspondinp,

to (2.5) we seek an approximate solution of the form of the truncated Fourier series wc(t) w().., t)

= .Q.,,.__

M s +}: (wc(t) coG mA+w (t) sin IDA) , m=l m rn

(3.6)

where for each Fourier cornponent

(3.7)

(wc(t) cos l!lA+ws(t) sin mA),

m

m

~.l,.

) --7

m is the zonal wavenumber r t h e nurnber of waves along the

latitude circle, and where U is the rnaxiMum wavenumber in the expansion.

retained

The number of time dependent expansion

coefficients wc(t) and ws(t) in (3.7) is seen to be N=2H+l. rn

m

Introducing the complex functions eimA (3.6) rnay be written as W(A,t)

=

M

l

w

m=-!f-1

(t)

e

imA

,

(3.8)

where the complex coefficients for rn > 0 are given by (3.9) w (t) = -1 (wc(t) - i ws(t)) , 2 m m m

- 139 -

defining w~

= 0.

The coefficients for negative and positive

values of mare related by

w

-m

=

(t)

(3.10)

(w (t))*

m

with the asterisks denoting the complex conjugate.

This

"

A

last relation follows from the faet that w is areal function. A

We note that w is completely specified if the complex coefficients w (t) are given for O m


~t) then (3.18) is the exact solution.

If this is not the case then (3.18) is only

an approxirnate solution.

Writing this approxirnate solution as

" wO,t) = f 0-yt), 11

where fuO.)

" _ w(A,0)

=rMm=-M wm(O)

0

irnA

and using the faet that the exact solution is given by (3.5),

the error function E~t)

=

"

w(A,t) - w(l,t) rnay be

written as E(l,t) = f(A-yt) - fM(l-yt) ~ E'(l'), where A' = A-yt.

Thus, E is independent of time in a coordinate

system rnoving with the constant angular velocity Y. It is therefore never larger than the error we cornmit by approxirnating the initial function, f(A), by its Fourier series truncated at wavenurnber M. Therefore, if the

Fourier

series of f (Al converges to f CA) then the approxirnate solution converges towards the exact solution, when M approaches infinity.

With the assumptions stated in subsection 3.2.1

satisfied for f (A) we know in faet that its Fourier series (3.16) does converge absolutely and uniforrnly.

The rate of convergence depends, of course, on f (A).

- 142 -

Generally, the "smoother" the function, the faster the convergence.

Concerning the arder of magnitude of the

Fourier coefficients, if f (\) has continuous derivatives up to the (p-1)-st arder and if its p-th derivative is piecewise continuous, then the coefficients, for

m ~ 1, can be

estimated by C


. m

aw

TI

is

'

whereas the exact expression is

... :~ =

I

m=-"'

imw eir.i>. m

where analogous to (3.28) 00

w

rn

=

I

q=-oo

wm+qK"

Although it is not necessary for the proof, let us assume

l,_/

- 148 -

wm = ~ m, which corresponds to neglecting the initial aliasing. (When this assumption is dropped, the proof may be carried out in a way analogous to that used by Kreiss and Oliger (1973) p. 45).

The error function is then

e:"P.)

=

aw aw ar-ar

J.

.

im).

=Tml>~rnwm e

•

Now, analogous to (3.20) we have lw

I< -C

,

m = mP

if w hasp continuous derivatives, so that we get the estimate CO

I e:"

0)

I

1

I

~2C

m=H+l


-

, l


0 m,n P~,n-1 ·,

Dm, n+l Pm, n+l + Dm,n Pm,n-1

(4.5a)

(4.Sb)

where

Dm,n

=

(n2 - m2)! 4n

2

-

1

(4.6) ·

Introducing the complex spherical harmonics

Ym, n(A,JJ) = Pm, n(µ) e

im).

(4.7)

I

I

- 178 -

we may write the expansion (4.1) as 00

L

1/1 (A , µ, t) =

m=-:00

~

m

(µ, t)

e im:l.

(4.8)

CO

a,

=r=_..,In= !ml ;m,n a). -

= -sa2f sµ,;

a(ucos~)) dS aa4>

cos(j>

dS

from which it follows that 2 s Js

dM_a dt -

a,; µat

(4.27)

as.

Using (4.24) an alternative expression for the integrand may be derived: -+

=-

µil at

-+

-+

µV• nV =- V•µnV + nV •Vµ w "w w 1 aw = -V•µnVtj, + a7" n TI

= = =

-+ 1 a an -V•µnVljJ + a7° -+ 1 a a -V•µnVtj, + a7°( »(nlji) - lji »V 21jl) -+ 1jJ aw 1 a -V• (µnVljJ + a7° V»> + i7 TI(ljin

+

!

v1jJ

2

).

Substituting the last expression into (4.27) and making use of Gauss' theorem it follows that dM/dt

= O.

A similar procedure is used in the proofs of (4.25b) and (4.25c):

We have

aK at = _L at

.!. V2. 2

w

= -ljiil + V• (lj,V aljl)

at

at

so that Gauss' theorem implies

.!.f

dK -- - s ,I,.,. !f. dS • -dt s at

(4.28)

We have furthermore

dE

dt

I

a,; = _1 1;-a s s t

dS

(4.29)

- 191 -

Using (4.24) the integrands in (4.28) and (4.29) may be written as -+-

~ =-'il • l/lnVI/I "' at

,,,

and

g= 11 g_

r;;

at

at

g =at

f

-+-

'il•

½11 2 v -2nu ~ 1/1

at

respectively, and from Gauss' theorem, (4.25a) and (4.27) (4.29) it follows that dK/dt = 0 and dE/dt = O. In Subsection 4.4.4 we shall see that the constraints (4.25) are valid also for the approximate solution to (4.24) determined by the corresponding truncated system of spectral equations, which will be derived in the following subsection.

4.4.3 The truncated spectral equations We write ( 4.24) in the form

Lat

"2,,._ v

.,,-

-

2n

::-T

a

a,,.

~

a"

+ F(·'·) "'

,

(4.30)

where F(ljl)

1 = i"T

( i l L 'i/21/1 - i l L'i/21/1) aµ a1t a" aµ •

(4.31)

We seek an approximate solution in termsofa truncated series of spherical harmonics and choose the triangular truncation (4.22), although any other type of truncation might have been

used.

The following derivations are dependent upon the choice

of truncation but they are easily modified to apply to any other type of truncation.

- 192 -

Generally a truncated series of the form (4.22) will not satisfy (4.24) exaetly.

This follows from the faet that the

equation ineludes a nonlinear term as well as linear terms. The situation is quite analogous to that for the nonlinear adveetion equation eonsidered in Seetion 3.

When substituting

(4.22) into the equation eaeh of the terms beeomes a truneated

series, but the nonlinear term ineludes more eomponents than the linear terms.

For the left-hand side term,by. using

(4.13),we get

N

" N 'v21/J= -

~ at

dljJ

l

l

m=-N

m,n y dt m,n

n=lml En

(4.32)

where E = n(n+l)/a 2 and for the linear term on the rightn hand side we get N

2n

~

- a7

l

iml/J m,n y m,n m=-N n=lml

a7"

a).

N

I

2n

(4.33)

For the nonlinear term on the ether hand we get 2(N-1)

"

t'

2(N-1)

l

F

F(t/1) =mi-2(N-l) IF!ml where F m,n

=

1 41T

rlT r 0

-1

Y

m,n m,n

,

,. F(t/1) y* m,n

dAdµ

We shall prove below that F(l/J) is in faet a truneated series with truneation limits as given in· (4.34). Onee this has been proven (4.35) follows from the orthonormality relation (4.14).

(4.34)

In order to prove (4.34) we substitute

(4.22) into (4.31) getting

(4.35)

- 193 -

"

F($) = 1

a7

, •

where

13

wo , 1 = -a7 l/Jo 1 1

(4.52)

.

Thus, conservation of M'. iMplies that

iµ

0

•1

or in other words

the solid rotating part of the velocity field is an invariant. 4.4.5 Exact solutions to the non-linear equation In Subsection 4.1 we have stated several reasons for choosing spherical harmonics as expansion functions on the sphere. One further reason is that each harmonic is an exact solution to the non-linear non-divergent vorticity equation (4.30) ( Craig (1945), Neamtan (1946)). This may, be shown by means of

- 200 -

~

the expression (4.37) for the non-linear term F(ijJ).

