A design of an oceanic GCM without the rigid lid approximation and its application to the numerical simulation of the circulation of the Pacific Ocean

Journal of Marine Systems, 1 (1991) 271-292 Elsevier Science Publishers B.V., Amsterdam

271

A design of an oceanic GCM without the rigid lid approximation and its application to the numerical simulation of the circulation of the Pacific Ocean Qing-Cun Zeng, Xue-Hong Zhang and Rong-Hua Zhang Institute of Atmoshperic Physics, Academia Siniea. Beijing~ P.R. China Received October 27, 1989: revised version accepted May 3, 1990

ABSTRACT Zeng, Q.-C, Zhang, X.-H. and Zhang, R.-H., 1991. A design of an oceanic GCM without the rigid lid approximation and its application to the numerical simulation of the circulation of the Pacific Ocean. J. Mar. Syst., 1: 271-292. Since the approximation of rigid lid of the ocean excludes the "available surface energy" and consequently, introduces errors in the computation of surface currents, propagation of very long waves and the variation of gigantic gyres, it is desirable to remove this approximation. Such an oceanic general circulation model has been designed in the Laboratory of Numerical Modelling for Atmospheric Sciences and Geophysical Fluid Dynamics, Chinese Academy of Sciences. In addition, the model differs from other oceanic GCM's (General Circulation Models) in the world by the following features: (1) subtraction of the standard stratification and introduction of departures of temperature, T', salinity, S', density, P' and pressure, p', from their standard values; (2) suitable transformation of coordinates and variables, which makes the energy equation more compact and the total "available energy" conserved, if the forcing and dissipation are neglected and some small reasonable modifications are introduced; (3) perfect conservation of total available energy (under the conditions mentioned in point 2) and no computational mode in the finite-difference scheme; (4) introduction of "flexible coefficients" and a modified splitting method for flexibility of computation and acceleration of convergence in time-integration. A four-level version of such oceanic GCM has been applied to simulations of annual mean circulations in the World Ocean and the annual cycle in the Pacific Ocean under the forcings of the climatological surface wind stress, surface heat flux and air pressure. More than 60 years integration has been made and the results show successful simulations of most observed large-scale features of the annual mean and annual cycle of currents, temperature, sea surface elevation and other features in the Pacific Ocean.

Introduction

e s p e c i a l l y , first t h r e e d i m e n s i o n a l u n s t e a d y o c e a n ic m o d e l s w e r e p r e s e n t e d b y S a r k i s y a n (1966) a n d

The numerical

simulation of oceanic general

c i r c u l a t i o n c a n b e t r a c e d b a c k to M u n k

(1950)

B r y a n (1969). S i n c e t h a t t i m e m o r e c o m p r e h e n s i v e oceanic GCM's

have been developed and applied

a n d S a r k i s y a n ' s (1954) w o r k s , w h e r e t h e c u r r e n t in

by Bryan

t h e o c e a n is r e p r e s e n t e d

1987), W a s h i n g t o n et al. (1980), H a n (1984), L a t i f

nondivergent

model

with

by a two-dimensional homogeneous

density

et al. (1975),

Philander

et al. (1985,

(1987) a n d m a n y o t h e r s f o r t h e i n v e s t i g a t i o n o f

u n d e r the a c t i o n o f w i n d stress. A f t e r that, w i t h

oceanic circulation

the r a p i d d e v e l o p m e n t of n u m e r i c a l w e a t h e r p r e -

modelling and experimental prediction of climate.

diction the oceanic modelling and the computa-

Oceanic GCM

tional method became more and more improved,

Ocean-Land

0924-7963/91/$03.50

© 1991 - Elsevier Science Publishers B.V.

and

dynamics

and

also the

as w e l l as C o u p l e d A t m o s p h e r e GCM

p l a y a m o r e a n d m o r e im-

272

portant role in the understanding of oceanic processes and climate. It is worth pointing out that most O-GCM's, currently adopted in research, introduce the rigidlid approximation, i.e., the approximation of nondivergence of the vertically integrated flow, which is an extension of the two-dimensional nondivergence to the three-dimensional case and also used in the early models for numerical weather prediction. All the models mentioned in the foregoing paragraph assume that the top surface of the ocean is a rigid plane, consequently, the fastest external gravity wave is excluded from the model. This approximation seems to be very reasonable as regards the reduction of computation time. However, meteorologists had revealed that the nondivergence of vertically integrated flow introduces undesirable serious errors to the propagation of ultralong waves of barotropic mode. Moreover, Zeng (1979) had pointed out that exclusion of "available surface energy" due to the approximation of nondivergence of vertically integrated flow also leads to the lack of its transformation from or into kinetic and available potential energy, hence results in a distortion of energetic cycle of motions. Therefore, the nondivergence approximation leads to not only the erroneous westward propagation of ultralong waves, but also a distortion of energy dispersion process, consequently resulting in a large error in the climate prediction. The problem might be more serious as an O G C M or a Coupled A - O G C M is concerned. In fact, due to the smallness of the Rossby deformation radius and very large meridian extension of the World Ocean the gigantic gyres and other relative long waves in the ocean belong to the category of ultra-long waves and hence the characteristics of their structures and propagations might be distorted by the approximation of vertically integrated nondivergence. Moreover, it is not difficult to show by simple calculation that in the ocean the "available surface energy" and kinetic energy have the same order of magnitude and the major part of kinetic energy consists of surface current. Hence this approximation might also result in erroneous simulation of surface oceanic current, which is very important in the formation and evolution of

