NCAR/TN-136+STR On the Interaction Between Mesoscale ...228/datastream/… · (i) the interaction...

(NCAR/TN-136+STRfNCAR TECHNICAL NOTE

April 1980 e /

On the Interaction betweenMesoscale Eddies and Topography

Bernard Durney ,i 0

ATMOSPHERIC ANALYSIS AND PREDICTION DIVISION

NATIONAL CENTER FOR ATMOSPHERIC RESEARCHBOULDER, COLORADO

Preface

In this study we describe the results of a variety of numerical experi-

ments performed with the help of Bretherton and Karweit's (1975) mid-ocean

model. Two layers in the vertical direction were included and the horizon-

tal dimension of the system was chosen to be 3091 km x 3091 km.

The following problems were considered:

(i) the interaction of a large-scale flow with a mesoscale topography.

The large-scale flow was assumed to satisfy the relation H1v1 + H2v2 = 0

(strictly valid in the limit of a uniform flow; Hr and vr are here the layer

thickness and north-south velocity, respectively), The object of these

experiments was to study the generation of eddies resulting from the inter-

action of the large-scale flow with the mesoscale topography. For south-north

large-scale flows there is a vigorous generation of eddies at the expense

of the potential energy of the large-scale flow. For east-west flows the

eddy generation is weak.

(ii) eddy-eddy interactions in the absence of topography.

As suggested by Rhines (1975, 1977) the energy cascades toward larger

scales. The energy becomes increasingly concentrated in one wave number

and this wave number decreases with time until, broadly speaking, the eddies

cease to be nonlinear. The energy spectrum can, however, show a marked

evolution in time even for weak eddies: the eddy energy becomes concen-

trated in wave vectors with low values of kx; that is, the flow tends to

become more zonal as a consequence of the s-effect.

(iii) eddy-eddy interactions in the presence of a mesoscale topographr.

The topography inhibits the tendency of the system to become baro-

tropic as well as the cascade of energy towards larger scales. For "large"

topographies the energy spectrum is considerably broadened. The actual

oceanographic topographies are of a magnitude likely to have a pronounced

effect on the eddy flow.

The numerical calculations presented in this study were performed

during the debugging period of the CRAY-1.

Acknowl edgements

The author is grateful to Drs. F. Bretherton and J. McWilliams for

discussions and comments and to Dr. S. Jackson for converting Bretherton

and Karweit's model to the CRAY-1 computer.

Table of Contents

Abstract ix

1. Description of the numerical model and initialconditions 1

(1) Topography 2(2) Initial Eddies 3(3) Initial Mean Flow 4

2. Description of the output and numerical experiments 5

3. Results 9

References 19

Figure captions 21

Figures 25

Abstract

Numerical experiments performed with the help of Bretherton and Karweit's

mid-ocean model are described. Two layers in the vertical direction were

included and the horizontal dimension of the system was chosen to be 3091 km

x 3091 km. The aim of these numerical experiments is the study of:

a) The interaction of a large-scale flow with a mesoscale topography;

b) Eddy-eddy interactions in the absence of topography; and

c) Eddy-eddy interactions in the presence of a mesoscale topography.

The flow was divided into a large-scale flow (small wave numbers) and

eddy flow. The kinetic energy and potential energy as well as the stream

functions were computed separately for the large-scale and eddy flow. This

makes it possible to study the tendency of the flow towards zonality

and the influence of a topography on the cascade of energy towards larger

scales. Perspective surfaces of the energy as a function of wave vector

(kx, ky) were plotted at appropriate time intervals as well as the energy

as a function of wave number k(= (kx' ykyas a function of wave number k(= (k ' + k )2).x y

1. Description of the numerical model and initial conditions.

For a two-layer model and in the quasigeostrophic approximation,

the potential vorticity for layer r(r = 1,2) can be written

DQr/Dt + Hr / x = 0 (la)

r r r r r r 22 2Qr = 6 rfhT4 ; qr = H v2 + (-1) r F(- 2) . (ib)

In Eq. (Ib), F = f2/g,(g (p2 - p)/p), Hr is the layer thickness, a2 is the2 rr

Kronecker delta and hT designates the height of the topographic features.

The remaining notation is standard. Equations (la) were integrated in time

with appropriate initial conditions and with the following values of Hr, g'

'2 3 s S 2 4 1H 1 km H k m; H 24 g'= 2 cx 10 3 km 1d 1; F f2/g= 2.644 10-4 km-11 2 4 n '

The horizontal dimensions of the system (L) were chosen as follows:

K = (F/H1) /8 (= 2.03 10- 3 km 1); L = 27/K = 3091 km . (3)

Equation (3) shows that L is about eight times 27Rd where Rd is the Rossby

radius of deformation. The model used to integrate Eqs. (1) was Bretherton

and Karweit's multi-layer model of the open ocean. This model (designated

hereafter as the BK model) has been described elsewhere (Bretherton and Karweit,

1975) and only the following points need to be mentioned here. The BK model is

quasigeostrophic, allowing for a maximum of six vertical layers. Quantities

are assumed to be periodic in x and y with a period L. The rectangular domain

of size L is divided into an N x N grid where the stream functions are defined

(for n,m = 1,2...N) by their Fourier series

N2r N/2 2:ri(nk + mk )/Np (n,m) = , . r(k ,k )e x Y + cc (4)

k= -N/2+1 ky= T/2+1 xx y

In the present study k and k always denote dimensionless wave numbers; the

x yusual wave vector is given by (kx,ky)K, where K is defined in Eq. (3). A

similar expansion holds true for the potential vorticity, which is integrated

forward in time with the help of fast Fourier transforms since the model is

fully nonlinear. The present calculations were performed with no bottom

viscosity and a small lateral viscosity which is introduced to control the

cascade of vorticity to large wave numbers; the value of N was chosen to be 96.

The BK model was integrated for several topographies and for a variety

of initial conditions corresponding to different eddy and mean flow energies.

