The Residual Circulation in the North Sea

219

THE RESIDUAL CIRCULATION IN THE NORTH SEA Jacques C.J. NIHOUL

1

and Yves RUNFOLA

Geophysical Fluid Dynamics, LiPge University, Belgium 'Also at the Institut d'Astronomie et de Ggophysique, Louvain University (Belgium) INTRODUCTION Hydrodynamic models of the North Sea are primarily concerned with tides and storm surges and the associated currents which can have velocities as high as several meters per second. However the period of the dominant tide is only about a half day and the characteristic life time of a synoptic weather pattern is of the order of a few days. The very strong currents which are produced by the tides and the atmospheric forcing are thus relatively transitory and a Marine Biologist will argue that over time scales of biological interest, they change and reverse so many times that they more or less cancel out, leaving only a small residual contribution to the net water circulation. The importance of tidal and wind induced currents for the generation of turbulence and the mixing

0.f

water properties is of course not denied but many biologists would

be content with some rough parameterization of the efficiency of turbulent mixing and, for the rest, some general description of the long term transport of "water masses". Although the concept of "moving water masses", and its train of pseudo-lagrangian misdoings, appeal to chemists and biologists who would like to find, in the field, near-laboratory conditions, it is impossible to define it in any scientific way and charts of the North Sea's waters like the one shown in figure 1 and reproduced from Laevastu (1963) are easily misinterpreted and often confuse the situation by superposing a flow pattern on an apparently permanent "geography" of water masses. The notion of "residual" circulation - which, at least, has an Eulerian foundation

- has long remained almost as vague.

Some people have defined it as the observed flow

minus the computed tidal flow. Such a definition is understandable from a physical point of view but one must realize that the residual flow so-defined contains all wind-induced currents, including small scale fluctuations. It is definitely not a steady or quasi-steady flow and some attempts to visualize it by means of streadlines are questionable. What it represents, in terms of marine chemistry or marine ecology is not at all clear.

Actually, if one wants to take the point of view of the marine ecologist, one should really look at the mean flow over some appropriate period of time of biological interest. It is customary for experimentalists to compute, from long series of observations, daily, weekly and monthly averages. What such averages actually represent is debatable. No doubt that tidal currents are essentially removed in this process. However with tidal velocities, one or two orders of magnitude higher than residual velocities and the latter of the order of traditional current-meters'errors, one may fear that, as a result of the non-linearities of the equipment, the error remains the same order of magnitude after averaging and leads to a 100

%

inaccuracy in the calculated mean re-

sidual. (e.g. Nihoul, 1980). Moreover the choice of the periods of time over which the averages are made is not obviois as it seems to rely more on the calendar than on physical processes. One must be quite clear of what one gets from such averages. With tides reversing four times daily and changes in the synoptic weather pattern taking several days, one may expect daily averages to remove tidal motions while still catching most of the residual currents responding to the evolving meteorological conditions. Monthly averages, on the other hand, will have a more "climatic" sense and will presumably represent the residual circulation which is induced by macroscale oceanic currents (such as the North-Atlantic current in the case of the North Sea) and the mean effect of non-linear interactions of mesoscale motions (tides, storm surges

...) .

Here, the terms "macroscale", "mesoscale" (and later on "microscale") are used in reference with the time scales of motion. In general time scales and length scales are related but it doesn't have to be so and no such assumption is made here at this stage. The role of residual currents and residual structures (fronts

...)

in the dynamics

of marine populations, the long term transport of sediments or the ultimate disposal of pollutants, for example, is universally recognized but different schools of theoreticians and experimentalists still favour different definitions which, in the case

of the North Sea, may have little in common,apartfrom the fact that the strong tidal oscillations have been removed. Obviously each definition addresses a particular kind of problem and if, as it is now universally agreed, the residual circulation is defined as the mean motion over a period of time sufficiently large to cancel tidal oscillations and transient windinduced currents, there is still the problem of choosing the time interval of averaging, taking into account the objectives of the study. In any case, it is not demonstrated that such a time average may be obtained with sufficient accuracy from experimental records. As pointed out before, the averaging takes away more than 90 the instrumental error.

%

of the signal and the final result is of the same order as

221

F i g . 1. Water types i n t h e North Sea according t o Laevastu ( 1 9 6 3 ) . [Adapted from F o l i o 4 of S e r i a l A t l a s of t h e Marine Environment with t h e permission of t h e American Geographical S o c i e t y . ]

222

In the following, one examines how the problem can be approached through mathematical modelling. THE GOVERNING EQUATIONS The three-dimensional hydrodynamic equations applicable to a well-mixed continental sea, like the North Sea, can be written (e.g. Nihoul, 1975)

0.v

0

=

av +

0 . (VV)

at

where

n

+

2

n

A

V

=

- Vq

+ Q.R

is the Earth's rotation vector, q =

P

+ gx, , p is the pressure,

p

the speci-

fic mass of sea water, x3 the vertical coordinate and R the turbulent Reynolds stress tensor (the stress is here per unit mass of sea water) resulting from the non-linear interactions of three-dimensional microscale turbulent fluctuations. The turbulent Reynolds stress tensor can be parameterized in terms of eddy viscosity coefficients. In microscale three-dimensional turbulence, these coefficients are of the same order of magnitude in the horizontal and vertical directions. Then, horizontal length scales being much larger than the depth, the last term in the righthand side of eq.(2) can be written simply, with a very good approximation

where C is the vertical eddy viscosity and K the turbulent Reynolds stress (vector). The residual flow is defined as the mean flow over a time T sufficiently large to cover at least one or two tidal periods. If the subscript

"o"

denotes such an average,

one may write

v

=

vo + v,

(4)

with (V), =

vo

(5)

What Vo and V, respectively include depends on the time of integration T. If T is of the order of one day (exactly two or three periods of the dominant Mp tide), T

- lo5

,

the averaging eliminates the tidal currents and smoothes out all

current fluctuations, - generated by variations of the wind field, for instance which have a characteristic time smaller than T.

,

223 However, as mentioned before, changes in the synoptic weather pattern often have time scales comparable with T

-

gible meteorological forcing, T

10'.