Z E N G ET AL.

climate anomaly. The approximation of vertically integrated nondivergence has already been removed in current atmospheric models. The design of oceanic G C M without the rigid-lid approximation is desirable. Based on the considerations mentioned above Zeng (1983) has developed an Atmosphere-Ocean Coupled G C M free from the approximation of vertically integrated nondivergence of both atmospheric and oceanic motions. Crowley (1969) has also developed another model without the same approximation but different in other aspects. The oceanic part of Zeng's model has been slightly simplified and completed by Zhang et al. (1989) in the simulation of annual mean circulations of the world ocean and by Zhang (1989) in the simulation of annual mean and the seasonal cycle of circulations in the Pacific Ocean. Dealing with an oceanic G C M without the approximation of vertically integrated nondivergence, the problem of how to guarantee the computational stability and to accelerate the convergence becomes crucial. In solving these problems the numerical methods and techniques developed in the Institute of Atmospheric Physics and other institutions such as the subtraction of standard stratification and introduction of departures as variables, the transformation of coordinates and variables into more convenient and compact forms, the introduction of flexible parameters and the splitting method (Zeng, 1979, 1983; Zeng and Zhang, 1982; Zeng et al., 1987; Liang, 1986) seem to be very helpful. The first experiments in application of these methods and techniques to the simulation of oceanic circulations were successful. The further modifications and improvements of these methods and techniques introduced by Zhang and Zeng (1988) and Zhang (1989) have made our oceanic G C M efficient and economical in numerical integration.

Governing equations, boundary conditions and the subtraction of standard stratification As shown in Fig. 1, the world ocean has a mean stratification which varies with depth and is governed by the global gravitational and thermal processes of the planet and the solar radiation. It

DESIGN OF OCEANIC

273

GCM

where /3, /5, 7~ and 2( are the standard vertical distributions of pressure, density, temperature and salinity respectively, 7~a is the standard vertical distribution of air temperature, Pao is the air pressure at the mean top surface of the ocean (z = 0); the three constants O0, To and So are the referent density, temperature and salinity respectively, and ato and aso are two positive constants, all of which depend on depth; 0 r and Q,s their mean source terms, K, and K s their vertical diffusion coefficients; k ,L and k 1' are the mean fluxes of downward and upward radiation at the mean top surface of the ocean respectively; /~ the mean rate of evaporation and L is the latent heat of evaporation; and H is the mean depth of the ocean, which can even be taken as infinite. Governing equations used commonly in the oceanic dynamics, but with subtraction of standard stratification, are now written as follows:

seems that the mean stratification could in principle be well simulated by a model, but is not easily simulated by current oceanic or coupled GCM's. Zeng (1983) suggested to subtract a standard mean stratification from the oceanic G C M and to compute the departures of all variables from their standards. The standards are taken as known from the observed climatological fields and satisfied the following governing equations and boundary conditions:

dp

dz-

d

tSg

(2.1)

(KdTI t-d--~] + Q r = O

d-7[

d (KdS 771 s-d-Tz)

(2.2)

Os=O

(2.3)

t5 = po[1 - ato(7 ~- To) + aso(2(- So)]

(2.4)

+

'/0 (0) -~-/~ao

(2.5)

dT

dT~a

PoCpKt-~z =

1

PaCpaKta-d~2 z=0+

,

d---7= - p o Vp - f * k x K + A m / ) +

~z Km-~"

I=0,

+ ( k $ - k 1' ) - L/~

(2.9)

(2.6)

d2(

-d-7 = 0

dT'dt_

_ _

(2.7)

z = - H , suitable conditions for T and S, 0

4

~ (K~T't F,W+A,AT'+ ~z~ Oz ]+QT, t

(2.10)

(2.8)

8

16

12

20

(T.E)

0

100 f

N

200

-g v Az

tm

':'

300 400 500 600

80(

1000 ~ 6

23 24 ~

25 26 ~ 10

27 ~

28 15

29 -

30 ~

31 20

33 25

(~-1000. k g - m -a) (N,, x lO-a, s-l)

Fig. 1. The standard vertical distribution used in the model. The data is taken from Bryan and Cox (1972).

274

ZENG ET AL.

dS' dt

aw 1 ( av 0 sin 0 av x az + a si--n-O, 30 + aX } = 0 ap'

az p'

=

(2.11)

and the boundary conditions at the lateral boundaries S r, which are assumed to be vertical walls, are the vanishing of the fluxes, i.e.

