Dynamically consistent shallow-atmosphere equations with a complete Coriolis force

2014 ◽  
Vol 140 (684) ◽  
pp. 2388-2392 ◽  
Author(s):  
M. Tort ◽  
T. Dubos
Keyword(s):  
Coriolis Force ◽  
Complete Coriolis Force

Multilayer shallow water equations with complete Coriolis force. Part 3. Hyperbolicity and stability under shear

2013 ◽  
Vol 723 ◽  
pp. 289-317 ◽  
Author(s):  
Andrew L. Stewart ◽  
Paul J. Dellar
Keyword(s):  
Shallow Water ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Three Dimensional ◽  
Direct Calculation ◽  
Shear Instability ◽  
Vertical Shear ◽  
Vortex Sheets ◽  
Multilayer Shallow Water Equations ◽  
Complete Coriolis Force

AbstractWe analyse the hyperbolicity of our multilayer shallow water equations that include the complete Coriolis force due to the Earth’s rotation. Shallow water theory represents flows in which the vertical shear is concentrated into vortex sheets between layers of uniform velocity. Such configurations are subject to Kelvin–Helmholtz instabilities, with arbitrarily large growth rates for sufficiently short-wavelength disturbances. These instabilities manifest themselves through a loss of hyperbolicity in the shallow water equations, rendering them ill-posed for the solution of initial value problems. We show that, in the limit of vanishingly small density difference between the two layers, our two-layer shallow water equations remain hyperbolic when the velocity difference remains below the same threshold that also ensures the hyperbolicity of the standard shallow water equations. Direct calculation of the domain of hyperbolicity becomes much less tractable for three or more layers, so we demonstrate numerically that the threshold for the velocity differences, below which the three-layer equations remain hyperbolic, is also unchanged by the inclusion of the complete Coriolis force. In all cases, the shape of the domain of hyperbolicity, which extends outside the threshold, changes considerably. The standard shallow water equations only lose hyperbolicity due to shear parallel to the direction of wave propagation, but the complete Coriolis force introduces another mechanism for loss of hyperbolicity due to shear in the perpendicular direction. We demonstrate that this additional mechanism corresponds to the onset of a transverse shear instability driven by the non-traditional components of the Coriolis force in a three-dimensional continuously stratified fluid.

Consistent shallow-water equations on the rotating sphere with complete Coriolis force and topography

2014 ◽  
Vol 748 ◽  
pp. 789-821 ◽  
Author(s):  
Marine Tort ◽  
Thomas Dubos ◽  
François Bouchut ◽  
Vladimir Zeitlin
Keyword(s):  
Shallow Water ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Primitive Equations ◽  
Conservation Of Energy ◽  
Rotating Sphere ◽  
Planetary Rotation ◽  
Dependent Change ◽  
Body Velocity ◽  
Complete Coriolis Force

AbstractConsistent shallow-water equations are derived on the rotating sphere with topography retaining the Coriolis force due to the horizontal component of the planetary angular velocity. Unlike the traditional approximation, this ‘non-traditional’ approximation captures the increase with height of the solid-body velocity due to planetary rotation. The conservation of energy, angular momentum and potential vorticity are ensured in the system. The caveats in extending the standard shallow-water wisdom to the case of the rotating sphere are exposed. Different derivations of the model are possible, being based, respectively, on (i) Hamilton’s principle for primitive equations with a complete Coriolis force, under the hypothesis of columnar motion, (ii) straightforward vertical averaging of the ‘non-traditional’ primitive equations, and (iii) a time-dependent change of independent variables in the primitive equations written in the curl (‘vector-invariant’) form, with subsequent application of the columnar motion hypothesis. An intrinsic, coordinate-independent form of the non-traditional equations on the sphere is then given, and used to derive hyperbolicity criteria and Rankine–Hugoniot conditions for weak solutions. The relevance of the model for the Earth’s atmosphere and oceans, as well as other planets, is discussed.

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