On the effect of bottom topography on two eddies in the Sardinia and Sicily Straits region

Deep-SeaResearch,Vol. 29, No. IA, pp. 77-86, 1982. Printed in Great Britain.

0198-0149/82/010077-10 $03.00/0 © 1982PergamonPress Ltd.

On the effect of bottom topography on two eddies in the Sardinia and Sicily Straits region SILVIA GARZOLI,* VALERIO PARISIf and ELIO PASCHINIt (Received 23 September 1981; in revisedform 27 May 1981; accepted 22 June 1981)

Abstrael--An hypothesisthat eddies in the Straits of Sardinia have a topographic planetary origin is tested with a model. We consider a two-layer system, each layer of variable thickness. The equations are solved numerically and the theoretical values are in good agreement with the observations. INTRODUCTION IN WINTER a convective, homogeneous water layer about 150 m thick is formed in the Rhodes-Cyprus area. At intermediate depths, this layer can be found in the whole eastern basin. The Intermediate Levantine Water flows through the Straits of Sicily and forms the intermediate layer in the Western Mediterranean Sea. The topography of this region is complicated (Fig. 1) and the communication between the Eastern and Western Mediterranean basins mainly takes place through two narrow channels (FRESSE'I'ro, 1964;

MORELLI, GANTAR and PISANI, 1975). Towards the west the water bifurcates,with one branch flowing through the Straitsof Sardinia, a narrow valley with an average m a x i m u m depth of about 2000 m. NIELSEN (1912), Wt~ST (1959), TCHERNIA (1960), LACOMBE (1977), and MOREL (1971), among others, have studied the circulationin this region. A complete study of the winter circulationin the Sardinia and SicilyStraitsregion has recently been done by GARZOLI and MAILLARD (1977, 1979). The data analysed came mainly from two French cruises,Amalthee 1967 and Hydromed 1968. They show that, down to 400 dbar, the circulationin the Straits of Sardinia is dominated by an anticycloniceddy in the southern part and a cyclonic eddy in the northern part (Figs 2, 3). The two eddies form an obstacle to the Intermediate Water flowing to the west. The hypothesis that both eddies may have a planetary origin (first proposed by SAINT-GuILY, 1961) is developed in a simple analyticalmodel. The purpose of the present paper is to extend the model to a two-layer ocean, considering the effectsof coastal and bottom topography. THE MODEL GARZOLIand MAILLARD(1979) studied the system of two eddies for a steady, frictionless, vertically homogeneous fluid in an ideal channel of infinite length and constant width.

* Laboratoire d'Oceanographique Physique Museum d'Histoire NatureUe, Paris, France. Present address: Lamont-Doherty Geological Observatory, Palisades, NY 10964, U.S.A. t Istituto de Fisicae delrAtmosfera, Roma, Italy.

77

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SILVIA GARZOLI, VALERIO PARISI a n d ELIO PASCHINI

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Eddies in the Sardinia and Sicily Straits region

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Fig. 2. Dynamic topography of the 101Ndbar surface relative to the 800-dbar surface. Units in dynamic centimetres (from GARZOU and MAmLARD, 1977).

Under these conditions, it is possible to introduce a stream function, ~k, such that I c~VJ.

i c~k

ho d y '

ho Ox'

where ho is the constant depth of the bottom and V = (u, v) is the horizontal velocity. The results are shown in Fig. 4. In our new model we will consider a two-layer system: a surface and an intermediate layer, each of variable thickness. The equations are solved numerically. In order to introduce the realistic effect of the variable thickness, hi, for each layer, we introduce the stream functions, ~bi, where

1 a@i. Ui = hi d y '

1 (~@i

v. . . .

hi d x '

i = 1,2.

80

SILVIAGARZOLI,VALERIOPARISIand ELlO PASCHINI

7°

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by0.5dyn.cm I

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7* 8" 9* 10" Fig. 3. Dynamic topography of the 400-dbar surface relative to the 800-dbar surface. Units in dynamic centimetres (from GARZOUand MAILLARD,1979).