These were chosen as follows:

(1) Topography

One run was made with a topography given by

h = 0.5 e-(x/100 km)km (5)T

which corresponds to a south-north ridge with a half width of 100 km and a

height of 0.5 km at x = 0. The topography for the other runs was defined as

follows: if

hT() = fh() e1i x d (6)

then from an analysis of abyssal hill profiles, Bell (1975) finds that

L2A A* L_< h () h (5 ) > P(() ;

~ ~- (2f)4

2 3/2P(Z) = 5 10-3ykm/(2+ T) ;T

-1 (7)T =0.1 km . (7)

In Eq. (7), Y is a dimensionless numerical factor, t is in km 1 and P(z)

is in km4 . In Bell's expression for P(Z), Y equals one. The numerical

calculations were performed with several values of Y; also P(a) was set

equal to zero for small P's (no mean topography) as well as for large V's

(for numerical convenience), i.e.,

P(z) = 0 if _1 4K or if a > 18K (8)

where K = 27r/L. A realization of a topography with < h() * (a) > satisfying

Eqs. (7) and (8) was obtained as follows.

In a suitable annulus centered at Q, h(Q) was chosen so that

A.h(Q) = aP(k) 2 (9)

where a is a complex random variable with |a| gaussianly distributed; the

real and imaginary part of a are chosen uncorrelated (< Real (a) Im(a) > =0)2

and furthermore <|a1 > 1.

It can be readily shown from Eqs. (7) (with Y = 1) and (8), that

h2 = fh2dxdy = 21m. If P(a) is not cut at small and large V's, then

it follows from Eq. (7) that h = 89m. Small scales, therfore, play an

important role in the ocean topography. Our cutoff at large I's, 18K (= 0.036 km- 1)

is smaller than PT in Eq. (7); therefore, the spectrum <R(a) *(Q) > is

essentially flat for 4K< a < 18K.

(2) Initial Eddies

The eddy energy as a function of wave number, E(k) is given in Fig. 1.

Only the ratio of numbers in the vertical scale aremeaningful. The source for

such an energy distribution was the initial eddy stream function used by

Bretherton and Karweit (1975) in their numerical simulation of the MODE area

data. With a knowledge of E(k) an actual realization of the stream function

was obtained by a random process similar to that described for the topography.

Care was taken to use different random number generators so as not to introduce

correlations between the eddies and topography. Figures 2 and 3 are perspective

surfaces of E(k ,k ) 2 where E(k ,k ) is the energy in wave numbers (k ,k ).xy xy xySince one of our aims is to study the energy cascade towards the largest

scales of the system no energy was put initially in wave numbers (k ,k ) ifx yk ' 4 and ky 4 as is apparent from Figs. 2 and 3. This small wave numberx y

flow was defined as the mean flow, therefore,

Mean flow: Ikx < 4 and Ik I 4 . (10)

For wave numbers outside the domain defined in Eq. (10), the flow was called

eddy flow.

(3) Initial Mean Flow

Some runs were made with a large-scale flow initially present. This

mean flow was taken to be purely meridional or zonal. That is, at t = 0

= ar cos 2Trx/L (lla)

-r =5 cos 27ry/L . (lib)r r

The only non-vanishing components of the meridional flow, for example, are

the wave numbers (1,0) and (-1,0).

2. Description of the output and numerical experiments.

Two important quantities which allow an easy visualization of the

evolution of the large-scale and eddy flows are the kinetic and potential

energy contained in the Fourier mode (k ,k ). Let the total energy per unit

surface be defined by -

E 2S [H1 iu dxdy + H2 f2 dxdy + Ff )2 dx dy (12)

where S is the surface of integration. It can be readily shown that the kinetic

and potential energy in the mode (k ,ky) are respectively given by (r = 1,2)x y

E(k ky) = K2H(k 2 + k2) (kxky)2 (13a)r x y rx r xy

P(k ky) 2 Fji(kxky) - (kxk 12 (13b)

The total energy, E, is then the sum of E (k ,k ) and P(k ,k ) over layers oner x y x y

and two and over wave numbers:

E r= , ((k,ky) + P(kkYk k r x y x y

Perspective surfaces of E (k ,k )2 and P(k ,k )2 (cf. Figs. 2 and 19, for example)r x y x yclearly show in which wave numbers the kinetic and potential energy are con-

centrated.

The energy as a function of wave number, E (k ), was defined as the sum ofr 0

all Er(kxky) such that k - 0.5 < k < ko + 0.5, where k = (k2 + k2 )½.

Therefore,

E (ko) = E(k );k r yx y

the sum is over all k , k such that (14)

05 ( + k k + 05 (k is an integer)k - 0.5 -< (k2 + k 2)½ < k + 0.5 (k is an integer)

x y o o

It should be noticed that E (ko ) is only defined for integer values of ko.

The equations for r(k ,ky) (r = 1,2) are:

2= (yrr'y' 2 x'at [ K (k 2 + k2 Hrr (kX ,k + (-1)r F 1(kXk) - 2(kxky]

+ i K kxHr r (k xky) + Jr (Kx'ky =0 (15)

where r(k ,k ) is the Fourier transform of J(Qr, r ). It is instructive

in some cases, in particular for zonal flows, to know the behavior with timeA ii

of J (k ky). Therefore, if TJ(kx k ) = Ir(kk )le r (16)r x y r X y r x y

IJr(kx ky)I and r were sometimes plotted as a function of time for k = 0,r x y r X

k = 1, corresponding to a zonal flow with the lowest wave number.

The experiments discussed here are tabulated in Table 1. In this table,

blanks denote zeros. With the exception of experiment 3 which was run with a

topography given by Eq.(5) , all experiments were performed with a topo-

graphic realization satisfying Eqs. (7) and (8). In experiment 1, the mean

flow was purely meridional at t=O and purely zonal in experiments 2 and 3.