We may

in faet show that even a field determined by a solid rotation and a superimposed linear combination of all admissible spherical harmonics of fixed n is a solution.

In arder to prove this

we substitute such a field n'

ijJ' = Wo lPO 1(µ) + l Wm n' ym n•O 'µ) ' ' m=-n' ' '

(4.53)

into the equation ( 4.30) getting

na w = n2

'

v

2n aX"a a7

w'

+

FCw ').

(4.54)

The non-linear term F (ijJ') is determined by (4.37) and we see that all contributions with n 1 =n 2 =n' vanish because of the factor e:n - e:n . The only non zero contributions are 2 1 therefore those involving 0 1 , as one of the factors and as

w

'

Lm o =Lom= -(n'(n'+l) n' 1 1 .n' a

y

- ~) m/3 a 2

m,n'

we get F(ijJ I) =13 i"

n' . ,,, y (2-n I (n I +l))wo > 1 r=-n I 1.m'l'm,n' m,n

(4.55)

For the two remaining terms in (4.54) we get

.Lv2;' at

_

2

dijJ0,1 n'(n'+l) - - i7° dt P0,1 a

n'

l

m=-n'

d

1jJ

m n' y (4.56) m,n' dt'

and 2!]

dijJ

I

-

2n

n'

-a7a5:- - a2" l

m=-n'

"-~-"=----·

imijJ m,n I Ym,n f

,

(4.57)

- 201 -

When (4.55) - (4.57) are substituted into (4.54) it becomes, after rearranging and when using ( 4.52), 2 n'(n'+l)

d,i,o 1 p .::...L.::.

where

0, 1

,2(n +w

Yn• .wo, . 1 -

0

(µ) +

0

l

I

(

d

m=-n'

~I

m,n' + imyn,lj,m n')ym n,(A,µ)=O' , , dt

, 1)

(4.58)

n'(n'+l)

As the spherical harmonics are linearly independent it follows tbat

d

{4.59)

dt Wo,1 = O

and

d lj, -dt ,,. n' m,n' + i myn•'l'm ,

=

0 for

< m= < n' -n' =

(4.60)

•

We note that (4.59) is a consequence of (4.25a) which is valid for any lj,. The solution of (4.60) shows that all spherical harmonics components in (4.53) with m

+0

propagate towards the

east with the constant angular velocity y n , and with constant amplitude. y , is known as the Rossby-Haurwitz wave speed found n by Haurwitz (1940) for the linearized non-divergent barotropic vorticity equation. For the special cases of initial fields of the form (4.53), the truncated system of spectral equations (4.43) gives the exact solution to the partial differential equation(4.30) if all ~

components in lj,' fall within the truncation limits.

If, on the

other band, non-zonal components with different n's are included initially then these components will interact and as a consequence the residual determined by (4.49) will sooner or later become non-zero.

Even in this case, however, the beta-

- 202 -

term and the advection of retained components in the solid rotating part of the zonal flow will be computed in each time step without space truncation. Asa consequence these linear contributions to the phase velocities of retained components, i.e. the Rossby-Haurwitz wave speeds, are computed exactly, apart from round off errors and time truncation and a cause of computational dispersion is hereby eliminated.

4.4.6

The interaction coefficient method

The linear term in (4.43) can easily be computed, but the problem is how to compute the non-linear term.

The original

method was presented by Silberman (1954) and it was treated. extensively in the papers by Platzman (1960) and Baer and Platzman (1961).

This method, the so-called interaction

coefficient rnethod, builds upon the expressions (4.35) and

(4.37).

When the latter is inserted into the former we get N Fm n = il/J 1/J Lrn l'1lm2 m =-N n 2 -lm 1 m ,n m ,n 2 n n n ' m1 =-N n1=lmll 2 2 1 1 2 2 1 n2 > nl

~

r

I

f

f_

(4.61) where ½(E

n

-E

2

1 dP dP )JP (inP m2,n2_mp m1,n1)d n 1 -l m,n 1 m1 ,n 1 dµ 2. m2,n 2 dµ U form= m +m 1 2

Lmm1m2_ nn 1 n 2

(4.62) t)

form+ m1 + m2

In the derivation of (4.61) - (4.62) we have taken advantage of the symmetry of (4.37), thus the summation in (4.61) is taken over all distinct combinations (rather than permutations) of (m ,n ) and (m ,n ); this can be assured, for example, 2 2 1 1 by requiring n >n as indicated below the last summation sign. 2 1

,

- 203 -

Even for moderately truncated representations the non-linear term is very laboriousto compute from these ,expressions. As especially the numerical computation of the interaction coefficients defined in (4.62) is very time-consuming, these are stored in the computer.

Besides the interaction coefficients

form+ m1 +m 2 also a large number of those form= m1+m 2 becomes zero and of course only the non-zero coefficients need to be stored, and to be included in the computation of (4.61). Rules which sort out interaction coefficients which become zero are given by Baer and Platzman (1961). referred to as selection rules. these rules.

They are generally

We have already used two of

Namely, that all interaction coefficients are

zero for which nand m fail to satisfy

m=~+~

and nl

+ n2.

Concerning the remaining rules the reader is referred to the paper mentioned above.

Even if the full advantage

is taken of the selection rules the required storage and the required number of arithmetic operations involved in the computation of the non-linear term each time step increase

very fast with increasing resolution.

Orszag (1970)

5 estimates this increase to be approximately as N . As explained in the introduction to this chapter, this very fast increase was the reason why the spectral method for several years was considered not tobea realistic alternative to the grid point method for higher resolution integrations. It was mentioned in the introduction that Robert (1966) developed an alternative method.

In this method no interaction

coefficients are used, so that the storage problem is

- 204 -

,~,

reduced substantially. Gm p

=

He uses the function

(l-µ2) 2 µP eimA,

where pis an integer larger than or equal to zero. call these functions Robert functions.

We shall

As any spherical harmonic

may be written as a sum of these functions a truncated representation in terms of spherical harmonics rnay be transformed to a truncated representation in terms of Robert functions.

The

advantage of using these functions is that the produet of two functions can be expressed in a simple form in comparison with the produet of spherical harmonics.

In Robert's rnethod the

calculation of each term in the equations is

carried out

separately and the results are added giving an array of coefficients for the time derivatives, which due to the nonlinear terms contain coefficients outside the limit of the original truncation.

This array is then truncated in such

a way that the result is exactly equivalent to non-aliased truncations of the corresponding series of spherical harmonics. By

this procedure no interaction coefficients are needed,

and as mentioned above the storing space needed is reduced. A very rapid increase of the required arithmetic operations with increasing resolution remains, however, a problem. 4.4.7

The transform method

As explained in the introduction to this chapter, the next step in the evolution of the spectral method was the introduction of the transform method, developed independently by Orszag (1970) and Eliasen et al (1970).

In this method

the increase with increasing resolution of both storage and the nunber of arithmetic operations is reduced substantially.

We shall illustrate the method by considering

--"

-----~------

""--

- 205 '--

the computation of the non-linear term in (4.43). This term is determined by (4.35), which may be split up into the following two integrals

F

-----

Il-1 Fm{µ)

m,n = i

=

Fm(µt

2'11'

1

21r

f

Pm n{µ) dµ

(4.63)

,

"

•

F(~(A~µ)) e-imA dA

(4.64)

0

where and

(11"

"

1 FO) -_ -2 a 3µ

" r;

"

li 3A

il 3A

"

li)

(4.65)

3µ

= v2~"

(4

.66)

Having chosen the triangular truncation Fm,n is to be < < < computed for O = m = n = N. The idea was then to evaluate the above integrals with the aid of quadrature formulae, noting that this evaluation can be done exactly when proper quadrature formulae are chosen. As we shall see the integrand in(4.64) fora certain 1J is a truncated Fourier series in A.

