WAVETRISK-OCEAN: an adaptive dynamical core for ocean modelling

<p>This talk introduces WAVETRISK-OCEAN, an incompressible version of the atmosphere model WAVETRISK.  This new model is built on the same wavelet-based dynamically adaptive core as WAVETRISK, which itself uses DYNAMICO's mimetic vector-invariant multilayer shallow water formulation. Both codes use a Lagrangian vertical coordinate with conservative remapping.  The ocean variant solves the incompressible multilayer shallow water equations with a Ripa type thermodynamic treatment of horizontal density gradients.  Time integration uses barotropic-baroclinic mode splitting via an implicit free surface formulation, which is about 15 times faster than explicit time stepping.  The barotropic and baroclinic estimates of the free surface are reconciled at each time step using layer dilation. No slip boundary conditions at coastlines are approximated using volume penalization.  Results are presented for a standard set of ocean model test cases adapted to the sphere (seamount,  upwelling and baroclinic jet) as well as  turbulent wind-driven gyre flow in simplified geometries.  An innovative feature of WAVETRISK-OCEAN is that it could be coupled easily to the WAVETRISK atmosphere model, providing a simple integrated Earth system model using a consistent modelling framework.</p>

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

We 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.

Multilayer shallow water equations with complete Coriolis force. Part 2. Linear plane waves

We investigate the behaviour of linear plane waves in multilayer shallow water equations that include a complete treatment of the Coriolis force. These equations improve upon the conventional shallow water equations, based on the traditional approximation, that include only the part of the Coriolis force due to the locally vertical component of the rotation vector. Including the complete Coriolis force leads to dramatic changes in the structure of long linear plane waves. It allows subinertial waves to exist with frequencies below the inertial frequency, the minimum frequency for which waves exist under the traditional approximation. These subinertial waves are characterized by a distinguished limit in which the horizontal pressure gradient becomes comparable to the upwellings and downwellings driven by the non-traditional Coriolis term in the vertical momentum equation. The subinertial waves connect wave modes that remain separate in the conventional multilayer shallow water equations, such as the surface and internal waves in a two-layer system. Eastward-propagating surface waves in a two-layer system connect with westward-propagating internal waves, and vice versa, via the long subinertial waves. The long subinertial waves cannot be classified as either surface or internal waves, due to the phase difference between the disturbances to the interfaces in these waves.