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142 CHAPITRE 4. ETUDES DE PROCESSUS OCÉANOGRAPHIQUES 562 Ocean Dynamics (2009) 59:551–563 tel-00545911, version 1 - 13 Dec 2010 downslope transport in his simulations using a Mellor– Yamada parametrisation of vertical mixing. The vertical resolution in today’s OGCMs is too sparse to explicitly resolve the two-layer structure of the gravity current dynamics. The friction layer is, at best, a few metres thick, and its explicit representation asks for a resolution of the order of 1 m at the bottom, a resolution that is a few hundred times smaller than the actually employed vertical resolution at depth in OGCMs. General circulation models are, thus, not able to correctly predict the two-part structure of gravity currents, underestimate the detrainment and overestimate the descent of the vein (when neglecting other difficulties which inhibit the descent of the vein). Or, even worse, the detrained water will be found diluted in the grid-box downstream of the vein; as the thickness of the grid-box is much larger than that of the Ekman layer, the water will continue to move with the vein, and the diluted water will be counted as entrained rather than detrained. Following these arguments, it seems difficult to correctly represent the dynamics of a gravity current when the vertical extension of the bottom gridbox is smaller than the Ekman layer thickness. Large-scale instabilities are explicitly suppressed in the research presented here by choosing a 2.5- dimensional geometry. Our results are, however, key to the research concerning the evolution of such instability as it gives the basic state from which large-scale instabilities will grow. So far, in the research on the instability of the vein of a gravity current, the basic state was conjectured or chosen to simplify the calculations (Meacham and Stephens 2001). The stable law of the shape of a gravity current, presented in Subsection 3.3, can now be used as the basis of analytical or numerical three-dimensional stability analysis. In the experiments presented here, the Ekman number (the vertical viscosity ν v ) was fixed, leading to a laminar Ekman layer dynamics in our experiments, when a low value of the reduced gravity is considered. I exploited this dynamics and obtained that the evolution of the shape of the vein is governed by the heat equation with a diffusion coefficient, which can be calculated from the external parameters of the problem. For higher values of the reduced gravity, or lower values of the Richardson number, the vertical velocity within the Ekman layer leads to a change in its dynamics, and the linear friction laws are no longer applicable but are replaced by a quadratic drag law. I showed that, when a quadratic drag law applies, the dynamics is governed by a nonlinear heat equation. It would be desirable to perform the 2.5-dimensional Boussinesq simulations with an explicitly resolved turbulent Ekman layer, which requires resolutions in the centimetre scale in the vertical and horizontal direction, but this is beyond our actual computer resources. In the ocean, the vertical turbulent viscosity or even the friction law depends on unobserved quantities as the bottom roughness. Ongoing research is directed towards the determination of the friction laws and coefficients by data assimilation (Wirth and Verron 2008); by doing so, I hope to be able to avoid the explicit resolution and determination of small-scale processes and, nevertheless, obtain solid estimates for the small-scale turbulent fluxes, which will allow us to concentrate on the study of large-scale features. This work is complementary to the theories of Killworth (2001) and others, who suppose that the thickness of the gravity current is determined locally by the entrainment or detrainment at the upper interface through a local Froude number criteria. The theory presented here determines the thickness non-locally by the convergences and divergences of the Ekman fluxes at the lower boundary and determines non-locally the evolution of the overall shape of the gravity current, using a (non-local) heat equation. This proves that there cannot be a purely local parametrisation of gravity current thickness. In realistic gravity currents, the two mechanisms are likely to act simultaneously. Furthermore, it was found in laboratory experiments on nonrotating gravity currents that the dissipative processes at the interface have a negligible role compared to those due to bottom friction (Ermanyuk and Gavrilov 2007). In oceanic gravity currents, the mixing and entrainment is enhanced by the roughness of the ocean floor, the change of slope and large-scale instabilities, features which are not considered here and which are the subjects of future research. Acknowledgements I am grateful to Bernard Barnier, Yves Morel, Joel Sommeria and Jacques Verron for discussion and to two anonymous reviewers for their remarks which have greatly improved the paper. This work is part of the COUGAR project funded by ANR-06-JCJC-0031-01. Appendix: Force balance in a rotating gravity current In the x direction (upslope), the dominant force balance is between the Coriolis force and reduced gravity. In the y direction, the reduced gravity vanishes and the dominant force balance is between friction and the Coriolis force (Fig. 7). Using linear Ekman layer theory, I obtain: f H v cosθ sin θ − ν√ 2 v cos(θ + π/4) = 0. (8) δ cos θ
