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The role of boundary layers in the large- scale ocean circulation

The role of boundary layers in the large- scale ocean circulation. Laure Saint- Raymond ENS & Université Paris 6. Western intensification of currents The Gulf Stream case. In the Atlantic ocean , average velocity in the gyres 1 to 10 cm/s

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The role of boundary layers in the large- scale ocean circulation

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  1. The role of boundarylayers in the large-scaleocean circulation Laure Saint-Raymond ENS & Université Paris 6

  2. Western intensification of currentsThe Gulf Stream case In the Atlantic ocean, averagevelocity in the gyres 1 to 10 cm/s On the West boundary (Florida, Cap Hatteras), averagevelocity of the order of 100 cm/s US Army, 1943

  3. RSMAS, University of Miami

  4. A 2D mathematical model:the Munk equation

  5. Seawater, an incompressible and weaklyviscousfluid. Seawaterisessentiallyincom-pressible, and homogeneous . The equation for the conservation of mass states The kinematicviscosity of water isnegligible, and does not account for the energy dissipation. A turbulent viscosityhas to beintroduced to model the effect of smallscales.

  6. The Coriolis force The Coriolis force takesintoaccount the Earth rotation (non Galileanreference frame). In bidimensionalmodels, the main contribution (f-plane) modifies only the pressure. The next contribution isdue to inhomogeneities

  7. Experimental observations show thatcurrents are stronglycorrelated to the wind. An analytical computation of the forcing wasproposed by Ekman. (pumpingmechanism) A more realistic model shouldtakeintoaccount a real couplingwith the atmosphere. North-East monsoon The role of wind South-West monsoon

  8. A balance equation • For a stationary flow, the accelerationvanishes • This system of partial differentialequations of order 2 issupplemented by someboundary condition. The no-slip condition states

  9. The boundary layer phenomenon.A recentdiscovery.

  10. The pionneeringwork of Prandtl International Congress of Mathematicians, 1904 ÜberFlüssigkeitsbewegungbeisehrkleinerRibung Flow around an obstacle : - inviscidexterior component satisfying a non penetration condition - boundary layer localized in the vicinity of the wall

  11. The boundary layer restores the no-slip condition on the wall. It isdominated by viscouseffects. It isexpected to split from the wallbehind the obstacle. The decompositionisactually not stable (and has no mathematical justification).

  12. Boundarylayers in oceanography The explorer Nansen hadnotedthat icebergs drift with an angle of 30 to 40 degreeswith respect to the wind direction. Ekman’s computation (1905), based on the balance between the Coriolis force and the viscosity, predicts an angle of 45 degrees.

  13. Withdepth, the currentdecreases and twists. This isthe Ekman spiral. A number of mathematical contributions have completedEkman’sanalysis : - stability issues -couplingwithothereffects (topography, nonlinear transport, resonant forcing,…)

  14. Multiscale expansionsA simple example

  15. A differentialequation of order 4 As the velocityfield u is divergence-free, one canintroduce the streamfunction Wethenstudy the singular perturbation problem For simplicity, computations willbedone in 1D.

  16. The Sverdrup relation To describe the asymptoticbehaviour of for westudythe limit Integrating by parts leads to the followingenergyestimate, givingsomeuniformbound In weaksense, converges to the solution of the Sverdrup equation :

  17. The boundary layer equation The Sverdrup equationis not compatible with the no-slip condition. Wethusintroduce a corrector: - The boundary layer restores boundary conditions - It isdominated by viscouseffects

  18. East/West disymmetry • The thickness of the layer isgiven by the scaling • - In the East, decaying solutions are of the form • - The space of West solutions is of dimension 2 • The boundary condition for the Sverdrup equationisthereforeprescribed on the East side.

  19. Influence of the geometrySomeremarkablefeatures

  20. Northern/Southerndegeneracy • In the vicinity of North and South boundaries, the transport termis not singular : • The size of the boundary layer isdifferent • The equation for the boundary layer isnon local • (of parabolic type) • The propagation iswestwards.

  21. Discontinuity zones In non convexdomains, the solution to the Sverdrup equationisgenerallydiscontinuous (jump condition). To get an approximation of , • A regularizationisneeded; • The errortermisdealtwithlike a boundary layer.

  22. Complex transitions… • No matching, but a superposition : • localized East and West boundarylayers, • extinction of North and South boundarylayers. • Construction startingfrom the East boundary.

  23. Towards more physicallyrelevant models?

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