Then, unless one considers periods of negli-

- lo5

does not correspond to a valley in the energy

spectrum of the currents. In that case, one cannot derive an equation for V, by averaging eq.(2) and assuming that, as for an ensemble average, the averaging commutes with the time derivative. Furthermore, V, defined in this way, depends very much on time and doesn't correspond to the quasi-steady drift flow the biologists have in mind when they talk about residuals. One might argue that such a time dependent daily mean is still worth calculating to follow the response of the sea to the evolving w e a t h e r p a t t e r n , e s p e c i a l l y in storm conditions. This however would be equivalent to modelling a storm surge with a time step of 10

5

and the results cannot be very accurate. It is much wiser, in that case, to solve equations (I) and ( 2 ) , without averaging for tides and storm surges simultaneously. Actually, "daily" residuals (i.e. mean currents over exactly two or three tidal periods) are meaningful only when the atmospheric forcing is negligible or exceptionally persistent. In the first case, they represent the so-called "tidal residuals" which result from in - and out-flowing macroscale oceanic currents and from the residual effect of non-linear tidal interactions. Tidal residuals represent a major contribution to the total residual circulation and, with very much less computer work needed, they already give a good idea of the long term residual circulation, such as the climatic circulation described below, where asubstantial part of the atmospheric contribution is actually removed by averaging over a variety of different weather conditions. If one takes, for instance, the averaging time T between l o 6 (- two weeks) and lo7 (-four months), one may expect, over such a long time, a diversity of meteorological conditions resulting in an almost random atmospheric forcing on the sea. The current patterns will reflect the atmospheric variability and, on the average, there will be only a small residue. The mean flow over a time T

- l o 6 , lo7

may be regarded as the "climatic residual"

flow which affects the dynamics of biological populations, the long term transport of sediments and the slow removal of pollutants. As

pointed out before, one may conceive a third kind of residuals obtained by ave-

raging over two or three tidal periods

(T 2 lo5

s)

in conditions of exceptionnally

persistent atmospheric forcing. The persistence of the meteorological conditions drives off the atmospheric energy input to small frequencies and a time of the order of

lo5 is acceptable for averaging

as it corresponds again to a valley in the energy spectrum. This kind of residuals may be called in brief "wind residuals". One must be aware, however, that they give a rather limited view of wind-induced currents in the sea.

224 If a typical weather pattern has a characteristic time of a few days, one must either determine the time dependent wind-induced and tidal flow described by equations ( l ) and ( 2 ) or the climatic residual flow which contains only the macroscale residue of changing weather patterns. Still, with moderate computer work needed, wind residuals may perhaps provide a first idea of various wind effects on the residual flow pattern which would not be apparent in the climatic or tidal residual pattern but which might occasionally be spotted by instruments in the field. In the following "residual circulation" will refer to the climatic residual circulation

(T

2 lo6

s)

or the tidal and wind residual circulations

(T 2 lo5 s ) with

the restrictions made above. The equations for the residual flow may be obtained by taking the average of equations (1) and ( 2 ) over the chosen time T. The time derivative in the left-hand side of equation ( 2 ) gives a contribution

One may argue that, since the time T is always a multiple of the main tidal period, the numerator of ( 7 ) is of the same order as the residual velocity V,. Then, for T 2 105

,

The average of the Coriolis acceleration is

One may thus neglect the contribution of the time derivative in the equation for v,. The residual circulation is then given by the steady state equations

v.v,

where

= 0

Since V,

is one or two orders of magnitude smaller than V, which contains in

particular the tidal currents, the first term in the left-hand side of eq.(12) is completely negligible. The tensor N

in the right-hand side plays, for mesoscale

motions, a role similar to that of the turbulent Reynolds stress tensor R

in eq.(2)

and may be called the "mesoscale Reynolds stress tensor". The last term in the righthand side of eq.(11) represents an additional force acting on the residual flow and resulting from the non-linear interactions of mesoscale motions (tides, storm surges

... ) .

The importance of this force was discovered, first, by depth-integrated numerical models of the residual circulation in the North Sea (Nihoul 1974, Nihoul and Ronday 1975) and the associated stress was initially referred to as the "tidal stress" to emphasize the omnipresent contribution of tidal motions. THE MESOSCALE REYNOLDS STRESS TENSOR The tensor N

can be computed explicitly by solving eqs.(I) and (2) for mesoscale

motions and taking the average of the dyadic V,V,

.

In fact the solution of eqs.(l) and (2) with appropriate wind forcing and open sea boundary conditions yields

v

=

v, + v,

and one may reasonably ask the question why one must go through the process puting N

and solving eqs. (10) and ( 1 1 ) to obtain the residual velocity V,

why one cannot solve (1) and ( 2 ) for the total velocity V from V

d i r e c t l y , by averaging the solution of eqs.(1) and 6V

on V

of, say, 10

%,

1.e.

and simply derive V, (2).

The problem here, again, is that, in the North Sea, V, represents 90 If one allows for an error

of com-

%

of V

.

resulting from the imprecision

of open-sea boundary conditions and from the approximations of the numerical method,

.

the error is of the same order of magnitude as the residual flow V, Because of non-linearities, one may fear that, in the averaging process, this error does not, for the essential, cancel out as V, does. Thus averaging the solution V

of eqs. (1) and (2), one gets V,

an error which may be as large as

100

%

+

(6V),

i.e. the residual velocity with

(Nihoul and Ronday, 1976a).

The procedure is conceivable when modelling a very limited area (near a coast, for instance) where the mesh size of the numerical grid can be reduced and where the open-sea boundary conditions can be determined with greater accuracy by direct measurements. Then

6V can be made small enough for the average V,

a satisfactory evaluation of the residual flow V,

.

+

(6V),

to provide

226 In the case of the North Sea or, even, the Southern Bight or the English Channel,

models of such a high accuracy are prohibitively expensive and cannot be considered for routine forecasting.

v, to compute the mesoscale stress tensor N However, the classical models give

with a fair accuracy and they can be used

.

The latter can be substitutdd in eq.(11) and the system of eqs.(10) and (11) can be solved very quickly to obtain V,

.

One can show that, in this way, one can determine V,

with good accuracy.

Typical values for the North Sea show that, in general, the two terms and

8 . (- V,V,),

If

2

n

A V,

are of the same order of magnitude.

6V, is the error on V,

This error induces an error

, one

has

6Vo on V,

given by

i.e.

Hence the r e l a t i v e error is the same on error as before. Thus if

v,

and on

v, and not the absoZute

v, can be computed with, say, a

90 % precision, the solu-

tion of the averaged eqs.(10) and (11) will give the residual circulation with the same 90

8

precision.