(2.12)

62"Am-~-n + 8 a ' " F = 0, F - i f = 0

~_~_~ S ' l] + Q; F~w + A~AS' + az l K ~ aaz

o'g

(2.13)

p0(--Otto • T ' + a ~ o - S ' )

(2.14)

where f * = (2w cos 0 + vxctgO/a ) is the apparent Coriolis parameter; /9 is the eddy diffusivity of m o m e n t u m in the horizontal direction; Kin, Am, Kt, A t K~, A~, are the vertical and horizontal diffusion coefficients respectively; and F t and F~ are the vertical gradient of standard distribution of temperature and salinity respectively, from which the Brunt-V~iis~il~i frequency N02 is determined as follows

N;=

dz ° ~

Z

= '8 Z0, J K m - ~ = ~0

aT'

/ 1

(2.19) aS'

,

[ K s - - ~z- = F~ '

W=

dz o dt

(2.20)

(2.21)

ab"

z = --Zb,

6 " . K m - ~ + 8v'" 17= 0 aT' aT as'

as' a~- -

a~ On = 0

(2.27)

aTa'

I P o C p K t ~ = PaCpaKta--~ - z=6,zo+O /

(2.26)

Transformation of coordinates and variables

(2.16)

/p' = [if(o)- ~a(0)] "gz0

O7~ On = 0

(2.18)

+ asoFs)

The other symbols are commonly used. The boundary conditions at the oceanic surface [z = z0(0, ~, t)] and the bottom [z = --Zb(O, ~)] surface are now taken as follows w=6.

aT' a~ -

(2.17)

(2.15)

g dr5 Po -~-~ = -g~-(-atoFt

(2.25)

where ff is the unit vector normal to the surface S r or z = --Zb; 6 ' " = 0 but 6 " ~ 0 is the vanishing of m o m e n t u m flux, 6 " = 0 but 6 ' " 3 0 is the vanishing of velocity itself; 8 and 6' should be equal to 1, however, 8 = 6' = 0 results in the rigidlid approximation and 6 = 1 but 6 ' = 0 is a more reasonable consideration which is adopted in our works and implies that the variation of oceanic surface during the motion is taken into account but the oceanic surface elevation itself is neglected due to the smallness of I z0 I. Note 1. The method of subtraction of a standard stratification is also widely introduced in Soviet works (see: Marchuk and Sarkisyan, 1980), but their most works take the rigid-lid approximation. Note 2. Due to the lack of good data, we first add (2.6) to (2.19) together to calculate OoCpKt(~T/az) from observed data according to the surface heat flux formula given by Haney (1971) and then calculate a T ' / a z by subtraction of d T / d z from aT/~z.

d7~ F , = dz dS rs = -d-~

OF

(2.22)

=

az

(2.23)

=

a~? 0z

(2.24)

According to Zeng and Zhang (1982) and Zeng (1983), it is desirable to make a proper transformation of coordinates and variables, which leads to a compact form of boundary conditions and the energy equation. Instead of z, we introduce a new vertical coordinate

~' . Zo-- Z = -- 8 ' . Z 0 qt- Zb

(3.1)

hence the top (z = 6 'z 0) and the bottom (z = - z b ) become ~"= 0 and f = - 1 respectively. Hereafter we will let 6'z 0 = 0 due to the neglectable small-

DESIGN OF OCEANICGCM

275

ness of I z01 except for its variation with time and take A,,,, At, A s as constants. Denoting ~-

g~-b

(3.2)

d~" ~ - dt

(3.a3)

Cpt = [ p ( 0 ) -- Pa(0)] "gz 0

aY Km'-~-~ = ~'=0,

aO T

zb,i,.

(3.14)

P0

Zb

= poCp

F,

,

(3.15)

(3.3) aO s Zb K s~ = poCp ~ . F~'

and introducing new variables 17- ~b', W - ~ OT-~ST,Os-~S

,

(3.4)

we transform (2.9)-(2.14) into the following equations: a-'-'g-= --0~m" L 3 ( V ) -- [~" -~o v p

-v'f*k×

q- -~0 ~gp VZb

Y 1

O {

alT~

(3.16)

The other boundary conditions have almost the same form, but with V T ' and V S ' approximated by qS-lX70T and ~-117Os respectively. Note that the same approximations are also introduced to eqns. (3.5)-(3.7) because A m, A t and A S have to be chosen more or less artificially. Operator L 3 is given by the combination of advection and divergence of fluxes, L3(F)

(3.5)

1

[

(

OF

3VosinOF)

- 2a sin--------0 al V° sin 0-~ff +

OaT at

at"

30

L3(OT)

-

OZ0 + fiT. ~ Z b ] d#(1 + f ) • ~--~-

1 a (KaOTI + ~ t ' A t A O T + C t ' z - - ~ - ~ ~ t a~ ]

(3.6)

OOs

at = - a s ' L 3 ( O ~ ) az °

3-7 "1-ffV. VZb] 1

a [

aO~

+ ,,~. A~AOs + ,~" ~ ~ ~ K~-ff~ )

(3.7)

83Zo a~W at + vq~lT+ = 0

(3.8)

ap' a~

(3.9)

= - p'gzb

P' = P0(-- -~-at° ¢'~ as° OQ\ "-"T + -~--

(3.10)

and the boundary conditions (2.16)-(2.21) into the following: W It=0 = 0

(3.11)

W I~= -1 = 0

(3.12)

