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Storm effects on the baroclinic tidal field in the Bay of Biscay
Journal of Marine Systems, 4 (1993) 327-347 Elsevier Science Publishers B.V., Amsterdam
327
Storm effects on the baroclinic tidal field in the Bay of Biscay J.Y. Le T a r e a u and R. Maze Laboratoire de Physique des Ocdans, Facult~ des Sciences, Av. V. Le Gorgeu, 29287 Brest Cedex, France (Received October 10, 1992; revised and accepted March 1, 1993)
ABSTRACT The effect of eastward travelling storms over the baroclinic tidal field of the Bay of Biscay is studied. A three-layer bidimensional model of generation and propagation of internal tides is constructed over the whole area of the Bay of Biscay. This model is forced by the M 2 and S 2 barotropic tidal components. Mixed layer physics and thermocline mixing are incorporated in the model. Numerical experiments show the development of a temperature field in the surface layer. The most important feature of this temperature distribution is the creation of patches of cold water near the shelf-break, in the north of the studied domain. These patches which are geographically localized in areas of large baroclinic tidal intensity, are conspicuous in shelf-break regions. This is consistent with satellite infra-red images taken from April-May to September-October which often show persistent ribbons of cool water along the European continental shelf, from west Ireland to south-west Brittany. The geographical location of these cool surface ribbons is well predicted by the model. Such results suggest that interactions between internal tides and surface fluxes can play an important role in the shelf-break cooling process.
Introduction It is now well known that continental slopes are likely places for the generation of high amplitude internal tides (Baines, 1982; MazE, 1983). These waves, which appear as the result of interaction between barotropic tides and sloping topography, have often been observed along the European continental slope. Dynamically active areas are found from the Bay of Biscay (Pingree and Mardell, 1981; Pingree et al., 1983; MazE, 1987; Serpette and MazE, 1989) to the Malin Shelf, north of Ireland (Sherwin, 1988). Field data exhibit internal tides propagating from the shelf edge toward the coast and toward the deep Ocean. On the continental shelf non-linearities lead to a deformation of the internal tide (Pichon and MazE, 1990). Layered numerical models have been developed to investigate tidally forced nonlinear internal wave generation over a one-dimensional sloping topography. Two-layer models have been described by Pingree et al. (1983) and by Maze
(1987). The three-layer model presented by Willmott and Edwards (1987) is an extension of the model due to Pingree et al. (1983). A three-layer numerical model has also been developed by Maze and Le Tareau (1990). These one-dimensional layered models predict an internal tide field which is in good agreement with environment observations. Satellite infra-red images taken from AprilMay to September-October often show persistent patches of cool water along the Celtic Sea shelfbreak (Pingree et al., 1982; Le Tareau et al., 1983; Pingree, 1984; Maze et al., 1986). As shown in Fig. 1, intensification of the cooling is observed northwestwards. This feature appears to be connected with the increase in intensity of the baroclinic internal tide field (Langlois et al., 1990). In previous studies, it has been suggested that interaction between internal tides and mixing processes gives a possible explanation for the observed spatial distribution of some water properties (temperature, nutrients, phytoplankton abundance . . . . ) in these regions. Thus, the im-
0924-7963/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
328
portance of interaction between wind induced mixing and linear baroclinic tides has been conjectured by Maz6 et al. (1986) to explain surface cooling in shelf-break regions. On the other hand, internal mixing near the continental shelf-break in the Bay of Biscay has also been investigated by employing a linear model of internal tides (New, 1988). In a more recent p a p e r (Maz6 and Le Tareau, 1990) we have demonstrated the role of interaction between internal tides and mixing processes
J.Y. LE T A R E A U A N D R. MAZI~
in the cooling of surface waters, resulting in cold spots of water consistent with those observed on satellite images in the neighbourhood of the continental slope. In this study a one-dimensional, three-layered model of generation and propagation of non-linear internal tides in a rotating environment is used. M o m e n t u m and continuity equations are solved on a transect perpendicular to the shelf-break. Results show internal tides propagating onshore and towards the abyssal plain from the edge. Different wave shapes are ob-
Fig. 1. Example of satellite infra-red image illustrating shelf-break cooling near the Celtic Sea and the Armorican continental slopes (NOAA7 satellite image--1026 GMT, 4 September 1981).
STORM
EFFECTS
ON BAROCLINIC
TIDAL
FIELD
329
IN BAY OF BISCAY
49 °
associated with Kelvin-Helmholtz instabilities and bottom drag mixing. By imposing a wind stress on the transect, the model shows the generation of a patch of cold water at the top of the slope. Later on, this patch is advected alternatively over the shelf and over the slope by tidal currents. This model, being one-dimensional, is not appropriate for simulation of geographical variations of the phenomenon, in particular along the slope. In the present study, the above described model is extended to two dimensions. The full non-linear equations consistent with a threelayered ocean structure including earth rotation effects are solved over the oceanic area represented in Fig. 2. At each grid point on the numerical lattice and at each time step, energetic fluxes accross the a t m o s p h e r e - o c e a n interface are included in the model, leading to a modification of both the internal tide and the temperature fields. Model results are compared with environment observations collected near the shelf-break and on the continental shelf during the Ondine 85 Cruise in the Bay of Biscay.
48*
47*
46 °
45 °
44 °
-8 °
-7 °
-6 °
-5 °
.4 °
-30
-2 °
_1 o
Fig. 2. Model coverage, topography of the continental slope and the location of points (A1, A 2. . . . T1, T2, etc.) at which model results have been compared.
served over the continental shelf and the abyssal plain. Three mixing processes are included in the model: surface wind mixing, thermocline mixing
6,
DENSITY hI° I 8L-h,
"' ~'~
P3(Z)N 0"(2~
h2o 111"
&
D Shelf
D
g
Shelf
h2
P2 t>c
1'=0
p
X•]Io ~\\\\\\\\\\\\-" Offshelf
H, = h,. n~
~\\\\\\\\\-%.
~\\\\\\\.~
H2= D-ha
Onshelf --~
Fig. 3. Schematic diagram of the three-layer model. (for notations, see first part of the second section).
330
J.Y. LE TAREAU AND R. MAZt~
coefficient of thermal expansion of seawater; horizontal eddy viscosity coefficient; horizontal eddy diffusion coefficient; nabla operator in the horizontal plane; heat budget across the oceanatmosphere interface; specific heat of seawater; wind speed at sea surface; surface wind stress; bottom tidal friction; vertical eddy viscosity coefficient in mixed layer i; vertically integrated wind induced current in the surface mixed layer; "Equivalent" vertically integrated current induced by bottom tidal friction in the lower mixed layer; linear damping coefficient for flows D w and DB; overall Richardson number in mixed layer i; local Richardson number in the thermocline; entrainment rate in mixed layer i; Brunt-V~iis~il~i frequency in the thermocline [N 2 = g ' / ( h 2 h0].
Theoretical background
As shown in Fig. 3, a three-layered vertical ocean structure is assumed: - a well mixed surface layer, - a well mixed bottom layer, - an intermediate thermocline layer. The internal tide model in its one-dimensional form has been described intensively by Maz6 and Le Tareau (1990). Here we summarize its main caracteristics and present the governing equations used in the two-dimensional extension of the model. For further details the reader should refer to Maz6 and Le Tareau (1990).
A: K: 7:
Q:
cp: W:
TW: TB:
Avi:
Symbols Dw: Using subscript i = 1 for the layer, i = 2 for the bottom mixed for the intermediate thermocline for variables and parameters are
surface mixed layer and i = 3 layer, symbols defined as fol-
Du:
lOWS:
~: "01: "/"/2('03):
hi(h2):
D:
Hi: hl0(h20): f: z0: U(U,V): Ui(u,v/): Pi:
Pi: Ti: g: g':
Sea surface elevation induced by the barotropic tide; Sea surface elevation induced by the baroclinic tide; baroclinic vertical displacement of the upper (lower) boundary of the thermocline; upper (lower) interface depth with respect to sea surface at rest in the only presence of the internal motion; mean water depth; instantaneous mixed layer thickness; mean upper (lower) interface depth; inertial frequency; vertical unit vector in the upward direction; barotropic tidal current; baroclinic tidal current; density; pressure; temperature; gravitational acceleration; reduced gravity;
/~:
Rii: RiT: Wei: N:
Assumptions In each layer currents are assumed to include the barotropic tidal component and a baroclinic perturbation: U(i) = U + U i
Sea surface elevation and interface displacements are accordingly written:
so2 -
~(D - h , ) D
+ "02
~3
~(D -hz) D
+ "03
331
STORM EFFECTS ON BAROCLINIC TIDAL FIELD IN BAY OF BISCAY
Upper and lower mixed layer depths are therefore: H1=
1+~
gradient terms and assuming ~/D << 1, the following system of equations which describe the internal tide is derived:
hl +rl I DU
at aU 1
In these expressions: ~/D << 1 (except near the coast). In the thermocline layer, density and baroclinic currents are depth-dependent and assumed to vary linearly between their value in the two mixed layers. The pressure is assumed to be hydrostatic. A closure assumption is defined in order to relate the baroclinic elevation of the sea surface to the depth of the thermocline:
at
-
(6)
[(u + ui). v]ui - (u, . v ) u - fzo A Ul - g ' ( 1 - ~-~)gh + Ag2U,
(7) DU-V
at
(8)
+ V " [ ( D - h2)U2]
aU: 0--7- = - [(U -4- U2). V]U 2 - (U 2 -V)U - f z 0 A U 2 + g ' ~ V h +AV2U2
with h = (h 1 + h2)/2 Momentum and continuity equations in the two well mixed layers define the motion according to the three-layered ocean structure.
Basic conservation equations Total current and depth of the mixed layers: aU(i) - -
at
aHi at
-~- - - ( U ( / ) • V ) U ( i )
-fz
0 A U(i) -
= -V'(HiU~i))
1 --VP i Pi
(2) (3)
where i = 1, 2.
Barotropic current and connected sea surface elevation aU -- = -(U'V)U-fz at
o A U-gV~
(4)
0£ -- = -V"
at
[ ( D + s~)U]
(5)
where g' = g f ( T a - T2). In the momentum Eqs. (7) and (9) the last term has been incorporated on the right hand side. This term represents attenuation by eddy viscosity with coefficient A and also assists with numerical stability. The source term for the internal motion appears in the continuity Eqs. (6) and (8) (first term on the right hand side). Because of the above prescribed assumptions the model only retains the first baroclinic mode. On the other hand, the structure of the baroclinic tide model does not permit a feedback effect of the baroclinic motion upon the barotropic tidal forcing. Therefore the barotropic tide Eqs. (4) and (5) and the internal tide Eqs. (6), (7), (8) and (9) are solved independently.
Temperature equations Temperature variations in the two mixed layers are governed by the following heat conservation equations:
Q
aT1 Baroclinic tide model Equations for the baroclinic motion are obtained by substracting Eqs. (4) and (5) from Eqs. (2) and (3) respectively. By taking into account the closure assumption [Eq. (1)] in the pressure
(9)
at
[(U-F
U1)*V]T
1 -f- -
-
PlCphl
+ KV2T1
(10) 0t
[(v + u:) -v] T: + KV:T:
(11)
332
J.Y. L E T A R E A U
and by the mixing processes which will be described below. The second term on the right hand side of Eq. (10) accounts for heat budget at the sea surface. However in the numerical experiments described in this p a p e r (fourth section), the heat conservation equation is solved without any heat input from the atmosphere (Q = 0). The last term on the right hand side of Eqs. (10) and (11) represents horizontal diffusion and plays a stabilizing effect on the numerical scheme. Mixing processes
In the model Eqs. (6), (7), (8) and (9), vertical fluxes across the thermocline boundaries are ignored. In the same way, the heat conservation Eqs. (10) and (11) only account for t e m p e r a t u r e variations due to advection, heat fluxes at the ocean surface and horizontal diffusion. Vertical mixing plays a primary role in thermocline evolution. In connexion with vertical fluxes, it governs mixed layer temperature, thermocline depth and thermocline thickness. T h r e e vertical mixing mechanisms are defined and introduced in the model.