THE EQUATION FOR THE HORIZONTAL TRANSPORT If one writes

u

v = u + w e 3

=

u, +

emphasizing the horizontal velocity vector u transport as

uo -

u, dx, -h

=

-

H, u,

(131 i (13')

U(

,

one defines the residual horizontal

227

co

where

is the depth-averaged velocity, H, = h

the residual surface elevation. (H,

h

because

+ 5,

,

L o << h)

.

h

is the depth and

5,

The derivation of equations for the residual transport by integration of eqs.(10) and (11) over depth is quite straightforward (e.g. Nihoul, 1975a). One finds, after some reordering, V.U,

=

0

f e3

A

U,

(15) =

- n o Vq, - K U, +

e

(16)

where

u,

denoting the depth-mean of U , and Q

standing in brief for

Tz

f

TI - Ti

where (i) Tg

is the residual wind stress

(ii) T:

is the mesoscale Reynolds stress

(iii)

$A

is the mesoscale “friction stress”

The friction stress is the part of the residual bottom stress (the first part is

-

K U,)

which results from the non-linear interactions of mesoscale motions. It is

analogous to the Reynolds stress t,” and represents an additional forcing on the residual flow. Since

u,

is a two-dimensional horizontal vector, eq.(15) suggests the introduc-

tion of a stream function $(x,,x2)

such that

228 Eliminating

q

between t h e two h o r i z o n t a l components of e q . ( 1 6 ) , one obtains

then a s i n g l e e l l i p t i c equation f o r

K ; v 2 J ,+

~a

$ [ a xa ,

(K) + ax,

( -hf) I

, viz.

J,

+

[using (20)1

= ax, [ a a x , - ( hK )

This equation must be solved with a p p r o p r i a t e boundary conditions. p l y take

J, = const.

I f one can s i m -

along t h e c o a s t s , t h e c o n d i t i o n s on t h e open-sea boundaries are

much more d i f f i c u l t t o a s s e s s . One has e s t i m a t e s of t h e t o t a l inflows through t h e 3 -1 S t r a i t s of Dover 7400 km . y (Van Veen, 1938; C a r r u t h e r s , 1935)1, t h e Northern 3 boundary [- 23000 krn .y-' (Kalle, 1949; Laevastu, 196311, through t h e Skagerrak 3 .-1 [- 479 km .y (Svansson, 1968; Tomczak, 196811 a s well a s of t h e c o n t r i b u t i o n of the 3 main r i v e r s [- 245 km .y-l ( M c Cave, 1974)l but t h e d i s t r i b u t i o n of t h e s e flows

c-

along t h e boundaries are p o o r l y known and one must r e s o r t t o i n t e r p o l a t i o n formulas which may o r may n o t r e p r e s e n t adequately t h e c o n t r i b u t i o n , t o t h e r e s i d u a l c i r c u l a t i o n of t h e North Sea, of inflowing o r outflowing macroscale c u r r e n t s . The open-sea boundary c o n d i t i o n s used i n t h e p r e s e n t model a r e derived from t h e above e s t i m a t e s following Ronday (1965). Ronday (1965) has shown t h a t they r e p r e s e n t t h e a v a i l a b l e observations reasonably well and t h a t eventual d e v i a t i o n s from t h e i n t e r p o l a t e d v a l u e s a t t h e open-sea bound a r i e s d o n o t g e n e r a t e errors which could propagate f a r i n t o t h e North Sea. S t i l l , a b e t t e r determination of t h e c o n d i t i o n s along open-sea boundaries i s needed and should be considered with t h e h i g h e s t p r i o r i t y i n t h e near f u t u r e . Eq.(22) shows t h e i n f l u e n c e on t h e r e s i d u a l flow of t h e r e s i d u a l f r i c t i o n c o e f f i c i e n t K and i t s g r a d i e n t , of t h e d i s t r i b u t i o n of depths, of t h e c u r l of t h e r e s i d u a l wind s t r e s s and of t h e mesoscale s t r e s s e s

T," and

T:

.

I n r e l a t i v e l y c o a r s e g r i d models of t h e whole North Sea (where t h e v a r i a t i o n s of K

and

h

a r e p a r t l y smoothed o u t ) , t h e e f f e c t of t h e mesoscale stresses appears t o

be t h e most s p e c t a c u l a r . T h i s i s i l l u s t r a t e d by f i g s . ( 2 ) , (3) and ( 4 ) , f i g u r i n g t h e r e s i d u a l c i r c u l a t i o n i n n e g l i g i b l e wind conditions.

F i g . ( 2 ) shows t h e r e s i d u a l flow

p a t t e r n assuming a c o n s t a n t depth of 80 m and n e g l e c t i n g

T," and

Ti

.

F i g . ( 3 ) shows t h e flow p a t t e r n t a k i n g t h e depth d i s t r i b u t i o n i n t o account and neglecting

T:

and

T:

.

F i g . ( 4 ) shows t h e flow p a t t e r n t a k i n g t h e depth d i s t r i b u t i o n i n t o account and including

T:

and

T ! ,

computed from t h e r e s u l t s of a preliminary time dependent model

of mesoscale flows. The d i f f e r e n c e s between f i g s . ( 2 ) and ( 3 ) a r e s m a l l . They both reproduce t h e broad t r e n d of t h e r e s i d u a l c i r c u l a t i o n induced by t h e in-and out-flow of two branches of t h e North A t l a n t i c c u r r e n t b u t they f a i l t o uncover r e s i d u a l g y r e s which c o n s t i t u e

229

----%---'

b'

3'

2'

I'

0'

I'

2'

3'

4'

5'

6'

7'

8'

9'

Ib

1'1

Ih

I$

Fig. 2. Residual c i r c u l a t i o n i n t h e North Sea c a l c u l a t e d i n n e g l i g i b l e wind c o n d i t i o n s assuming a c o n s t a n t depth and n e g l e c t i n g t h e mesoscale s t r e s s e s . (Streamlines i n l o 3 m 3 . s - ' )

230 61'

6d

5 9'

5 6'

57'

56'

5 5'

5i'

53'

5 2'

5 1'

5 0'

't9'

Fig. 3. Residual circulation in the North Sea calculated in negligible wind conditions with the real depth distribution, neglecting the mesoscale stresses. (Streamlines in l o 3 rn3.s-')

231 6 1'

60'

5s'

5s'

57.