Then, from the steady Euler equation we obtain for each layer: J FV(V~//--hi)-f, ] l_ h~ ~k~ = 0,

(1)

where J is the Jacobian operator and f the Coriolis parameter. A solution of (1) is of the form V v ( - h T ) - f = hiF,(~b),

(2)

where Fi are unknown functions. According to GARZOLI and MAILLARD(1979), we assume that F~ is a linear function of ~k, Fi = ai+bi~k,

Eddies in the Sardinia a n d Sicily Straits region

!y

Fig. 4.

81

C~5

Theoretical stream lines ~ = - u sin O[r + a J, (r/ro)] = constand axes are the dimensionless variables X = x/ro ; Y = y/ro (from GARZOLI and MAILLARD, 1979).

where a~ and bi are unknown parameters, and we have to solve the linear elliptical equation V V(~-i)- f =

hiai+hibi~k.

(3)

As the available data form a closed domain (i.e., we can estimate the value of ~, at the boundaries) we use the Dirichlet boundary condition ~b = ~bc(x,y) for both layers. The numerical procedure used is described in Appendix 1. Knowing the parameters ai and bi at the boundaries, we can compute the unknown function in the domain within the boundary. Once we know the function in the whole domain, we then make the best-fit test on ai, bi, and ~ki at the boundary, based on the hypothesis that the experimental data satisfy the differential equation with a gaussian error. Considering the experimental data at the boundary (values of the geostrophic fluxes) and within the boundary (some values of the fluxes and the variable thickness, hi), and using the best-fit test, we obtain the theoretical values a, b, and ~b that minimize the mean square of the difference between the theoretical and experimental data. Concerning the mathematical method employed to obtain the stream lines, and in view of future application to other cases and other regions of the sea, we must emphasize two points: (1) In the case we considered, the experimental points were nearly uniformly distributed in the region. The dynamical method is then sufficient to give the pattern of the circulation (without recourse to the present scheme) in the region. If, however, there are discontinuities in the distribution of the data, the dynamic method is not sufficient to describe the circulation. In such a case, simply given the values of the fluxes at the boundaries, we can estimate the general circulation using our method. (2) The numerical method that we use to solve equation (3) (see Appendix 1) does not need a regular grid for the boundary conditions. Such a characteristic is obviously useful because, in general, the experimental values to be used as boundary conditions are not in a regular grid.

82

SILVIA GARZOLI,VALERIOPARIS[and ELIO PAscvir~i

RESULTS TO study the effectof the bottom topography of the two eddies in the Sardinia and Sicily Straits region, equation (1) was solved for two cases: (1) h i = constant for both layers. For the upper layer (surface water) we took hi = 200 m and for the lower boundary of the intermediate water layer h2 = 600 m. Those values correspond to the mean upper and lower interfaces of the intermediate water as inferred from the data. The transition between layers was considered at a salinity of 38.5 x 10 -3. (2) h i = hi(x,y). At each (x,y) point, equation (1) is solved considering the data providing depth (hi or h2). Again, the transition between layers was considered at a salinity of 38.5 x 10-a. To supply input to our numerical model, (boundary conditions) geostrophy was applied to a closed domain defined by the hydrographic stations. To compare theoretical and observational results, we compute the dynamic topography from the Hydromed 1968 cruise data. Solutions are calculated for both layers and for both cases. In the first case, hi = constant, we calculate the dynamic topography for the surface layer at hi = 200 m (0/200 dbar), and for the intermediate layer at h2 = 600 m (200/600 dbar). In the second case [hi = hi(x, y)] the dynamical topography was calculated for both layers at the depth of the 38.5 x 10 -3 isohaline. The stream function was calculated from the preceding calculations and from the model at each point in the domain. Results are given in Figs 5, 6, and 7. Figure 5 shows, as an example, the results for a north-south section at the center of the eddies along 8°30'E (see Fig. 2). The upper part of the figure (a, b) corresponds to the cases solved for constant depth: (a) surface water, (b) intermediate water. The lower part of the figure (c, d) shows the results for variable depth : (c) surface water, (d) intermediate water. The agreement between the observations and the model is better for the intermediate water layer and for h i variable. The general circulation pattern of the whole region is given in Figs 6 and 7. Figure 6 shows the results obtained for h = constant: (a) and (c) are the dynamic topography as