Shear in the surface mixed layer generates turbulence which is available for thermocline deepening. This process is taken into account by defining an entrainment velocity of water from the thermocline layer into the surface mixed layer. This entrainment rate, which depends on wind current shear at the bottom of the surface mixed layer is parameterized by the expression: dh 1 dt
Wel
Z
i' O/
Pl
Pz
•
I'
/|
(12)
Av 1 - plNv/~
= g g ' h l hl + h 2
The wind induced flow in the surface mixed layer D w is given by the linear depth integrated m o m e n t u m equation: c3Dw -
-
at
Thermocline erosion at the bottom of the surface mixed layer is assumed to result from wind stress action on the sea surface. This stress creates a wind induced current in the surface layer.
Au~ 2 a hi Ri~ a
where a and a are two dimensionless coefficients. Their numerical values result from fitting Eq. (12) to laboratory experiments of Kato and Phillips (1969); a = 0.205, a = 0.9. The vertical eddy viscosity coefficient A v 1 and the bulk Richardson Ri I number are given by:
-
Wind mixing
A N D R. M A Z I ~
'r W fzo A Dw + --
Pl
- txD w
(13)
Where ~'w is the surface wind stress. A linear drag force - / z D w is introduced on the right hand side of equation (13) in order to damp wind induced inertial oscillations (/x = 4 . 1 0 -6 s - l ) .
P
° II I
L
t.
I I
(I)
(11)
(111)
(rv)
Fig. 4. Evolution of the vertical density structure as a result of mixing. (I) Initial stratification; (II) Surface mixing (wind stress on the sea surface); (III) Thermocline mixing (K.H. instabilities); (IV) Bottom mixing (tidal drag on the ocean floor) (dashed line: before mixing; continuous line: after mixing).
333
S T O R M E F F E C T S ON BAROCL1NIC T I D A L F I E L D IN BAY O F BISCAY
The wind induced flow D w therefore is assumed to take the same form as in an ocean initially at rest: no interaction term with the tidal motion is introduced in Eq. (13). The flow D w will only be used in parameterizing vertical mixing. This process is illustrated schematically in Fig. 4 by the vertical density profiles (II).
Thermocline mixing Kelvin-Helmoltz instabilities lead to mixing in the thermocline layer if the local Richardson number Ri r falls below 0.25 (Thorpe, 1971).
gi T
g'(hz-hl)
(14)
(AU) 2
where AU -- U 1 - U 2 + D w / h 1. Shear instabilities induce a thickening of the thermocline layer until the local Richardson number reaches a critical value equal to 0.3 (Corcos and Sherman, 1976). Therefore, after thermocline mixing, the new thermocline thickness is given by: d = h2- hI
0.3(AU) 2 g'
(15)
This expression is consistent with conservation of a linear vertical density profile in the thermocline layer. Density profiles (III) in Fig. 4 illustrate the thickening of the thermocline layer schematically as the result of this mixing mechanism.
In the bottom mixed layer a mixing mechanism similar to the one in the surface mixed layer is assumed, the wind stress on the sea surface being replaced by the tidal friction on the sea floor. Therefore, to parameterize bottom mixing, an entrainment rate at the top of the layer is defined: dh2
dt
Av2
a
2 a ~ - h2gi2
(16)
where A/-,' 2 and RiEare given by:
Au2
0DB 8t
-fz 0 A D a +
TB
~D a
(17)
P2
where ~'B is the bottom tidal friction and - / ~ D B a linear damping term. Mixing in the bottom layer vanishes over the deeper part of the continental slope and on the abyssal plain. The bulk Richardson number associated with the bottom layer (Ri z) is large in these regions. This implies a very weak entrainment from the thermocline layer into the bottom layer. On the other hand shelf and shelf-break areas will be sensitive to bottom mixing. In particular this process can induce a warming effect on the bottom temperature. In shelf-break areas, the bottom tidal drag will account for increasing turbulence connected to high level internal wave energy which plays an important role in sediment transport in the north of the Bay of Biscay (Castaing et al., 1981; Heathershaw et al., 1987). The interaction of internal tides with sediments can also lead to the formation of large-scale sand ridges on continental shelves (Boczar-Karakiewicz et al., 1991). Density profiles (IV) in Fig. 4 illustrate the bottom mixing mechanism schematically. Numerical model
Bottom mixing
We2
D B is an "equivalent" flow solution of the linear equation:
P2N~/~ 3D - 2h I - h 2 { D - h 2 ~2
Ri2=lg'(O-h2) 2D_h1-~2 I - ] - ~
)
Barotropic tidalforcing The system of Eqs. (4) and (5) governing the barotropic tide has been linearized and solved numerically in the Bay of Biscay by means of a relaxation method (Serpette, 1989). Results of this integration produce the barotropic components of the M 2 and S 2 tide. These harmonic components are used in the calculation of the barotropic tidal currents which appear in the forcing terms for the generation of internal tides. Barotropic currents are available at grid points as defined in Fig. 5a. The actual geographical distribution of these points in the north of the Bay of Biscay is shown in Fig. 5b.
334
J.Y. LE TAREAU
Baroclinic tide Integration of the model Eqs. (6) to (11) is performed numerically using of a "leapfrog" scheme for approximation of the time derivatives, and centered finite difference quotients at the nearest points for approximation of space derivatives. The scheme is applied to the staggered spatial grid shown in Fig. 5a. This grid is an Arakawa C-lattice for internal motion (Mesinger and Arakawa, 1976). A square grid space Ax = Ay = 2.5 km and a time step At = 532 s are taken. These values are
/,,"
6,
i
J
J
.4/
7/ _/
_
__s//.. / / f k-1
V-
f k
k~
AND
R. MAZIE
consistent with numerical stability. A larger time step could be used. However, here we want to reach a good resolution for the internal tide evolution versus time. Therefore integration over a M 2 tidal period is achieved after 84 time steps. Implicit Orlanski boundary conditions are applied to the whole set of dependent variables, at the edges of the domain (Orlanski, 1976). In addition a sponge layer is applied over 10 grid intervals all around the study area. This procedure allows elimination of false reflections at the boundaries crossed by sloping topography. In order to eliminate spurious numerical waves introduced by the "leapfrog" scheme, time filtering is necessary. A simple three-point centered filter is applied at each time step to the whole time-dependent variables. Let X be one of these variables. Then:
=x(t) + S [ X ( t - At) - 2 X ( t ) +X(t + At)] S is the filter parameter taken here as S = 0.08 and the overbar denotes the filtered value of X.
f k+!
I
®
variables [
0
ul
I
•
Vl
i iiiil i l
......
:::::: :::::: :::::: :::::: :::::: :::::: :::::: ::::::
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiiiii!iiii! i!iiiiii!iii:
i iiiiiiii
iiiiiiiiiiiiiiiii i!iiiiliiiiiiiiiii iiiiiiiiiii i!iiii!i
Fig. 5. a. Horizontal numerical grid. Dashed lines connect the equivalent points of the calculation (same indices k, l in the mesh) for the various quantities. (for notation see first part of second section), b. Geographical distribution of the grid points in the North of the Bay of Biscay.
Mixing At the end of each time step and throughout the integration of the internal tide equations, the depths of the upper and lower boundary of the thermocline (h I and h 2 respectively) are modified according to the entrainment relationships Eqs. (12) and (16), and the stability requirement in the thermocline layer. The following procedure therefore occurs: a first evaluation of h 1 at time t + At is given from time and space differencing of Eq. (6). Let h ~ be that value. A second estimation h~* is obtained from time integration of the entrainment Eq. (12). In the same way we get a first evaluation h i of h 2 at time t + At from Eq. (8) and a second one, h i * from Eq. (16). To get h~* and h i * , integration of flow Eqs. (13) and (17) is also performed on the mesh shown in Fig. 5. This method is also applied to variables U~ and T t in the upper mixed layer and to U 2 and T2 in the lower one. Equations (7), (10) and (9), (11) lead to a first evaluation of the concerned vari-
STORM
EFFECTS
ON BAROCLINIC
335
TIDAL FIELD IN BAY OF BISCAY
able denoted Xi* (i = 1, 2). Conservation of heat content or conservation of momentum in a fluid column then gives a second evaluation X/** after mixing. Knowledge of variables Xi** and h * * leads to the calculation of the thermocline Richardson number Ri r given by Eq. (14). Two cases can arise: - If Ri T > 0.25, no internal mixing occurs and the estimates of Xi** and h** are the final values of the dependent variables at time t + At; - If Ri r < 0.25, shear instabilities induce mixing in the thermocline and an associated thickening of the intermediate layer. Assuming symetrical variations of h I and h 2, the final values of these depths become:
hl(t + At) =
h~* + h ~ * - d 2
h2(t + At)
h~*+h~*+d 2
=
OUTLINE FLOWCHART ]
I
BAROTROPI ~DE CI ~,~
!l
BAROCUNI ~DE C ENTRAI MIXINNGMENT MECHANI CAL B (wiFORCI ntiddfrial~frllNocG ntlon)
I
boHom
THERMAL FORCI NG (surface heat fluxes)
where d is given by Eq. (15). The variables Xi** are unaffected by thermocline mixing so:
I
X/(t + At) =Si** As previously mentioned, a more detailed analysis of these procedures may be found in Maz6 and Le Tareau (1990). The outline flowchart of the global model is represented in Fig. 6. Results and discussion
Internal tides in the Bay of Biscay Simulations are performed by forcing the model with the M 2 and S 2 components of the barotropic tide. In the results discussed below, two kinds of initial conditions are used: - T10=15°C, - TI0=16°C,
T2o=12°C, T2o=12°C,
hlo=30m, h10=35m,
h2o=38m hzo=60m
(I) (II)
where Tlo, T20, hi0 and h2o a r e respectively the values of the surface temperature, the bottom temperature, the depth of the upper and lower limit of thermocline. These conditions can be
RESULTS (u,,v.h.T,) I
Fig. 6. Outline flowchartof the model.
found in the north of the Bay of Biscay in late summer and in autumn. Using the initial conditions (I) or (II) in the model, results show the generation of internal tides in the shelf-break region. These waves propagate over the shelf and toward the abyssal plain. A few areas of maximum internal tide amplitude can be observed in the Celtic Sea and Armorican region in the vicinity of the 200-250 m isobath. Interference between internal tides propagating from different locations at the shelf-break can occur, giving local stationary waves. Maximum values of the internal tide range are found at spring tides. Figure 7, which corresponds to the initial values (I), illustrates these characteristics. Another original feature on this diagram is the succession of regions of maximum amplitude in the vicinity
336
J.Y. LE TAREAU AND R. MAZI~
of the shelf-break. The maximum range for the baroclinic oscillation of the lower boundary of the thermocline is close to 54 m. The associated maximum range for the oscillation of the u p p e r limit of the thermocline reaches about 44 m. The maximum internal tide range is located near the Celtic shelf-break, in regions where persistent and localized bands of cool water are often observed from spring to autumn. The internal tide range in a particular source area depends on the stability p a r a m e t e r g ' , the initial thickness of the u p p e r layer ht0 and the initial depth of the lower boundary of the thermocline h20. For a given value of g ' the amplitude of the internal oscillation increases with increasing values of hlo (he0). On the other hand, an increase of the stability p a r a m e t e r g ' reduces the internal tide range. Figure 7 also shows that the internal tide field intensity is lighter in the south of the Bay. This is consistent with the existence of weak tidal currents in these regions.
49*
48*
47*
46*
45*
44'
-8*
-7 °
-6*
-50
-40
-30
-2 °
-10
Fig. 7. C o n t o u r s of s i m u l a t e d i n t e r n a l tide r a n g e of the l o w e r b o u n d a r y of the t h e r m o c l i n e at s p r i n g t i d e in m e t r e s w i t h initial c o n d i t i o n s (I):
T10 = 15°C, T20 = 12°C, hi0 = 30 m, h20 = 38 m.
