Post on 13-Oct-2020
(NCAR/TN-136+STRfNCAR TECHNICAL NOTE
I
April 1980 e /
On the Interaction betweenMesoscale Eddies and Topography
j,/
Bernard Durney ,i 0
ATMOSPHERIC ANALYSIS AND PREDICTION DIVISION
NATIONAL CENTER FOR ATMOSPHERIC RESEARCHBOULDER, COLORADO
4
67
iii
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
iv
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.
v
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.
vii
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
ix
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
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'
and S
'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 '
(2)
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
2
(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)
3
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
4
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).
5
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
1 1
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
6
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
, .
1
2
3
4
5
6
7
8
9
10
11
12
13
14
105m
105m
Eq. 5
21m
42m
84m
126m
53m
105m
105m
ct1
6 i = 5'V2/K
62
s 2
2
= -a1/4
=-6 1/4
6 6.95 km
13.9 km
6.95 km
3.475km
3.475km
13.9 km
13.9 km
13.9 km
13.9 km
6.95 km
6.95 km
1.74 km
6.95 km
10.4 km
6.95 km
3.475km
1.74 km
10.4 km
10.4 km
10.4 km
10.4 km
3.475km
6.95 km
0.87 km
d-1
d-1
d- 1d-1
dd-1d-1
d-1
d- 1d- 1
d-1
d
-1d
-1d
d- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~
d-1d-
d-1d-
1.d
d-1
d-
d-11
d-
d-I
d-
d-1
8
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.
9
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.
10
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).
Experiment 2 (Fig. 13 to Fig. 18)
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
11
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.
Experiment 3 (Fig. 19 to Fig. 25)
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
12
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 5 (Fig. 40 to Fig. 46)
Experiment 6 (Figs. 47 to Fig. 52)
Experiment 7 (Fig. 53 to Fig. 57)
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
13
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
14
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 8 (Fig. 58 to Fig. 63)
Experiment 9 (Fig. 64 to Fig. 66)
Experiment 10 (Figs. 67 to Figs. 75)
Experiment 11 (Fig. 76 to Fig. 77)
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
that
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.
15
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.
16
Experiment 12 (Figs. 78 to Fig. 86)
Experiment 13 (Figs. 87 to Fig. 97a)
Experiment 14 (Figs. 98 to Fig. 99)
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.
17
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).
19
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).
21
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
22
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)).
23
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.
24
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
25
ENER (K)
0 5 10 15 20 25 30 35 40 45 50
k
Fig. 1
6
4
2
0
26
48 48
0
Fig. 2
-47
Fig. 3
48
48
0
27
Fig. 4
28
5.5 . i'i' i'
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
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
k
Fig. 5
t=120d:; M=2
Fig. 6
Expl
29
ROOT M SQ (EDDY) Exp I20
(a)18 -
VELI ,16
14
12
10
8
6
4
2 -
16 -(b)
VEL214
12
.I 0
8
6
4
2
0 60 120 180 240 300 360 420 48<
t(d)
0
Fig. 7
30
(POT EN(I.O))/2 Exp I
0 60 120 180 240 300 360 420 480
t(d)
Fig. 8
24
22
20
18
16
14
12
10
8
6
KA (I,0)/2
0 60 120 180 240 300 360 420 480
t(d)
Fig. 9
31
M=I Exp2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
I
32
MEAN KIN EN M=2 Exp I280 ......... i'" ' ,) , ,,ir
260
240
220
200
180
160
140
120
100
80
60
40
20
00 60 120 180 240 300 360 420 480
t(d)
Fig. 10
ENER (K) /440/2 Exp I150
140
130120I1O
10090
807060504030 -20IO
00 4 8 12 16 20 24 28 32 36 40 44
kFig. 11
33
t=440 d M=2 ~ Exp I
0, l- -- -................._...
:-2.. ... . .. ...., -,.!: , . 1, 1 .
....... .2
Fig. 12
o-- ,'~~~~~~~~~~~~
34
ENER(K) /120/20.140.13
0.12
0.11 I
0.100.0 A A lQ090.08
0.07
0.06
0.050.04
0.03
0.020.01
0 4 8 12 16 20
k
Exp 2
24 28 32 36 40 44
Fig. 13
35
KA (0,I) / 2
0 10 20 30 40 50 60
t (d)
Fig. 14
Exp 2
M =I (a)
-- ~~~~~~~~~~
3.10
3.00
2.90
2.80
2.70
2.60
2.501.3
1.2
I .