Since K = 27r/L, = 5 2/K corresponds to a zonal flow in layer 1 having a

-1root mean square velocity of 5 kmd . With the exception of a multiplicative

factor, the initial eddy stream functions i'(t=O) in layers 1 and 2 were the

same for experiments 3 to 1'. The energy as a function of wave number and

perspective surfaces of E(k ,k )2 corresponding to p' (t=O) have been plottedx y

in Figs. 1, 2, and 3. Figure 4 shows contours of contrast i'. This figure

suggests the existence of a south-north difference in i'. However, plots of

(' (x,y0 ) and i' (x ,y) as a function of x (for several values of yo) and as

a function of y (for several values of xo) reveal that i' (t=O) does not contain

a meaningful north-south statistical difference. To obtain the initial eddy

root mean square velocities in layers 1 and 2, tabulated in Table 1, p' was

TABLE I

Experiments discussed in this study

Experiment Root Mean Square Topography Initial mean flow. Values Initial Eddy Flow: root Mean Square Vel.Number (obtained by adjusting Y in of a, 6 in Eqs. (11) r = 1 r = 2

Eq. (7))r= 1 r = 2

6 i = 5'V2/K

= -a1/4

=-6 1/4

6 6.95 km

13.9 km

6.95 km

3.475km

13.9 km

6.95 km

1.74 km

6.95 km

10.4 km

6.95 km

3.475km

1.74 km

10.4 km

3.475km

6.95 km

0.87 km

d- 1d-1

dd-1d-1

d- 1d- 1

d- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~

multiplied by appropriate real factors.

The numerical integration of a two-layer model with a grid of 96 x 96

points is time-consuming. This is the more so because the time interval

(At) has to be chosen quite small (at -0.2 days). The difficulty does not

lie in the usual Courant-Friedrichs-Lewy stability criterion which is amply

satisfied but in the fact that the time integration scheme of the BK model

is not strictly energy-conserving. For At = 0.4 days, for example, energy

is fed into meridional flows with low wave numbers I( ±1,0) in particular)and after '800 time steps, the stream function shows a noticeable and spurious

meridional flow. This problem is linked to the dimensions of the system; this

is the reason why it did not arise in previous numerical simulations with the

BK model where L " 1000 km.

3. Results

Experiment 1 (Fig. 5 to Fig. 12)

As Table 1 shows, the mean flow is purely meridional and satisfies

the relation H1a /ax + H2 a 92/a x = 0 . (17)

This relation must be strictly satisfied in the limit of a uniform flow

(Bretherton and Karweit, 1975, Eq. (10)). In the present case, the meridional

flow is given by Eq. (lla) and, of course, no contradiction arises if Eq. (17)

is not satisfied. It should be noticed, however, that if H1 i 1 + H2 2 ~ 0,

rapid oscillations are present since the Rossby frequency of the barotropic

component of the mean flow is given by m = L/27r~1 day.

In Fig. 5, E2(k) is plotted for t = 120 days. At t = 0 no eddies are

present and all the energy is in the mean flow. Since a2 = K (K= 2/L),

it follows from Eq. (lla) that initially E2(1) = H2(5/4)2 = 3.125 km3d 2.

Figure 5, therefore, shows that the meridional flow has decreased somewhat

and that there is a vigorous generation of eddies as a consequence of the

interaction of the mean flow with the topography. Figure 6 is a perspective

surface of Ed (kxky) at t =120 days showing clearly the meridional flow at

wave number (1,0) and the eddies. The eddy root mean square velocities in

layers one and two are plotted in Figs. 7a and 7b as a function of time.

Essentially all this eddy energy comes from the potential energy of the mean

flow as Fig. 8 indicates. Figure 8 is a plot of P(1,0)½ (defined in Eq. (13b))

12 3/2-1versus time; P(1,0)½ is in km/ d . At t=0 the energy of the system is3 -2 3 2

1265 km3d- and almost all of it is in the form of potential energy (-1250 km3d-).

In Fig. 9, E1(1,0)½ is plotted versus time. It can be readily seen that if

Eq. (17) is valid, i.e., if 2(1,0) =- = 1(1,0)/4, then the ratio

P(1,0)/E1 (1,0] = 10. A comparison of Figs. 8 and 9 shows that this is indeed[ ~~~1.

the case: the barotropic component of the meridional flow remains small

as time increases. Figure 10 shows the mean kinetic energy versus time for

M = 2 and Fig. 11 is a plot of E2(k) for t = 440 days. These figures can be

readily interpreted: the system is baroclinically unstable and the topography

helps to convert the potential energy of the mean flow into eddy energy.

Energetic eddies are generated at wave number ^12, and due to the nonlinear

interactions there is a cascade of energy towards larger scales as time

increases. Figure 11 shows a considerable amount of kinetic energy

("22 km3d- 2) in wave number 1 (larger, in fact, than the kinetic energy of

the initial meridional flow). This kinetic energy belongs to a zonal flow,

as is apparent from Fig. 12, which is a plot of contour lines of constant .2'

In opposition to the initial meridional flow which was strongly baroclinic,

this zonal flow is mainly barotropic (P(0,1)^0.16 km3d 2 at t = 480 days).

As Table 1 indicates, the difference between experiments 1 and 2 is that

in Experiment 2 the flow is zonal instead of meridional. The initial potential

energy is the same in both bases. In Fig. 13, E2(k) is plotted for t = 120 days.

A comparison with Fig. 5 reveals the following striking differences: there is

not, in this case, a vigorous generation of eddies and the zonal flow in layer

2 has weakened considerably. Figures 14a and 14b are a plot of E2(0,1)2

and E1(0,1) 2 versus time. The root mean square velocities of the zonal flows

in layers 1 and 2 are related to E2(0,1)½ and E1(0,1) by

u1 = 2 E.1 (0,1) ; U2 = E2 (0,1) . (18)

Figures 14 and 15 show that the zonal flow increases in layer 1 and decreases

in layer 1 in such a way that the potential energy of the system remains

approximately constant (Fig. 15 is a plot of the square root of the potential

energy in wave number (0,1),P(0,1)½, as a function of time). As Fig. 16

shows, the total eddy energy remains small. The system evolves rapidly during

the first ^ 25 days and slowly thereafter, as is apparent from Fig. 17 where

the eddy root mean square velocity in layer 2 is plotted versus time.A

The behavior of J2(0,1) as a function of time is of interest. (T2(0,1)

is the Fourier transform corresponding to wave number (0,1) of the JacobianA ^

J(Q2,2)). Let J 2(0,1) = I 2(O,l)|e1,; in Figs. 18a and 18b 1J2(0,1)l and e

are plotted versus time. The numerical calculations show that

I|1(0,1)| <« |J2(0,1)I. Neglecting J1(0,1) we can write Eqs. (15) for 1(0,1)

and 2((0,1) as follows:

K2A A o^

K a^ = F at (2(0,1) - 1(0,1)) (19a)

2 t( 19b)H2 K a 2( 1)=-F a (~2(0,1) - 1(0,1)) + J2(0,1) (19b)

2 at 7t2 a__since K2 << F, Eq. (19X) shows that a Wl(0,1)/a t »> a (2(0'1) - (0,1)).