The integral can therefore

be computed exactly by the trapezoidal quadrature formulae (3.26) if a sufficient number of quadrature points are used. For the

evaluation of the integral in (4.63) on the other band,

we choose the Gaussian quadrature formula.

l

J-1

K g(µ

> dµ

-

I

G(K) • g(µk) ' k

k=l

where µk are the roots of the Legendre polynomial P 0 ,K(µ) and the Gaussian coefficients are defined by G(K)= 2(1-1Jk2)(U = 3N-l .

(4.71)

By substituting F($), given by (4.37), into (4.64), using mlm2 the expression (4.42) for L and the orthonormality of the nln2 trigonometric functions, given by (3.11), we find N

1 I Fm(µ) =::T a m =-N

N

N

l

l

N

I

n1=/m1I m2=-N

1

(i(E

n2=\m2I

E

)

n2- nl

(ml+m2=m) m

X

(1-µ 2

)2 $

$ Q(3) (µ)) m ,n , m ,n 2 n +n -1-m 2 1 2 1 1

•

Using this expression and the expressien (4.4) for P we see that the integrand in (4.63), F (µ)P

(µ), m,n m ~ 0 is a polynomial of degree at most 2N - 2 + n. m

to compute F

m,n

only forl


(µ) =

im

m

I

t

n=lmlm,n

P

m,n

(µ>

N 1/1(µ)(µ) = m

l

n= 1ml

1/Jm n Hm n(µ) ' ·'

N

i: (A ) ( µ ) m

=

r(µ)(µ) ,m

=' n;lml

im

I t m,n Pm,n(µ) n=lml

N

i;;

m,n

H (µ) m,n

(4.76)

- 209 -

Here n{n+l) 1jJ a m,n

r;m,n =

(4.77)

and Hm,n{µ) = - (l-µ2) dPm,n dµ The expressions(4.75) and (4.76) follow

(4.78) from application of

the operators (4.73) to the expansions (4.22) and (4.47) of ~

~

1jJ and i;;

The functions H (µ), defined in (4.78), may be m,n

•

computed from the Legendre functions P (µ) using (4.5a). m,n We have now established all formulae necessary for the computation of the coefficients F of the non-linear term m,n from the coefficients 1jJ

m,n

by the transform method.

The

procedure may be summarised as follows:

Step_l

The vorticity components i;; for O ~ m ~ n ~ N using

are computed m,n (4.77).

In the following steps the contributions to (4.68) from the Gaussian latitudes µk; k = 1,2, ... ,K 2 are accumulated successively.

The contribution from a certain Gaussian

latitude µ=µkis computed using the following steps:

Step 2

The Fourier coefficients 1)J~A)(µ~),1)Jm(µ)(µk), r;~A)(µk) and r;~µ)(µk) are computed for 0 ~ m ~ N using (4.76).

Step 3

The grid point values ${A)(A j'µk) ... , ~(µ)(Aj,µk) are computed for j = 1,2, ... ,K

1

using (4.75).

Step 4

The non-linear term F(A,µ) is computed at the grid points (Aj,µk); j = 1,2, .... ,K 1 using (4.74) .

The folowing page 210 is missing. Instead it is kopied at the end of this manuscript, before the references.

- 211 -

the number of arithmetic operations involved in each FFT is proportional to K1 log 2 K1 . This implies that if K is a power of 2 1 and if Nis the maximum value satisfying (4.72) then when using FFT the number of operations involved in step 3 and step 5 becomes 2 proportional to N log2 N for large values of N. So far it has not been possible to find an algorithm which, in analogy with the FFT, cuts down the rate of increase of the number of operations involved in the Legendre transforms in step 2 and step 6.

A certain reduction in the number of operations can be

obtained in global models, at the expense of extra storage, by taking advantage of the faet that the Legendre functions are either symmetric or antisymmetric with respect to the equator. The rate of increase for large Nis, however, unchanged so that the total number of operations involved in the computation of the non-linear term will be dominated by the number of operations involved in

the Legendre transformations when

3 Nis large and therefore become approximately proportional to N • This is still a considerably reduced rate of increase compared to that of the interaction coefficient method. Instead of an increase proportional to N5 we have an increase proportional to N3 for large values of N. It should be noted that at more moderate 3 N a somewhat slower increase than N must be expected.

This is

especially the case for more complicated models in which a parameterization of physical processes is included in step 4. For the sake of completeness it should be rnentioned that for very small values of N the interaction coefficient rnethod becornes the most efficient method, in terms of number of operations involved. The main arrays that must be available in core when the transform method is used are the spherical harmonic expansion

- 212 -

coefficients of the variables and their time derivatives, that is

~ n· The number m,n ,~m,n and 1-0t m, 2 of these is proportional to N In addition it is a computational in the case considered

~

advantage to store the Legendre functions Pm,n (µk) and the functions Hm,n (µk). The number of elements in these arrays is proportional to N 3 •

If this is not done additional

computations of the Legendre function must be carried out at each time step, and the values of the functions Hm n(µk) must '

be computed from equation (4.5a).

The Legendre functions can be

computed by recurrence by the method described by Belousov (1962).

The number of operations required to compute the Legendre functions is proportional to N3 Merilees (1973) has developed a method by which the Legendre transformations are carried out without the explicit use of the Legendre functions.

The method turns out to be much more

efficient than computing Legendre functions each time step and using (4.68) and (4.76) directly.

Unfortunately the method

suffers from a precision problem and breaks down when the resolution is too high. Whilst the Legendre transforms can be considered inefficient relative to the FFT the direct use of (4.68) and (4.76) greatly facilitates computer coding of the transform method, using the successive accumulation at each Gaussian latitude described above.

Even if a fast Legendre transform

could be found it seems unlikely that it would be used in more complex multi-level models, as it would necessitate simultaneous grid point representation of the full two dimensional grid or alternately substantial peripheral device usage.

::;;;;;;;;;...

.

~

- 213 -

4.5 Extensions to more general models 4.5.1 Introduction It was shown by Merilees (1968) that the spectral method in principle may be applied to multi-level pressure coordinate models if the differentiated forms of the equation of motion are used.

In this case stream function $ and

velocity potential X may be used as the prognostic variables describing the horizontal velocity fields.

Neglecting forcing

and friction terms Merilees shows that three types of nonlinear terms appear in the equations, namely terms of the form

....

k x VA·VB, VA • VB and AB, where A and B are scalar variables.

Corresponding to these different terms three types of interaction coefficients arise, one of which is the coefficients nun m L 1 2 defined in (4.62). The storage problem with the internn1n2 action coefficient method fora more general model than the non-divergent barotropic model considered in Subsection 4.4 becomes therefore even more obvious.

The transform method

may, however, be used also for such models as each of the three types of non-linear terms becomes truncated series of spherical harmonics

(cf.

Eliasen et al. (1970)).

If the triangular

type of truncation is used for all variables then the non-linear terms become truncated series with O ~

1ml

~ n ~ 2N.

Consequently the number of Gaussian latitudes, K , and 2

equally spaced longitudes, K1 , to be used in the transform grid must satisfy K 1

=>

3N+l

K ~ 3N+l 2 2

and

(4.79)

respectively .. When using parallelograrn.mic truncations, the numbers must satisfy K ~ 3M+l 1

and

K2 -~ 2M+3J+l 2

(4.80)

- 214 -

Concerning integral constraints of pressure-coordinate models it follows from ~erilees (1968) that when the lower boundary is assumed to be an isobaric surface at which w =

*

=

0 and an energetically consistent vertical

discretization is used then the spectral truncation does not disturb the energy consistency. This property of the spectral method

is due to the faet that the total energy for such

models, just as for the non-divergent barotropic model, is a quadratic quantity in variables represented by spherical harmonics.

The non-ø.liased truncation of the non-linear terms,

therefore,implies quasi-conservation of energy in adiabatic friction-free integrations in the same way as for the nondivergent barotropic model. Host non-balanced spectral models used at present are based on the general system of equations in sigr.:ia-coordinates. For such models the mean kinetic energy is a cubic quantity in variables, which are expanded in terms of spherical harmonics and consequently a non-aliased truncation of non-linear terms does not automatically imply quasi-conservation of the total energy.