1 .0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.170
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
Exp 2
0 10 20 30 40 50 60 70t(d)
Fig. 16
36
Exp 24.0
3.6
3.2
2.8
2.4
2.0
1.6
1.2
0.8
0.4
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
t(d)
Fig. 17
500 600
37
(EDDY) Exp 21.35
1.25
1.15
1.05
0.95
0.85
0.75
065
0.55
0.45
0.350 100 700
38
M=2 Exp 2
0 10 20 30 40 50 60 70
t(d)
Fig. 18
0.00048
0.00042
0.00036
0.00030
0.00024
0.00018
0.00012
0.00006
03.5
3.0
2.5
2.0
1.5
1.0
0.5
0
-0.5
- 1.0
-1.5
-2.0
-2.5
-3.0
-3.5
39
Exp3t=1Od
Fig. 19
40
INTERFACE DISPLACEMENT
Fig. 20
t=20d Exp 3
t = 20d M = 2
41
Exp 3
Fig. 21
42
TOTAL M ENERGY Exp 365
60
55
50
45
40
35
30
25
200 40 80 120 160 200 240 280 320
t (d)Fig. 22
TOTAL EDDY ENERGY Exp 3165
160
155 -
150
145
140
135
130
1200 40 80 120 160 200 240 280 320
t (d)Fig. 23
43
t = 300d M = 2 Exp 3
Fig. 24
44
t = 300d Int. Displacement Exp 3
Fig. 25
45
ENERGY (K)/2/55
50
45
40
35
30
25
20
15
10
160
I) , I * l I0 5 10 15 20 0 5 I0 15
k k
Fig. 26
60
55
50
45
40
35
30
25
20
15
10
5
0
Exp 4
0 5 10 15
k
ENERGY (K)/2/ Exp 4
440
0 5 10 15 20 25 0 5 0 15 20
k k
Fig. 27
30
28
26
24
22
20
18
16 -
14
12
10
8
6
4
2
0
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
t = 40d1;_ M=2
t = 160dk?& M=2
46
Exp 4
Fig. 28
Exp 4
Fig. 29
t = 300dI_ M=2
t = 440d~E, M=2
47
Exp 4
Fig. 30
Exp 4
Fig. 31
t = 620dIb M=2
48
Exp 4
Fig. 32
49
t = 20d M = 2 Exp 4
Fig. 33a
t 600 M = 2
50
V, Exp 4
Fig. 33b
t= 600
51
M = 2 Exp 4
Fig. 34
52
100
90
80
70
60
50
40
30
20
10
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)
180
170
160
150
140 -
130
120
110 _-IO
100
90320 360 400 440 480 520 560 600 640
t (d)
Fig. 35
53
TOTAL EDDY ENERGY Exp 4
0 40 80' 120 160 200 240 280 320
210 ,i I | I
200
190
180 -
170
160
150
140 \
130 -
120
110 -
100 -
90 I0 I 0 I I .320 360 400 440 480 520 560 600 640
t (d)
Fig. 36
320
310
300
290
280
270
260
250
240
230
220
210
200
54
EPOT ENERGY13
12
.
I0
9
8 -
7-
6
5
4 -
0 40 80 1
8.0 r, I , * i ,,
7.2
6.8
6.4
6.0
5.6
5.2
4.8
Exp 4
20 160 200 240 280 320
4.4 I ' I- 1 -i ' 1 '320 360 400 440 480
t (d)
520 560 600 640
Fig. 37
55
ROOT M SQ VEL I (EDDY) Exp 414.0 * | |
13.5 (a)
13.0
12.5
12.0
11.5
11.0
10.5
10.0
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
t (d)
Fig. 38
)
56
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
t (d)
Fig. 39
10.6
10.4
10.2
10.0
9.8
9.6
9.4
9.2
9.0
8.8
8.6
57
ENERGY (K)/2/ Exp 5
12 3 6
32I0
k9 2840 960
8 ' 24
720
6
~~~~~~5 - 16
4 12
38
2
0 00 4 8 12 16 20 24 0 4 8 12 16 20
k k
Fig. 40
Exp
Fig. 41
t =960 dM=2
58
t=960d M=2 Exp 5~~~~~II)!IU J
t "' '~~~~~~,, ~q'q r':- <'....-· · r , ·o, ...--, ~ · ~ ,,•'
I~ ' ' ~ el / e I
dI
, .,. ~,, ,4 ir~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -.,'O·: · r,,:,u..:::''::..'· r~~~~~~~~: ;' . . ... ::r '·a rrr~~~~ rr rr ·~:' .. . . .. ·
`--~~~Q, ....- . . ,......· · ' : .:.: ... .o..-'I" . '. . ..,...,.-"~~~~~~~~~~ '""- <:~:: .....-''::'' ... "4'i... .. (
,.../ ,,' ,' .