(3.17) Some flexible parameters a, fl, )', v, E are also introduced in the equations. They can be chosen in different ways to obtain more efficiency during the integration and do not violate the energy relationship. In fact, no matter what coefficients a, fl, 7, P and e are, we have the following energy equation in the whole ocean [(0, X ) ~ - E , 0 ~< ~< -11: dE dt = forcing and dissipation terms and those related to vertical variation of stratifications. where

e=poff +~" P0(0)~00 ~a(0)

+

+

g" azg} zsinOd2td02

(3.19/

In particular, we have available energy conservation (d E/d t - 0) if the forcing and diffusion terms are omitted and the standard stratification is taken such that No is a constant. Different u and ~ lead

276

ZENG ET

to different rate of generation and dissipation of total energy, however, the oceanic v and c are still not well known in the present. In addition, we also have the mass conservation

AL.

Finite-difference scheme

A C-grid finite-difference scheme was proposed by Zeng (1983), Zeng et al. (1987) and Zhang and Zeng (1988) and a B-grid scheme by Zeng and Zhang (1987). Since the C-grid has been realized and applied to the simulation of oceanic circulation by Zhang and Liang (1989) and Zhang (1989), we here will briefly describe only the C-grid scheme, whose staggered difference scheme is shown in Fig. 2. Readers who are interested in the design of such schemes might refer to the original papers mentioned above. The vertical and horizontal structure of C-grid is shown in Fig. 2. Note that the real coastal lines and bottom surface have to be properly modified and adjusted to the grids of the model. Figure 3 shows the modified bottom topography (meter) of the Pacific Ocean in our simulations. In the fine-

y.

which is obtained directly by integrating (3.8). (3.19) can be taken as a norm in Lz-space (in our case, L2-space consists of all functions, every one of which is quadratically integrable in the domain [(0, X) ~ Y,, - 1 ~< ~"~< 0]), hence is very compact and convenient. It can be seen in (3.19) and (3.20), that if rigid lid is taken (8 = 0), the available surface energy [iS0(0) -~,(O)]gzZ/2Po is absent, and some uncertainty of the elevation of oceanic surface z 0 which is an arbitrary function of time is introduced unless we check mass conservation all the time.

~__T=o

l~x 1;y Q , p~S

i V:O

/

/

V=O

/

f

V-O

U:O

u=o

.

V=O

,40.,.,"~ V

K

1/2

1 11/2 2 21/2 3 31/2 4 41/2

-(7

0--~ 0.009 0.018 0.054 Q089 0.225 0 760 06715 1

U= 0

,

z 7 ~ - ~ ~ , ,

_,-

. . . . . .

-

U,V,T _

_~j-~_,~

....

~

U.V,T--_

b',

25m 5Ore

-r-

.~yj ______

----T"~.J&2

--

150m 250m t 630m L 1000 m

"~'

"A°R

o/y

~j'1-I/2 C-Grid

Zo.b:o.Po,

hs ._

~ ~ -ON,

T----

~ . . . .

-~--

--

!800 m f b-: o, p ' , O ~ _ _ 6 r

-o

_.~-

J

, ,# ,#1

f

AO:I ° AX:5 ° 4 - layer

Fig. 2. T h e v e r t i c a l a n d h o r i z o n t a l s t r u c t u r e s in t h e l A P 4 - l a y e r P a c i f i c O G C M .

DESIGN OF OCEANICGCM

277

resolution models, a unique f-coordinate defined by (3.1) is not suitable, but a blocking-technique [Zeng, 1982 (unpublished) and 1983; Mesinger et al., 1985] or a hybrid coordinate (Zhang and Zeng. 1988) is desirable. We need also a nonuniform arrangement of vertical grids. This can be done by a second transformation of vertical coordinates by taking

o =f(f)

=--O/m

and f ( - 1 ) = - l ,

"Lg(gx) - B

P0

qb , )x 8xzb + -pOfgO a sin 08X

(4.1)

with f ( 0 ) = 0 and

OF Of = f ' ( f )

our finite difference scheme. For example, we have

a sinOSX ---X

+v-(f'v0)

Ao=constant

3F 3o

' S o8 0 + v m A m D x , + f'm " f~2 "b

(4.2)

Km f t,j,k

(4.5)

where f ' ( f ) is a known function of o (or f). Introducing

0t ]i+'2,j,k

(SxF),,j,k--:-- ( F ~ + l z - F ~ _ ' z ) j , k

[

( 8o F ) ,. j.k -- ( Fj + ~ - Fj_ , ) ,. k

(4.3)

= I --O~t" L3(OT)

( 8oF ) ,.j,k =- ( F~,.+½- Fk- '2),,j

0z °

- f l . r t. w + ~ ( l + f ) . 8 .

0--7-

--X

( F ) i . j . k = ½(F,+_~ + F, '2)j,k

_.1_f FX

(ff0)',J, k -- ½(6+~ + FJ-'~),.k

(

(4.4)

-t- Pt" A t " '~OT

),.j.k--~(Fk+'+Fk-'~),.j

f ' 3o( t 8°OT ~} +¢t" z 2 80 K t f ~ ) ,+':.s.k

and applying the quasi-central differences to the right hand sides of all eqns. (3.5)-(3.9), we obtain

60N

8xZb

a sin 08X + p°°

;-Z

"

EQ

305

60S 120E

150E

180

150W

120W

90W

Fig. 3. The model domain and smoothed oceanic topography used in the OGCM.