56 '

55'

5c'

53'

52.

51'

50'

b9'

I b'

3'

2'

I'

d

I'

2'

3'

4'

5'

6'

7'

R'

9'

tb

1'1

I5

I

15

Fig. 4. Residual circulation in the North Sea calculated in negligible wind conditions with the real depth distribution, taking the mesoscale stresses T : and T: into account. (Streamlines in l o 3 m3.s-')

232

essential features of the residual flow pattern and which have been traced in the field by observations ( e . g . Zimmerman, 1976; Riepma, 1977; Beckers et al, 1976).

Fig. 5. Residual circulation in the Southern Bight calculated in negligible wind conditions, with the real depth distribution, taking into account the mesoscale (Streamlines in lo3 m3.s-’) stresses T,” and

‘ri .

A comparison between fig.(4) and fig.(I) shows a good agreement between the predictions of the model and the expected circulation of water masses in the North Sea. However, as mentioned before, a model covering the whole North Sea does not have a sufficiently fine resolution (of bottom topography, for instance) and cannot detect all the existing gyres. For that reason, three models were run simultaneously, one covering the North Sea, another one, the Southern Bight and the third one, the Belgian coastal waters; the lar.ge scale models providing open-sea boundary conditions for the smaller scale models. Fig.(5) shows the residual circulation in the Southern Bight. One notices in particular a gyre off the Belgian coast which was not apparent on fig.(4). This gyre is produced by the mesoscale stresses in relation with the spatial variations of the depth and of the residual friction coefficient K (Nihoul and Ronday, 1976).

233 The presence of the gyre has been shown to play

an important role in the distri-

bution of sediments (Nihoul, 1975b) and in creating o f f the Northern Belgian coast the conditions of an outer-lagoon with specific chemical and ecological characteristics (Nihoul, 1974; Beckers et al., 1976). THE ENERGY EQUATIONS Using eq.(3), one can write the equations for V

+

0. (V,V,)

av, a t

+

2

V.cV,V,

V,

A

+

V,V,

=

+

- Vq, V,V, -

+

+ ax3

,

and

V,

v, in the form

V.N

( V I V l ) , 1+ 2 CJ

A

V,

with

v.v

=

v.v,

= V.VI = 0

One can see that the equation for V

v 1 is essentially the same as the equation for

. They only differ by terms which are orders of magnitude smaller. It is the reason

why, dne can, with the appropriate boundary conditions, determine the mesoscale velocity

v, ,

in a first step, and the residual velocity

,

V,

in a second step, ta-

king the coupling between the two types of motion into account in the calculation of

v, only. Taking the scalar products of eqs. (231, and

V,

, using

( 2 4 ) and (25) respectively by

( 2 6 ) , (27) and ( 2 8 ) , and averaging 3ver

T

,

V

one finds, neglecting

again the contributions from the time derivatives under the assumption that sufficiently large

;

, Vb T

is

234 The terms i n t h e left-hand s i d e s of eqs. (291,

( 3 0 ) and (31) a r e of t h e divergence

form. They r e p r e s e n t f l u x e s of energy i n p h y s i c a l space. The terms i n t h e right-hand s i d e s r e p r e s e n t r a t e s of energy production o r d e s t r u c t i o n o r energy exchanges between s c a l e s of motion. I n t e g r a t i n g over depth, one can see f o r i n s t a n c e t h a t t h e f i r s t terms r e p r e s e n t t h e average r a t e of work of t h e wind s t r e s s

where

Vs

T’

,

i.e.

denotes t h e s u r f a c e v e l o c i t y .

The second term i n t h e right-hand s i d e of e q . ( 2 9 ) i s r e l a t e d t o t h e average d i s s i p a t i o n of energy. using e q . ( 3 ) , one has, indeed - T . -

where

av

ax3

3

-

- 3

av IIax,II

(33)

i s t h e eddy v i s c o s i t y .

The depth-averaged d i s s i p a t i o n r a t e

can be s p l i t i n t w o p a r t s , a s seen from eqs. (30) and ( 3 1 ) , i.e.

THE CONTRIBUTION OF THE RESIDUAL STRESS

T o TO THE ENERGY BUDGET

The second term i n t h e right-hand s i d e of e q . ( 3 5 ) i s obviously r e l a t e d t o t h e energy d i s s i p a t e d by t h e mesoscale motions. I t i s , by f a r , t h e e s s e n t i a l c o n t r i b u t i o n t o E

and may serve a s a f i r s t approximation of it. I t i s however t h e f i r s t term one i s

i n t e r e s t e d i n , h e r e , t o e x p l a i n t h e p h y s i c a l mechanisms which c o n t r i b u t e t o shape the residual circulation. I n e v a l u a t i n g t h i s t e r m , one can obviously r e s t r i c t a t t e n t i o n t o t h e h o r i z o n t a l components of t h e v e c t o r s d u c t of t h e form

av

T.2x3

To and

v o , and t h i s i s a l s o t r u e f o r any s c a l a r pro-

235 One has indeed, from the continuity equation, aw

5

x3

where

v.u L

-

o);(

is the characteristic scale of horizontal varia ions. On the other hand

Since h << L

,

the contributions from the vertical velocity are completely negli-

gible in the integrals of eq.(35). The application of the three-dimensional equation (1) and (2) to the North Sea (Nihoul, 1977; Nihoul et al., 1979) shows that (i) the turbulent stress can be written

where

T'

and

T~

are respectively the surface stress and the bottom stress (per

unit mass of sea water),

5

= H-'

(x,

+

h)

,

H = h

the surface elevation, the A n ' s are functions of Tb

and their time derivaties, K

and the functions fn(5)

a,

+

5

,

h

t , x, and

5

is the depth and

x2 involving T s

,

is the Von Karman constant,

are the eigenfunctions of the problem

being the corresponding eigenvalue. The last term in the right-hand side of eq.(36) plays an important role in the de-

termination of the velocity field v periods of weak currents

but its effect is limited to relatively short

(at tide reversal, for instance) (Nihoul, 1977; Nihoul

et al., 1979) and it contributes very little to the residual turbulent Reynolds stress

obtained by averaging over a time

T

covering several tidal periods.

236 Hence, setting

+

z = x3

(ii) the bottom stress T~ and the time derivaties of

h

, one

may write, with a good approximation

is a function of T S

u .