Weigle (1972) has made a detailed study of the

conservation properties of a shallow water model (the simplestsigma-coordinate model).

He demonstrated that the

time derivative of the total energy, determined by the truncated set of spectral equations, in general is non-zero. In ether words, quasi-conservation of total energy is not automatically ensured in a spectral shallow water model. In practice, however, experiments with this model as well as with more general sigma-level models have shown that the total energy is very nearly conserved during adiabatic friction- free integrations

(cf. Eliasen et al. (1970), Bourke (1972; 1974),

Hoskins and Sir.unons (1975)and Baede et al. (1976)).

- 215 -

When using the differentiated forms of the equations of motion, that is, the vorticity and divergence equations, only true scalar variables are involved and no prohlerns are encountered in representing the variables in terms of spherical harmonics. Concerning systems of equations with the equations of motion in the Drimitive form the question arises as to how the horizontal velocity field should be representec in the spectral domain.

As

discussed in Subsection 4.1 the velocity components, u and v, themselves should not be represented by truncated series of spherical harmonics, because both components generally are discontinuous at the poles. Robert (1966) succeeded in making the first spectral integration of a model based on the equations of motion on the primitive form.

He uses a representation of the velocity components,

u and v, which is equivalent to the stream function

~

a truncated representation of

and the velocity potential x in terms of

surface spherical harmonics.

Such a representationautomatically

fulfils the proper boundary conditions at the poles. Using this representation of u and v he computes at each time step the time derivatives of u and v, without any truncation in the meridional representation.

These

untruncated tendencies are then converted to the equivalent tendencies in~ and X, which are finally truncated.

The

spectral computations are made using the Robert functions mentioned in Subsection 4.4.6 and the computations are rather time consuming as all the components in the non truncated meridional representation of the tendencies have to be cornputed.

He had, however, shown that the spectral method

could be used also for the primitive equations. Eliasen et al. (1970) used the same principles as Robert

- 216 -

in the integration of a spectral barotropic, primitive equation model, i.e. a shallow water model.

But here the variables

were represented directly in spherical harmonics.

In the

following Subsection the method introduced by these authors will be presented. 4.5.2

Primitive equations models

Given the representation of a velocity field by the truncated series of the strearn function X

N

A

1jl and.

the velocity ~otential

N

•·Im•-N n=lml I •m,n

Y , m,n

(4 .81)

N

N

x-rm=-N n=lml I A

y

X

m,n

m,n ,

the equivalent representations of u and v, determined from the definitions

!__ . [~ -

u • _

are ,.

,.,

u

A

i l • COS4)

,.

ff)

;

u•

N+l

N rm•-N rn•lml

A

V

v ··cos•

(4.82)

(fr + cos+~}

! __ .

v = _

cos+

;

"

V•

N

l

m=-N

N+l

1

n=lml

um,n ym,n

9

(4.83)

y

m,n

y

m,n

,

where the coefficients Um,n and Vm,n are determined from the coefficients 1/lm ,n and Xm,n by the relations 1

. .

um,n = i {(n-l) 0 m,n 1/lm,n-1 + imXm,n - (n+ 2 >0 m,n+11/lm,n+l} ' (4.84)

vm,n =

½{(l-n)

0 m,n xm,n-1 + imilim,n +(n+ 2 ) 0 m,n+l Xm,n+ll •

- 217 -

The relations (4.83) and (4.84) are easily derived by substituting (4.81) into (4.82) and by rnaking use of the relation It is seen from (4.83) that u and vare represented by

(4.5).

A

A

truncated series of spherical harmonics, U and V, divided by cosp A

We note that the series U and V extend to one more degree above "

that for 1jl and

"

A

X

------------

A

When the series ip and X are gi ven then the

;(•

series U and V can easily be computed from the above relations (4.84) and the velocity field determined by (4.83) will then

automatically be a smooth continuous field all over the globe, including the pole points.

It is obvious that this must

imply certain relations between the coefficients Um,n and V as the number of these coefficients is larger than the number m,n of coefficients $ m, nand Xm,n from which they are determined.

Such relations were derived by Eliasen et al. (1970),

who called them truncation relations. Orszag (1974) and Byrnak (1975) have later shown that these relations are equivalent to boundary conditions at the poles and that such /

relations must be satisfied even for infinite series, if the

'--

velocity field to be described is a smooth (infinitely differentiable) vector field.

We shall derive these relations

in a slightly different manner. Supposing that 1jl and X can be expanded in the infinite series

~

!J

...

$

=I

m=-'"'

...

X =

I

m=-m

;meim). =I m=-. xme

""

I

n=lml

... ... =Im=- 0 m,N+l dt xm"N+t '

where the uncorrected values d (Um, N)/dt and d (Vl!l, N)/dt can be determined from the primitive equations.

Thus, if the

time derivatives d(~m,N+l)/dt and d (Xm,N+l)/dt can be determined in some other way then the corrected coefficients in (4.96) can be computed and the remaining two corrected coefficients can be computed from the relations corresponding to (4.91). The method used by Eliasen et al. (1970) to compute d(~m,N+l)/dt and d(Xm,N+l)/dt, is based on the following relations l

{a(A,B)}m,n ,.. i f[imAm(µ) Pm,n(µ) + Bm(µ) Bm,n(µ)] e.µ2 ' -1 1

{a(B,-A) lm,n •

if

2 [im Bm(µ) Pm,n(µ) - Am(µ) Bm,n(µ)] $!. 1-ll -t

(4.97) •

Here A and Bare some fields which are known to be zero at the poles.

Amand Bm are the Fourier coefficients

O A(A,µ) e -imAdA ' Am(µ) = 12,r J2,r (4.98)

Bm(µ)

=

1 f2,r B(A,µ) e -imAdA ' 2,r 0

- 224 -

corrected tendencies are determined by the time derivatives d

d

dt um,n and dt vm,n for n

< N.

L

For m ,.. 0, however, we have only

two relations, and we, therefore, have to correct two of the time derivatives in another way. From (4.95) it is seen that the d cor d cor . corrected values dt um,N and dt vm,N are given by

d

cor

d

dt um,N • dt um,N

+ 1

i

(N+2 >0 m,N+l

d

=

3N+l and

N+l when triangular truncations are used and K ~ 3M+l 1 ~ M+½J+l when parallelogramrnic truncations are used.

l

- 226 -

4.5.3

Vorticity and divergence equations models

When using the grid point method it is a big advantage to use the primitive equations instead of the vorticity and divergence equations.

The main reason for this faet is that

it is a time consuming process

to salve a Helmholtz equation

when using the grid point method.

With the spectral method

based on spherical harmonics the b~sis functions have been chosen as eigensolutions of the Helrnholtz equation and the solution therefore becomes a simple operation.

Bourke (1972) has

shown that a very efficient spectral model can be formulated using special forms of the vorticity and divergence equations. For the shallow water model he uses the prognostic equations

an at = ao at a4i•

-+

V ·n

= k•'v

at

-+

v, -+

xnv

-V

-+ = -V• 4>'V

2

(

v2

2 +4>'))

(4.102)

'få ,

where n = ~+f = 'v 2 ~+ 2Qsin~ is the absolute vorticity, 2

å='v X is the divergence and 4> =~+4>' is the geopotential at the free surface; ~ denotes a time-independent global mean and 4>' denotes the deviation from the mean.

The advantage

of using the equations in this form is that the terms on

the

right-band sides involve only the Laplacian operator or the a

operator defined in (4.99), both of which are simple to

handle when using spectral algebra.

We shall consider a

slightly changed version of Bourke's model following Hoskins and Simmons (1975) in the use of absolute vorticity and divergence instead of stream function and velocity potential as dependent variables. The equations (4.102) may be written

- 227 -

l!l: _ !. a.(nU, nV) at

ae.

l i : la.(nV, -

at

ct.