Replacing in the left-hand side of Eq. (19b), ?2 by 1 we obtain

(0,1)= J(0,1) ( (1) )- ( 1(0,1))/5F (20)

Since ¢2(0,1) - 1 (0,1)-25I 2/8K it is readily seen that, as Fig. 15 shows,

the variations in the potential energy are small.

In this experiment the topography was given by Eq. (5). At t = 0, a

barotropic zonal flow and eddies were present with root mean square velocities

equal to 5 km d 1 and 6.95 km d 1 , respectively. Figure 19 is a perspective

surface of the potential energy P2(kx k ) (t = 10 d) and Fig. 20 is the

vertical interface displacement at t = 20 d. Figure 21 is a plot of contour

lines of constant values of the stream function for r = 2, t = 20 d. The

initial zonal flow is readily apparent. Figures 22 and 23 give the total

mean energy and eddy energy versus time. There is a rapid decrease of the

total mean energy (giving rise to an increase in the eddy energy) and a slow

increase thereafter as a consequence of the energy cascade towards larger

scales. Figures 24 and 25 show the stream function (r = 2) and the interface

displacement at 300 days.

Experiment 4 (Fig. 26 to Figs. 39)

Experiment 6 (Figs. 47 to Fig. 52)

The aim of these numerical simulations is to study the transfer of energy

towards larger scales for different values of the initial eddy energy. Both

barotropic and baroclinic systems were considered. Since the figures speak

for themselves, we offer here only the following remarks. The energy gets

increasingly concentrated in one wave number and this wave number decreases

with time. Since these processes are a consequence of nonlinear interactions

they proceed more slowly as the eddy energy decreases. This is also apparent

from the plots of perspective surfaces of E2 (kx,k ): Fig. 32 shows a con-

siderable amount of energy in the lowest wave numbers whereas this is not the

case in Fig. 48d.

A measure of the nonlinearity of the eddies is the ratio R1 of the root

mean square velocity, u, divided by the phase speed corresponding to the

dominant scale, that is, R1 = 2u (kK)2/3 . (Following our convention, we

measure wave vectors in units of K; k is dimensionless). It has been

advocated by Rhines (1975) that the evolution to larger scales ceases at a

wave number k such that R1= 1, i.e., k = (3 /2uK 2)½. If we take

u = 10 km d , corresponding to Experiment 4, we obtain k 5 in good

agreement with Fig. 27. For u = 3.475 km d-1 (Experiment 6), k = 8.5

(cf. Fig. 47).

Plots of the stream function also show the increase in eddy scale with

time (cf., for example, Figs. 33a, b). Figures 33b and 49 are a plot of the

total stream functions, at t = 600 d, for Experiments 4 and 6 and Figs. 34

and 50 are a plot of the mean stream function (t = 600 d) for the same

experiments. In the case of energetic eddies (Experiment 4) p2 and 2 are

remarkably similar: in fact, as Figs. 35 and 36 show, the mean flow is more

energetic at t = 600 days than the eddy flow (the potential energy of both

the mean and eddy flow are small). In the case of weak eddies (Experiment 6)

there is little energy in the mean flow at 600 d (cf. Figs. 51 and 52 and

Fig. 49 (for i2) and Fig. 50 (for i2 )). It is important, however, to notice

(cf. Figs. 48a and 48d) that the energy spectrum shows a pronounced evolution

with time. It is only perspective surfaces of the energy spectrum that show

this evolution clearly. It is apparent from Fig. 47 that, as time increases,

E2(k) narrows and the value of k for which E2(k) is maximum, decreases somewhat

But this gives little insight into the main evolution of EM(k ,k ) with time,Xynamely a concentration of energy in wave vectors with low values of k . A

pronounced evolution of the energy spectrum is again revealed by Experiment 7,

which is baroclinic with larger energies in the upper layer; a comparison of

Fig. 54a with Figs. 54b and 54c reveals the more rapid evolution of the energy

spectrum in the upper layer when compared with the energy spectrum in the

lower layer. However, as is clear from Fig. 54d, in 300 days the energy

spectrum in the lower layer has evolved considerably (cf. Fig. 54a) in spite

of the low energies in this layer. The flow has become markedly more

zonal.

Figures 45 and 46 are a plot of |J2(0,1)1 and E (0,1) versus time for

Experiment 5; J2(0,1) determines A2(0,1) according to Eq. (15). The behavior

of IJ2(0,1)1 and E2 (0,1) are typical; the Jacobian always shows large time

oscillations as, in general, does the energy in a given wave vector.

Experiment 10 (Figs. 67 to Figs. 75)

In these numerical simulations a topography of increasing strength was

added to Experiment 4. Comparison of these experiments reveal the following

facts: the topography broadens the energy spectrum (E (k)) and inhibits the

cadcade of energy towards larger scales.

The nonlinearity ratio (R1) was defined above; we define R2 and R3 such

R = 2u(kK)2/R ; R fkThT/H2k Ku; R = fhTkTK/B H2 (21)

where kT and hT are typical values of the topographic wave number and height,

respectively; kT (as well as k) are dimensionless. The choice of k for the

eddy flow is, in general, straightforward: it is the k corresponding to the

dominant scale. The topography, however, does not have a dominant scale. It

appears natural to define hTand kT byT T

= hT ; KkThT = (h)2> + < (hTy)> . (22)h h.: · I; KkT h <(h.Tx)Ty) 2 -]" (22)

where hTx and hTy denote the derivatives of hT with respect to x and y.

The value of kT can be readily estimated from Eqs. (6), (7), and (8). It is

found that kT - 13. The ratios R2 and R3 are a measure of the strength of

the topographic term compared with the nonlinear and S-term, respectively.