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38000'N -

37o30'N -

Fig. 5. Stream function values for a cross-s¢ction along 8°30'E; (O ©) calculated from Hydromed 68 data; (©. . . . C)) theoreticalvaluesfrom the model. (a) Surfacewater, hi = constant. (b) Intermediate water, h2- constant. (c) Surface water, hi = ht(x,y). (d) Intermediate water, h2 = h2(x,y). Units in 106 m3 s -1.

Eddies in the Sardinia and Sicily Straits region

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84

SILVIA GARZOLI, VALERIOPARISIand ELIO PASCHINI

obtained from the observations for the surface water (0/200 dbar) and for the intermediate water (200/600 dbar), respectively; (b) and (d) are the same, but for theoretical results. Figure 7 is the same as Fig. 6, but for hi = hi(x, y). In both cases two eddies, one cyclonic and one anticyclonic, appear. For the surface layer, the difference between results with h, = constant and h 1 = variable is slight. For the intermediate water layer, when h2 = constant, theoretical results show a third eddy and differences in position and shape. When solving for h 2 = variable, we find good agreement between theoretical and experimental results. Solutions dearly show the north branch of the intermediate water. We can conclude that whether we consider h = constant or not, the effect is more critical for the intermediate water layer.

CONCLUSIONS

The solution of the equation for the conservation of potential vorticity, solved for a twolayer ocean and considering that the depth of each layer can vary with position, reproduces the general pattern for the circulation in the Straits of Sardinia. In the intermediate water layer the circulation is dominated by the anticyclonic eddy in the southern part and a cyclonic eddy in the northern one. This shows that the hypothesis of the planetary origin of the two eddies is valuable. The two eddies form an obstacle to the intermediate water flowing to the west and this may be why it is divided into two branches. The northern branch parallels the Sardinian Continental Shelf and this appears in the solutions when it is considered that the depth of the interface between layers is variable (a function of position). The fact that this does not appear in solutions when h = constant suggests that the northern branch of intermediate water flowing to the west is related to a varying interfacial layer.

REFERENCES FRASSFrro R. (1964) A study of the turbulent flow and character of the water masses over the SicilianRidge in both summer and winter. Rapports et Proces-Verbaux, Commission Internationale pour l'Explication Scientifique de la Met Mediterranee, 18, 812-815. GARZOLI S. and C. MmLLAgD (1977) Hydrologie et circulation hivernales dans les canaux de Sicile et de Sardaigne. Rapport internal Laboratoire d'Oceanographie Physique du Museum, Paris. GARZOLI S. and C. MAmLAP,D (1979) Winter circulation in the Sicily and Sardinia Strait region. Deep-Sea Research, 26, 933-954. LACOMBE H. (1977) The meso-scale and local response of the Mediterranean to the transfer and exchange of energy across the sea surface. In: Physics of oceans and atmosphere, Vol. 1, International Centre for Theoretical Physics, Trieste (Italic), pp. 211-278. MOREL A. (1971) Caracteres hydrologiques des eaux echangees entre le bassin oriental et le bassin occidental de la Mediterranee. Cahiers Oceanographiques, 23, 329-342. MORELLI C., C. GA~TAR and M. PISANI (1975) Bathymetry, gravity and magnetism in the Straits of Sicily and in the Ionian Sea. Bolletino de Geofisica Teorica ed Applicata, 17, 39-58. NmI.SEN J. (1912) Hydrography of the Mediterranean and adjacent waters. Report on the Danish Oceanographical Expeditions 1908-1910, Copenhagen, pp. 72-191. SAI~T-GumY B. (1961) Influence de la variation avec la latitude du parametre de Coriolis sur les mouvements plans d'un fluide parfait. Cahiers Oceanographiques, 12, 184-198. TcrmRNm P. (1960) Hydrologie d'hiver en Mediterranee occidentale. Cahiers Oceanographiques, 12, 184-198. WOST G. (1959) Remarks on the circulation of the intermediate and deep water masses in the Mediterranean Sea and the methods of their further exploration. Annali Istituto Universitario Navale, Napoli, 28, 3-16.