:iiii!iiiiiiiiii iiJiiiiiiiiiiiiiii!!!!!!!iiiii!!!iiiiiiiii iii i ....................... iii-"~i........ ~::ii...............iiii:::iiii :'1
. . ~ s -~ t
::!ii!i!: !??ii::;iiiiiiiiiii!!d!~;[email protected]~iiiiii!ii!}~.:~i~.~-!!::i
i]i i?iii~-.::i::iiiiii ii]i;i~]i~;::~gjj'-'.~jjjtj]~!llflr
::
,
Fig. 8. Simulated baroclinic tidal currents in the surface mixed layer one day after neap tide (T = 24.6 days of integration). This chart corresponds to initial conditions (I).
Figure 8 [initial conditions (I)] represents the baroclinic current field in the surface layer in the north of the Bay at time t = 24.6 days (one day after neap tide). The maximum velocities are close to 13 cm s - l . The wavelength of the internal tide appears clearly on this chart. The propagation off-slope towards the ocean tends to become linear, but on-shelf, across the Celtic Sea and the Armorican shelf, the non-linear character of the wave field is well marked. Another interesting feature which is conspicuous on this chart is the existence of a succession of areas of weak and strong currents in the alongslope direction. These structures seem to be connected to the topography of the slope which shows the existence of submarine canyons in this region. In Figs. 7 and 8, the wavelength of the internal tide is close to 30-32 km over the abyssal plain and 25-30 km over the shelf. Figure 9, with initial conditions (I), shows the time evolution over 15 days of the two boundaries of the thermocline at different locations along a transect crossing the shelf-break (Points T 1 t o T 9 located on latitude 47.38°N in Fig. 2). The variation in shape of the internal wave and its decrease in amplitude with increasing distance from the shelf-break can be observed.
STORM EFFECTS ON BAROCLIN1C TIDAL FIELD IN BAY OF BISCAY DAYS D
T1
0
DAYS
DAYS
~ ~ so 3','
"
~
DAYS
-zo
"
T4 DAYS
-ao
~,,, DAYS
DAYS
~ -~0
~
T0
ii,f
~)AYS
i,
DAYS
,,
,,,
,,
,,
,,
,,,
,,
Fig. 9. T i m e v a r i a t i o n of the d e p t h of t h e r m o c l i n e b o u n d a r i e s at d i f f e r e n t p o s i t i o n s a l o n g an e a s t - w e s t t r a n s e c t across the c o n t i n e n t a l slope (see Fig. 1) w i t h initial c o n d i t i o n s (I).
337
These general features are also found in Fig. 10 which shows results obtained with the initial conditions (II). Initially, the thermocline is deeper and thicker, and the heat content larger than in case (I). Comparison between Figs. 9 and 10 indicates a significant enhancement of the internal tide amplitude in case (II). The oscillation of the lower boundary of the thermocline at spring tide is larger. Another conspicuous feature is the increase in the variation of the thermocline thickness over a M 2 tidal period in particular near the edge. Figures 11 and 12 show the time evolution of the limits of the thermocline at selected locations on the shelf-break with the initial conditions (I) and (II). In these two figures, a large set of wave shapes appears, showing the importance of the topography of upper part of the continental slope in the generation of the internal tide. In particular a steepening of the continental slope enhances the amplitude of the generated baroclinic waves. In the southern part of the Bay of Biscay the internal tide field is weaker. The role of the initial conditions can be observed by comparison of Figs. 11 and 12. It appears that the amplitude of the lower limit of the thermocline is larger with the set of initial values (II) than with the set (I). It is now known that the generation mechanism for internal tides is sensitive to numerous parameters: the depth of the thermocline, the vertical temperature gradient in the thermocline, the steepness of the slope, the barotropic tidal current intensity (Maze and Le Tareau, 1990). Thus a deep interface [case (II)] will oscillate with larger amplitude than a shallow one [case (I)]. During the Ondine 85 Cruise (September-October 1985) accurate field data was obtained in the north of the Bay of Biscay. The aim of this experiment was to study internal tides and their action on the spatial and temporal evolution of different physical, chemical and biological parameters. Moorings were deployed in the Celtic Sea. The moorings P1 and P4 (Fig. 2) provided temperature data between 10 and 100 m depth. These data were obtained by Aanderaa thermistor
338
J.Y. L E T A R E A U
TIME )days)
Ti lIME )days)
TIME (days)
TIME )days)
T4
TIME (days(
= ~so
i
TIME { cilys) 121#
I
2
)
4
5
i
I
I
I
~1
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IZ
Jl
i
2
1
,
s
I
r
l
*
Im
,I
*I
11
L4
IS
,: Z o ~!i)le~ .iii
TIME (days)
H
Ls
A N D R. M A Z I ~
chains. The vertical displacement of the thermocline is illustrated by the time evolution of the isotherms 13.5°C and 15°C (Figs. 13a and 14a). The amplitude of the vertical displacements is larger at mooring P~ (near the shelf-break) than at mooring P4 (on the shelf). This feature is enhanced during spring tides. The model response at locations P1 and P4 is shown in Figs. 13b and 14b. Comparison with data is difficult here because of the heterogeneity of the surface t e m p e r a t u r e field in the Celtic Sea region. Large horizontal t e m p e r a t u r e gradients were observed near the shelf-break on satellite images during Ondine experiment. The isotherms 13.5°C and 15°C lie in the thermocline but do no represent its lower and u p p e r boundaries. This choice was made because only these values remained in the range of depth of the records. However one can see that qualitatively the range of the thermocline displacements given by the model is in agreement with the data. To explain the difference between the model and the data, it must also be r e m e m b e r e d that only two components of the barotropic tide are used in the model, and that the model retains only the first baroclinic mode as previously mentioned. The topographic data used in the model (Fig. 2) are digitized from nautical charts and available at each grid point. Therefore topographic discontinuities of horizontal scale less than one mesh size are note taken into account. This causes a slight distortion of the bottom shape in shelfbreak regions particularly. An increase of the horizontal resolution of the computational grid in shelf-break areas would improve the model resuits.
TIME (days)
Model response to a travelling storm
',,,
~' ~' ~' ,'
~' ,'
~' ,'
,'
lIME (days)
The action of the wind and the associated mixing processes are introduced in the internal tide model as external forcing terms.
,; ,; ~ ,~ ,i ,~
~)'.
ii:I
Fig. 10. Same as Fig. 9 with initial conditions (II): T10= 16°C, 7"20= 12°C, hi0 = 35 m, h20 = 60 m.
STORM
EFFECTS
ON BAROCLINIC
TIDAL
FIELD
339
IN BAY OF BISCAY
DAYS
0 .++88
o~ -Z8 Ill -86 -tO ++0 +tO
+TO +85
Al
-I08
O +++-~ +.tO ~+
I
2I
I
41
I
I
I
,1
! ....
~
I ......
DAYS
I
-2+ II ++10 -58 -tO
+70 -80
A2 DAYS 10... .,,,+
-ZO
i!
+4G -50
-.
A3
-+88
~AYS 8
~
2~
i
~
f
t
i
tt
I
,,5~
t
i
L
|
t
1
+... o+ -ZO ,m -30
:~
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~t ~
-|0
-i08
8
I
, #
,,
DAYS I
I
+.,., o: -20 "t~
-30
-10
""
A 5
-100
Fig. 11. Time variation of the depth of thermocline boundaries at five selected points at the top of the continental slope [Initial conditions (I)].
340
J.Y. LE T A R E A U A N D R. M A Z E
TIME (days) .100
1
2
)
4
5
t
l
#
9
11
!
~
10
11
12
13
14
15
=
-60
-99 -IQ0 110
"
A1
TIME (days) .1o0
ua
1
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S
4
li
~
l!
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13
14
15
-20
T: -50
-, o" f
. . . . .
,v
~v IV IV |V
F ~ v" ~
-
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TIME (days) i.. e.a
lit
z
-lil~
~0
-~1,
A3
TIME (days)
0 .10 0 oa
w
-l# -]0 -40
1...
-69 -89 -$e -llt~
A4 t
0
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!
2
a
4r
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TIME
t, t
-I0 -40 =
-IO -eD
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-110
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Fig. 12. Same as Fig. 11 with initial conditions (II).
(days)
341
S T O R M E F F E C T S O N B A R O C L I N I C T I D A L F I E L D IN BAY OF BISCAY
therefore:
First a theoretical atmospheric depression of circular shape is assumed. The associated wind velocity defined in a polar coordinate system moving with the storm is given by the expression:
W(rm)
=
W o R--~oe - ( 1 - " / R°':j
Wx
where: j: unit vector normal to the radius vector Ro: "radius" of the depression r: radius vector W0: characteristic velocity The maximum intensity of the wind is found on a circle of radius rm:
=
-W o
Y --Yo
e-O
_~/x_Co(t_to_tl)/Ro)2
Ro
wy=w0
x - C o ( t - t o - ti)
X e -(1
R0 - ~[x - C o ( t - t o - t
1 ) / R 0 )2
where: Yo: ordinate of the storm centre; tl: time at which the storm is imposed in the model;
R0(1 + V~-) rm
1.19 W0
If we consider a storm moving eastward with a celerity C O, the components of the wind velocity in a Cartesian coordinate system are given by:
F
W(r)
=
2
P1
,o! 2O :5O
~4o
~ so
•
80-
W
90IO0 •
®
3'o'''','
''~
SEPT 85
''~'''~,'''~'
'~''
;'
OCT 85
2O
~ 3o ~4o o
/j
6C 7C 8C
®
J
Fig. 13. Time variation of the depth of: - two isotherms in the thermocline deduced from Ondine 85 data (a) - the boundaries of the thermocline calculated by the model (b) at location Px near the shelf-break.
342
J.Y. LE T A R E A U AND R. MAZIE
In order to see, under the most favorable circumstances, the influence of the moving storm on the whole baroclinic tidal field, time t 1 has been chosen such that the storm travels the domain during spring tides. Our interest in this section is focused on the evolution of the surface temperature field during the storm action. The wind field and the surface temperature structure at four selected times are displayed in Figs. 15, 16, 17 and 18. The time interval between the successive charts is equal to a M 2 tidal period (12.42 h). Initially the surface temperature is constant over the whole of the domain. At time t = 719.47 h the wind starts to blow over the domain (Fig. 15a), inducing surface mixing and the beginning
to: travelling time of the storm centre from its initial position at time t 1 to the west boundary of the domain. In our calculations we have chosen t 0 = 3 R o / C o which means that at time t 1 the storm centre is at a distance 3R o from the western boundary of the domain. The following numerical values are taken to define the storm: R o = 300 km Co = 8 m s -1 W0 = 2 0 m s -1 Y0 = 865 km Y0 corresponds to the latitude 51°N with the origin of the coordinate system taken in the south-west corner of the domain. The initial conditions (I) are prescribed.
1 I0~
2O
f
3C 4C £ 5O
/
b
6o
I
I
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®
I00 j I10-t
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'
'
30
I I
'
'
'
I 2
'
'
'
I 3
'
'
'
I 4
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I 5
'
'
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I 6
'
'
'
I
'
7
SEPT
OCT
85
85
2O E
30
cn I
40
uJ
50 60 70 8O
®
Fig. 14. Same as Fig. 13 at location P4 on the shelf, at 40 km from the shelf edge.
STORM
EFFECTS
ON
BAROCLINIC
TIDAL
FIELD
IN BAY
~ 4 ~ 4 ~ 4 4 ~
. . . . . . . . . . .
~~44444444444~qv'*-4~,~ ~ ~ 4 4 4 4 4 4 4 4 4 4 4 ~ t ~ t , 3 , , ~ ~.~444444444444,4a~*~,~
. . . . . . . . . . . . . . . . . . . .
~
t
. . . . . . . . . . . . . . . . . . . . .
..................... ................. P. . . . . . .
.
.
:::::::::::::::::::::::::::::::::::::: 4¢4~¢~,~,*1 ¢~¢*,,
J I
r < 15°c
. . . . . . . . . . . . . . . . . .