,

the depth-averaged velocity 3

If one excepts, again, short periods of weak currents,

Tb

can be approximated by

the classical "quadratic bottom friction law"

where

D

is the drag coefficient.

Averaging over a time

T

as before and neglecting small order terms, one obtains

then

i.e., using eqs. (17) and (19)

Changing variable to

z

and using (39), one can write

The horizontal velocity u

,

however it may for the rest vary with depth, always

has a logarithmic profile near the bottom. This implies that near

z = 0

.A

au az

behaves like

z-'

first consequence of this asymptotic behaviour of the velocity pro-

file is that integrals over depth are, strictly speaking, not taken from the surface but from some very small height

zo

z = 0

to

(the "rugosity length") to the

surface. This has been taken into account in eq.(43).

A

second consequence is that t h e

first integral in the right-hand side of eq.(43) is largely dominant, the singularity at

z = 0

being cancelled in the second integral by the factor

z

.

Since the second integral is only a small correction, one may make the approximation

237

Eq. (43) can thus be rewritten

Integrating by parts, one gets

i.e., using eq. ( 4 2 )

The first two terms in the right-hand side of eq. (30) integrated over depth give then

i.e., the same result one would have obtained from the depth integrated transport equation for the contribution of the wind stress and the bottom stress, by taking the scalar product of eq.(16) by

Go

.

One notes that, in the right-hand side of eq.(46), only the first term can be associated without ambiguity with the dissipation of energy. The second term represents the rate of work of the mesoscale friction stress. Although its "bottom friction" origin is clear, its sign cannot be set a p r i o r i and there is no reason why it could not actually provide energy to the residual flow. The same can be said for the last term in the right-hand side of eq.(30). This term appears with the opposite sign in eq.(31). It thus represents an exchange of energy

between residual and mesoscale flows: this term can be either positive or

negative. There is no way of knowing a p r i o r i wether the energy is extracted from the mean flow and goes from macroscales to mesoscales or if it is supplied to the mean flow by mesoscale motions.

238 THE EXCHANGE OF ENERGY BETWEEN SCALES OF MOTION The exchange of energy between macroscales and mesoscales can be characterized, at each point of the North Sea, by the depth-averaged rates of energy transfer

These quantities, which may be positive or negative should be compared with the rate of energy dissipation by the residual motion ED

:

i.e.

- 2

(50)

uo

= K

The mesoscale Reynolds stress tensor N

also contributes to the term

V.(- N.V,)

in the left-hand side of eq. ( 3 0 ) . This is a completely different effect because it implies a flux of energy in physical space while

N

: VV,

,

appearing in boths eq.(30) and (31), represents a trans-

fer of energy between scales, i.e., a flux of energy in Fourier space. This effect cannot however be ignored if one wants to understand the mechanisms by which the mesoscale stresses act on the residual flow. One shall write

It is convenient to normalize the rates of energy transfer using the rate of energy dissipation

E~

as the normalizing factor, Let

S = - 6

(52)

ED

EN

=

EN -

(53)

EF

=

EF -

(54)

2,

ED

ED

..= 6

+ i, + i, +

1

(55)

In non-dimensional forms, (i) S

is the rate of change of kinetic energy due to energy divergence or convergence

in physical space.

239 (ii) E N is the rate of energy exchange between macroscale and mesoscale motions resulting from the action of the mesoscale Reynolds stresses. (iii) t F is the rate of energy exchange between macroscales and mesoscales resulting from the action of the mesoscale friction stress. (iv)

+ 2,

is the rate of work of the mesoscale Reynolds stresses on the residual

flow.

E T is the net rate at which energy is extracted from the residual flow

(v)

by the combination of energy fluxes in physical space, energy transfers between scales of motion and energy dissipation by bottom friction. positivevalue of

A

eN

or

CF

implies a transfer of energy from the mean flow to

the mesoscale motions. At the opposite, negative values indicate a transfer of energy from the mesoscales to the mean flow. Using turbulence terminology, these situations will be referred to as cases of positive eddy viscosity and negative eddy viscosity respectively. [The word "eddy" is here used in an extended sense referring to mesoscal e non-linear waves and turbulence. A similar definition was proposed by Rhines and

Holland (1979). 1

TIDAL R E S I D U A L S A detailed study of the tidal residuals in the North Sea was made by means of two coupled three-dimensional models, one for tides and storm surges (Nihoul, 1977; Nihoul et al., 1979) and one for the residual circulation described by eqs.(10) and (11).

The data of the three-dimensional models were used to compute the depth-averaged cir&lation EN

,

EF

and the spatial distributions of the depth-averaged transfer functions and

.

*

The results are presented in figs. ( 6 ) to (12)

.

Fig.(6) shows the residual streamlines. One can see that there is, in general, a good agreement between the result of the three-dimensional models and those of the depth-integrated model represented in fig.(4). The main differences are observed in regions of very weak residual circulation where the three-dimensional models have a better resolution. The existence of extended regions of very small residual currents (less than 1 cm

s-')

is one important characteristic of the tidal residual flow pattern. It has been found

*

In interpreting these figures, and those which follow in the next section, one must remember that, having to calculate horizontal gradients, the model can only provide results one grid point away from the coast. One cannot say anything from the figures about the coastal fringe.

240

2'

I

0

I

2'

9'

4

5'

6'

7

a'

9'

Fig. 6. Tidal residuals in the North Sea. Residual flow pattern. Streamlines IL = const. in l o 3 m3.s-'.

241

58

57

56

55

54

53

52

51

Fig. ?a. Tidal residuals in the North Sea. Map of positive values of the non-dimensional function 8 representing the rate at which the residual kinetic energy is redistributed in physical space. (Positive values indicate regions of residual energy divergence.) 8 = 50 Heavy line Slender line - 6 = 1 0 Broken line Z = O

----

58

52

56

55

54

5s

12

5!

Fig. 7b. Tidal residuals in the North Sea. Map of negative values of the non-dimensional function 8 representing the rate at which the residual kinetic energy is redistributed in physical space. (Negative values indicate regions of residual energy convergence.) $ = - 50 Heavy line Slender line $ = - 10 Broken line 6 = 0

-----

243

I - sa

- 57

- 56

- 55

54

5s

52

51

Map of positive values of the non-dimensional function EN representing the rate of energy transfer between residual and mesoscale flows by the mesoscale Reynolds stresses. (Positive values indicate regions of positive eddy viscosity.) Heavy line EN = 5 0 EN = l o Slender line Broken line EN = 0

----

244

58

57

56

55

sb

59

52

5!