4.7. ON THE BASIC STRUCTURE OF OCEANIC GRAVITY CURRENTS 143 Ocean Dynamics (2009) 59:551–563 563 tel-00545911, version 1 - 13 Dec 2010 θ u FC or iolis Fg Ffr ict ion θ + π 4 Fig. 7 Force balance in a gravity current descending at an angle θ to the horizontal at a constant speed u. The Coriolis force is at an angle of 3π/2 to the direction of propagation and the frictional force at an angle of 5π/4. In a stationary state, these two forces balance the gravitational force: F Coriolis + F friction + F g = 0. Please note the turned coordinate system As the angle of descent is small to leading order sin θ ≈ θ and cos(θ + π/4) ≈ 1/ √ 2, Eq. 8 then gives: θ = ν √ √ ν Ek H fδ = 2 f H = 2 4 ≈ 1.1 · 10−2 . (9) Please note that this result depends only on the Ekman number and is independent of the velocity of the gravity current. The analysis presented here does not apply to the friction layer as the Ekman spiral is not complete and θ is not small in this case, but can be extended to cases with a turbulent Ekman layer using a quadratic drag law. References Baringer MO, Price JF (1997) Momentum and energy balance of the mediterranean outflow. J Phys Oceanogr 27:1678–1692 Emms PW (1998) Streamtube models of gravity currents in the ocean. Deep-Sea Res 44:1575–1610 Ezer T (2005) Entrainment, diapycnal mixing and transport in three-dimensional bottom gravity current simulations using the Mellor–Yamada turbulence scheme. Ocean Model 9:151–168 Ezer T, Weatherly GL (1990) A numerical study of the interaction between a deep cold jet and the bottom boundary layer of the ocean? J Phys Oceanogr 20:801–816 Ermanyuk EV, Gavrilov NV (2007) A note on the propagation speed of a weakly dissipative gravity current. J Fluid Mech 574:393–403 y x Frisch U, Kurien S, Pandit P, Pauls W, Ray SS, Wirth A, Zou J-Z (2008) Hyperviscosity, Galerkin truncation, and bottlenecks in turbulence. Phys Rev Lett 101:144501 Garrett C, MacCready P, Rhines P (1993) Boundary mixing and arrested Ekman layers: rotating stratified flow near a sloping boundary. Annu Rev Fluid Mech 25:291–323 Gill AE (1982) Atmosphere-ocean dynamics. Academic, London, pp 662 Griffiths RW (1986) Gravity currents in rotationg systems. Annu Rev Fluid Mech 18:59–89 Killworth PD (1977) Mixing on the Weddell Sea continental slope. Deep-Sea Res 24:427–448 Killworth PD (2001) On the rate of descent of overflows. J Geophys Res 106:22267–22275 Killworth PD, Edwards NR (1999) A turbulent bottom boundary layer code for use in numerical ocean models. J Phys Oceanogr 29:1221–1238 Lane-Serf GF, Baines PG (1998) Eddy formation by dense flows on slopes in a rotating fluid. J Fluid Mech 363:229–252 Lane-Serf GF, Baines PG (2000) Eddy formation by overflows in stratified water. J Phys Oceanogr 30:327-337 Legg S, Hallberg RW, Girton JB (2006) Comparison of entrainment in overflows simulated by z-coordinate, isopycnal and non-hydrostatic models. Ocean Model 11:69–97 MacCready P (1994) Frictional decay of abysal boundary currents. J Mar Res 52:197–217 Meacham SP, Stephens JC (2001) Instabilities of currents along a slope. J Prog Oceanogr 31:30–53 Nof D (1983) The translation of isolated cold eddies on a sloping bottom. Deep-Sea Res A 30:171–182 Pedlosky J (1998) Ocean circulation theory. ISBN: 978-3-540- 60489-1. Springer, Berlin Heidelberg New York, p 453 Price JF, Baringer MO (1994) Outflows and deep water production by marginal seas. Prog Oceanogr 33:161–200 Price JF, Yang J (1998) Marginal sea overflows for climate simulations. In: Cassignet E, Verron J (eds) Ocean modeling and parameterization. Kluwer, Dordrecht, pp 155–170 Shapiro GI, Hill AE (1997) Dynamics of dense water cascades at the shelf edge. J Phys Oceanogr 27:2381–2394 Shapiro GI, Zatsepin AG (1997) Gravity current down a steeply inclined slope in a rotating fluid. Ann Geophys 15:366–374 Smith PC (1975) A stream tube model for bottom boundary currents in the ocean. Deep-Sea Res 22:853–873 Smith PC (1977) Experiments with viscous source flows in rotating systems. Dyn Atmos Ocean 1:241–272 Stull RB (1988) An introduction to boundary layer meteorology. Kluwer, Dordrecht Sutherland BR, Nault J, Yewchuk K, Swaters GE (2004) Rotating dense currents on a slope. Part 