WVAdaptiveDamping

Adaptive small-scale damping

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Declaration

WVAdaptiveDamping < WVForcing

Overview

This damping operator is a linear closure that dynamically changes its amplitude to keep the Reynolds number at the grid scale equal to one. This closure is ideal for a spin-up problem where the amplitude of the flow is changing.

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.

Usage

Assuming there is a WVTransform instance wvt, to add this forcing,

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

• Initialization
  • WVAdaptiveDamping initialize the WVAdaptiveDamping
• Properties
  • assumedEffectiveHorizontalGridResolution effective resolution used in the damping calculation
  • damp spectral matrix that multiplies Ap,Am,A0 to damp
  • dampingTimeScale Computes the minimum damping time scale
  • j_damp wavenumber at which the significant scale damping starts.
  • j_no_damp wavenumber below which there is zero damping
  • k_damp wavenumber at which the significant scale damping starts.
  • k_no_damp wavenumber below which there is zero damping
• Internal
  • buildDampingOperator Builds the damping operator
  • spectralVanishingViscosityFilter Builds the spectral vanishing viscosity operator
• CAAnnotatedClass requirement
  • classRequiredPropertyNames Returns the required property names for the class

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