The values of R1, R2, and R3 for Experiments 8, 9, 10, and 11 are given in

Table II.

TABLE II

Values of R1, R2, R3 For

Experiments 8, 9, 10, and 11

EXPERIMENT R1 R2 R

8 4 0.2 0.4

9 4 0.4 0.810 4 0.9 1.711 4 1.3 2.6

The values of Table II were obtained with u = 10 km d 1 and an eddy wave

number of 10 (cf. Fig. 1). In Experiment 8, the topographic term is small

and there is a cascade of energy towards larger scales which ceases at a

wave number k.-5. For this wave number, R1= 1 and R2= 0.8. A comparison

of Figs. 27 and 58 gives an idea of the effect of the topography on these

eddies. As R2 (or R3) increases, the energy spectrum broadens. Figures 67

and 76 show that there is a significant decrease in the energy for wave

numbers larger than 18, which is the cut-off wave number for the topography.

In this context, it should be remembered that Bretherton and Haidvogel (1976)

and Herring (1977) have argued (for one-layer models) that the Fourier trans-

forms for the steady-state stream function and topography should be relatedA A 2 2K2

by i = h/(p2 + k K) where u is a constant.

As expected, the presence of a topography increases the eddy potential

energy considerably (Figs. 37a, 73); if hT= 0, the flow tends to become

baratropic as time increases; this process is strongly hindered by the

topography.

Experiment 13 (Figs. 87 to Fig. 97a)

The values of R1, R2, and R3 are given in Table III.

TABLE III

Values of R1, R2, R3 For

Experiments 12, 13, and 14

EXPERIMENT R1 R2 R3

12 1.4 1.6 1

13 2.8 1.5 2

14 0.35 12 2

In Experiments 12 and 13 we took u = 3.475 km d 1 and 0.87 km d

respectively, to obtain the values of Table III. In Experiments 12, 13, and

14, the eddies were less energetic than in the cases discussed above. Figure

86 shows an increase in the eddy root mean square velocity of layer two. This

is not a topographic effect: if hT= O, < u2> also increases initially with

time. Figures 95 and 96 are typical examples of the behavior of E (0,1) and

|J2(0,1)f with time. The energy in wave numbers (kx,ky) strongly oscillates,

not only for small values of (kx,ky), as is apparent from Fig. 97a, which is

a plot of Ei(8,8) versus time for Experiment 13. These oscillations are not

a topographic effect: in Fig. 97b, E2(8,8) is plotted versus time for a

case with the same initial eddy values as Experiment 13, but with no topo-

graphy (hT= 0); the energy in wave numbers (8,8) decreases with time due to

the transfer of energy towards larger scales.

It is clear from Table III that Experiment 14 is topographically

dominated. Figures 99 show that in this case there is not a sharp

decrease of the eddy energy for wave numbers larger than the topographic

cutoff (18).

References

Bell, T. H.: J. Geophys. Res., 80, 320 (1975).

Bretherton, F. P. and M. Karweit in Numerical Models of Ocean Circulation.

Pub. National Academy of Sciences, 237 (1975).

Bretherton, F. P. and D. Haidvogel: J. Fluid Mech., 78, 129 (1976).

Herring, J. R.: J. Atmos. Sci., 34, 1731 (1977)

Rhines, P. B.: J. Fluid Mech., 69, 417 (1975).

Rhines, P.B. in The Sea, Vol. 6, p. 189 (E.D. Goldberg, I.N. McCave, J.J.

O'Brien, and J. H. Steele, editors). J. Wiley and Son, New York

(1977).

Figure Captions

The mean flow is defined in Eq. (10); it contains all wave vectors

(kxky) such that Ikxlj 4 and |ky 4. The eddy flow contains the wave

vectors (kx, ky) outside the above domain. The units of length and time are

kilometers and days. The exception is the interface displacement, which is

measured in meters. The experiment number is written in the lower left corner

of each figure.

A list follows of the quantities which have been plotted and their

corresponding figures.

1) The energy as a function of wave number-Figures 1; 5; 11; 13; 26; 27; 40;

47a,b; 53; 58; 64; 67a,b; 76; 78a,b,c,d; 87a,b,c,d,e,f; 98a,b; 99a,b,c.

The energy as a function of wave number, EM(k), is defined in Eqs. (13a)

and (14). The units of EM(k) are km d-2 In the top of these figures

the following labels appear: ENER(K)/t/M or ENER(K)/M/; M = 1,2 denotes

the upper and lower layer, respectively, and t is the time in days. If

the time does not appear in the top label it is written near the corresponding

curve.

2) Perspective surfaces of the square root of the energy in wave numbers k ,kx-y

Figures 2; 3; 6; 28; 29; 30; 31; 32; 41; 48a,b,c,d; 54a,b,c,d; 59a,b; 65;

68a,b,c; 77; 79a,b,c,d; 88a,b,c,d.

EM(kxk ) is defined in Eq. (13a). What is plotted are perspective surfaces

of E(kx,ky) (km3/2 d1 ) for k and k in the range 0 to 48. (The exception

is Fig. 3, where k varies from -47 to 48.) The axes are defined inx

Fig. 2. Each figure contains the time (in days) and the layer number.

3) Perspective surfaces of the square root of the potential energy in wave

numbers k , k Figures 19; 80.x-y

P(k ,k ) is defined in Eq. (13b); perspective surfaces of P2(k ,k )3/2 -1(km3/2 d ) are plotted for k and k in the range 0 to 48. The axesx y

are defined in Fig. 2 and each figure contains the time in days.

4) Contour lines of constant values of the stream function-Figures 4; 21

(contour from -320 to 320, contour interval of 40); 24 (-220 to 240,

c.i. of 20); 33a (-200 to 200, c.i. of 40); 33b (-360 to 330, c.i. of

60); 42 (-180 to 160, c.i. of 20); 49 (-72 to 96, c.i. of 8); 57 (-30

to 30, c.i. of 6); 60 (-280 to 350, c.i. of 70); 69 (-200 to 200, c.i.

of 40); 81a (-70 to 80, c.i. of 10); 81b (-60 to 60, c.i. of 10); 89a

(-90 to 120, c.i. of 10); 89b (-110 to 100, c.i. of 10). In these

figures *M/10 is plotted (pM is in km2 d-l). The time (in days) and

the value of M are on the top of each figure.