60W

(4.6)

278

ZENG

( 3Zo ) 3

•

--3T

r-l ( l ,+L,' = - E k=l as-q

( 3o~ sin O" Vo a0

ET AL.

TABLE1 Timeintegration methodsbasedontheflexiblecoefficients I

+ 3x3~)}

(4.7)

I

the value o f f l e x i b l e l

time integration

t

++½,Y,k

methods

t i m e steps

adopted

I

~ = v - v = e - 1/6

I

coefficients

k

and so on, where h is the Laplacian term in finite-difference form and £3 is the approximation of L 3 and is very simple in its form:

I

accelerating

I

algorithm

b. . . . .

a

i I

T

5 t - t

hr.

5t

-6

hr.

6t

=6

hr.

5t

-I

hr.

5t

ffi6

min.

I f

J

] f o r c i n g and

a-<+=v=O

I

1

I dissipation

uffie-1

I

[ L s ( F ) ] ,+½.;./,

= { a sinOs[3x(Vx'+~'F'+~ - Vx'-tF~-'~)JJ + Ot 2

+ ~ -

(V0,+~ sin 0j+½ • Fj+a

sin 0;_½-Fj_,),+½.k]

Vo;_,.. OC3

~

•

+~-do "f "(~k++'Fk+!--~k-'+'Fk-t),+,j]

I

l advections

13 - v = u = e = 0

[

I modified

I

all

I

I splitting

F . . . . . . .

Pscheme

I

baroclinic

I

modes

?

I

I barotropic

I

I

I

modes

a-ufe=0

I

B- ~ = !

b t a

)

(4.8) It is not difficult to transfer boundary conditions to the finite difference scheme• The finite difference equations with these boundary conditions satisfy the finite analogues of (3.18) and (3.20). Time integration methods

If the temporal derivative 3/3t in the left hand side of (4.5)-(4.7) and so on is approximated by the associated time difference 3t( )/3t, and the right hand side denoted as R for simplicity, is taken as the time mean, i.e. R t, we have the exact four-dimensional finite difference analogy to that of the energy equation. Note that the scheme is a time-consuming one if the full terms of right hand side are computed in every time step. However introduction of proper flexible coefficients, splitting methods and some other considerations for accelerating convergence greatly reduce the computation and make it economical. The flexible coefficients represent a set of indicators which distinguish different physical processes with different time scales. Usually, the values of all these coefficients should be 1, however, in a certain case all or some of them can be changed to perform an accelerating algorithm without violating the energy relationship. Moreover, by setting some of them equal to zero at

different time steps in the time integration, we can carry out the computation of different processes separately. In the present numerical simulations with 32~= 5 ° and 3 0 = 4 °, the time integration methods we use are shown in Table 1, i.e., (1) the algorithm of accelerating convergence by reducing artificially some flexible coefficients as less than 1 with some similarity to the method given by Bryan (1984); (2) modified splitting scheme (i.e., splitting the adjustment process further into barotropic and baroclinic modes) with the following choice of the coefficients and time steps for different physical processes: (a) forcing and dissipation are computed with 3 t = 6 h by taking a = f l = ' y = 0 , but v : ~ 0 and ~:~0; (b) advections are computed with 3t = 6 h by taking/3 = y = v = c = 0, but (~ 4= 0 (if one or two of a 1, a 2 and a 3 are zero, then we have the splitting scheme in alternative directions); (c) evolution of inertio-gravity waves and the adjustment process are computed with 3t = 1 h for the baroclinic modes, and 6 min for the barotropic modes (surface waves) by taking a = v = e = 0 but fl =/=0 and v =/=O. Simulations of annual mean and seasonal variation of circulations in the Pacific Ocean

The first successful simulation of annual mean circulation of the World Ocean by using our model

DESIGN

OF

OCEANIC

279

GCM

[annual

.an

'seas°nal-vEar'ing] i atmos, f3rcing .~'-~

I

i att~s, forcing I

coupled m~de~

r

anomalous

--gtsimulat i°I~

t--O.

/

t=N yr.

yr.

t=53

]

simulationS--

t=65 ym

splitting scheme of adjuctment process further into barotropic and baroclinlc modes

algor ith.

frith flexible | coefficients

Fig. 4. Time integration sequence in the simulations of Pacific Ocean circulation.

neglecting the salinity (Zhang and Liang, 1989) stimulated us to make further improvements of the model and apply it to the simulation of annual cycle of oceanic circulation. Here we will present R.H. Z h a n g ' s results in the simulations of annual

mean and annual cycle of circulations in the Pacific Ocean. In the simulations, 7 is taken from H a n and Lee's data (1981), the atmospheric thermal forcings from Esbensen and Kushnir's (1981) and the

"jan

4~N I y

~

.

