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 On the West boundary (Florida, Cap Hatteras), average velocity of the order of 100 cm/s US Army, 1943

RSMAS, University of Miami

A 2D mathematical model: the Munk equation

Seawater, an incompressible and weakly viscous fluid. Seawater is essentially incom- pressible, and homogeneous. The equation for the conservation of mass states The kinematic viscosity of water is negligible, and does not account for the energy dissipation. A turbulent viscosity has to be introduced to model the effect of small scales.

The Coriolis force The Coriolis force takes into account the Earth rotation (non Galilean reference frame). In bidimensional models, the main contribution (f- plane) modifies only the pressure. The next contribution is due to inhomogeneities

The role of wind North-East monsoon South-West monsoon Experimental observations show that currents are strongly correlated to the wind. An analytical computation of the forcing was proposed by Ekman. (pumping mechanism) A more realistic model should take into account a real coupling with the atmosphere.

A balance equation For a stationary flow, the acceleration vanishes This system of partial differential equations of order 2 is supplemented by some boundary condition. The no-slip condition states

The boundary layer phenomenon. A recent discovery.

The pionneering work of Prandtl Flow around an obstacle : - inviscid exterior component satisfying a non penetration condition - boundary layer localized in the vicinity of the wall International Congress of Mathematicians, 1904 Über Flüssigkeitsbewegung bei sehr kleiner Ribung

The boundary layer restores the no-slip condition on the wall. It is dominated by viscous effects. It is expected to split from the wall behind the obstacle. The decomposition is actually not stable (and has no mathematical justification).

Boundary layers in oceanography The explorer Nansen had noted that icebergs drift with an angle of 30 to 40 degrees with 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.

With depth, the current decreases and twists. This is the Ekman spiral. A number of mathematical contributions have completed Ekman’s analysis : - stability issues - coupling with other effects (topography, nonlinear transport, resonant forcing,…)

Multiscale expansions A simple example

A differential equation of order 4 As the velocity field u is divergence-free, one can introduce the streamfunction We then study the singular perturbation problem For simplicity, computations will be done in 1D.

The Sverdrup relation To describe the asymptotic behaviour of for we study the limit Integrating by parts leads to the following energy estimate, giving some uniform bound In weak sense, converges to the solution of the Sverdrup equation :

The boundary layer equation The Sverdrup equation is not compatible with the no-slip condition. We thus introduce a corrector: - The boundary layer restores boundary conditions - It is dominated by viscous effects

East/West disymmetry The thickness of the layer is given 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 equation is therefore prescribed on the East side.

Influence of the geometry Some remarkable features

Northern/Southern degeneracy In the vicinity of North and South boundaries, the transport term is not singular : -The size of the boundary layer is different -The equation for the boundary layer is non local (of parabolic type) The propagation is westwards.

Discontinuity zones In non convex domains, the solution to the Sverdrup equation is generally discontinuous (jump condition). To get an approximation of, -A regularization is needed; -The error term is dealt with like a boundary layer.

Complex transitions… No matching, but a superposition : -localized East and West boundary layers, -extinction of North and South boundary layers. Construction starting from the East boundary.

Towards more physically relevant models?