,

nU) - 'i/

a~ at = - {i a. ( ~'u , ~ ' V) I

I

Using the triangular

2[U2+v2 2 2(1-JJ

)

(4.103)

+ ~·),

-~ o

truncation, the following expansions

are introduced N

n =

N

I

I

n

m=-N n=lml N

0 =

Y

m,n

N

I

I

0

m=-N n=lml N

~·=

m,n

m,n

y

m,n

(.4

' ,

(4.104b)

N

l

l

~

m=-N n=fml

m,n

y

(4.104c)

m,n

The expansions for U and V corresponding to (4.104a) (4.104b)

.104a)

and

are of the form (4.83) where relations between the

coefficients

n

m,n and

om,n

and the coefficients Um , nand Vm , n

are obtained from the relations (4.84). These relations may be written

um,n

D im + 1 D ~ ) = a ( - .!. n m,n ~m,n-1 - n(n+l) Om,n n+l m,n+l m,n+l >

(4.105)

V

m,n

1 =a ( -D n m,n °m,n-1

im -=:.__n(n+l)

~m,n

1D ---

n+l

) m,n+l 0m,n+l >

where we have used (4.13) together with the faet that A

A

~ = 'i/ 2iµ and

A

o

= 'i/ 2 X,which implies the relations

sm,n

n(n+l) 1/1 a

m,n

- 228 -

and

o

= - n(nil) X

m,n

a

The -coefficients ,;;

m,n

m,n

for the

relative vorticity are identical to those for the absolute vorticity, n

, except for (m,n) = (0,1).

m,n

=,;; + 2 Q P

the relation n =,;; + 2rlJJ

13

2g

- t'3

m,n

,which implies that

for (m,n)

f

(0,1).

for (m,n)

=

(0,1).

~m,n

.,;;m,n {

0 ,1

This follows from

(4.106)

The truncated spectral equations are obtained as usual by A

A

A

inserting the truncated expansions n,o.~' for the basic A

A

variables and those U,V for the auxiliary variables and then applying the operator [ }rn,n on both sides of each equation, utilising the orthonormality of the spherical harmonics. The result is

!!..._ n = A dt m,n m,n d -o dt m,n

=B

m,n

!!..._t = C dt m,n m,n

t

+e:~ n m,n - t

6

(4.107)

t

,

m,n

where the non-linear terms are determined by t A

m,n

,..,.. ,..,..

=--{::.a(nU,nV)}

=

.~

AA

m,n

,

u2 A

AA

A

+ y2

( 4 .108)

Bm,n:.l¾dCnV, -nU)}m,n + e:nS,(1- JJ2Ym,n" C

m,n

=

-{-ia(;'U ;'V)} m,n «!I: '

These terms are readily computed by the transform method using (4.11J4a),

(4.104c),

(4.105),

(4.83),

(4.93) and (4.97) where

as usual the quadrature formulae (3.26) and (4.67) are used in

- 229 -

the evaluation of the integrals involved.

In arder to

avoid aliasing K and K2 must satisfy (4.79) or (4.80) if the 1 truncation chosen is triangular or parallelograrrmic. For comparative purposes Bourke (1972) made a number of integrations with different rhomboidal truncations (H=J) using the transform method as well as the interaction coefficient method.

The computation time per time step was measured for

both methods and the results are presented in Figure 10, which clearly illustrates the computational advantage of the transform method. A comparison of Bourke's transform model with the U and V model described in the preceding Subsection shows that approximately the same nurnber of arithmetic operations is involved.

The two methods are of course equivalent methods of

integration as the U and V model simulates the truncation that would pertain to prognostics

for vorticity and divergence.

The truncation procedure is, however, more straightforward in Bourke's model, since the prognostic variables are true scalars.

Another advantage with Bourke's model is that the

implementation of a semi-implicit time scheme is facilitated by the explicit prognostic for divergence.

Although more

complicated, the semi-implicit time scheme may,however, also

.

1~

be incorporated in a

U and V model (Byrnak(l975)).

The semi-implicit time scheme has been considered in

k

Volume I Chapter 3 fora shallow water grid point model. For comparative purposes we shall consider the implementation of this time scheme to Bourke's model.

We introduce a

representation at discrete times and use the standard notation

6 X = (Xt+tit - Xt-tit)/2tit

t

t

(4.109) Xt = ½ (Xt+tit + Xt-tit) .

- 230 -

Ina manner similar to that used in Volume I in (4.107) we use an averaging in time for the gravity wave terms and a centered time differencing for the time derivatives.

This

scheme gives

ot nm n -,

At

m, n

_ t + ,,,..-t ot om,n - 8 m,n En ~m,n'

ot

4>

m,n

= etm,n

(4.110)

~ o~t

m,n •

As the vorticity equation does not involve any gravity wave terms, the values n!+~t are explicitly determined ,

from the first equation.

Using the relation otX =(Xt - xt-~t)/~t

and eliminating o-t from the last two equations, we get m,n

~t

m,n

=

1 { 4>t-M +~t (Ct _ ~ 0 t-M) _ ~t2~ 8 t }• l+E ~t2~ m,n m,n m,n rn,n n

(4.111) Thus, for all spectral components -4>-t m,n

is determined

explicitly by quantities known at the time considered. Once t+~t these values have been computed the values 4> and m,n åt+~tcan be obtained directly from (4.109) and (4.110). m,n Thus, for this model the semi-implicit scheme involves only very little extra computations each time step compared to an explicit scheme.

On the other band, for the grid point model

considered in Volume I the extra computations are much more extensive. For the grid point model we get instead

of (4.111)

an elliptic finite difference equation (equation (6.8) in Volume I Chapter 4) which is of the Helmholtz type. This equation must be solved every time step.

The simplicity of

(4.111) compared to the corresponding finite difference

equation is obviously due to the faet that the spherical harmonics are eigensolutions to the Helmholtz equation.

.___

.

- 231 -

4.5.4

The structure of present sigma-coordinate models

At the present time several groups have developed or are in the stage of developing non-filtered baroclinic spectral models ( see references in Section 1).

The spectral

representation in the horizontal direction has usually been combined with a discrete representation in the vertical direction, although spectral representations as well as a finite element representation have also been considered (see references at the end of Section 2). We shall not here consider the extension of the spectral representation to the vertical direction as this subject has already been treated in Chapter 1 of this Volume.

Fora

discussion of the difficulties connected with such an extension the reader is also referred to the papers Machenhauer (1974) and Eliasen and Machenhauer (1974). When using a discrete representation in the vertical direction the extension to baroclinic models of the spectral method described above is rather straightforward.

The equations

in sigma-coordinates are used and generally temperature, T, and the logarithmof surface pressure, tnp*, are chosen as the prognostic variables.

If p* was chosen as a

variable, and represented by a truncated series of spherical harmonics, the horizontal variation of the terms which include division by p* would not in general be given by truncated series of these functions, and hence a truncation without aliasing would not be possible by the transforrn method. ether band

When on the

tnp* is chosen as a variable,aliasing may be avoided.

In the models the geopotential i at discrete sigma-levels is determined by the:, hydrostatic relationsbip

- 232 -

~

= ~* + R

J:

(4.112)

TdR.no ,

where ~* is the time independent geopotential of the lower boundary surface, which is represented by a series of spherical harmonics (truncated as the series of T).

As usual the

integral is approximated by a quadrature formulae and~ is seen to become a truncated series as T. vorticity

~

and divergence

Following Bourke (1972)

o ( or stream function

~

and velocity

potential X ) are usually chosen as the prognostic variables describing

the wind fields and the equivalent fields for

U = u cos~

and V= v cos~

are used as auxiliary fields in the

computation of the non-linear terms. The diagnostic equation for

cr

= o

I

Adcr -

l

0

where A

=o

r

cr=

do/dt may be written

Ada ,

0

+ V•~R.np*

= o +e1.(1-u _1_ 2f U Hnp* 3A

+

v

(l-µ2)

cHnp*) d\.l

and where the right-hand side is approximated by some finite difference analogue in the vertical direction.

We note that

å is

Asa consequence

determined by quadratic non-linear terms.

the non-linear terms in the prognostic equations which involve

å,

as for example the vertical advection term

produets.

