WVAdaptiveDamping

Adapt small-scale spectral damping to the current flow speed.

Declaration 

WVAdaptiveDamping < WVForcing

Overview

This closure rebuilds its spectral shape when the transform’s effective resolution changes and scales its coefficient tendency by the current maximum horizontal speed. It is useful when the flow amplitude evolves substantially, such as during spin-up.

This closure has a number of noteworthy features:

  • It does not mix geostrophic and wave modes, which requires setting the diffusivity equal to the viscosity.
  • The properties k_no_damp and j_no_damp indicate the wavenumber and mode below which there is zero damping, due to the spectral vanishing viscosity filter.
  • The properties k_damp and j_damp are estimates of the wavenumber and mode above which significant damping will occur.

The damping operator acts in the spectral domain, directly damping the wave-vortex coefficients.

\[\begin{align} \partial_t A_\pm^{k\ell j} =& - \nu (k^2 + \ell^2 ) A_\pm^{k\ell j} - \nu_z \lambda_j^{-2} A_\pm^{k\ell j} \\ \partial_t A_0^{k\ell j} =& - \nu (k^2 + \ell^2 ) A_0^{k\ell j} - \nu_z \lambda_j^{-2} A_0^{k\ell j} \end{align}\]

where

\[\nu_z = \nu \lambda^2_\textrm{min} k^2_\textrm{max} = \nu \lambda^2_\textrm{min} \left( \frac{\pi}{\Delta} \right)^2\]

is chosen to make the damping isotropic. The notation here is that \(\Delta\) is the horizontal grid resolution and \(\lambda^2_\textrm{min}\) is the smallest resolved radius of deformation. The value of \(\nu\) is set as

\[\nu = \frac{U \Delta}{\pi^2}\]

where \(U\) is the maximum fluid velocity.

Example 

wvt = WVTransformConstantStratification([40e3,30e3,2e3],[8,6,5],N0=5.2e-3,latitude=45,isHydrostatic=true);
wvt.addForcing(WVAdaptiveDamping(wvt));

Notes

This currently damps the non-hydrostatic wavemodes the same as the hydrostatic geostrophic modes. The non-hydrostatic modes would have a smaller deformation radius, and thus would be damped more strongly. So arguably they’re under-damped in a non-hydrostatic simulation.

Topics

  • Create the forcing
  • Inspect forcing or damping scales
    • k_no_damp Horizontal wavenumber below which damping is exactly zero.
    • k_damp Estimated horizontal wavenumber for significant damping.
    • j_no_damp Vertical mode number below which damping is exactly zero.
    • j_damp Estimated vertical mode number for significant damping.
    • assumedEffectiveHorizontalGridResolution Effective horizontal resolution used to construct damp, in meters.
    • dampingTimeScale Return the inverse maximum unit-speed damping coefficient.
    • damp Unit-speed spectral damping operator in inverse meters.

Developer Topics

These items document internal implementation details and are not part of the primary public API.

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