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4444444444444444 44~4444444¥44444
343
OF BISCAY
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.........................
S ii!i!!iiii!;;ii!;;il)i!!!; i~i!ii ~
'~
!iiiiiiiii;iiiiii;iiiii!
!
T = 15 °C
Fig. 15. Effect of a travelling storm on the surface t e m p e r a t u r e field at time t = 7 ] 9 . 4 7 hours: (a) w i n d field; (b) surface t e m p e r a t u r e field.
~.,-~r~MMHX~X~.~.~,,~tI'~ itlllIlll r~444
- -
. ~4
p ~ ~
~
~ ~ ~
' '
4 ~
4 ~
.
4 '
4 ~
~
4
4
4
4 ¢44444444 ~ I f44444444 ¢444444¢4 4444444444 4 444444 ~44~4qq
?ii?ii!iii??iiii!i?i?i!iii Fig. 16. S a m e as Fig. 15 at time t = 731.89 h.
344
J.Y. LE T A R E A U A N D R. MAZI~
o
o
0
T=15
Fig. 17. S a m e as Fig. 15 at t i m e t = 744.31 h. (spring tide).
Fig. 18. S a m e as Fig. 15 at t i m e t = 756.73 h.
°c
(
,I
STORM
EFFECTS
ON BAROCL1NIC
345
TIDAL FIELD IN BAY OF BISCAY
of surface cooling in the north-west of the Bay of Biscay where the temperature falls below 15°C (Fig. 15b). The 15°C isotherm propagates towards the south-east. The spatial oscillation of this isotherm is connected to the baroclinic tidal field during the mixing event. One tidal period later, the storm practically takes place over the whole of the domain (Fig. 16a). The surface temperature field at this time is displayed in Fig. 16b. The 15°C isotherm is located in the southern part of the Bay. In the north, a rather complicated temperature field occurs, in particular on the shelf. This feature is connected to the baroclinic tidal field in this region. Time t = 744.31 h corresponds to spring tides. The wind stress reaches its maximum intensity in the north of the Bay (Fig. 17 a). Surface mixing is therefore very active. In Fig. 17b the final temperature field is practically achieved. The last charts indicate the storm leaving the domain (Fig. 18 a) and the horizontal temperature structure induced by interaction between the travelling storm and the baroclinic tides (Fig. 18b). Eddy diffusion and advection by the tidal current will affect this temperature field, until a new gust of wind blows again over the domain. The most important feature which appears on the temperature charts is the creation of patches of cold water near the top of the continental slope. Moreover it appears that the cooling is greater in regions of large internal tide amplitude, in particular along the Celtic Sea shelfbreak. Such cold bands or spots of surface waters can be observed on satellite infra-red images taken in these regions (Fig. 1). These bands of cold water which follow the continental edge can be observed from April-May to September-October. The cold waters are generally associated with high nitrate and increased chlorophyll concentrations. These features have been observed during Ondine 85 Cruise (Le Corre, 1990, pers. commun.) These characteristic features have been described by numerous authors: Pingree and Mardell (1981), Dickson et al. (1980) from infra-red images by Satellites NOAA6 and TIROS N respectively. Field observations have also been made
by Maz6 et al. (1986), Langlois et al. (1990) and Maz6 and Le Tareau (1990). The good agreement between the geographical location of the patches of cold water and the area of large internal tide amplitude suggests that the process studied in this paper plays an important role in the shelf-break cooling in the Bay of Biscay. Conclusion and summary
A non-linear three-layer hydrostatic model has been used to predict the baroclinic tidal structure over the whole of the Bay of Biscay. This model is an extension to a two dimensional level of a previous work by Maz6 and Le Tareau (1990). It takes into account mixed layer physics (surface wind mixing, bottom drag mixing) and thermocline mixing due to Kelvin-Helmholtz instabilities. The model predicts regions at the shelf-break where internal tide amplitude, currents and connected vertical mixing are likely to be important. Irregularities of the depth contours near the shelf-break interact with the internal wave field forcing the internal wave to have an irregular, wavy, alongshore pattern in addition to the across-shore wave-line propagation of internal tides. The action of a moving storm over a baroclinic tidal field is studied. Results display the development of a surface temperature field with a shelf-break front. Patches of colder surface water are predicted along this front in the neighbourhood of the Celtic Sea slope. Results are in good agreement with oceanographic data and satellite infra-red images. This suggest that the action of a travelling storm on a baroclinic tidal field is an attractive explanation for the shelf-break cooling which is observed to occur in the north of the Bay of Biscay. Alternative explanations for shelf-break cooling have been put forward. Thus, Dickson et al. (1980) considered that shelf-break cooling was the result of the interaction between Kelvin waves propagating along the continental slope and the slope topography. Heaps (1980) suggested that the shelf-break cooling phenomenon was due to an upwelling which appears along the shelf edge as the result of a cross-shelf wind action. New
346
(1988) argued that cooling was due only to mixing created by internal motion that results from the interactions of the surface tide with bottom topography. Shelf-break cooling has been observed in other oceanic areas. In particular, it has been studied by Wolansky and Pickard (1983) in the Great Barrier Reef regions. These authors suggest that the cooling process is also connected to internal tides and presumably to internal Kelvin waves. All these mechanisms may probably contribute to shelf-break cooling. However in the Bay of Biscay, satellite imagery indicates that the phenomenon is more conspicuous in areas of strong tidal currents and therefore, of large internal tide amplitude and strong baroclinic currents. Shelfbreak cooling is also often enhanced at spring tides (Langlois et al., 1990). These observations are in good agreement with our results and suggest that the mechanism described here is an important factor in the cooling process. This work also shows the necessity of taking into account the two-dimensional aspect when modelling internal tides and the associated phenomenon in actual regions. The response of the model to other kinds of surface forcing may be analysed and in particular, we intend to focus on the calculation of residual baroclinic currents on the shelf and in slope areas. Observations collected during the Ondine 85 survey show vertical shear of residual currents and a rotation of these currents from the surface to the bottom. How well can these observations be described by the model? This question will be addressed in further studies.
Acknowledgements This research was carried out with the support of French "Service Hydrographique et Oc6anographique de la Marine" (S.H.O.M.), of the "Direction des Recherches et Etudes Techniques" (D.R.E.T.). Miss Christine Gu6renne carefully typed the manuscript and Mr. Pierre Doar6 drafted the figures.
J.Y. LE T A R E A U A N D R. M A Z E
We thank the referees for their valuable comments and remarks.
References Baines, P.G., 1982. On internal tide generation models. DeepSea Res., 29: 307-338. Boczar-Karakiewicz, B., Bona, J.L. and Pelchat, B., 1991. Interaction of internal waves with the seabed on continental shelves. Cont. Shelf Res., 11(8-10): 1181-1197. Castaing, P., Philippe, I. and Weber, O., 1981. Processus hydrodynamiques en bordure des plateaux continentaux: effets de la turbulence et des remont6es d'eau sur les s~diments. C.R. Acad. Sci. Paris, 292(II): 1481-1484. Corcos, G.M. and Sherman, F.S., 1976. Vorticity concentrations and the dynamics of unstable shear layers. J. Fluid Mech., 73: 241-264. Dickson, R.R., Gurbutt, P.A. and Pillai, N.V., 1980. Satellite evidence of enhanced upwelling along the European Continental Shelf. J. Phys. Oceanogr., 10: 813-819. Heaps, N.S., 1980. A mechanism for local upwelling along the European Continental Slope. Oceanol. Acta, 3(4): 449454. Heathershaw, A.D., New, A.L. and Edwards, P.D., 1987. Internal tides and sediment transport at the shelf-break in the Celtic Sea. Cont. Shelf. Res., 7(5): 485-517. Kato, H. and Phillips, O.M., 1969. On the penetration of a turbulent layer into a stratified fluid. J. Fluid Mech., 37: 643-655. Langlois, G., Gohin, F. and Serpette, A., 1990. Refroidissements locaux aux abords du talus continental armoricain. Oceanol. Acta, 13(2): 159-169. Le Tareau, J.Y., Maze, R., Le F~vre, J., Billard, C. and Camus, Y., 1983. ENVAT-81, Campagne de recherche multidisciplinaire en Atlantique: Aspects m~t~orologiques, chimiques, biologiques, hydrologiques et thermodynamiques. Met-Mar, 118(1): 6-25. Maze, R., 1983. Mouvements internes induits dans un golfe par le passage d'une d6pression et par la mar~e. Th~se de Doctorat d'Etat, U.B.O., Brest, 320 pp. Maze, R., 1987. Generation and propagation of non-linear internal waves induced by the tide over a continental slope. Cont. Shelf Res., 7(9): 1079-1104. Maz6, R. and Le Tareau, J.Y., 1990. Interactions between internal tides and energetic fluxes across the atmosphere ocean interface over a continental shelf-break. J. Mar. Res., 48: 505-541. Maze, R., Camus, Y. and Le Tareau, J.Y., 1986. Formation de gradients thermiques ~ la surface de l'Oc~an, au-dessus d'un talus, par interaction entre les ondes internes et le m61ange dfi au vent. J. Cons. Int. Explor. Mer, 42: 221-240. Mesinger, F. and Arakawa, A., 1976. Numerical methods used in Atmospheric models. GARP Publications Series, 17(1). WMO-ICSU Joint Organizing Committee, 64 pp. New, A.L., 1988. Internal tidal mixing in the Bay of Biscay. Deep-Sea Res., 35(5): 691-709.
STORM EFFECTS ON BAROCLINICTIDAL FIELD IN BAY OF BISCAY Orlanski, I., 1976. A simple boundary condition for unbounded hyperbolic flows. J. Comput. Physics, 21: 251-269. Pichon, A. and Maz6, R., 1990. Internal tides over a shelfbreak. Analytical model and observations. J. Phys. Oceanogr., 20(5): 657-671. Pingree, R.D., 1984. Some applications of remote sensing to studies in the Bay of Biscay, Celtic Sea and English Channel. In: J.C.J. Nihoul, (Editor). Remote Sensing of Shelf Sea Hydrodynamics. Elsevier Oceanography Series. Elsevier, Amsterdam, pp. 287-315. Pingree, R.D. and Mardell, G.T., 1981. Slope turbulence, internal waves and phytoplankton growth at the Celtic Sea shelf-break. Philos. Trans. R. Soc. London, A 302: 663682. Pingree, R.D., Griffiths, D.K. and Mardell, G.T., 1983. The structure of the internal tide at the Celtic Sea shelf-break. J. Mar. Biol. Assoc. U.K., 64: 99-113. Pingree, R.D., Mardell, G.T., Holligan, P.M., Griffiths, D.K. and Smithers, J., 1982. Celtic Sea and Armorican current structure and the vertical distributions of temperature and chlorophyll. Cont. Shelf Res., 1(1): 99-116.
347 Serpette, A., 1989. Mar6es internes dans le Golfe de Gascogne: un module bi-dimensionnel. Th~se de Doctorat Univ. Bretagne Occidentale, Brest, 242 pp. Serpette, A. and Maz6, R., 1989. Internal tides in the Bay of Biscay: a two-dimensional model. Cont. Shelf Res., 9(9): 795-821. Sherwin, T.J., 1988. Analysis of internal tides observed on the Malin shelf, North of Ireland. J. Phys. Oceanogr., 18: 1035-1050. Thorpe, S.A., 1971. Experiments on instability of stratified shear flows: Miscible fluids. J. Fluid Mech., 46: 299-320. Willmott, A.J. and Edwards, P.D., 1987. A numerical model for the generation of tidally forced nonlinear internal waves over topography. Cont. Shelf Res., 7(5): 457-484. Wolanski, E. and Pickard, G.L., 1983. Upwelling by internal tides and Kelvin waves at the Continental shelf-break on the Great Barrier Reef. Aust. J. Mar. Freshwater Res., 34: 65-80.