~

-~

o

-

>

:,-

.7 ~.~--::r~

~DN L':::,. ~ , .

-~-~:,l~-..~lk_:~-~-

-4,- " ~

~

-~

"~- -b.

4,-

)i..k.

*.

I

J

-~}-~

,

~

r,,uAr-

I

~)N 10 N B~ ,:. ~ ~ ~ - - . ~ . a "-

.

?--2.

~..,, . , ~ . ~ . . ~ :

_ ~:~_

~" ,w, 4 .

_~_~_~....~.~..~~--~_~~..,-

". . . . . -~

,¢

"~

~

=--~-~-~'-~-.---~YS-----'K2----aY'5[~r-.. . . . . "~

~

- . ~ - ' - ~ 2 , 0 ~-g-.

"~

~-

..-..~ ,_

-~--------

....

"}'-

-'~-"--'~=-~Z

"~-"~.

"

=

~'~'""

"-

'

' '~

"~

~S ~S

% "~.~_-''% "=:

i

I

120E

140E

<

"=:

!

160

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DESIGN OF OCEANIC GCM

281

standard stratification from Bryan and Cox (1972) with small modification (see: Fig. 1). There are four levels in the vertical. The spatial and time steps have been mentioned in the previous section. For accelerating the convergence, the time integration sequence we have made is displayed schematically in Fig. 4. The sequence is divided into three stages. The first stage, under the annual mean atmospheric forcing, can be further divided into two parts: a spin-up integration for 24 years by means of the acceleration algorithm with flexible coefficients shown in Table 1 from a resting, uniformly stratified state and the annual mean integration for 29 years with the splitting scheme of adjustment process further into barotropic and baroclinic modes. After the time integration for 53 years, the model has almost reached the quasi-equilibrium state under the annual mean atmospheric forcing. The second state was then carried on using the seasonally-varying atmospheric forcing for another 12 years. During the last four years of the seasonal integration, the

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282

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DESIGN OF OCEANIC G C M

283

narrow band along the equator and the weaker and broader downwelling immediately to the north and south, are the same. The difference lies in the fact that the simulated equatorial upwelling is broader and stronger than the observed one, which results in the colder surface water of the model in that region (Fig. 8A). Fig. 8A and Fig. 8B show the simulated surface layer temperature and the climatological S S T from Esbensen and Kushnir (1981). The agreement between them, including the zonal asymmetry, the tongue of cold surface water extending westward from the south American coast along the equator and the warm water pool in the equatorial western Pacific, is remarkably good. The basic thermal structure observed in the real ocean (e.g., Levitus, 1982) is also qualitatively reproduced at the level 2 (150 m) and level 3 (600 m). (The figure is not given here.) One of the characteristics of our O G C M is that the model can directly predict the sea-surface elevation. The simulated annual mean sea-surface

resolution of the model, however, some of the observed narrow currents are highly distorted in the simulation; the equatorial countercurrents are absent and the equatorial currents themselves are too broad and weak. The subsurface current simulated at the level 2 (about 150 m) is shown in Fig. 6A, which is an essentially geostrophic depiction of most of the major ocean gyres related with the sea-surface elevation (Fig. 9). Interesting equatorial features in the central and eastern Pacific are the eastward-flowing equatorial undercurrents and the associated equatorward convergence. The simulated currents vectors at the level 4 (about 2000 m) are displayed in the Fig. 6B, in which one of the interesting phenomena is the western boundary countercurrents with respect to the surface boundary current. The vertical velocity (~, 10 -1° m s -1) computed at the bottom of the surface layer is shown in Fig. 7. Compared with the upwelling calculated from wind stress data (cf. Bryan et al., 1975), the main features, such as the strong upwelling in a

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284

Z E N G ET AL.

elevation in the Pacific Ocean is shown in Fig. 9, while Fig. 10 displays the annual mean dynamic topography of the Pacific Ocean sea surface relative to 1000 dba (from Wrytki, 1974). A comparison shows that in both observation and simulation the main features are: anticyclonic circulation in the North and South Pacific; the highest sea level elevation located respectively at the mid-latitude regions in the western Pacific of the both hemispheres; in the tropical region, the steady increase of sea level from east to west is clearly seen, which represents the general east-west pressure gradient that balances the mean easterly wind and at the same time produces and maintains the eastwardflowing undercurrent. It is interesting to point out that along the western boundary observations show

a difference of 120 cm between the subarctic and subtropical gyres in the Northern Hemisphere and our simulation predicts 140 cm. It is a good agreement. The advantage of the O G C M without rigid-lid approximation is evident even in the simulation of annual mean (steady) oceanic circulation due to its capability of direct computation of sea surface elevation and the improvement of the simulated surface current.

The simulations of seasonal uariation Much of the variations of the surface currents is modulated by changes in the surface wind stress. Some of the characteristics of the seasonal varia-

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Fig. 10. Annual mean dynamic topography of the Pacific Ocean sea surface relative to 1000 dbar in dynamic centimetres (Wyrtki, 1974).