5) Contour lines of constant values of the mean stream function- Figures 12

(contour from -4200 to 6000, contour interval of 600); 34 (-3000 to 3300,

c.i. of 300); 50 (-240 to 160, c.i. of 20); 61 (-3000 to 3000, c.i. of

300); 70a (-1200 to 1100, c.i. of 100); 70b (-1300 to 1100, c.i. of 100).

In these figures ~M is plotted (M is in km2 dl1). The time (in days)

and the value of M are on the top of each figure.

6) Contour lines of constant values of the interface displacement- Figure 20

(contour from -63 to 56, contour interval of 70); 25 (-100 to 70, c.i. of

10). The interface displacement is in meters. The time) in days is on

the top of each figure.

7) Total mean energy versus time - Figures 22; 35a,b; 43; 51; 55; 62; 71; 82; 90.

In these figures the value of E (km3 d 2 ) is plotted versus time; E is

defined as in Eq. (12) but with 1PM and uM in place of 1M and uM (we recall

that the mean flow is defined in Eq. (10)).

8) Total eddy energy versus time - Figures 16; 23; 36a,b; 44; 52; 56; 63;

72; 83; 91. In these figures the value of E-E (km3d 2) is plotted versus

time; E is defined in Eq. (12) and E has been defined above.

9) Eddy root mean square velocity versus time - Figures 7a,b; 17; 38a,b;

39a,b; 74a,b; 75a,b; 85; 86; 93a,b; 94a,b. In these figures

[fU2 dxdy/S (km d ) is plotted versus time; S is the integration

surface and u' is the eddy velocity (we recall that the eddy flow has

been defined following Eq. (10)). The value of M is at the top of each

figure.

10) Eddy potential energy versus time - Figures 37a,b; 66; 73; 84; 92. In these

figures F/2Sj (p - i )2 dxdy (km3 d 2) is plotted versus time; i and A

are the eddy stream function.

11) Mean kinetic energy versus time - Figure 10. In this figure

-2 -2H2/2Sf u2 dxdy (km3 d 2 ) is plotted versus time.

12) E(l1,O) or E"i(0,1) versus time - Figures 9; 14a,b; 46; 95. EM(kx,ky) has

been defined in Eq. (13a) (r=M). In these figures E (1,0) or EM (0,1)

(as indicated on the top of each figure) is plotted versus time; the units

3/2 -1are km 2 d . These figures give the kinetic energy in purely meridional

or zonal flows with the lowest wave numbers.

13) P2(1O) or P2(0,1) versus time - Figures 8; 15. P(k ,k ) has been definedx y

in Eq. (13b). In these figures, P2(1,0) or P2(0,1) (as indicated on the

top of each figure) is plotted versus time; the units are km3/2 d1 .

14) JM(0,1) versus time - Figures 18a,b; 45; 96. JM(kxky) is the Fourier

transform of the Jacobian eJQM We write J(01) = IJ(0,1) etransform of the Jacobian J(QM,ipM). We write J3M(0,1) : 3JM(0,1) e

IJM(0,1)l (km d-2) is plotted in Figs. 18a, 45, and 96, and e is plotted

in Fig. 18b.

15) E (8,8) versus time - Figures 97a,b. In these figures, E (kx,ky)

(K = k= 8) is plotted versus time; the units are km d .x y

ENER (K)

0 5 10 15 20 25 30 35 40 45 50

Fig. 1

Fig. 2

Fig. 3

Fig. 4

5.5 . i'i' i'

0 ,-,1 I , I I II I I I I I ! I I I I i I . .

0 4 8 12 16 20 24 28 32 36 40 44

Fig. 5

t=120d:; M=2

Fig. 6

ROOT M SQ (EDDY) Exp I20

(a)18 -

VELI ,16

16 -(b)

VEL214

0 60 120 180 240 300 360 420 48<

Fig. 7

(POT EN(I.O))/2 Exp I

0 60 120 180 240 300 360 420 480

Fig. 8

KA (I,0)/2

0 60 120 180 240 300 360 420 480

Fig. 9

M=I Exp2.6

MEAN KIN EN M=2 Exp I280 ......... i'" ' ,) , ,,ir

00 60 120 180 240 300 360 420 480

Fig. 10

ENER (K) /440/2 Exp I150

130120I1O

807060504030 -20IO

00 4 8 12 16 20 24 28 32 36 40 44

kFig. 11

t=440 d M=2 ~ Exp I

0, l- -- -................._...

:-2.. ... . .. ...., -,.!: , . 1, 1 .

....... .2

Fig. 12

o-- ,'~~~~~~~~~~~~

ENER(K) /120/20.140.13

0.11 I

0.100.0 A A lQ090.08

0.050.04

0.020.01

0 4 8 12 16 20

24 28 32 36 40 44

Fig. 13

KA (0,I) / 2

0 10 20 30 40 50 60

Fig. 14

M =I (a)

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

2.501.3

I TVVII IIIII IIIIV IIIII T I F I I I I I I I I I .I .I I Irrr r rr T T · I I r r - I ------

A- I---- I - -.-- L .I . II -I I. . .I... I ... ...I.............'·C-L- LI

TOT EDDY ENERGY

0 10 20 30 40

t(d)Fig. 15

(POT EN (0,I)) 1/2

50 60 70

0 10 20 30 40 50 60 70t(d)

Fig. 16

Exp 24.0

24.995

24.985

24.975

24.965

24.955

24.945

24.935

24 9 f2L- T.%, -.