... o ,, ,,, .. ·
~· r, . . .. .,, ,.. ~. t-· · ~ ~~~~~~~~~ :-'---~/r ,· ' .I-._. .....
Fig. 42
59
TOTAL M ENERG Exp 538
36
34
32
30
28
26
24
22
20 -
18600 700 800 900 1000 1100 1200 1300
t(d)
Fig. 43
TOT EDDY ENERGY Exp59292 M,,,M ... |, . ... |U. |.M.,.M,..',...|.....' ' . . '"'
90
88
86
84
82
80 -
78
76 \74
74 -
72
600 700 800 900 1000 1100 1200 1300
t(d)
Fig. 44
60
0 100 200 300 400 500 600
t(d)
Fig. 45
Exp 5
0 100 200 300 400 500 600
t(d)
Fig. 46
28XI0- 6
24X 10 6
20XIO- 6
16xlO- 6
12XO1- 6
8X10-6
4x10-6
0.
0.44
0.40
0.36
0.32
0.28
0.24
0.20
0.16
0.12
0.08
0.04
0
w
KA (0, O'12 M=2
61
ENER (K)/2/3.0
2.8 (a)
2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
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
k
Exp 6
0 4 8 12 16 20k
0 4 8 12 16 20 24
Fig. 47
62
t= 40dM=2
(a)
t= 160 dM=2
(b)
Fig. 48
Exp6
Exp 6
t= 440 dM=2
(C)
t=1000dM=2
(d)
Fig. 48 (cont.)
63
Exp 6
Exp 6
64
t =600 d M=2 Exp6
Fig. 49
65
t=600 d M=2 Exp6
Fig. 50
66
TOTAL M ENERGY Exp 61.15
1.05
0.95
0.85
0.75
0.65
0.55
0.45650 750 850 950 1050 1150 1250
t(d)
Fig. 51
TOT EDDY ENERGY Exp 6
27.2
27.0
26.8
26.6
26.4 -
26.2
26.0
25.8 I
650 750 850 950 1050 1150 1250
t(d)
Fig. 52
67
ENER (K)/2/1.3
1.2
1. 1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
00 5 10 15 20 25 0 5 10 15
k
Fig. 53
Exp 71.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
t =40 dM=2
(a)
t =160de,, M=l
(b)
Fig. 54
68
Exp7
Exp 7
t =160dM=2
(C)
t = 300 dM=2
Fig. 54 (cont.)
69
Exp 7
Exp 7
(d)
70
TOTAL M ENERG Exp 7
) 40 80 120 160 200 240 280 320
Fig. 55
TOT EDDY ENERGY Exp 7
0 40 80 120 160 200 240 280 320
Fig. 56
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0
-0.02
12.7
12.6
12.5
12.4
12.3
12.2
12.1
12.0
11.9
71
t=300d M=2 Exp
Fig. 57
7
72
ENER (K)/2/
0 5 10 15 20
55
50
45
40
35
30
25
20
15
10
5
0
Exp 8
0 5 10 15 20 25
k
Fig. 58
55
50
45
40
35
30
25
20
15
10
5
0
t =440 dM=2
(a)
t = 600 dM=2
(b)
73
Exp 8
Exp 8
Fig. 59
74
t =600d M=2 Exp8
Fig. 60
-75
t=600d M=2 Exp 8
Fig. 61
76
TOTAL M ENERG Exp 8128
126
124
122
120 -
118 -
116
114
112 I i
460 480 500 520 540 560 580 600 620 640
t (d)
Fig. 62
TOT EDDY ENERGY Exp 8178
176
174
172
170
168
166 -
164
162
160460 480 500 520 540 560 580 600 620 640
t (d)
Fig. 63
77
I , i I .I I I , I I I
II I 1 I I I
40
i I I I I I I I I 1i I i I I I I .1 I
5 10
k
15 20
26
24
22
20
18
16
14
12
10
8
6
4
2
0
Exp 9
0 5 10 15 20 25
k
Fig. 64
ENER (K)/2/
I
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
00
I I - -,-I- _ -_ _
- - ---
78
Fig. 65
EPOT ENERGY Exp 9
0 20 40 60 80 100 120 140 160
t (d)
Fig. 66
Exp 9t=320d
- M=2
18
16
14
12
10
8
6
4
2
79
ENER (K) /40/2 Exp 1020 .