DESIGN

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285

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286

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tions in this field can be seen from Fig. 11, the simulated surface currents at 25 m for winter and summer respectively. For instance, the Kuroshio shows a marked seasonal variation with m a x i m u m speed in July and minimum in January as analyzed by Taft (1972) although the simulated speed is much lower than the observed one. In the tropical region, the North Equatorial Current (north of

1 0 ° N ) appears through the year with relatively small variations in magnitude; the North Equatorial Countercurrent in the eastern Pacific is discontinuous or nonexistent in winter and spring but it is strong, broad and extended eastward to the coast of Costa Rica in summer and fall, which indicates the close relation to the shift of the ITCZ. In the subsurface layer (150 m), the sea-

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DESIGN OF OCEANIC GCM

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sonal variations of the eastward undercurrent is clearly discerned for all seasons as shown in Fig. 13. Out of the phase with Southern Equatorial Current on the equator (Fig. 12, the simulated seasonal variation of zonal current on the equator at 25 m depth), the model undercurrent is strong in the spring and fall, but weak in the winter and summer.

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The simulated vertical velocity at 50 m depth for January and July is shown in Fig. 14. It can be seen that during fall when the southeast trade winds are at their most intense, the equatorial and the coastal upwelling at Southern America are at a maximum. And during the winter season, the downwelling in the subtropic region and upwelling in the subarctic region are strong.

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Figure 15 displays the simulated S S T as a function of time along the equator, together with Fig. 16, the observed counterpart from Esbensen and Kushnir (1981). The simulated seasonal variations are in good agreement with observations. The S S T in the eastern tropical Pacific Ocean is at a minimum during the fall when the equatorial upwelling is at a maximum (Fig. 14). Towards the end of calendar year, as the ITCZ moves equator-

ward and the intensity of the southeast trade winds decreases, the S S T increases. It is at its highest during the spring when the southerly cross-equatorial surface winds are at their weakest, and the ITCZ makes its closest approach to the equator. The zonal averages of computed sea surface heat flux from S S T are shown in Fig. 17 for the simulation and in Fig. 18 for observation as a

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DESIGN OF OCEANIC GCM

function of latitude and time. In general, there is good correspondence in the latitudes and months of the maxima and minima between observation and simulation although there exist some differences in the magnitude between them. For example, in the extratropics, simulated heating during summer and cooling during winter are of nearly the same magnitude; in the tropics, heating occurs throughout the year, and the heating undergoes a semi-annual variation with maximum heating in spring and fall. Based on the assumption that the heat content of upper 275 m of the ocean represents the seasonally-varying content (Oort and Vonder Haar, 1976; Levitus, 1987), the zonal average of simulated heat storage rate in the Pacific Ocean is shown as function of latitude and time in Fig. 19, along with the observed estimate in Fig. 20 (Levitus, 1987). Compared with the sea surface heat flux (Figs. 17, 18), the time variation of heat storage rate in the extratropical regions can be explained with that of sea surface heat flux. The heat storage rate in the tropics, however, represents some interesting features. For example, at 15 ° N, there is a curious phase difference with a maximum of heat storage rate in March and minimum in September, which can also be seen in the calculation performed by Levitus (1987). Near the equator, the simulations of the maximum oceanic heat storage rate during September and October, minimum during December and January are also in agreement with the observation and the propagation of the phase is evident with the strong semi-annual variation in the eastern Pacific Ocean (the figure omitted).

Concluding remarks By subtraction of standard stratification from the dynamic equations, transformation of coordinates and variables and introduction of flexible coefficients, splitting methods and additional algorithm of accelerating convergence, the oceanic model without rigid lid approximation can be realized and not a time-consuming one. The experiments by using our oceanic G C M (IAP OGCM) show that at least the simulation of surface current is improved indeed by the remov-

291

ing of the rigid lid approximation. The simulations of annual mean and seasonal variations of other elements such as ocean surface elevation, SST, heat flux and temperature and current in the deep ocean are successful or satisfactory. Further improvements of such model and its application to the simulations and coupling with Atmospheric G C M are desirable. The following works are currently in progress. (1) improvement of vertical and horizontal resolutions of the model and the use of blocking technique or hybrid coordinates in order to have a reasonable representation of bottom topography: (2) taking salinity into account: (3) design of sea-ice model, improvement of the surface layer dynamics in order to resolve the mixing layer process and their coupling to the oceanic GCM; (4) full coupling to the atmospheric GCM.