ROOT M SQ VEL2

200 300 400

Fig. 17

500 600

(EDDY) Exp 21.35

0.350 100 700

M=2 Exp 2

0 10 20 30 40 50 60 70

Fig. 18

0.00048

0.00042

0.00036

0.00030

0.00024

0.00018

0.00012

0.00006

Exp3t=1Od

Fig. 19

INTERFACE DISPLACEMENT

Fig. 20

t=20d Exp 3

t = 20d M = 2

Fig. 21

TOTAL M ENERGY Exp 365

200 40 80 120 160 200 240 280 320

t (d)Fig. 22

TOTAL EDDY ENERGY Exp 3165

1200 40 80 120 160 200 240 280 320

t (d)Fig. 23

t = 300d M = 2 Exp 3

Fig. 24

t = 300d Int. Displacement Exp 3

Fig. 25

ENERGY (K)/2/55

I) , I * l I0 5 10 15 20 0 5 I0 15

Fig. 26

0 5 10 15

ENERGY (K)/2/ Exp 4

0 5 10 15 20 25 0 5 0 15 20

Fig. 27

t = 40d1;_ M=2

t = 160dk?& M=2

Fig. 28

Fig. 29

t = 300dI_ M=2

t = 440d~E, M=2

Fig. 30

Fig. 31

t = 620dIb M=2

Fig. 32

t = 20d M = 2 Exp 4

Fig. 33a

t 600 M = 2

V, Exp 4

Fig. 33b

t= 600

M = 2 Exp 4

Fig. 34

TOTAL M ENERGY Exp 4

0 80 120 160 200 240 280 320 I0 40 80 120 160 200 240 280 320

200 i | . | i .

190 - (b)

110 _-IO

90320 360 400 440 480 520 560 600 640

Fig. 35

TOTAL EDDY ENERGY Exp 4

0 40 80' 120 160 200 240 280 320

210 ,i I | I

90 I0 I 0 I I .320 360 400 440 480 520 560 600 640

Fig. 36

EPOT ENERGY13

0 40 80 1

8.0 r, I , * i ,,

20 160 200 240 280 320

4.4 I ' I- 1 -i ' 1 '320 360 400 440 480

520 560 600 640

Fig. 37

ROOT M SQ VEL I (EDDY) Exp 414.0 * | |

13.5 (a)

9.50 40 80 120 160 200 240 280 320

9.69.49.2 - (b)9.08.8-8.68.48.28.07.87.67.47.27.06.86.6

320 360 400 440. 480 520 560 600 640

Fig. 38

ROOT M SQ VEL 2 (EDDY) Exp 4

0 40 80 120 160 200 240 280 320

8.88.68.48.28.07.87.67.4

7.27.06.86.66.46.26.05.8 3 6 . I 4, . I , I .