(a) I18-
16
10 5 15 5 0 35 40 45 50k
18 14(b) 13-
16 2
14 II180 ,0 -A 320
12 991 0 5 10 1 85 20
78- 6 -6
2
4 -I
0 00 5 10 15 20 0 5 10 15 20
k k
Fig. 67
t=160 d
P- M=2
(a)
t=320dM=l
(b)
t=320dM=2
(c)
Fig. 68
Exp 10
80
t=320d Exp 10
81
M=2 'I
Fig. 69
t=320d Exp 10
82
M=l
Fig. 70a
83
t=320d M=2 Exp 10M=2~
Fig. 70b
84
36.5
35.5
34.5
33.5
32.5
31.5
30.5
29.5
TOTAL M ENERG Exp 10
140 160 180 200 220 240 260 280 300 320
t(d)
Fig. 71
TOT EDDY ENERGY274 - .
272
270 -
268 -
266 -
264
262
260
258
256
254
252
250 ' I I140 160 180 200
Exp 10
220 240 260 280 300 320
t(d)
Fig. 72
85
EPOT ENERGY
0 20 40 60 80
t(d)
Exp 10
100 120 140 160
Fig. 73
26
24
22
20
18
16
14
12
10
8
6
4
2
86
ROOT M SQ VEL I (Eddy) Exp 1014.0 , , , . , , , ,. -
f\ (a):13.5
13.0
12.5
12.0
11.5
11.0
10.5
10.0- *0 20 40 60 80 100 120 140 160
10.2
10.1 - (b)
10.0
9.9 -
9.8 -
9.7 -
9.6-
9.5 -
9.4
9.3 -
9.2
9.1
9.0
8.9140 160 180 200 220 240 260 280 300 320
t(d)
Fig. 74
87
ROOT M SQ VEL 2 (Eddy) Exp 10
10.4
10.3
10.2
10 .I
10.0
9.9
0 20 40 60 80 100 120 140 160
10.02(b)
9.98
9.94
9.90
9.86
9.82
9.78
9.74
140 160 180 200 220 240 260 280 300 320
t(d)
Fig. 75
88
ENER (K)/2/
; ] \ 6 - 120
4
2
0 5 10 15 20 25 0 5 10 15 20
k k
Fig. 76
t=140d
wsn\ M=2
Fig. 77
16
14
12
10
8
18
16
14
12
'10
8
6
4
2
0
Exp II
I
I
89
ENER (K) /40/2 Exp 12
0 4 8 12 16 20 24 28 32 36 40 44k
Fig. 78
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
ENER (K) /160/2
90
ENER (K)/300/10.9 m . .,'r .i i
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0ENER (K)/300/2
1.6
1.4
1.2
1.0.
0.8 -
0.6
0.4
0.2
O ,0 4 8 12 16 20 24 28 32 36 40 44
k
Fig. 78 (cont.)
t=40di_ M=2
(a)
t=160dN A M=2
(b)
Fig. 79
91
Exp 12
Exp 12
Exp 12
Exp 12
t=300d_ MM=l
(c)
t=320dA M=2
(d)
Fig. 79 (cont.)
92
Exp 12t=320d3a PE
93
Fig. 80
94
t=300d M=l I
Fig. 81a
Exp 12
t=300d Exp 12
95
M=2
Fig. 81b
96
TOTAL M ENERG2.2 -, , I i i
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0 .- 0 .2 I ' I I , I I I .
51.0
50.0
49.0
48.0
47.0
46.0
45.0
44.0
0 40 80 120 160
t(d)
Fig. 82
TOT EDDY ENERGY
Exp 12
200 240 280 320
Exp 12
40 80 120 160 200 240 280
t(d)
Fig. 83
I
L~~~~~I I 1I I ~ I I 1
0 320a . . . . . . . . . . . . . . . . . . . . . .