å~~ , are triple

Using triangular truncation such terms become

truncated series with n ~ 3N and in order to avoid aliased interactions the numbers of longitudes K1 and latitudes K2 in the transform grid must satisfy K ~ 4N+l, 1

K ~ (4N+l)/2 2

(4.113)

For parallelogramrnic truncations the corresponding numbers becorne K 1

'·ø,.':

=

K 2

4M+l,

.,,=, ~--,,,,.~--~---'-·--~- --,,. ...-

~

= ( 4M+4J+l) /2

•

(4.114)

- 233 -

The remaining non-linear terms in the non-adiabatic frictionless part of the prognostic equations are only quadratic produets and require only numbers of longitudes and latitudes which satisfy (4.79) or (4.80). When moisture is included in the model specific humidity q (or the water vapour mixing-ratio) is most aften chosen as prognostic variable.

An exception is the Canadian spectral

model (Daley et al. (1976)) where dewpoint depression is chosen as moisture variable.

This choice is made on the basis of

experiments conducted by Simmonds (1975), which indicate that a spectral approximation to the horizontal variation of dewpoint depression contains more information than a similar spectral approximation to relative humidity or rnixing ratio.

Furthermore,

a spectral approximation to dewpoint depression can never be physically unrealisable due to spectral truncation as can the two other measures considered. The effect of moisture on the density of mass may be included by the use of the virtual temperature Tv= T (.622+i)/.622(l+q). In the hydrostatic equation (4.112), for instance, we should then use Tv instead of T.

With Tand q being truncated series, Tv

becornes an infinite series and so does including T

V

V

. Consequently, terms

or ~. which are evaluated by the transform method,

imply a certain aliasing.

as T

~

This aliasing is probably insignificant

is quasi-linear in Tand q.

It can be avoided by using

a linearised expression for Tv which probably will be sufficiently accurate in most cases. As mentioned in the introduction it is a big advantage of the transform method that the parameterization of physical processes, which depends upon locally determined decisions, can be included in physical space in the transform grid points.

- 234 -

Utilising this possibility spectral multi-level models with advanced parameterization of physical and sub-truncation scale processes have been developed.

A detailed description of two

such models, which at present are used in routine forecasting, may be found in the papers Bourke (1974), Bourke et al, (1977) and Daley et al, (1976 ).

Aversion of the model discussed by

Bourke et al, (1977) has been used also in general circulation experiments. Based on experiments by Bourke (1974) and Hoskins and Sir.unons (1975) the numbers of grid points in the transform grid are generally chosen to satisfy (4.79) and (4.80) and not (4.113) or (4.114).

That is, only linear and quadratic terms

are calculated alias-free whereas aliasing is allowed for triple terms and for the pararneterization of physical processes, which introduce a non-linearity of higher degree than quadratic. Theabove mentioned experiments indicate that the effect of this aliasing is acceptable.

According to Bourke et al. (1977) who

uses the rhomboidal truncation an increase of the nurn!:>er of points above that determined by the minimum satisfying (4.80) has a negligible effect.

->

of the number

K_Lof

equation

On the other hand a decrease

points along all Gaussian latitudes below

the minimum determined by (4.80) was found to give unsatisfactory results. The effect of reducing the number of Gaussian latitudes below the limits determined by (4.80) was not investigated. If both K and K are reduced to the numbers required to give 2 1 alias-free calculations of linear terms only, that is with 2N+l, K = N+l, then one should 2 expect non-linear instability at least in the absence of triangular truncations to K 1

artificial damping

=

of smal! scale waves.

This is indeed

- 235 -

found to be the case.

So, a shallow water model i.e. the model

described by Machenhauer and Rasmussen (1972) is found to "explode" within a few time steps, when K1 and K2 are set equal to these values. Although a general reduction of the number of points along the Gaussian latitude circles seems to give unsatisfactory results, experiments by the author with a 9 level spectral model developed at ECMWF indicate that at least when the triangular truncation is used a certain reduction of points along latitude circles below the value 3N+l can be made in rniddle to higher latitudes ,,,ithout any significant change of the integration results.

This could be expected, since the resolution given

by a triangular truncated spectral representation is isotropic and uniform all over the globe, whereas the transform grid, with the same number of points at all latitude circles, is highly nonisotropic

and non-uniform due to the convergence of the

meridians towards the poles. As explained in Subsection 4.4.4 the truncated spectral equations for inviscid non-divergent flow conserve the mean kinetic energy and the mean enstrophy.

In numerical integrations

of these equations non-conservation of the invariants can be due only to round-off errors and time truncation.

In practice,

it is found that the effect of these errors is very small even for very long period integrations ( Ellsaesser (1966)).

As

a consequence the average two-dimensional wavenumber ·R.av is very nearly conserved and an unlimited systematic energy cascade towards the highest wavenurnbers

n is not possible.

Nevertheless, as noted in Subsection 4.4.4, a certain blocking of energy in the pighest wavenumbers

is not excluded if at

the same time energy is transferred to low wavenurnbers.

- 236 -

That such a blocking does in faet occur was shown by Puri and Bourke (1974).

Two 8 day integrations with rhomboidal

truncations, M=15 and M=36, were compared.

In both integrations

the same initial field, derived from observed data, was used. The total kinetic energy in each zonal wavenumber considered as a function of m.

m was

A considerable blocking of energy

in the M=15 integration in the region m = 10 tom= 15 was found already after 4 days of integration.

The blocking is

illustrated in Figure 11, which shows the energy spectra averaged over the whole 8 day integrations period.

As expected,

this blocking shows up even more strongly in the enstrophy spectrum.

It was shown by Gordon and Stern (1974) that the

blocking phenomenon occurs in a spectral shallow water model as well and apparently it was found also by

Bourke (1974)

in his multi-level model . .The blocking phenomenon is a result of neglecting interactions involving components outside the truncation limits. It is observed as a gradual increase of energy in the small scale components and occurs most strongly and at an earlier time in low resolution models than in high resolution models. Apparently a damping influence on the smallest scales retained is missing. When the amplitudes of the small scale components have grown to unrealistically large values, one should expect that eventually, through non-linear interactions, this will lead to serious errors of the large scale components. It seems likely that some parameterization of the scale selective damping influence of the components outside the truncation limits will improve the integration results and that such a parameterization is needed especially in longer range low resolution models.

· : : , · · · · · · - - · . , , , . ~ - - ~ ~ - ·-....,, -.,-' ..-N*. In arder to reduce this aliasing one tries to obtain as high a value of N* as possible fora given number of grid points to be used in the quadrature formulae. Writing (4.122 ) as

f

1 Tm n = ½ Tm(µ) Pm n(µ) dµ'

,

(4. 125a)

'

-1

J

1 2n -imA Tm{µ) = 2 n T(A,µ) e dA , 0

(4.125b)

we want to choose quadrature formulae for the integrals which are exact for T = fN*)(A,µ) and lml~ n ~ N* where T(N*)(A,µ)

N*

=

I m=-N*

T(N*)(µ) e imA m

t

(4.126)

T(N*)(µ) m

N*

=

I

n=lml

T

m,n

p

rn,n

(µ)

•

The first step, the Fourier transform, at certain latitude circles µ= vk fork= 1, ... ,K 2 is common to all methods. Here the trapezoidal quadrature (3.26) is used, giving the expression 1 K1 imA Tm (µk) = Kl !=lT(Aj,µk) e j (4.127) 2 n j, j = 1,2, ... ,K and O ~a ~ ~ where A. = a+ K 1 J 1 1 When T = T(N*) the integrand in (4. 125b) becomes a truncated Fourier series with maximum wavenumber

equal to 2N* so that

Tm(µk) = Tm(µk) if
2 dµd).= minimum,

O

.._,~,~

r'(r /~ ' \ ,,z,,,q - 4> ( ::z h. 'I -: ,1 I

The coefficients determined by(4.135) are identical to those

!1r

:~:KK

....

as Lr(~) Pm,n (~) is a polyDDmial inµ of degree r + n.

where

.

N* = N* l l

T'{A,µ)

m=-N r=O

( r )L ( ~) im). e Tm r

is determined as a least squares fit to the given grid point values with equal weight to each point value.

Ina slightly

changed form the method may be used also for analysis of data at equally spaced latitudes including the Equator and the pole points.

