boundary layer

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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 incompressible, 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 (fplane) modifies only the pressure.
The next contribution is due to inhomogeneities
The role of wind
Experimental observations show
that currents are strongly
correlated to the wind.
North-East monsoon
South-West monsoon
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
International Congress of Mathematicians, 1904
Über Flüssigkeitsbewegung bei sehr kleiner Ribung
Flow around an obstacle :
- inviscid exterior component
satisfying a non penetration
condition
- boundary layer localized in the
vicinity of the wall
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
we study the limit
for
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.
North boundary layer, size ν1/4
Discontinuity
boundary layer,
size ν1/4
West boundary layers, size ν1/3
Towards more physically
relevant models?
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