Fig. 8b. T i d a l r e s i d u a l s in t h e North Sea. representing t h e r a t e Map of n e g a t i v e values of t h e non-dimensional f u n c t i o n S, o f energy t r a n s f e r between r e s i d u a l and mesoscale flows by t h e mesoscale Reynolds s t r e s s e s . (Negative values i n d i c a t e regions of negative eddy v i s c o s i t y . ) SN = - 50 Heavy l i n e Slender l i n e - i, = - 10 Brpken l i n e ---- ZN = 0

-

245 almost impossible to give any comprehensible representation of the flow field using the traditional methods which consist in drawing the velocity vector at each grid point. To display the residual flow in some regions, it has been necessary to draw streamlines 5 lo3 m3 s-'

apart while, in other regions, the difference between two

consecutive streamlines is

20

lo3

m3 s-' or more.

Although the small scale gyres are essential to understand the hydrodynamics of the North Sea, especially in coastal zones, they contain little of the total residual energy of the North Sea and the tidal residual circulation appears to be constituted essentially of two main energetic streams corresponding to the penetration in the North Sea of two branches of the North-Atlantic current. These are the analogue of the macroscale gyres of oceanic circulation and, in this sense, the residual circulation which has been qualified as "macroscale" with reference to its time scale (i.e. its quasi steady character) may be classified also among macroscale motions with respect to length scales (with a peak of energy in the small wave numbers range in a spectral analysis of the North Sea energy). The maps of the functions 6

, EN

and

EF

(figs. 7, 8 and 9) show a marked

patchiness with alternating positive and negative values. Absolute values are one or two orders of magnitude larger for for

2,

and

ED

(ED

=

8

and

EN

than

1, in non-dimensional form). Thus the mesoscale Reynolds

stresses play the main role in the energetics of the tidal residual flow. They are responsible for a transfer

EN

"in Fourier space", i.e. an exchange of energy between

macroscale residual flow and mesoscale motions, and for a flux "in physical space", i.e. a transport of energy from one region of the North Sea to another. These two effects tend to compensate each other and the sum

a^

iE N

is about one order of

magnitude smaller than each of its terms (fig. 10). Thus, when energy is supplied to the mean flow in some region (negative viscosity effect), it is, to a large extent, exported to other regions where energy is extracted from the mean flow by the mesosca-

le motions. The pattern of streamlines (fig. 6) shows that the regions of negative eddy viscosity contain little residual energy (the residual currents are in general very small as indicated by the wide spacing of the streamlines) and most of the residual vorticity (the streamlines are curved and often closed, forming secondary gyres). This is confirmed by the vorticity pattern (fig. 12) showing vorticity scales as small as 10 km associated with the gyres. In the regions of negative eddy viscosity where the mesoscale stresses transfer energy from the mesoscale motions to the residual flow, they also generate vorticity in the residual flow. What is actually happening is that the energy supplied to the residual flow in these regions is immediately exported away to the regions of positive eddy viscosity related to the main streams. This energy supply thus contributes to enhance the large scale currents and, in this sense, the energy is truly going from mesoscales to macroscales, referring now indifferently to time scales or length scales.

246

-

58

. 57

' 5 6

55

54

58

52

51

Fig. 9. Tidal residuals in the North Sea. Map of positive and negative values of the non-dimensional function EF representing the rate of energy transfer between residual and mesoscale flows by the friction stress,

247

58

57

56

55

59

52

51 2'

1

0

1

2

9

i

5

6

7

6

9

Fig. 10. Tidal residuals in the North Sea. Map of positive and negative values of the non-dimensional function d + C N representing the rate o f work of the mesoscale Reynolds stresses on the residual f l o w .

248

58

57

56

58

54

53

52

51

Fig. 11. Tidal residuals in the North Sea. Map of positive and negative values of the non-dimensional function Z T = 6 + EN + S, + 1 representing the rate at which energy is extracted from the residual f l o w (positive values) or supplied to the residual flow (negative values).

249

58

57

56

55

54

53

52

51

Fig. 12. T i d a l r e s i d u a l s i n t h e North Sea. Map of t h e function 110 A U o l l w =

-

..

11TnIl ,. I

-

indicating the scale Heavy l i n e 3 Slender l i n e - 3 Broken l i n e --- 3

of t h e r e s i d u a l vorticity. =

m-'

=

m-l

= 5 10-5

,,-I

250 The secondary flows which are generated in the regions of negative eddy viscosity contain little energy but most of the vorticlty. This vorticity is characterized by length scales smaller in many places than the typical length scale of tidal motions (referred to as the mesoscale motions before) i.e. the energy cascade to larger scales is paralleled by an enstrophy cascade to smaller scales. It is tempting to see here a case of two-dimensional turbulence and identify the patches of intense energy exchanges between scales of motions as the "synoptic eddies" of the North Sea. The fact that,in these eddies, the bottom dissipation is comparatively completely negligible is perhaps an argument in favor of such interpretation. As

a final remark, it is interesting to note that

gT

,

the net rate at which

energy is extracted from the residual flow (or supplied to it) is not positive everywhere. A rather extensive patch of negative values (in the range -1, -10) spreads out from the Western North Sea, off the coasts of Scotland and Northern England, into the central part of the North Sea. Going back to eq. ( 3 0 ) where the term

VZ

8. (v, 2) is always completely negligible, 2

one sees that, in the absence of wind forcing, negative values of V.(V, 9,)

= v,.vq,

iT imply

'0

i.e. the mesoscale stresses are actually driving the residual flow "up the residual slope and pressure gradient".

WIND RESIDUALS As pointed out in the introduction, the tidal residuals considered in the preceding sections can only constitute a first approximation of what a real climatic residual circulation is. The atmospheric forcing has been neglected both in determining the mesoscale motion and in computing the resulting residuals. The advantage was that the time of averaging could be limited to two or three tidal periods. This is not possible if one includes the effect of a wind field which itself evolves with a characteristic time of the same order. In some cases, one may have to go to averaging over several weeks to ensure that the average is meaningful. This implies that the mesoscale velocity field must be calculated over the same period of time and the cost of operating the model becomes rapidly prohibitively large. Although an effort of this size is now being considered to model the residual circulation during the period of the JONSDAP 76 experiment, it has not been possible so far to apply the three-dimensional model with real atmospheric conditions. However, to have some idea of the effect of wind forcing, two cases of constant uniform wind fields have been considered.