Concerning these changes and further details

of the method the reader is referred to the original paper. The analysis methods described above may be used for the analysis of the true scalar fields inp*, Tand S. Concerning the computation of expansion coefficients for vortici ty

r; and divergence o from grid point values of

the wind ~omponents u and v the most reasonable approach , fs I~i!//"'

' v-(!

'k~:

KR-:

z l~'I --

96 -/· 1/0

-·-------·- --·-·--

{f(, ,.J., ~.:~->~ -

...,_

--,

-

bf; /~P" ·'

.c:J(:(,., _,,,,

llfa7o

(,7i~,I ,_ ! I- .

"r"',-z) ,~ I-: 1=,'1/ !' ...,...~,,)/, ! t{_;t.;-$JI/ (i?.1..tf"ld /t~-w.r~ i

{"'

,j

- 248 -

seems to be that used by Bourke (1974).

He uses the

definitions (4.100) which when transformed to spectral form by means of (4.97) give

r-

1 ,;;m,n = 2a

-

{imV ( µ ) P . ( µ ) - U ( µ ) H ( µ ) } d/ 2 , m,n m m,n -µ m -1

r -

1 0m,n -- 2a

_ 1{im Um(µ) Pm,n (µ)+

(4.139)

Vm(µ) Hm,n(µ)} ~~µ2.

Here we have assumed that,;; and o can be expanded in infinite series.

-

-

Um and Vm are the Fourier coefficients of the

corresponding fields of U = u cos~ and V= v cos~ , that is

2 TI

m

" (µ) = 1 U 2 ID

f

2

V(µ)= 1

TI

0

TI

V(JJ,A) e -imA dA

f02TIU(µ,A)

(4.140)

e -imA dA.

Using the Gaussian and the trapezoidal quadrature forrnulae in the evaluation of the integrals in (4. 139) and (4. 140) .., respectively we can compute approximate values of,;; and o • m,n m,n When the number of Gaussian latitudes, K2 , is half the number

.

of grid points at each latitude circle then the aliasing is limited to the small scale components with n >N* = K2 - 1. Or in ether words, then the estimated expansion coefficients are exact for

,;; and o being truncated series with maximum degree

Once the coefficients ,;;m , nand crn, n are computed the corresponding coefficients Um,n and Vm,n can be

equal to N*.

determined from the relations (4.105). The nrocedure described above implies that the data are given at Gaussian latitudes.

If given at equally spaced

L

- 249 -

latitude circles an evaluation of the integrals in (4.139) by a generalisation of the method of Machenhauer and Daley (1972) seems to be the most reasonable approach.

In doing so,

the same accuracy as with Gaussian latitudes is achieved, whereas an interpolation to Gaussian latitudes may result in a decrease in accuracy. 4.7

Inclusion of orography

Simulation of orographic effects in sigma-coordinate models implies determination of truncated expansions of ~. from aset of grid point values of

~ •.

The point values

used are, in faet, mean values over the area surrounding each grid point.

As an example grid point values of ~.

in a regular 1° longitude - latitude grid may be obtained from Gates and Nelson (1975).

We shall refer to these data

as the Rand topography. Usually, however, spectral modellers have used values given in amore coarse-mesh grid.

The actual

topography of the earth's surface is very rough and a very slow convergence of expansions of ~. is obtained by spherical harmonic analysis of the grid point values.

This is the case

especially when fine mesh data, such as the Rand topography, are analysed.

The deviations are especially large in the neighbour-

hood of the narrow transition zones with steep slopes between mountain areas or plateaus and flat low-lands or oceans, for example at the coasts of Greenland and Antarctica, of North Himalayas.

and South

at the west coasts

America and at the southern edge of the

In these areas where the topography has quasi-

discontinuities in the first derivative Gibb's phenornena do occur in the truncated series representations of

~ •.

This

is manifested as short waves whose amplitudes decrease with increasing distance from the transition zones.

- 250 -

As an example Figure 12 shows the Rand topography and two triangular truncated spectral representations, truncated at wavenumber

N

=

40 and N

=

80 (T40 and T80), along latitude

39.5°N. ( The T80 curves are interrupted over part of the oceans where the nurnerical values are smaller than 40 meters). The global spectral representations were computed from the Rand topography using the method of Machenhauer and Daley (1972) described in the preceding Subsection. It is seen that even the T80 representation deviatesconsiderably from the Rand topography in certain areas. The quasi-discontinuities mentioned above are eliminated in the spectral representations, but at the same time Gibb's waves are introduced which, of course, are aesthetically displeasing.

Different approaches have been used in order to

reduce the amplitude of these small scale waves.

One method

is to apply some smoothing to the data before the spherical harmonic analysis.

This approach has been suggested by

Arpe (1976) and in a somewhat different version by Bourke et al, (1977).

As an example Figure 13 shows the representations

corresponding to those in Figure 11 obtained by applying a 25 point filter to the Rand topography before spherical harmonic analysis. (The smoothed value at a certain point (A ,$)

=

(A 0 ,$ 0 ),

was obtained as a weighted sum of the value at the point itself and the surrounding 24 points, using for

points with

((A-A ) 2 + ($-$ ) 2 )! = 0,1,/2,2,/5,/8 the weights 0.138499, 0

0 0,090212, 0.069794, 0.030215, 0.018348, - 0.011543, respectively). Extremes in the topography and its slope are reduced and as a consequence a faster convergence of the series and a reduction of the Gibb's waves are achieved. the original data and

The overall

deviation between

the truncated series representations

- 251 -

is, however, increased.

Except for an increase in aliasing

corresponding effects are obtained when analysing more coarsemesh grid point data.

The application of a smoothing operator

to the grid point data before analysis may be considered equivalent to multiplying the original spectral amplitudes by certain response functions, which decrease in magnitude with decreasing scale.

Therefore a similar effect may be

achieved directly by selective damping of the original spectral amplitudes.

Such an approach is used by Gordon

and Stern (1974). The question of how to include large scale orographic effects in the most ideal way in sigma-coordinate spectral models is still an unsolved problem.

When unsmoothed

representations of ~* , which deviate as little as possible from the actual topography, are used then due to the horizontal stratification and large vertical variations in the atmosphere the fields of the dependent variables in the models constant sigma-surfaces become rough fields.

Asa consequence a slow

convergence of the spectral series is achieved and large truncation errors are made, initially as well as during the numerical integration.

When on the other hand smoothed

representations of ~* are used these errors are reduced but the deviations between the model and the actual topography become larger.

As explained in Chapter 1, similar problems

are encouritered in grid point models.The influence of topography is a rather explicit problem in a spectral model formulation. There is an explicit requirement in the spectral model to predetermine the representation of topography that is compatible with the model resolution; this requirement is less explicit in a grid point model, although just as necessary.

- 252 -

4.8

Performance and Efficiency

We have seen tr..at the spectral method has several desirable properties compared to the traditional grid point method. Pure linear terms in the equations are evaluated exactly for a given spectral representation of the variables. Here, by pure linear terms we mean terms w~j_ch can te represented by series with the same truncation as that used for the prognostic variables. That is, when using spherical harmonics as e:xpansion functions, for instance A

terms like the 6-term, -2n/a 2 (a~/aA), and the advection by the solid lx>dy rotation,

-w 0 1 a(IJ 2 ~)/aA,

linear part

' of

in thevorticityequation. In particular the pure

the phase speed of retained spectral components is computed

exactly, apart from round-off errors and time trtmcation errors, which are generally small. Asa consequence no space errors due to the evaluation of pure linear terms are introduced in the group velocities and especially no space computational dispersion(as that described in Volume I, Chapter 3) occurs in spectral models. The errors introduced by spectral truncation, initially as well as due to quadratic nonlinear processes during the integration, are minim.ized in an area weighted least squares sense and the unresolved scales of the dominating quadratic terms are not misrepresented in terms of the scale resolved.

In grid point models, on the other

hand, phase speed errors, computational dispersion and aliasing introduce errors, especially in the shortest resolvable scales. Because of these differences we should expect that a spectral model with a certain nurnber of degrees of freedom will produce forecasts that are equally as accurate as a similar traditional grid point model with a much higher nurnber of degrees of freedom. This is certainly the case for simple linearised models as indicated by the results obtained in Chapter 9, Section 4. The question of efficiency of a spectral model compared to a similar grid point model becomes therefore a question of the balance between the fewer degrees of freedom and the greater

- 253 -

arnount of cornputation associated with each degree of freedorn in the spectral model. J

~-

The very first meteorologically relevant comparison between the two methods was made by Ellsaesser (1966), who compared hemispheric 36-hour

forecasts computed from filtered barotropic

models using areal initial data set.