I-- ,"I
t j
97
EPOT ENERGY Exp 12
0 40 80 120 160 200 240 280 320
t (d)
Fig. 84
ROOT M SQ VEL I (EDDY) Exp 12
0 40 80 120 160 200 240 280 320
t (d)
Fig. 85
7.0
6.6
6.2
5.8
5.4
5.0
4.6
4.2
3.8
3.4
3.0
2.6
7.0
6.6
6.2
5.8
5.4
5.0
4.6
4.2
98
ROOT M SQ VEL 2 (EDDY)
0 40
Exp 12
80 120 160 200 240 280 320
t(d)
Fig. 86
3.86
3.82
3.78
3.74
3.70
3.66
3.62
3.58
3.54
3.50
3.46
99
ENER (K) /40/2 Exp 13I I I 1' 1 , i , l l , l l , 1 , 1 1 .I
7.0 (a)
6.0
5.0
4.0
3.0
2.0
I .0
-' 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
4.5
4.0
3.5
3.0
2.5
2.0
1.5
I .0
0.5
00 4 8 12 16 20 24 28 32 36 40 44
k
Fig. 87
1"00
ENER (K) /300/2 Exp 13
4.4 (c)
4.0
3.6
3.4
.2.8 -
2:-4
2.0
i.6
i.2 -
0.8
0.4
ENER (K) /440/23.8 I T T r"--r-
(d)3.4
3.0
2.6
2.2
1.8 !1.0
0.6
0.2.0 4 8 12 16 20 24 28 32 36 40 44
k
Fig. 87 (cont.)
101
ENER (K) /600/1 Exp 13
0.8 I A (e)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
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
(a)
t= 160d
t M=2
(b)
Fig. 88
102
Exp 13
t=300dM=2
(c)
t=440dA M=2
(d)
Fig. 88 (cont.)
103
Exp 13
104
t=440d M=l 4 Exp 13
Fig. 89a
t=440d
105
M=2 I Exp 13
Fig. 89b
106
TOTAL M ENERG Exp 137.0
6.0
5.0
4.0
3.0
2.0
1.0
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
96
94-
92
90
88
86I l I I I i I I
310 330 350 370 390 410 430 450 470
t (d)Fig. 91
107
EPOT ENERGY Exp 13
0 40 80 120 160 200 240 280 320
t(d)Fig. 92
9.0
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0
108
7.0
6.8
6.6
6.4
6.2
6.0
5.8
5.6
5.4
5.2
5.0
4.8
4.84
4.80
4.76
4.72
4.68
4.64
4.60
4.56
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
t(d)
Fig. 93
109
ROOT M SQ VEL 2 (EDDY)
6.9
6.8
6.7
6.6
6.5
6.4
6.3
6.2
Exp 13
0 40 80 120 160 200 240 280 320
6.22 b)
6.18
6.14
6.10
6.06
6.02
5.98
5.94
5.90
5.86
310 330 350 370 390 410 430 450 470
t(d)
Fig. 94
110
KA(0,1)1/ 2 M=2 Exp 13
40 80 120 160 200 240 280 320
t (d)
Fig. 95
0 40 80 120 160 200 240 280 320
t (d)
Fig. 96
0.36 -
0.32
0.28
0.24
0.20
0.16
0.12
0.08 -
0.04
0 l0
0.00024
0.00020
0.00016
0.00012
0.00008
0.00004
0
111
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
t(d)
Fig. 97
0.44
0.40
0.36
0.32
0.28
0.24
0.20
0.16
0.12
0.08
0.04
0
0.60
0.50
0.40
0.30
0.20
0.10
112
ENER (K)/160/1
ENER (K)/300/I
0 4 8 12 16 20 24 28 32 36 40 44
k
Fig. 98
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
113
ENER (K) /60/2 Exp 14
ENER (K) /160/2
0 4 8 12 16 20 24 28 32 36 40 44
k
Fig. 99
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0.06
0.05
0.04
0.03
0.02
0.01
0
114
ENER (K)/300/2 Exp 140.07
(c)
0.06
0.05
0.04
0.03
0.02
0.01
0, .0 4 8 12 16 20 24 28 32 36 40 44
k
Fig. 99 (cont.)