327
Storm effects on the baroclinic tidal field in the Bay of Biscay J.Y. Le T a r e a u and R. Maze Laboratoire de Physique des Ocdans, Facult~ des Sciences, Av. V. Le Gorgeu, 29287 Brest Cedex, France (Received October 10, 1992; revised and accepted March 1, 1993)
ABSTRACT The effect of eastward travelling storms over the baroclinic tidal field of the Bay of Biscay is studied. A three-layer bidimensional model of generation and propagation of internal tides is constructed over the whole area of the Bay of Biscay. This model is forced by the M 2 and S 2 barotropic tidal components. Mixed layer physics and thermocline mixing are incorporated in the model. Numerical experiments show the development of a temperature field in the surface layer. The most important feature of this temperature distribution is the creation of patches of cold water near the shelf-break, in the north of the studied domain. These patches which are geographically localized in areas of large baroclinic tidal intensity, are conspicuous in shelf-break regions. This is consistent with satellite infra-red images taken from April-May to September-October which often show persistent ribbons of cool water along the European continental shelf, from west Ireland to south-west Brittany. The geographical location of these cool surface ribbons is well predicted by the model. Such results suggest that interactions between internal tides and surface fluxes can play an important role in the shelf-break cooling process.
Introduction It is now well known that continental slopes are likely places for the generation of high amplitude internal tides (Baines, 1982; MazE, 1983). These waves, which appear as the result of interaction between barotropic tides and sloping topography, have often been observed along the European continental slope. Dynamically active areas are found from the Bay of Biscay (Pingree and Mardell, 1981; Pingree et al., 1983; MazE, 1987; Serpette and MazE, 1989) to the Malin Shelf, north of Ireland (Sherwin, 1988). Field data exhibit internal tides propagating from the shelf edge toward the coast and toward the deep Ocean. On the continental shelf non-linearities lead to a deformation of the internal tide (Pichon and MazE, 1990). Layered numerical models have been developed to investigate tidally forced nonlinear internal wave generation over a one-dimensional sloping topography. Two-layer models have been described by Pingree et al. (1983) and by Maze
(1987). The three-layer model presented by Willmott and Edwards (1987) is an extension of the model due to Pingree et al. (1983). A three-layer numerical model has also been developed by Maze and Le Tareau (1990). These one-dimensional layered models predict an internal tide field which is in good agreement with environment observations. Satellite infra-red images taken from AprilMay to September-October often show persistent patches of cool water along the Celtic Sea shelfbreak (Pingree et al., 1982; Le Tareau et al., 1983; Pingree, 1984; Maze et al., 1986). As shown in Fig. 1, intensification of the cooling is observed northwestwards. This feature appears to be connected with the increase in intensity of the baroclinic internal tide field (Langlois et al., 1990). In previous studies, it has been suggested that interaction between internal tides and mixing processes gives a possible explanation for the observed spatial distribution of some water properties (temperature, nutrients, phytoplankton abundance . . . . ) in these regions. Thus, the im-
0924-7963/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
328
portance of interaction between wind induced mixing and linear baroclinic tides has been conjectured by Maz6 et al. (1986) to explain surface cooling in shelf-break regions. On the other hand, internal mixing near the continental shelf-break in the Bay of Biscay has also been investigated by employing a linear model of internal tides (New, 1988). In a more recent p a p e r (Maz6 and Le Tareau, 1990) we have demonstrated the role of interaction between internal tides and mixing processes
J.Y. LE T A R E A U A N D R. MAZI~
in the cooling of surface waters, resulting in cold spots of water consistent with those observed on satellite images in the neighbourhood of the continental slope. In this study a one-dimensional, three-layered model of generation and propagation of non-linear internal tides in a rotating environment is used. M o m e n t u m and continuity equations are solved on a transect perpendicular to the shelf-break. Results show internal tides propagating onshore and towards the abyssal plain from the edge. Different wave shapes are ob-
Fig. 1. Example of satellite infra-red image illustrating shelf-break cooling near the Celtic Sea and the Armorican continental slopes (NOAA7 satellite image--1026 GMT, 4 September 1981).
STORM
EFFECTS
ON BAROCLINIC
TIDAL
FIELD
329
IN BAY OF BISCAY
49 °
associated with Kelvin-Helmholtz instabilities and bottom drag mixing. By imposing a wind stress on the transect, the model shows the generation of a patch of cold water at the top of the slope. Later on, this patch is advected alternatively over the shelf and over the slope by tidal currents. This model, being one-dimensional, is not appropriate for simulation of geographical variations of the phenomenon, in particular along the slope. In the present study, the above described model is extended to two dimensions. The full non-linear equations consistent with a threelayered ocean structure including earth rotation effects are solved over the oceanic area represented in Fig. 2. At each grid point on the numerical lattice and at each time step, energetic fluxes accross the a t m o s p h e r e - o c e a n interface are included in the model, leading to a modification of both the internal tide and the temperature fields. Model results are compared with environment observations collected near the shelf-break and on the continental shelf during the Ondine 85 Cruise in the Bay of Biscay.
48*
47*
46 °
45 °
44 °
-8 °
-7 °
-6 °
-5 °
.4 °
-30
-2 °
_1 o
Fig. 2. Model coverage, topography of the continental slope and the location of points (A1, A 2. . . . T1, T2, etc.) at which model results have been compared.
served over the continental shelf and the abyssal plain. Three mixing processes are included in the model: surface wind mixing, thermocline mixing
6,
DENSITY hI° I 8L-h,
"' ~'~
P3(Z)N 0"(2~
h2o 111"
&
D Shelf
D
g
Shelf
h2
P2 t>c
1'=0
p
X•]Io ~\\\\\\\\\\\\-" Offshelf
H, = h,. n~
~\\\\\\\\\-%.
~\\\\\\\.~
H2= D-ha
Onshelf --~
Fig. 3. Schematic diagram of the three-layer model. (for notations, see first part of the second section).
330
J.Y. LE TAREAU AND R. MAZt~
coefficient of thermal expansion of seawater; horizontal eddy viscosity coefficient; horizontal eddy diffusion coefficient; nabla operator in the horizontal plane; heat budget across the oceanatmosphere interface; specific heat of seawater; wind speed at sea surface; surface wind stress; bottom tidal friction; vertical eddy viscosity coefficient in mixed layer i; vertically integrated wind induced current in the surface mixed layer; "Equivalent" vertically integrated current induced by bottom tidal friction in the lower mixed layer; linear damping coefficient for flows D w and DB; overall Richardson number in mixed layer i; local Richardson number in the thermocline; entrainment rate in mixed layer i; Brunt-V~iis~il~i frequency in the thermocline [N 2 = g ' / ( h 2 h0].
Theoretical background
As shown in Fig. 3, a three-layered vertical ocean structure is assumed: - a well mixed surface layer, - a well mixed bottom layer, - an intermediate thermocline layer. The internal tide model in its one-dimensional form has been described intensively by Maz6 and Le Tareau (1990). Here we summarize its main caracteristics and present the governing equations used in the two-dimensional extension of the model. For further details the reader should refer to Maz6 and Le Tareau (1990).
A: K: 7:
Q:
cp: W:
TW: TB:
Avi:
Symbols Dw: Using subscript i = 1 for the layer, i = 2 for the bottom mixed for the intermediate thermocline for variables and parameters are
surface mixed layer and i = 3 layer, symbols defined as fol-
Du:
lOWS:
~: "01: "/"/2('03):
hi(h2):
D:
Hi: hl0(h20): f: z0: U(U,V): Ui(u,v/): Pi:
Pi: Ti: g: g':
Sea surface elevation induced by the barotropic tide; Sea surface elevation induced by the baroclinic tide; baroclinic vertical displacement of the upper (lower) boundary of the thermocline; upper (lower) interface depth with respect to sea surface at rest in the only presence of the internal motion; mean water depth; instantaneous mixed layer thickness; mean upper (lower) interface depth; inertial frequency; vertical unit vector in the upward direction; barotropic tidal current; baroclinic tidal current; density; pressure; temperature; gravitational acceleration; reduced gravity;
/~:
Rii: RiT: Wei: N:
Assumptions In each layer currents are assumed to include the barotropic tidal component and a baroclinic perturbation: U(i) = U + U i
Sea surface elevation and interface displacements are accordingly written:
so2 -
~(D - h , ) D
+ "02
~3
~(D -hz) D
+ "03
331
STORM EFFECTS ON BAROCLINIC TIDAL FIELD IN BAY OF BISCAY
Upper and lower mixed layer depths are therefore: H1=
1+~
gradient terms and assuming ~/D << 1, the following system of equations which describe the internal tide is derived:
hl +rl I DU
at aU 1
In these expressions: ~/D << 1 (except near the coast). In the thermocline layer, density and baroclinic currents are depth-dependent and assumed to vary linearly between their value in the two mixed layers. The pressure is assumed to be hydrostatic. A closure assumption is defined in order to relate the baroclinic elevation of the sea surface to the depth of the thermocline:
at
-
(6)
[(u + ui). v]ui - (u, . v ) u - fzo A Ul - g ' ( 1 - ~-~)gh + Ag2U,
(7) DU-V
at
(8)
+ V " [ ( D - h2)U2]
aU: 0--7- = - [(U -4- U2). V]U 2 - (U 2 -V)U - f z 0 A U 2 + g ' ~ V h +AV2U2
with h = (h 1 + h2)/2 Momentum and continuity equations in the two well mixed layers define the motion according to the three-layered ocean structure.
Basic conservation equations Total current and depth of the mixed layers: aU(i) - -
at
aHi at
-~- - - ( U ( / ) • V ) U ( i )
-fz
0 A U(i) -
= -V'(HiU~i))
1 --VP i Pi
(2) (3)
where i = 1, 2.
Barotropic current and connected sea surface elevation aU -- = -(U'V)U-fz at
o A U-gV~
(4)
0£ -- = -V"
at
[ ( D + s~)U]
(5)
where g' = g f ( T a - T2). In the momentum Eqs. (7) and (9) the last term has been incorporated on the right hand side. This term represents attenuation by eddy viscosity with coefficient A and also assists with numerical stability. The source term for the internal motion appears in the continuity Eqs. (6) and (8) (first term on the right hand side). Because of the above prescribed assumptions the model only retains the first baroclinic mode. On the other hand, the structure of the baroclinic tide model does not permit a feedback effect of the baroclinic motion upon the barotropic tidal forcing. Therefore the barotropic tide Eqs. (4) and (5) and the internal tide Eqs. (6), (7), (8) and (9) are solved independently.
Temperature equations Temperature variations in the two mixed layers are governed by the following heat conservation equations:
Q
aT1 Baroclinic tide model Equations for the baroclinic motion are obtained by substracting Eqs. (4) and (5) from Eqs. (2) and (3) respectively. By taking into account the closure assumption [Eq. (1)] in the pressure
(9)
at
[(U-F
U1)*V]T
1 -f- -
-
PlCphl
+ KV2T1
(10) 0t
[(v + u:) -v] T: + KV:T:
(11)
332
J.Y. L E T A R E A U
and by the mixing processes which will be described below. The second term on the right hand side of Eq. (10) accounts for heat budget at the sea surface. However in the numerical experiments described in this p a p e r (fourth section), the heat conservation equation is solved without any heat input from the atmosphere (Q = 0). The last term on the right hand side of Eqs. (10) and (11) represents horizontal diffusion and plays a stabilizing effect on the numerical scheme. Mixing processes
In the model Eqs. (6), (7), (8) and (9), vertical fluxes across the thermocline boundaries are ignored. In the same way, the heat conservation Eqs. (10) and (11) only account for t e m p e r a t u r e variations due to advection, heat fluxes at the ocean surface and horizontal diffusion. Vertical mixing plays a primary role in thermocline evolution. In connexion with vertical fluxes, it governs mixed layer temperature, thermocline depth and thermocline thickness. T h r e e vertical mixing mechanisms are defined and introduced in the model.