The grid point integrations

were performed using the standard practices at the National Meteoroloeical Centre in the U.S.A. in 1962.

Second-order

finite differences on an octagon grid in a polar-stereographic projection with 1977 grid points (corresponding to about a 400 km grid length at 45° N)

were used.

The spectral

integrations were performed using the interaction coefficient method with rhomboidal truncated representations at M=l2 (Rl2) and M=lB (R18), corresponding to 150 and 333 degrees of freedom, respectively. time scheme was used.

In all integrations the leap frog Due mainly to its larger phase speed

errors Ellsaesser found that the grid point integration was inferior to the spectral R18 integration.

He concluded that

for comparable phase speed errors and for integrations spanning the whole hemisphere, the spectral model was more efficient

by a factor of about 2 in terms of consumption of computing time.

With the interaction coefficient method it could,

however, be foreseen that the computational advantage of the spectral model would be rapidly reversed as the minimum resolvable scale was reduced in the models. Since the introduction of the transform method further comparisons have been made.

Doron et al. (1974) and

Simmons and Hoskins (1975) have compared integrations with primitive equations models; the former authors using barotropic models and the latter using baroclinic models for

- 254 -

a dry atmosphere.

No parameterization of orography, turbulent

diffusion, convection, surface fluxes or radiation was included in the models.

In both studies simple analytical, though

meteorologically significant, initial conditions were chosen.

The models compared were similar except for the

representation of the horizontal fields.

In the grid point

models second-order finite differences on spherical grids were used.

The experiments demonstrated that, when increasing

the resolution, the solution obtained by the grid point and the spectral models converged to a comrnon solution.

Low

resolution spectral and grid point integrations were then compared to the almost identical solutions obtained by the high resolution models.

The low resolution grid point models

had a regular 5° longitude by 3° latitude grid and the low resolution spectral models used a rhomboidal truncation at wavenumber

M=16 (R16).

The resolutions were chosen so that

the spectral models required less or the same computing time and less storage than the corresponding grid point models. The particular initial conditions used by Doron et al.(1974) are a Rossby-Haurwitz wave of zonal wavenumber

m=4, which

moves with little change in shape or amplitude, and a RossbyHaurwitz wave of zonal wave number m=8 which undergoes large changes within 5 days.

Both low resolution models perforrned

equally well for the rn=4 integrations, whereas the spectral model was clearly superior for the m=B integrations up to day 5.

It describes the rapid changes well, whereas the

grid point model failed to predict them.

A closer analysis

showed that these changes depend crucially on the phase differences between two waves with zonal wavenumber different meridional structure.

"""

-~-

. _-__

8, but a

The low resolution grid point

- 255 -

model treats the development poorly because of its inaccurate zonal phase velocities, despite there being 9 poin1Sper zonal wavenurnber

and even more points per wave length in the

meridional direction.

The low resolution spectral model

somewhat overestimated the development; an effect which was explained by the inclusion of the modes which are important in the instability but the exclusion of modes which have a modifying effect.

After about 5 days the spectral blocking

phenomenon becaræ serious and caused divergence of the low resolution spectral model solution fromthose of the high resolution models. In the study by Simmons and Hoskins (1975) a growing baroclinic wave is simulated using five-level primitive equation models.

Small perturbations with wavenurnber

m=S are

initially superimposed upon a baroclinic unstable zonal flow

and the integrations are carried out to a stage at

which fronts have formed in the high resolution models with temperature differences of the order of 10°c across a distance of about 200 km.

In this case the conclusions drawn from a

comparison of the low resolution models areless clear than in the study by Doron et al. (1974).

An inspection of the

dominant spectral components clearly illustrates the superior behaviour of these modes in the spectral model.

Generally the

lower order harmonics ancluding those describing the largescale zonal mean state) of the solution are predicted to a greater accuracy by the spectral model during the whole development.

At the last stage, however, in which sharp

fronts are developed in the high resolution models, the resolution of the low resolution spectral model is insufficient and a spectral blocking in high wavenurnbers

shows up.

The

- 256 -

lack of a parameterization of the transfer of energy to

unresolved scales in the spectral model is evident at this stage.

The grid point model on the ether band is able to

resolve the smaller scale details in the last stage, at least in a qualitatively correct manner.

It is remarkable that even

though sharp fronts cannot be formed due to lack of resolution, and even though the prediction of the small scale components deterioratesbecause of blocking in the low resolution spectral model,the large scale components are still simulated more accurately in this model than in the low resolution grid point model. The studies referred to above indicate that the spectral method is superior with regard to the prediction of large scale developments, but that grid point models have a better capability of resolving small scale features such as steep fronts. The question is, however, if this capability of a grid point model is of any value in actual practice as the position and/or time of the developrnent of the small scale features may well be ratherpoorly predicted. Studies of performance of hemispheric spectral primitive equations models in short range weather forecasting and comparisons with grid point models have been published by Bourke (1974) and Bourke et al. ),;

Daley et al.

TJ:ie

(1977) for the Southern Hemisphere and by

(1976) for the Northern Hemisphere.

extensive -quali tative and. ciuarit.itative comparisons

made by Bourke et al., with the hemispheric operational system hitherto employed,

led

in January 1976 to the adoption of

a 7-level R15 spectral model for daily operational prediction in the Southern Hemisphere at the Australian Bureau of Meteorology.

Shortly after, in February 1976, a 5-level

- 257 -

R20 spectral model was chosen as the operational large scale forecast model for the Northern Hemisphere at the Canadian Meteorological Centre.

The various subjective and quantitative

evaluations and comparisons with the grid point models, which were made prior to the adoption

of the spectral model in

Canada, are described in detail by Daley et al. (1976) and we shall only quote a few of the main points of the paper here. The spectral model has 5 vertical levels (equally spaced in sigma).

All vertical operations are handled by finite

differencing and a semi-implicit time scheme is used.

Physical

parameterization processes include orography, pre~ipitation and latent heat release, dry and moist convective adjustment and a simple boundary layer parameterization.

With the horizontal

resolution chosen for the operational model, rhomboidal with M=20 (R20), this model was found capable of simulating a case

of cyclogenesis successfully during a 36 hour

forecast, although

a R30 model was found to be even better in this respect. The R20 model was compared, among others, with a grid point model which has

the same vertical structure, semi-implicit

algorithm and physical parameterization as the spectral model and which uses a fourth-order finite difference scheme on a regular 2805-point polar-stereographic ~rid ( 381 km grid length at 60°N).

The grid point model's lateral

boundary is a rectangle on a polar stereographic projection and the boundary conditions are as follows : the relative vorticity at inflow points is zero and the normal wind and normal pressure gradient are specified initially and are independent of time.

The spectral model, on the other hand, has a free-slip

wall boundary condition at the Equator.

These differences

in boundary conditions should,however, not be important in

l - 258 -

the area of verification over North Arnerica during 36 hour forecasts.

Asa result of the comparisons of many forecasts

with the two models they were found to produce approximately "equivalent" forecasts and the spectral model was found to be the most efficient, both with respect to computation time and storage requirements.

Although these results are computer dependent and depend upon the degree of programme optimization, they show that the spectral method is a highly competitive method, in terms of both accuracy and computational efficiency; at least when applied to short range weather forecasting and with those resolutions used operationally

in

1976.

It remains, however, to be seen

if this is the case also for higher resolution models and for extended range forecasts.

-, - 196 -

a"i:;

Tt

=

2n

- a7

! m=-N

N +

l

M=-N

N

l

n=lml

im lj,m n y , m,n (4.45)

N

l

n=lml

F m,n

y

m,n

Comparing (4.44) and (4.45) we see that the linear term, the so called beta-term, is evaluated exactly from the truncated series representation of

lj, whereas the components F

nonlinear term with N< n