320 360 400 440 480 520 560 600 640

Fig. 39

ENERGY (K)/2/ Exp 5

12 3 6

k9 2840 960

8 ' 24

~~~~~~5 - 16

0 00 4 8 12 16 20 24 0 4 8 12 16 20

Fig. 40

Fig. 41

t =960 dM=2

t=960d M=2 Exp 5~~~~~II)!IU J

t "' '~~~~~~,, ~q'q r':- <'....-· · r , ·o, ...--, ~ · ~ ,,•'

I~ ' ' ~ el / e I

, .,. ~,, ,4 ir~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -.,'O·: · r,,:,u..:::''::..'· r~~~~~~~~: ;' . . ... ::r '·a rrr~~~~ rr rr ·~:' .. . . .. ·

`--~~~Q, ....- . . ,......· · ' : .:.: ... .o..-'I" . '. . ..,...,.-"~~~~~~~~~~ '""- <:~:: .....-''::'' ... "4'i... .. (

,.../ ,,' ,' .

... o ,, ,,, .. ·

~· r, . . .. .,, ,.. ~. t-· · ~ ~~~~~~~~~ :-'---~/r ,· ' .I-._. .....

Fig. 42

TOTAL M ENERG Exp 538

18600 700 800 900 1000 1100 1200 1300

Fig. 43

TOT EDDY ENERGY Exp59292 M,,,M ... |, . ... |U. |.M.,.M,..',...|.....' ' . . '"'

76 \74

600 700 800 900 1000 1100 1200 1300

Fig. 44

0 100 200 300 400 500 600

Fig. 45

0 100 200 300 400 500 600

Fig. 46

28XI0- 6

24X 10 6

20XIO- 6

16xlO- 6

12XO1- 6

8X10-6

4x10-6

KA (0, O'12 M=2

ENER (K)/2/3.0

2.8 (a)

00 4 8 12 16 20

4.2 ,,,i 4.

4.0 (b) 4.3.8 4.3.6 3.3.4 33.2 33.3.02.82.6 440 2.

2.2.4 2.2.2

2.2.0 2.1.81.6 I.1.4 -1.2 I-1.0 I.0.8 0.0.6 0.0.4 -0

0.2 .0

0 4 8 12 16

0 4 8 12 16 20k

0 4 8 12 16 20 24

Fig. 47

t= 40dM=2

t= 160 dM=2

Fig. 48

t= 440 dM=2

t=1000dM=2

Fig. 48 (cont.)

t =600 d M=2 Exp6

Fig. 49

t=600 d M=2 Exp6

Fig. 50

TOTAL M ENERGY Exp 61.15

0.45650 750 850 950 1050 1150 1250

Fig. 51

TOT EDDY ENERGY Exp 6

26.4 -

25.8 I

650 750 850 950 1050 1150 1250

Fig. 52

ENER (K)/2/1.3

00 5 10 15 20 25 0 5 10 15

Fig. 53

Exp 71.0

t =40 dM=2

t =160de,, M=l

Fig. 54

t =160dM=2

t = 300 dM=2

Fig. 54 (cont.)

TOTAL M ENERG Exp 7

) 40 80 120 160 200 240 280 320

Fig. 55

0 40 80 120 160 200 240 280 320

Fig. 56

t=300d M=2 Exp

Fig. 57

ENER (K)/2/

0 5 10 15 20

0 5 10 15 20 25

Fig. 58

t =440 dM=2

t = 600 dM=2

Fig. 59

t =600d M=2 Exp8

Fig. 60

t=600d M=2 Exp 8

Fig. 61

112 I i

460 480 500 520 540 560 580 600 620 640

Fig. 62

160460 480 500 520 540 560 580 600 620 640

Fig. 63

I , i I .I I I , I I I

II I 1 I I I

i I I I I I I I I 1i I i I I I I .1 I

0 5 10 15 20 25

Fig. 64

ENER (K)/2/

I I - -,-I- _ -_ _

- - ---

Fig. 65

EPOT ENERGY Exp 9

0 20 40 60 80 100 120 140 160

Fig. 66

Exp 9t=320d

ENER (K) /40/2 Exp 1020 .

(a) I18-

10 5 15 5 0 35 40 45 50k

18 14(b) 13-

14 II180 ,0 -A 320

12 991 0 5 10 1 85 20

78- 6 -6

0 00 5 10 15 20 0 5 10 15 20

Fig. 67

t=160 d

P- M=2

t=320dM=l

t=320dM=2

Fig. 68

Exp 10

t=320d Exp 10

M=2 'I

Fig. 69

t=320d Exp 10

Fig. 70a

t=320d M=2 Exp 10M=2~

Fig. 70b

140 160 180 200 220 240 260 280 300 320

Fig. 71

TOT EDDY ENERGY274 - .

250 ' I I140 160 180 200

Exp 10

220 240 260 280 300 320

Fig. 72

EPOT ENERGY

0 20 40 60 80

Exp 10

100 120 140 160

Fig. 73

ROOT M SQ VEL I (Eddy) Exp 1014.0 , , , . , , , ,. -

f\ (a):13.5

10.0- *0 20 40 60 80 100 120 140 160

10.1 - (b)

8.9140 160 180 200 220 240 260 280 300 320

Fig. 74

ROOT M SQ VEL 2 (Eddy) Exp 10

0 20 40 60 80 100 120 140 160

10.02(b)

140 160 180 200 220 240 260 280 300 320

Fig. 75

ENER (K)/2/

; ] \ 6 - 120

0 5 10 15 20 25 0 5 10 15 20

Fig. 76

t=140d

wsn\ M=2

Fig. 77

Exp II

ENER (K) /40/2 Exp 12

0 4 8 12 16 20 24 28 32 36 40 44k

Fig. 78

ENER (K) /160/2

ENER (K)/300/10.9 m . .,'r .i i

0ENER (K)/300/2

O ,0 4 8 12 16 20 24 28 32 36 40 44

Fig. 78 (cont.)

t=40di_ M=2

t=160dN A M=2

Fig. 79

Exp 12

t=300d_ MM=l

t=320dA M=2

Fig. 79 (cont.)

Exp 12t=320d3a PE

Fig. 80

t=300d M=l I

Fig. 81a

Exp 12

t=300d Exp 12

Fig. 81b

TOTAL M ENERG2.2 -, , I i i

0 .- 0 .2 I ' I I , I I I .

0 40 80 120 160

Fig. 82

TOT EDDY ENERGY

Exp 12

200 240 280 320

Exp 12

40 80 120 160 200 240 280

Fig. 83

L~~~~~I I 1I I ~ I I 1

0 320a . . . . . . . . . . . . . . . . . . . . . .

I-- ,"I

EPOT ENERGY Exp 12

0 40 80 120 160 200 240 280 320

Fig. 84

ROOT M SQ VEL I (EDDY) Exp 12

0 40 80 120 160 200 240 280 320

Fig. 85

ROOT M SQ VEL 2 (EDDY)

Exp 12

80 120 160 200 240 280 320

Fig. 86

ENER (K) /40/2 Exp 13I I I 1' 1 , i , l l , l l , 1 , 1 1 .I

7.0 (a)

-' I :. I 1la I * Ir I * I, I, I *,Il * I * I I L I, I : , 1 1 ! I v I . I I ' '

ENER (K) /160/25.5 ,

(b)5.0

00 4 8 12 16 20 24 28 32 36 40 44

Fig. 87

ENER (K) /300/2 Exp 13

4.4 (c)

.2.8 -

ENER (K) /440/23.8 I T T r"--r-

(d)3.4

1.8 !1.0

0.2.0 4 8 12 16 20 24 28 32 36 40 44

Fig. 87 (cont.)

ENER (K) /600/1 Exp 13

0.8 I A (e)

0ENER(K)/600/2

3.2 , , ' I -- ' '

3.02.82.62.42.22.01.81.61.4

1.2I.O0.80.6

0.4 I-el. .A , It i- id id i

0 4 8 12 I16 za vZ e-o oe aP-u '-V -k

Fig. 87 (cont.)

t= 40d

y:. M=2

t= 160d

Fig. 88

Exp 13

t=300dM=2

t=440dA M=2

Fig. 88 (cont.)

Exp 13

t=440d M=l 4 Exp 13

Fig. 89a

t=440d

M=2 I Exp 13

Fig. 89b

TOTAL M ENERG Exp 137.0

0 40 80 120 160 200 240 280 320

t(d)Fig. 90

TOT EDDY ENERGY Exp 1398 T rr - --r ,-1- I I I I r I I

86I l I I I i I I

310 330 350 370 390 410 430 450 470

t (d)Fig. 91

EPOT ENERGY Exp 13

0 40 80 120 160 200 240 280 320

t(d)Fig. 92

ROOT M SQ VEL I (EDDY) Exp 13

0 40 80 120 160 200 240 280 320

4.52 \

310 330 350 370 390 410 430 450 470

Fig. 93

ROOT M SQ VEL 2 (EDDY)

Exp 13

0 40 80 120 160 200 240 280 320

6.22 b)

310 330 350 370 390 410 430 450 470

Fig. 94

KA(0,1)1/ 2 M=2 Exp 13

40 80 120 160 200 240 280 320

Fig. 95

0 40 80 120 160 200 240 280 320

Fig. 96

0.36 -

0.08 -

0.00024

0.00020

0.00016

0.00012

0.00008

0.00004

EKD"/ 2 (8,8)/M=2 Exp 13

0 40 80 120 160 200 240 280 320

Eddies as Exp 13. No topography

0 100 200 300 400 500 600

Fig. 97

ENER (K)/160/1

ENER (K)/300/I

0 4 8 12 16 20 24 28 32 36 40 44

Fig. 98

ENER (K) /60/2 Exp 14

ENER (K) /160/2

0 4 8 12 16 20 24 28 32 36 40 44

Fig. 99

ENER (K)/300/2 Exp 140.07

0, .0 4 8 12 16 20 24 28 32 36 40 44

Fig. 99 (cont.)

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