251 The concept of a uniform wind f i e l d over t h e whole North Sea i s c e r t a i n l y i d e a l i s -

tic. Moreover, t h e d i r e c t e f f e c t of t h e wind o n t h e r e s i d u a l c i r c u l a t i o n , which i s a s s o c i a t e d with t h e c u r l of t h e wind s t r e s s , i s not taken i n t o account. Going back t o e q . ( 2 4 ) , one can see t h a t if one t a k e s t h e c u r l of t h i s equation t o eliminate

vqo

,

t h e second term i n t h e right-hand s i d e w i l l g i v e a c o n t r i b u t i o n

(56)

Eq.(36) shows t h a t t h e Reynolds s t r e s s

T

c o n t a i n s a term

Ts5

where

Ts is

t h e wind stress. Hence, a c o n s t a n t wind stress c o n t r i b u t e s t o ( 5 6 ) a t e r m

(57)

I f t h e wind f i e l d i s uniform, t h e f i r s t term of t h e right-hand s i d e of e q . ( 5 7 ) drops o u t . The e f f e c t of t h e wind s t r e s s on t h e r e s i d u a l c i r c u l a t i o n i s then due, i n p a r t , t o t h e l a s t t e r m i n t h e right-hand side of e q . ( 5 7 ) and, i n p a r t , t o t h e modif i c a t i o n of t h e mesoscale s t r e s s e s r e s u l t i n g from t h e a c t i o n of t h e wind f i e l d on t h e time dependent mesoscale motions. There i s t h u s n o " d i r e c t " t r a n s f e r of v o r t i c i t y from t h e wind f i e l d t o t h e r e s i d u a l circulation. In t h e p r e s e n t c a s e , it i s perhaps p r e f e r a b l e s i n c e t h e o b j e c t i v e

of

t h e study

i s t o e l u c i d a t e t h e r o l e played by t h e mesoscale s t r e s s e s i n determining t h e r e s i d u a l flow p a t t e r n and i n p a r t i c u l a r i n generating secondary g y r e s . The hypothesis t h a t t h e wind i s c o n s t a n t i n time ( i . e . , t h a t it has a very l a r g e c h a r a c t e r i s t i c time of e v o l u t i o n ) i s of course l i m i t a t i v e . I t j u s t i f i e s however t h e averaging over a few t i d a l p e r i o d s a s t h i s corresponds again t o a v a l l e y i n t h e

P

energy spectrum of t h e c u r r e n t s . In t h i s p r o c e s s , t h e t i d a l c u r r e n t s a r e a l s o removed but they a r e e v i d e n t l y taken i n t o account i n t h e time dependent mesoscale flow which determines t h e mesoscale

stresses. One should n o t be mislead by t h e name "wind r e s i d u a l s " . The e f f e c t of t h e wind cannot be d i s s o c i a t e d from t h e e f f e c t of t h e t i d e s . A l l "wind r e s i d u a l s " a r e wind and t i d a l r e s i d u a l s . Two c a s e s of reasonably s t r o n g wind have been considered

a c o n s t a n t wind of

15 m s-'

from t h e North-West,

(ii)a c o n s t a n t wind of

15 m s-'

from t h e South-West.

(i)

:

The r e s u l t s a r e presented i n f i g s . (13) t o ( 1 9 ) f o r t h e f i r s t case and i n f i g s . ( 2 0 ) t o (26) f o r t h e second case. I n both c a s e s , t h e s t r e a m l i n e s p a t t e r n ( f i g . 13 and 20) show a d e f l e c t i o n , under t h e i n f l u e n c e of t h e wind f i e l d , of t h e main streams o r i g i n a t i n g from t h e North

252

58

57

56

55

54

5s

52

51 2

I

I

I

1

0

I

I

1

2

I

3

c'

I

I

I

5

6

I

7

Fig. 13. Wind residuals in the North Sea. from the North-West.) (Uniform constant wind of 15 m s-' Residual flow pattern. Streamlines $J = const. in l o 3 1 n ~ 5 - I

I

I

8

S'

253

58

57

56

55

54

5s

52

S! I

I

2'

I

0'

I

2'

3'

b

5

6'

7

a'

9'

Fig. 14a. Wind residuals in the North Sea. from the North-West:) (Uniform constant wind of 15 m s-' Map of positive values of the non-dimensional function 6 representing the rate at which the residual kinetic energy is redistributed in physical space. (Positive values indicate regions of residual energy divergence.) = 50 Heavy line Slender line - f; = 10 Broken line 6 = 0

-

----

254

5a

57

56

55

54

53

52

51 2

1

0

1

2'

3'

\

5'

6'

a'

S

Fig. 14b. Wind r e s i d u a l s i n t h e North Sea. 15 m s - ' from t h e North-West:) (Uniform c o n s t a n t wind of Map of p o s i t i v e values of t h e non-dimensional f u n c t i o n 6 r e p r e s e n t i n g t h e r a t e a t which t h e r e s i d u a l k i n e t i c energy i s r e d i s t r i b u t e d i n p h y s i c a l space. (Negative values i n d i c a t e regions of r e s i d u a l energy convergence.) $ = - 50 Heavy l i n e Slender l i n e ___ 6 = - 1 0 Broken l i n e 8 = 0

-----

255

58

57

56

55

5L

5.3

52

51

Fig. 15a. Wind residuals in the North Sea. (Uniform constant wind of 15 m s - ’ from the North-West.) Map of negative values of the non-dimensional function ?, representing the rate of energy transfer between residual and mesoscale flows by the mesoscale Reynolds stresses. (Positive values indicate regions of positive eddy viscosity.) Heavy line ?, = 5 0 Slender line - C N = 10 Broken line EN = 0

-----

256

58

57

56

55

5\

5s

52

61

Fig. 15b. Wind residuals in the North Sea. from the North-West.) (Uniform constant wind of 15 m s - ' Map of negative values of the non-dimensional function Z N representing the rate of energy transfer between residual and mesoscale flows by the mesoscale Reynolds stresses. (Negative values indicate regions of negative eddy viscosity.) E N = - 50 Heavy line Slender line - EN = - 10 Broken line EN = 0

-----

251

sa

57

56

55

54

59

52

51 2'

1

0'

1

2

3

4'

5

6'

7'

9

Fig. 16. Wind residuals in the North Sea. (Uniform constant wind of 15 m s-' from the North-West.) Map of positive and negative values of the non-dimensional function E F representing the rate of energy transfer between residual and mesoscale flows by the friction stress.