1. Stability. J Fluid Mech 508:241–264 Umlauf L, Arneborg L, Burchard H, Fiekas V, Lass HU, Mohrholz V, Prandke H (2007) Transverse structure of turbulence in a rotating gravity current. Geophys Resea Lett 34:L08601. doi:10.1029/2007GL029521 Weatherly GL, Kelley Jr EA, (1982) Too cold bottom layers at the base of the Scotian Rise. J Mar Res 40:985–1012 Wirth A (2005) A non-hydrostatic flat-bottom ocean model entirely based on Fourier expansion. Ocean Model 9:71–87 Wirth A, Verron J (2008) Estimation of friction parameters and laws in 1.5D shallow-water gravity currents on the f-plane, by data assimilation. Ocean Dyn doi:10.1007/s10236-008-0151-8 Wu W, Danabasoglu G, Large WG (2007) On the effects of parameterized Mediterranean overflow on North Atlantic ocean circulation and climate. Ocean Model 19:31–52
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Mémoire de Recherche présenté pa
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Table des matières I Etudes de pro
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Chapitre 1 Avant-propos tel-0054591
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Chapitre 2 Comprendre la dynamique
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2.3. HIÉRARCHIE DE TYPES DE MODÈL
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2.3. HIÉRARCHIE DE TYPES DE MODÈL
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2.5. MA RECHERCHE DANS LE CONTEXTE
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Chapitre 3 Dynamique océanique et
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3.1. LA CIRCULATION OCÉANIQUE À G
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3.1. LA CIRCULATION OCÉANIQUE À G
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3.2. PROCESSUS SUBMÉSO ECHELLES ET
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3.4. RETOUR À LA GRANDE ECHELLE PA
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Bibliographie tel-00545911, version
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33 Curriculum Vitae du Dr. Achim WI
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Colloque bilan LEFE, Plouzané, Mai
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Chapitre 4 Etudes de Processus Océ
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4.1. VARIABILITY OF THE GREAT WHIRL
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193 4.2. FRESH WATER FLUX 19 water.
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195 Chapter 5 Dynamics of the Ocean
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197 5.2. THE LINEARIZED ONE DIMENSI
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199 5.4. TWO DIMENSIONAL STATIONARY
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201 5.6. THE CORIOLIS FORCE 27 Whic
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203 5.8. GEOSTROPHIC EQUILIBRIUM 29
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205 5.10. LINEAR POTENTIAL VORTICIT
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207 5.13. A FEW WORDS ABOUT WAVES 3
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209 Chapter 6 Gyre Circulation tel-
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211 6.1. SVERDRUP DYNAMICS IN THE S
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213 6.2. THE EKMAN LAYER 39 In the
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215 6.3. SVERDRUP DYNAMICS IN THE S
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217 Chapter 7 Multi-Layer Ocean dyn
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219 7.3. GEOSTROPHY IN A MULTI-LAYE
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221 7.5. EDDIES, BAROCLINIC INSTABI
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223 Chapter 8 Equatorial Dynamics t
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225 Chapter 9 Abyssal and Overturni
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227 9.2. MULTIPLE EQUILIBRIA OF THE
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229 9.3. WHAT DRIVES THE THERMOHALI
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231 Chapter 10 Penetration of Surfa
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233 10.2. TURBULENT TRANSPORT 59 If
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235 10.5. ENTRAINMENT 61 instabilit
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237 Chapter 11 Solution of Exercise
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239 65 Exercise 32: The moment of i
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241 INDEX 67 Transport stream-funct
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Annexe A Attestation de reussite au
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Annexe B Rapports du jury et des ra
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251 utilisés avec pertinence. Sur
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