Shear in the surface mixed layer generates turbulence which is available for thermocline deepening. This process is taken into account by defining an entrainment velocity of water from the thermocline layer into the surface mixed layer. This entrainment rate, which depends on wind current shear at the bottom of the surface mixed layer is parameterized by the expression: dh 1 dt
Wel
Z
i' O/
Pl
Pz
•
I'
/|
(12)
Av 1 - plNv/~
= g g ' h l hl + h 2
The wind induced flow in the surface mixed layer D w is given by the linear depth integrated m o m e n t u m equation: c3Dw -
-
at
Thermocline erosion at the bottom of the surface mixed layer is assumed to result from wind stress action on the sea surface. This stress creates a wind induced current in the surface layer.
Au~ 2 a hi Ri~ a
where a and a are two dimensionless coefficients. Their numerical values result from fitting Eq. (12) to laboratory experiments of Kato and Phillips (1969); a = 0.205, a = 0.9. The vertical eddy viscosity coefficient A v 1 and the bulk Richardson Ri I number are given by:
-
Wind mixing
A N D R. M A Z I ~
'r W fzo A Dw + --
Pl
- txD w
(13)
Where ~'w is the surface wind stress. A linear drag force - / z D w is introduced on the right hand side of equation (13) in order to damp wind induced inertial oscillations (/x = 4 . 1 0 -6 s - l ) .
P
° II I
L
t.
I I
(I)
(11)
(111)
(rv)
Fig. 4. Evolution of the vertical density structure as a result of mixing. (I) Initial stratification; (II) Surface mixing (wind stress on the sea surface); (III) Thermocline mixing (K.H. instabilities); (IV) Bottom mixing (tidal drag on the ocean floor) (dashed line: before mixing; continuous line: after mixing).
333
S T O R M E F F E C T S ON BAROCL1NIC T I D A L F I E L D IN BAY O F BISCAY
The wind induced flow D w therefore is assumed to take the same form as in an ocean initially at rest: no interaction term with the tidal motion is introduced in Eq. (13). The flow D w will only be used in parameterizing vertical mixing. This process is illustrated schematically in Fig. 4 by the vertical density profiles (II).
Thermocline mixing Kelvin-Helmoltz instabilities lead to mixing in the thermocline layer if the local Richardson number Ri r falls below 0.25 (Thorpe, 1971).
gi T
g'(hz-hl)
(14)
(AU) 2
where AU -- U 1 - U 2 + D w / h 1. Shear instabilities induce a thickening of the thermocline layer until the local Richardson number reaches a critical value equal to 0.3 (Corcos and Sherman, 1976). Therefore, after thermocline mixing, the new thermocline thickness is given by: d = h2- hI
0.3(AU) 2 g'
(15)
This expression is consistent with conservation of a linear vertical density profile in the thermocline layer. Density profiles (III) in Fig. 4 illustrate the thickening of the thermocline layer schematically as the result of this mixing mechanism.
In the bottom mixed layer a mixing mechanism similar to the one in the surface mixed layer is assumed, the wind stress on the sea surface being replaced by the tidal friction on the sea floor. Therefore, to parameterize bottom mixing, an entrainment rate at the top of the layer is defined: dh2
dt
Av2
a
2 a ~ - h2gi2
(16)
where A/-,' 2 and RiEare given by:
Au2
0DB 8t
-fz 0 A D a +
TB
~D a
(17)
P2
where ~'B is the bottom tidal friction and - / ~ D B a linear damping term. Mixing in the bottom layer vanishes over the deeper part of the continental slope and on the abyssal plain. The bulk Richardson number associated with the bottom layer (Ri z) is large in these regions. This implies a very weak entrainment from the thermocline layer into the bottom layer. On the other hand shelf and shelf-break areas will be sensitive to bottom mixing. In particular this process can induce a warming effect on the bottom temperature. In shelf-break areas, the bottom tidal drag will account for increasing turbulence connected to high level internal wave energy which plays an important role in sediment transport in the north of the Bay of Biscay (Castaing et al., 1981; Heathershaw et al., 1987). The interaction of internal tides with sediments can also lead to the formation of large-scale sand ridges on continental shelves (Boczar-Karakiewicz et al., 1991). Density profiles (IV) in Fig. 4 illustrate the bottom mixing mechanism schematically. Numerical model
Bottom mixing
We2
D B is an "equivalent" flow solution of the linear equation:
P2N~/~ 3D - 2h I - h 2 { D - h 2 ~2
Ri2=lg'(O-h2) 2D_h1-~2 I - ] - ~
)
Barotropic tidalforcing The system of Eqs. (4) and (5) governing the barotropic tide has been linearized and solved numerically in the Bay of Biscay by means of a relaxation method (Serpette, 1989). Results of this integration produce the barotropic components of the M 2 and S 2 tide. These harmonic components are used in the calculation of the barotropic tidal currents which appear in the forcing terms for the generation of internal tides. Barotropic currents are available at grid points as defined in Fig. 5a. The actual geographical distribution of these points in the north of the Bay of Biscay is shown in Fig. 5b.
334
J.Y. LE TAREAU
Baroclinic tide Integration of the model Eqs. (6) to (11) is performed numerically using of a "leapfrog" scheme for approximation of the time derivatives, and centered finite difference quotients at the nearest points for approximation of space derivatives. The scheme is applied to the staggered spatial grid shown in Fig. 5a. This grid is an Arakawa C-lattice for internal motion (Mesinger and Arakawa, 1976). A square grid space Ax = Ay = 2.5 km and a time step At = 532 s are taken. These values are
/,,"
6,
i
J
J
.4/
7/ _/
_
__s//.. / / f k-1
V-
f k
k~
AND
R. MAZIE
consistent with numerical stability. A larger time step could be used. However, here we want to reach a good resolution for the internal tide evolution versus time. Therefore integration over a M 2 tidal period is achieved after 84 time steps. Implicit Orlanski boundary conditions are applied to the whole set of dependent variables, at the edges of the domain (Orlanski, 1976). In addition a sponge layer is applied over 10 grid intervals all around the study area. This procedure allows elimination of false reflections at the boundaries crossed by sloping topography. In order to eliminate spurious numerical waves introduced by the "leapfrog" scheme, time filtering is necessary. A simple three-point centered filter is applied at each time step to the whole time-dependent variables. Let X be one of these variables. Then:
=x(t) + S [ X ( t - At) - 2 X ( t ) +X(t + At)] S is the filter parameter taken here as S = 0.08 and the overbar denotes the filtered value of X.
f k+!
I
®
variables [
0
ul
I
•
Vl
i iiiil i l
......
:::::: :::::: :::::: :::::: :::::: :::::: :::::: ::::::
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiiiii!iiii! i!iiiiii!iii:
i iiiiiiii
iiiiiiiiiiiiiiiii i!iiiiliiiiiiiiiii iiiiiiiiiii i!iiii!i
Fig. 5. a. Horizontal numerical grid. Dashed lines connect the equivalent points of the calculation (same indices k, l in the mesh) for the various quantities. (for notation see first part of second section), b. Geographical distribution of the grid points in the North of the Bay of Biscay.
Mixing At the end of each time step and throughout the integration of the internal tide equations, the depths of the upper and lower boundary of the thermocline (h I and h 2 respectively) are modified according to the entrainment relationships Eqs. (12) and (16), and the stability requirement in the thermocline layer. The following procedure therefore occurs: a first evaluation of h 1 at time t + At is given from time and space differencing of Eq. (6). Let h ~ be that value. A second estimation h~* is obtained from time integration of the entrainment Eq. (12). In the same way we get a first evaluation h i of h 2 at time t + At from Eq. (8) and a second one, h i * from Eq. (16). To get h~* and h i * , integration of flow Eqs. (13) and (17) is also performed on the mesh shown in Fig. 5. This method is also applied to variables U~ and T t in the upper mixed layer and to U 2 and T2 in the lower one. Equations (7), (10) and (9), (11) lead to a first evaluation of the concerned vari-
STORM
EFFECTS
ON BAROCLINIC
335
TIDAL FIELD IN BAY OF BISCAY
able denoted Xi* (i = 1, 2). Conservation of heat content or conservation of momentum in a fluid column then gives a second evaluation X/** after mixing. Knowledge of variables Xi** and h * * leads to the calculation of the thermocline Richardson number Ri r given by Eq. (14). Two cases can arise: - If Ri T > 0.25, no internal mixing occurs and the estimates of Xi** and h** are the final values of the dependent variables at time t + At; - If Ri r < 0.25, shear instabilities induce mixing in the thermocline and an associated thickening of the intermediate layer. Assuming symetrical variations of h I and h 2, the final values of these depths become:
hl(t + At) =
h~* + h ~ * - d 2
h2(t + At)
h~*+h~*+d 2
=
OUTLINE FLOWCHART ]
I
BAROTROPI ~DE CI ~,~
!l
BAROCUNI ~DE C ENTRAI MIXINNGMENT MECHANI CAL B (wiFORCI ntiddfrial~frllNocG ntlon)
I
boHom
THERMAL FORCI NG (surface heat fluxes)
where d is given by Eq. (15). The variables Xi** are unaffected by thermocline mixing so:
I
X/(t + At) =Si** As previously mentioned, a more detailed analysis of these procedures may be found in Maz6 and Le Tareau (1990). The outline flowchart of the global model is represented in Fig. 6. Results and discussion
Internal tides in the Bay of Biscay Simulations are performed by forcing the model with the M 2 and S 2 components of the barotropic tide. In the results discussed below, two kinds of initial conditions are used: - T10=15°C, - TI0=16°C,
T2o=12°C, T2o=12°C,
hlo=30m, h10=35m,
h2o=38m hzo=60m
(I) (II)
where Tlo, T20, hi0 and h2o a r e respectively the values of the surface temperature, the bottom temperature, the depth of the upper and lower limit of thermocline. These conditions can be
RESULTS (u,,v.h.T,) I
Fig. 6. Outline flowchartof the model.
found in the north of the Bay of Biscay in late summer and in autumn. Using the initial conditions (I) or (II) in the model, results show the generation of internal tides in the shelf-break region. These waves propagate over the shelf and toward the abyssal plain. A few areas of maximum internal tide amplitude can be observed in the Celtic Sea and Armorican region in the vicinity of the 200-250 m isobath. Interference between internal tides propagating from different locations at the shelf-break can occur, giving local stationary waves. Maximum values of the internal tide range are found at spring tides. Figure 7, which corresponds to the initial values (I), illustrates these characteristics. Another original feature on this diagram is the succession of regions of maximum amplitude in the vicinity
336
J.Y. LE TAREAU AND R. MAZI~
of the shelf-break. The maximum range for the baroclinic oscillation of the lower boundary of the thermocline is close to 54 m. The associated maximum range for the oscillation of the u p p e r limit of the thermocline reaches about 44 m. The maximum internal tide range is located near the Celtic shelf-break, in regions where persistent and localized bands of cool water are often observed from spring to autumn. The internal tide range in a particular source area depends on the stability p a r a m e t e r g ' , the initial thickness of the u p p e r layer ht0 and the initial depth of the lower boundary of the thermocline h20. For a given value of g ' the amplitude of the internal oscillation increases with increasing values of hlo (he0). On the other hand, an increase of the stability p a r a m e t e r g ' reduces the internal tide range. Figure 7 also shows that the internal tide field intensity is lighter in the south of the Bay. This is consistent with the existence of weak tidal currents in these regions.
49*
48*
47*
46*
45*
44'
-8*
-7 °
-6*
-50
-40
-30
-2 °
-10
Fig. 7. C o n t o u r s of s i m u l a t e d i n t e r n a l tide r a n g e of the l o w e r b o u n d a r y of the t h e r m o c l i n e at s p r i n g t i d e in m e t r e s w i t h initial c o n d i t i o n s (I):
T10 = 15°C, T20 = 12°C, hi0 = 30 m, h20 = 38 m.