58

57

56

55

5b

5s

52

51

Fig. 17. Wind r e s i d u a l s i n t h e North Sea. 15 m s - ’ from t h e North-West.) (Uniform c o n s t a n t wind of repreMap of p o s i t i v e and n e g a t i v e v a l u e s of t h e non-dimensional f u n c t i o n 8 + ?, s e n t i n g t h e r a t e of work of t h e mesoscale Reynolds s t r e s s e s on t h e r e s i d u a l flow.

259

58

57

56

55

54

59

52

51

Fig. 18. Wind residuals in the North Sea. from the North-West.) (Uniform constant wind of 15 m s - ’ Map of positive and negative values of the non-dimensional function ? T = 6 + 2, + ?, + 1 representing the rate at which energy is extracted from the residual flow (positive values) or supplied to the residual flow (negative values).

260

58

57

56

55

54

5s

52

51

I

2'

0

1

2

3

t'

5

6'

Fig. 19. Wind r e s i d u a l s i n t h e North Sea. 15 m s-' from t h e North-West.) (Uniform c o n s t a n t wind of Map of t h e f u n c t i o n

-

w =

IIV

A

Go11

..Ilii, II " 1 1

-w

i n d i c a t i n g t h e s c a l e of t h e r e s i d u a l v o r t i c i t y . = m-' Heavy l i n e Slender l i n e - 2 = 5 m-l Broken l i n e ---- 6j = m-t

7

8

9

261

5.9

57

56

55

54

59

52

51 2

1

0

1

2

3

4

5

6’

7

Fig. 20. Wind r e s i d u a l s i n t h e North Sea. 15 m s - ’ from t h e South-West.) (Uniform c o n s t a n t wind of Residual flow p a t t e r n . Streamlines J, = const. i n l o 3 m 3 s - ’ .

8

9

262

58

57

56

55

5c

59

52

51

Fig. 21a. Wind residuals in the North Sea. from the South-West:) (Uniform constant wind of 15 m s - ' Map of positive values of the non-dimensional function 6 representing the rate at which the residual kinetic energy is redistributed in physical space. (Positive values indicate regions of residual energy divergence.) S = 50 Heavy line Slender line - 6 = 10 Broken line ---- 8 = 0

-

263

58

57

56

55

5b

53

52

51

Fig. 21b. Wind residuals in the North Sea. from the South-West.) (Uniform constant wind of 15 m s - ’ Map of negative values of the non-dimensional function 6 representing the rate at which the residual kinetic energy is redistributed in physical space. (Negative values indicate regions of residual energy convergence.) fi = - 50 Heavy line Slender line - 6 = - 1 0 Broken line ----- X = O

-

264

58

57

56

55

54

5s

52

5!

Fig. 22a. Wind residuals in the North Sea. (Uniform constant wind of 15 m s-' from the South-West.) Map of positive values of the non-dimensional function E N representing the rate of energy transfer between residual and mesoscale flows by the mesoscale Reynolds stresses. (Positive values indicate regions of positive eddy viscosity.) Heavy line E N = 50 Slender line - E N = 10 Broken line ---- E N = 0

-

265

sa

57

56

55

54

59

52

51

Fig. 22b. Wind r e s i d u a l s i n t h e North Sea. 15 m s-’ from t h e South-West.) (Uniform c o n s t a n t wind of Map of n e g a t i v e v a l u e s of t h e non-dimensional f u n c t i o n E N r e p r e s e n t i n g t h e r a t e of energy t r a n s f e r between r e s i d u a l and mesoscale flows by t h e mesoscale Reynolds s t r e s s e s . (Negative v a l u e s i n d i c a t e r e g i o n s of n e g a t i v e eddy v i s c o s i t y . ) E N = - 50 Heavy l i n e Slender l i n e - E N = - 10 Broken l i n e EN = 0

----

266

Fig. 23. Wind residuals in the North Sea. (Uniform constant wind of 15 m s-’ from the South-West.) Map of positive and negative values of the non-dimensional function C F representing the rate of energy transfer between resldual and mesoscale flows by the friction stress.

267

58

57

56

55

54

5s

52

51

Fig. 24. Wind residuals in the North Sea. f r o m the South-West.) (Uniform constant wind of 15 m s - ' Map o f positive and negative values of the non-dimensional function 8 + C N representirig the rate of work of the mesoscale Reynolds stresses on the residual flow.

268

58

57

56

55

5k

5s

52

51

Fig. 25. Wind residuals in the North Sea. from the South-West.) (Uniform constant wind of 15 m s-' Map of positive and negative values of the non-dimensional function representing the rate at which energy is extracted from the E , = 8 + i, + E , + 1 residual flow (positive values) or supplied to the residual flow (negative values).

269

58

57

56

55

54

5s

52

51

Fig. 26. Wind residuals in the North Sea. from the South-West.) (Uniform constant wind of 15 m s - ’ Map of the-function i j = IIV A u,II I I i i O

II

-

indicating the scale Heavy line Slender line Broken line ----~

of the residual vorticity. 5 = m-l 6 = 5 l o + m-l ;= m-l

270

Atlantic current's inflows. The tracery of the streamlines is more contorted and encloses large (length) scale gyres associated with relatively strong residual currents. These gyres appear to determine the vorticity distribution and only few small scale secondary gyres are apparent in regions of weak residual currents. The maps of energy transfer functions reflect the streamlines patterns. The energy dissipation

E,,

is more important (stronger currents) than in the absence of wind

and the rates of energy transfer the normalized functions

8 ,

significantly larger than

1

.

EN

,

E~

and

E,

E~

... are

found generally comparable to

E

~

having only a few located peaks of values

One can still find "eddies" where the exchange of energy between scales of motion sharply dominates the energy dissipation by bottom friction and, induced to look for it by one's understanding of tidal residuals, one can still see a relationship between negative viscosity eddies and small (length) scale vorticity in secondary gyres but basically, bottom friction has gained control of energy and enstrophy cascades and an interpretation of the residual flow's energetics in terms of two-dimensional turbulence is no longer profitable.

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