References Bryan, K., 1969. A numerical method for the study of the circulation of the world ocean. J. Comput. Phys., 4: 347376. Bryan, K., 1984. Accelerating the convergence to equilibrium of ocean-climate models. J. Phys. Oceanogr., 14: 666-673. Bryan, K. and Cox, M., 1972. An approximate equation of state for numerical models of ocean circulation. J. Phys. Oceanogr., 2: 510-514. Bryan, K., Manabe, S. and Pacanowski, R.C., 1975. A global ocean-atmosphere climate model. Part II. The oceanic circulation. J. Phys. Oc~anogr., 5: 30-46. Crowley, W.P., 1969. A global numerical ocean model: Part I, J. Comp. Phys., 3: 111-147. Esbensen, S.K. and Kushnir, Y., 1981. Heat budget of the global ocean: estimates from surface marine observations. Rep. No. 29, Clim. Res. Inst., Oregon State Univ., Corvallis, Oreg., 271 pp. Han, Y.-J., 1984. A numerical world ocean general circulation model, Part I: Basic design and barotropic experiment. Part II: A baroclinic experiment. Dyn. Atmos. Oceans. 8: 107172. Han, Y.-J. and Lee, S.-W., 1981. A new analysis of monthly mean wind stress over the global ocean. Rep. No. 26, Clim. Res. Inst., Oregon State Univ., Corvallis, 148 pp. Haney, R.L., 1971. Surface thermal boundary condition for ocean circulation models. J. Phys. Oceanogr., 1: 241-248. Latif, M., 1987. Tropical ocean circulation experiments. J. Phys. Oceanogr., 17: 246-263. Levitus, S., 1982. Climatological Atlas of the World Ocean.

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NOAA Prof. Pap. 13. U.S. Govt. Printing Office, Washington, D.C., 173 pp. Levitus, S., 1987. Rate of change of heat storage of the World Ocean, J. Phys. Oceanogr., 17: 518-529. Liang, X.Z., 1986. The Design of IAP GCM and the Simulation of Climate and Its Interseasonal Variability. Diss. Inst. Atmos. Phys., Chin. Acad. Sci., 250 pp. Lydolph, P.E., 1985. Observed atmospheric and oceanic circulation patterns. Weather and Climate. Published in the U.S.A. Marchuk, G.I. and Sarkisyan, A.C., 1980. Mathematical models of circulations in ocean Novosibirsk Nauk, 341 pp. Mesinger, F. and Janjic, Z.I., 1985. Problems and numerical methods of the incorporation of mountains in the atmospheric model. Lectures in Applied Mathematics, Vol. 22: 81-118. Munk, W.H., 1950. On the wind-driven ocean circulation. J. Meteorol. 7:79-93. Oort, A.H. and Vonder Haar, T.H., 1976: On the observed annual cycle in the ocean-atmosphere heat balance over the Northern Hemisphere. J. Phys. Oceanogr., 6: 781-800. Philander, S.G.H. and Seigel, A.D., 1985. Simulation of E1 Nino of 1982-1983. In: J. Nihoul (Editor). Coupled Ocean-Atmosphere Models. Elsevier, Amsterdam, pp. 517-541. Philander, S.G.H., Hurlin, W.J. and Seigel, A.D., 1987. Simulation of the seasonal cycle of the tropical Pacific Ocean. J. Phys. Oceanogr., 17: 1986-2002. Sarkisyan, A.S., 1954. Calculation of the stationary wind-driven currents in an ocean. Izv. Akad. Nauk, USSR, Set. Geofiz., 6: 554-561. Sarkisyan, A.S., 1966. Theory and computation of oceanic currents. Gjdro-Meteo-Izdat., 123 pp. Taft, B.A., 1972. Characteristics of the flow of the Kuroshio

Z E N G ET AL.

south of Japan. In: H. Kuroshio, Stommel and K. Yoshida (Editors), Univ. Washington Press, 717 pp. (see Chap. 6). Washington, W.M., Semtner, A.J., MeehL G.A., Knight, D.J. and Mayer, T.A., 1980. A general circulation experiment with a coupled atmosphere, ocean, and sea ice model. J. Phys. Oceanogr., 10: 1887-1908. Wyrtki, K., 1974. The dynamic topography of the Pacific Ocean and its fluctuations. Rep. H-16-74-S. Hawaii Inst. Geophys., 19 pp. Zeng, Q.C., 1979. Physical-Mathematical Basis of Numerical Weather Prediction. Vol. 1, Science Press, Beijing, 543 pp. Zeng, Q.C., 1983. Some numerical ocean-atmosphere coupling models. Paper presented at the First Int. Symp. Integrated Global Ocean Monitoring. Tallinn, USSR (Oct. 2-10). Zeng, Q.C., Yuan, C.G., Zhang, X.H., Liang, X.Z. and Bao, N., 1987. A global gridpoint general circulation model. In: Collection of Papers Presented at the W M O / I U G G NWP Symposium, Tokyo, 4-8 August 1986, 421-430. Zeng, Q.C. and Zhang, X.H., 1982. Perfectly energy-conservative time-space finite-difference schemes and the consistent split method to solve the dynamical equations of compressible fluid. Sci. Sin. Set. B, XXV (8): 866-880. Zeng, Q.C. and Zhang, X.H., 1987. Available energy conservating schemes for primitive eqautions of spherical baroclinic atmosphere. Chin. J. Atmos. Sci., 11 (2): 121-142. Zhang, R.H., 1989. The numerical simulation studies for oceanic circulation in the Pacific Basin. Diss. Inst. Atmos. Phys., Chin. Acad. Sci., 184 pp. Zhang, X.H. and Liang, X.Z., 1989. A numerical world ocean general circulation model. Adv. Atmos. Sci., 6(1): 44-61. Zhang, X.H. and Zeng, Q.C., 1988. Computational design of world ocean general circulation model. Chin. J. Atmos. Sci., Spec. Iss., pp. 149-165.