:iiii!iiiiiiiiii iiJiiiiiiiiiiiiiii!!!!!!!iiiii!!!iiiiiiiii iii i ....................... iii-"~i........ ~::ii...............iiii:::iiii :'1
. . ~ s -~ t
::!ii!i!: !??ii::;iiiiiiiiiii!!d!~;[email protected]~iiiiii!ii!}~.:~i~.~-!!::i
i]i i?iii~-.::i::iiiiii ii]i;i~]i~;::~gjj'-'.~jjjtj]~!llflr
::
,
Fig. 8. Simulated baroclinic tidal currents in the surface mixed layer one day after neap tide (T = 24.6 days of integration). This chart corresponds to initial conditions (I).
Figure 8 [initial conditions (I)] represents the baroclinic current field in the surface layer in the north of the Bay at time t = 24.6 days (one day after neap tide). The maximum velocities are close to 13 cm s - l . The wavelength of the internal tide appears clearly on this chart. The propagation off-slope towards the ocean tends to become linear, but on-shelf, across the Celtic Sea and the Armorican shelf, the non-linear character of the wave field is well marked. Another interesting feature which is conspicuous on this chart is the existence of a succession of areas of weak and strong currents in the alongslope direction. These structures seem to be connected to the topography of the slope which shows the existence of submarine canyons in this region. In Figs. 7 and 8, the wavelength of the internal tide is close to 30-32 km over the abyssal plain and 25-30 km over the shelf. Figure 9, with initial conditions (I), shows the time evolution over 15 days of the two boundaries of the thermocline at different locations along a transect crossing the shelf-break (Points T 1 t o T 9 located on latitude 47.38°N in Fig. 2). The variation in shape of the internal wave and its decrease in amplitude with increasing distance from the shelf-break can be observed.
STORM EFFECTS ON BAROCLIN1C TIDAL FIELD IN BAY OF BISCAY DAYS D
T1
0
DAYS
DAYS
~ ~ so 3','
"
~
DAYS
-zo
"
T4 DAYS
-ao
~,,, DAYS
DAYS
~ -~0
~
T0
ii,f
~)AYS
i,
DAYS
,,
,,,
,,
,,
,,
,,,
,,
Fig. 9. T i m e v a r i a t i o n of the d e p t h of t h e r m o c l i n e b o u n d a r i e s at d i f f e r e n t p o s i t i o n s a l o n g an e a s t - w e s t t r a n s e c t across the c o n t i n e n t a l slope (see Fig. 1) w i t h initial c o n d i t i o n s (I).
337
These general features are also found in Fig. 10 which shows results obtained with the initial conditions (II). Initially, the thermocline is deeper and thicker, and the heat content larger than in case (I). Comparison between Figs. 9 and 10 indicates a significant enhancement of the internal tide amplitude in case (II). The oscillation of the lower boundary of the thermocline at spring tide is larger. Another conspicuous feature is the increase in the variation of the thermocline thickness over a M 2 tidal period in particular near the edge. Figures 11 and 12 show the time evolution of the limits of the thermocline at selected locations on the shelf-break with the initial conditions (I) and (II). In these two figures, a large set of wave shapes appears, showing the importance of the topography of upper part of the continental slope in the generation of the internal tide. In particular a steepening of the continental slope enhances the amplitude of the generated baroclinic waves. In the southern part of the Bay of Biscay the internal tide field is weaker. The role of the initial conditions can be observed by comparison of Figs. 11 and 12. It appears that the amplitude of the lower limit of the thermocline is larger with the set of initial values (II) than with the set (I). It is now known that the generation mechanism for internal tides is sensitive to numerous parameters: the depth of the thermocline, the vertical temperature gradient in the thermocline, the steepness of the slope, the barotropic tidal current intensity (Maze and Le Tareau, 1990). Thus a deep interface [case (II)] will oscillate with larger amplitude than a shallow one [case (I)]. During the Ondine 85 Cruise (September-October 1985) accurate field data was obtained in the north of the Bay of Biscay. The aim of this experiment was to study internal tides and their action on the spatial and temporal evolution of different physical, chemical and biological parameters. Moorings were deployed in the Celtic Sea. The moorings P1 and P4 (Fig. 2) provided temperature data between 10 and 100 m depth. These data were obtained by Aanderaa thermistor
338
J.Y. L E T A R E A U
TIME )days)
Ti lIME )days)
TIME (days)
TIME )days)
T4
TIME (days(
= ~so
i
TIME { cilys) 121#
I
2
)
4
5
i
I
I
I
~1
Jl
IZ
Jl
i
2
1
,
s
I
r
l
*
Im
,I
*I
11
L4
IS
,: Z o ~!i)le~ .iii
TIME (days)
H
Ls
A N D R. M A Z I ~
chains. The vertical displacement of the thermocline is illustrated by the time evolution of the isotherms 13.5°C and 15°C (Figs. 13a and 14a). The amplitude of the vertical displacements is larger at mooring P~ (near the shelf-break) than at mooring P4 (on the shelf). This feature is enhanced during spring tides. The model response at locations P1 and P4 is shown in Figs. 13b and 14b. Comparison with data is difficult here because of the heterogeneity of the surface t e m p e r a t u r e field in the Celtic Sea region. Large horizontal t e m p e r a t u r e gradients were observed near the shelf-break on satellite images during Ondine experiment. The isotherms 13.5°C and 15°C lie in the thermocline but do no represent its lower and u p p e r boundaries. This choice was made because only these values remained in the range of depth of the records. However one can see that qualitatively the range of the thermocline displacements given by the model is in agreement with the data. To explain the difference between the model and the data, it must also be r e m e m b e r e d that only two components of the barotropic tide are used in the model, and that the model retains only the first baroclinic mode as previously mentioned. The topographic data used in the model (Fig. 2) are digitized from nautical charts and available at each grid point. Therefore topographic discontinuities of horizontal scale less than one mesh size are note taken into account. This causes a slight distortion of the bottom shape in shelfbreak regions particularly. An increase of the horizontal resolution of the computational grid in shelf-break areas would improve the model resuits.
TIME (days)
Model response to a travelling storm
',,,
~' ~' ~' ,'
~' ,'
~' ,'
,'
lIME (days)
The action of the wind and the associated mixing processes are introduced in the internal tide model as external forcing terms.
,; ,; ~ ,~ ,i ,~
~)'.
ii:I
Fig. 10. Same as Fig. 9 with initial conditions (II): T10= 16°C, 7"20= 12°C, hi0 = 35 m, h20 = 60 m.
STORM
EFFECTS
ON BAROCLINIC
TIDAL
FIELD
339
IN BAY OF BISCAY
DAYS
0 .++88
o~ -Z8 Ill -86 -tO ++0 +tO
+TO +85
Al
-I08
O +++-~ +.tO ~+
I
2I
I
41
I
I
I
,1
! ....
~
I ......
DAYS
I
-2+ II ++10 -58 -tO
+70 -80
A2 DAYS 10... .,,,+
-ZO
i!
+4G -50
-.
A3
-+88
~AYS 8
~
2~
i
~
f
t
i
tt
I
,,5~
t
i
L
|
t
1
+... o+ -ZO ,m -30
:~
-50
~t ~
-|0
-i08
8
I
, #
,,
DAYS I
I
+.,., o: -20 "t~
-30
-10
""
A 5
-100
Fig. 11. Time variation of the depth of thermocline boundaries at five selected points at the top of the continental slope [Initial conditions (I)].
340
J.Y. LE T A R E A U A N D R. M A Z E
TIME (days) .100
1
2
)
4
5
t
l
#
9
11
!
~
10
11
12
13
14
15
=
-60
-99 -IQ0 110
"
A1
TIME (days) .1o0
ua
1
~
~
S
4
li
~
l!
1~
13
14
15
-20
T: -50
-, o" f
. . . . .
,v
~v IV IV |V
F ~ v" ~
-
-1~0
TIME (days) i.. e.a
lit
z
-lil~
~0
-~1,
A3
TIME (days)
0 .10 0 oa
w
-l# -]0 -40
1...
-69 -89 -$e -llt~
A4 t
0
.10 0
!
2
a
4r
5~
TIME
t, t
-I0 -40 =
-IO -eD
~IDO
-110
A5
Fig. 12. Same as Fig. 11 with initial conditions (II).
(days)
341
S T O R M E F F E C T S O N B A R O C L I N I C T I D A L F I E L D IN BAY OF BISCAY
therefore:
First a theoretical atmospheric depression of circular shape is assumed. The associated wind velocity defined in a polar coordinate system moving with the storm is given by the expression:
W(rm)
=
W o R--~oe - ( 1 - " / R°':j
Wx
where: j: unit vector normal to the radius vector Ro: "radius" of the depression r: radius vector W0: characteristic velocity The maximum intensity of the wind is found on a circle of radius rm:
=
-W o
Y --Yo
e-O
_~/x_Co(t_to_tl)/Ro)2
Ro
wy=w0
x - C o ( t - t o - ti)
X e -(1
R0 - ~[x - C o ( t - t o - t
1 ) / R 0 )2
where: Yo: ordinate of the storm centre; tl: time at which the storm is imposed in the model;
R0(1 + V~-) rm
1.19 W0
If we consider a storm moving eastward with a celerity C O, the components of the wind velocity in a Cartesian coordinate system are given by:
F
W(r)
=
2
P1
,o! 2O :5O
~4o
~ so
•
80-
W
90IO0 •
®
3'o'''','
''~
SEPT 85
''~'''~,'''~'
'~''
;'
OCT 85
2O
~ 3o ~4o o
/j
6C 7C 8C
®
J
Fig. 13. Time variation of the depth of: - two isotherms in the thermocline deduced from Ondine 85 data (a) - the boundaries of the thermocline calculated by the model (b) at location Px near the shelf-break.
342
J.Y. LE T A R E A U AND R. MAZIE
In order to see, under the most favorable circumstances, the influence of the moving storm on the whole baroclinic tidal field, time t 1 has been chosen such that the storm travels the domain during spring tides. Our interest in this section is focused on the evolution of the surface temperature field during the storm action. The wind field and the surface temperature structure at four selected times are displayed in Figs. 15, 16, 17 and 18. The time interval between the successive charts is equal to a M 2 tidal period (12.42 h). Initially the surface temperature is constant over the whole of the domain. At time t = 719.47 h the wind starts to blow over the domain (Fig. 15a), inducing surface mixing and the beginning
to: travelling time of the storm centre from its initial position at time t 1 to the west boundary of the domain. In our calculations we have chosen t 0 = 3 R o / C o which means that at time t 1 the storm centre is at a distance 3R o from the western boundary of the domain. The following numerical values are taken to define the storm: R o = 300 km Co = 8 m s -1 W0 = 2 0 m s -1 Y0 = 865 km Y0 corresponds to the latitude 51°N with the origin of the coordinate system taken in the south-west corner of the domain. The initial conditions (I) are prescribed.
1 I0~
2O
f
3C 4C £ 5O
/
b
6o
I
I
N 7e 8c 9c
®
I00 j I10-t
I
'
'
'
30
I I
'
'
'
I 2
'
'
'
I 3
'
'
'
I 4
'
'
'
I 5
'
'
'
I 6
'
'
'
I
'
7
SEPT
OCT
85
85
2O E
30
cn I
40
uJ
50 60 70 8O
®
Fig. 14. Same as Fig. 13 at location P4 on the shelf, at 40 km from the shelf edge.
STORM
EFFECTS
ON
BAROCLINIC
TIDAL
FIELD
IN BAY
~ 4 ~ 4 ~ 4 4 ~
. . . . . . . . . . .
~~44444444444~qv'*-4~,~ ~ ~ 4 4 4 4 4 4 4 4 4 4 4 ~ t ~ t , 3 , , ~ ~.~444444444444,4a~*~,~
. . . . . . . . . . . . . . . . . . . .
~
t
. . . . . . . . . . . . . . . . . . . . .
..................... ................. P. . . . . . .
.
.
:::::::::::::::::::::::::::::::::::::: 4¢4~¢~,~,*1 ¢~¢*,,
J I
r < 15°c
. . . . . . . . . . . . . . . . . .