Eddies in the Sardinia and Sicily Straits region

85

APPENDIX

The numerical procedure We have to solve the differential equation:

~h. ~

V2~

h

t- A + B~ = 0,

(1)

where

A = -h(f+ha);

B = -h2b

with ~, = ~kc(x,y) at the boundary. Because we know only a finite number of ~c, A, B, and h values, irregularly placed (see Figs 2, 3), we can make equation (1) discrete using an irregular grid. Let us call x~, y~ the coordinates of the ith point. For every ith internal point we consider the eight nearest points (internal and boundary points) that best surround it. To do that let us generate the integer function P(i, l) with 1 < l < 9 and i the index of an internal point, whose value is, for I = 1, P(i, 1) = 1 ; otherwise its value is the index of the ith point. To compute the first and second derivatives of a generic function, g(x, y), we approach it in the neighbourhood of the ith internal point using the quadratic form: 6

g(x, y) " Y. O~k ")fl~(X--Xi, Y--Yi);

(x,y) "~ (xl, yl),

k=l

where

ill(r, s) = 1,

fl4(r, s) =- r2/2,

p~(r, s )

P d r , s) -

- r,

fla(r, s) ----s,

rs,

fl6(r, s) -- s2/2,

and a~) are six constant values for every i point. To smooth any error, the a~~Jvalues were obtained by the least-square method applied to the nine P(i, 1) points. If we call gi = (xi, Yi) and ~

-

OP.,jj j=l

x,, ye,.~-y~)]

[~°pdxp,.i~k=l

the minimum condition is honored when --=0,

k = 1,6.

Such a condition is satisfied when 6

k - - 1,6, j=l

where

~l,~k = Z [flj(Xe(i,t)-- xi, Ye(t,z)- Yi) . flk(xe(i.,)- xi, Ye..,)- Yi)], j , k - - 1,6, 9

u~ ~ =- ~ [ f l d x e . , t j - x . y p ( i . i } - y i ) . a P . , j ,

k -- 1, 6.

/=1

If we define 6

~ -•

Y. {C('r 0 ) - 1 b,P,(x.,.~-x,,y.,.~-y,)}. l=l

j=l,6, k = 1,9,

86

SILVIA GARZOLI,VALERIOPARIS~and ELIO PASCHINI

we have that 9

• =

d~Oe.,j), j=l

O(x,Y) ~-

d Op(Lj)flk(X--Xt, y - - y i k=l kj=l

,

(x,y) ~-- (X,,y,).

J

So we can say that in the ith point the following analytical expressions are approached: 9 V2O (i) ~- Z

[ ( d 4(i)j + d 6 j(i) )gp(l,j) ]

j=l Og(i)

9 --

0x

j=l

Og(O

~ 2j~P(i,j)J

9 --

~ 3j~P(i,j)P

The other terms of equation (1) can be approximated using the same principle. Thus. for each internal point we have the relation: E

d(i) t¢2k ht~P(i, k) '."2j

j=1

X" d(i) I,

~ 1 k ¢~P(i, k)

k=1

E d(O/, ~ 3 k r'P(i,

k)

q k=l 9 Z

k=l

a.) o ,.3j + d<~jBe..j)

O) d/eo.j) + dtjAp(l,j)

= 0 (2)

d(i) l.,

"1 k'~P(i, k)

and all the i equations for the internal points must be simultaneously solved. Equation (2) can be rewritten in matrix form: F~,~+C~c+T

= 0.

(3)

The matrices F and C and the vector T are easily obtained from (2). So the values of ~,~ can be obtained inverting the linear relation (3). ~t = -F-IC~Pc - F - 1 T "

In this way, the values of ~ in the whole domain (the values of ~ at the boundaries, ~c, are given as data) can be obtained.