~ E ~ 4 4 4 4 4 4 4 4 4 4 4 4 ~ , ~ I ~ ~ 4 4 4 4 4 4 4 4 4 4 4 4 ~ 4 ~ , , ~ 444444444444444~,~,,,i,,
4444444444444444 44~4444444¥44444
343
OF BISCAY
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.........................
S ii!i!!iiii!;;ii!;;il)i!!!; i~i!ii ~
'~
!iiiiiiiii;iiiiii;iiiii!
!
T = 15 °C
Fig. 15. Effect of a travelling storm on the surface t e m p e r a t u r e field at time t = 7 ] 9 . 4 7 hours: (a) w i n d field; (b) surface t e m p e r a t u r e field.
~.,-~r~MMHX~X~.~.~,,~tI'~ itlllIlll r~444
- -
. ~4
p ~ ~
~
~ ~ ~
' '
4 ~
4 ~
.
4 '
4 ~
~
4
4
4
4 ¢44444444 ~ I f44444444 ¢444444¢4 4444444444 4 444444 ~44~4qq
?ii?ii!iii??iiii!i?i?i!iii Fig. 16. S a m e as Fig. 15 at time t = 731.89 h.
344
J.Y. LE T A R E A U A N D R. MAZI~
o
o
0
T=15
Fig. 17. S a m e as Fig. 15 at t i m e t = 744.31 h. (spring tide).
Fig. 18. S a m e as Fig. 15 at t i m e t = 756.73 h.
°c
(
,I
STORM
EFFECTS
ON BAROCL1NIC
345
TIDAL FIELD IN BAY OF BISCAY
of surface cooling in the north-west of the Bay of Biscay where the temperature falls below 15°C (Fig. 15b). The 15°C isotherm propagates towards the south-east. The spatial oscillation of this isotherm is connected to the baroclinic tidal field during the mixing event. One tidal period later, the storm practically takes place over the whole of the domain (Fig. 16a). The surface temperature field at this time is displayed in Fig. 16b. The 15°C isotherm is located in the southern part of the Bay. In the north, a rather complicated temperature field occurs, in particular on the shelf. This feature is connected to the baroclinic tidal field in this region. Time t = 744.31 h corresponds to spring tides. The wind stress reaches its maximum intensity in the north of the Bay (Fig. 17 a). Surface mixing is therefore very active. In Fig. 17b the final temperature field is practically achieved. The last charts indicate the storm leaving the domain (Fig. 18 a) and the horizontal temperature structure induced by interaction between the travelling storm and the baroclinic tides (Fig. 18b). Eddy diffusion and advection by the tidal current will affect this temperature field, until a new gust of wind blows again over the domain. The most important feature which appears on the temperature charts is the creation of patches of cold water near the top of the continental slope. Moreover it appears that the cooling is greater in regions of large internal tide amplitude, in particular along the Celtic Sea shelfbreak. Such cold bands or spots of surface waters can be observed on satellite infra-red images taken in these regions (Fig. 1). These bands of cold water which follow the continental edge can be observed from April-May to September-October. The cold waters are generally associated with high nitrate and increased chlorophyll concentrations. These features have been observed during Ondine 85 Cruise (Le Corre, 1990, pers. commun.) These characteristic features have been described by numerous authors: Pingree and Mardell (1981), Dickson et al. (1980) from infra-red images by Satellites NOAA6 and TIROS N respectively. Field observations have also been made
by Maz6 et al. (1986), Langlois et al. (1990) and Maz6 and Le Tareau (1990). The good agreement between the geographical location of the patches of cold water and the area of large internal tide amplitude suggests that the process studied in this paper plays an important role in the shelf-break cooling in the Bay of Biscay. Conclusion and summary
A non-linear three-layer hydrostatic model has been used to predict the baroclinic tidal structure over the whole of the Bay of Biscay. This model is an extension to a two dimensional level of a previous work by Maz6 and Le Tareau (1990). It takes into account mixed layer physics (surface wind mixing, bottom drag mixing) and thermocline mixing due to Kelvin-Helmholtz instabilities. The model predicts regions at the shelf-break where internal tide amplitude, currents and connected vertical mixing are likely to be important. Irregularities of the depth contours near the shelf-break interact with the internal wave field forcing the internal wave to have an irregular, wavy, alongshore pattern in addition to the across-shore wave-line propagation of internal tides. The action of a moving storm over a baroclinic tidal field is studied. Results display the development of a surface temperature field with a shelf-break front. Patches of colder surface water are predicted along this front in the neighbourhood of the Celtic Sea slope. Results are in good agreement with oceanographic data and satellite infra-red images. This suggest that the action of a travelling storm on a baroclinic tidal field is an attractive explanation for the shelf-break cooling which is observed to occur in the north of the Bay of Biscay. Alternative explanations for shelf-break cooling have been put forward. Thus, Dickson et al. (1980) considered that shelf-break cooling was the result of the interaction between Kelvin waves propagating along the continental slope and the slope topography. Heaps (1980) suggested that the shelf-break cooling phenomenon was due to an upwelling which appears along the shelf edge as the result of a cross-shelf wind action. New
346
(1988) argued that cooling was due only to mixing created by internal motion that results from the interactions of the surface tide with bottom topography. Shelf-break cooling has been observed in other oceanic areas. In particular, it has been studied by Wolansky and Pickard (1983) in the Great Barrier Reef regions. These authors suggest that the cooling process is also connected to internal tides and presumably to internal Kelvin waves. All these mechanisms may probably contribute to shelf-break cooling. However in the Bay of Biscay, satellite imagery indicates that the phenomenon is more conspicuous in areas of strong tidal currents and therefore, of large internal tide amplitude and strong baroclinic currents. Shelfbreak cooling is also often enhanced at spring tides (Langlois et al., 1990). These observations are in good agreement with our results and suggest that the mechanism described here is an important factor in the cooling process. This work also shows the necessity of taking into account the two-dimensional aspect when modelling internal tides and the associated phenomenon in actual regions. The response of the model to other kinds of surface forcing may be analysed and in particular, we intend to focus on the calculation of residual baroclinic currents on the shelf and in slope areas. Observations collected during the Ondine 85 survey show vertical shear of residual currents and a rotation of these currents from the surface to the bottom. How well can these observations be described by the model? This question will be addressed in further studies.
Acknowledgements This research was carried out with the support of French "Service Hydrographique et Oc6anographique de la Marine" (S.H.O.M.), of the "Direction des Recherches et Etudes Techniques" (D.R.E.T.). Miss Christine Gu6renne carefully typed the manuscript and Mr. Pierre Doar6 drafted the figures.
J.Y. LE T A R E A U A N D R. M A Z E
We thank the referees for their valuable comments and remarks.
References Baines, P.G., 1982. On internal tide generation models. DeepSea Res., 29: 307-338. Boczar-Karakiewicz, B., Bona, J.L. and Pelchat, B., 1991. Interaction of internal waves with the seabed on continental shelves. Cont. Shelf Res., 11(8-10): 1181-1197. Castaing, P., Philippe, I. and Weber, O., 1981. Processus hydrodynamiques en bordure des plateaux continentaux: effets de la turbulence et des remont6es d'eau sur les s~diments. C.R. Acad. Sci. Paris, 292(II): 1481-1484. Corcos, G.M. and Sherman, F.S., 1976. Vorticity concentrations and the dynamics of unstable shear layers. J. Fluid Mech., 73: 241-264. Dickson, R.R., Gurbutt, P.A. and Pillai, N.V., 1980. Satellite evidence of enhanced upwelling along the European Continental Shelf. J. Phys. Oceanogr., 10: 813-819. Heaps, N.S., 1980. A mechanism for local upwelling along the European Continental Slope. Oceanol. Acta, 3(4): 449454. Heathershaw, A.D., New, A.L. and Edwards, P.D., 1987. Internal tides and sediment transport at the shelf-break in the Celtic Sea. Cont. Shelf. Res., 7(5): 485-517. Kato, H. and Phillips, O.M., 1969. On the penetration of a turbulent layer into a stratified fluid. J. Fluid Mech., 37: 643-655. Langlois, G., Gohin, F. and Serpette, A., 1990. Refroidissements locaux aux abords du talus continental armoricain. Oceanol. Acta, 13(2): 159-169. Le Tareau, J.Y., Maze, R., Le F~vre, J., Billard, C. and Camus, Y., 1983. ENVAT-81, Campagne de recherche multidisciplinaire en Atlantique: Aspects m~t~orologiques, chimiques, biologiques, hydrologiques et thermodynamiques. Met-Mar, 118(1): 6-25. Maze, R., 1983. Mouvements internes induits dans un golfe par le passage d'une d6pression et par la mar~e. Th~se de Doctorat d'Etat, U.B.O., Brest, 320 pp. Maze, R., 1987. Generation and propagation of non-linear internal waves induced by the tide over a continental slope. Cont. Shelf Res., 7(9): 1079-1104. Maz6, R. and Le Tareau, J.Y., 1990. Interactions between internal tides and energetic fluxes across the atmosphere ocean interface over a continental shelf-break. J. Mar. Res., 48: 505-541. Maze, R., Camus, Y. and Le Tareau, J.Y., 1986. Formation de gradients thermiques ~ la surface de l'Oc~an, au-dessus d'un talus, par interaction entre les ondes internes et le m61ange dfi au vent. J. Cons. Int. Explor. Mer, 42: 221-240. Mesinger, F. and Arakawa, A., 1976. Numerical methods used in Atmospheric models. GARP Publications Series, 17(1). WMO-ICSU Joint Organizing Committee, 64 pp. New, A.L., 1988. Internal tidal mixing in the Bay of Biscay. Deep-Sea Res., 35(5): 691-709.
STORM EFFECTS ON BAROCLINICTIDAL FIELD IN BAY OF BISCAY Orlanski, I., 1976. A simple boundary condition for unbounded hyperbolic flows. J. Comput. Physics, 21: 251-269. Pichon, A. and Maz6, R., 1990. Internal tides over a shelfbreak. Analytical model and observations. J. Phys. Oceanogr., 20(5): 657-671. Pingree, R.D., 1984. Some applications of remote sensing to studies in the Bay of Biscay, Celtic Sea and English Channel. In: J.C.J. Nihoul, (Editor). Remote Sensing of Shelf Sea Hydrodynamics. Elsevier Oceanography Series. Elsevier, Amsterdam, pp. 287-315. Pingree, R.D. and Mardell, G.T., 1981. Slope turbulence, internal waves and phytoplankton growth at the Celtic Sea shelf-break. Philos. Trans. R. Soc. London, A 302: 663682. Pingree, R.D., Griffiths, D.K. and Mardell, G.T., 1983. The structure of the internal tide at the Celtic Sea shelf-break. J. Mar. Biol. Assoc. U.K., 64: 99-113. Pingree, R.D., Mardell, G.T., Holligan, P.M., Griffiths, D.K. and Smithers, J., 1982. Celtic Sea and Armorican current structure and the vertical distributions of temperature and chlorophyll. Cont. Shelf Res., 1(1): 99-116.
347 Serpette, A., 1989. Mar6es internes dans le Golfe de Gascogne: un module bi-dimensionnel. Th~se de Doctorat Univ. Bretagne Occidentale, Brest, 242 pp. Serpette, A. and Maz6, R., 1989. Internal tides in the Bay of Biscay: a two-dimensional model. Cont. Shelf Res., 9(9): 795-821. Sherwin, T.J., 1988. Analysis of internal tides observed on the Malin shelf, North of Ireland. J. Phys. Oceanogr., 18: 1035-1050. Thorpe, S.A., 1971. Experiments on instability of stratified shear flows: Miscible fluids. J. Fluid Mech., 46: 299-320. Willmott, A.J. and Edwards, P.D., 1987. A numerical model for the generation of tidally forced nonlinear internal waves over topography. Cont. Shelf Res., 7(5): 457-484. Wolanski, E. and Pickard, G.L., 1983. Upwelling by internal tides and Kelvin waves at the Continental shelf-break on the Great Barrier Reef. Aust. J. Mar. Freshwater Res., 34: 65-80.