WVBottomFrictionQuadratic

Quadratic bottom friction

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Declaration

WVBottomFrictionQuadratic < WVForcing

Overview

Applies quadratic bottom friction to the flow, i.e., \(\frac{du}{dt} = -\frac{Cd}{dz}\lvert \mathbf{u} \rvert\mathbf{u}(x,y,-D)\). Cd is unitless, and dz is (approximately) the size of the grid spacing at the bottom boundary.

To compare with linear bottom friction where \(\frac{du}{dt} = -r \cdot u(x,y,-D)\), note that \(- \frac{L_z}{dz} r = -\frac{C_d}{dz} \lvert \mathbf{u} \rvert\) and you will find a characteristic velocity \(\lvert \mathbf{u} \rvert\) of about 11.5 cm/s for Cd=0.002 and r=1/(200 days). If \(C_d=0.001\), then the damping time scale has to double to 1/(400 days) for and equivalent characteristic velocity.

For barotropic QG we want the scaling to work out similarly, but now have \(-r = -\frac{C_d}{D}\lvert \mathbf{u} \rvert\) where \(D\) works out to be \(L_z\). Thus, we will scale the barotropic QG quadratic bottom drag by 4000 m, to match typical oceanic scales.

Using the notation that

\[|\mathbf{u}(x,y,-D)| = \sqrt{u^2(x,y,-D) + v^2(x,y,-D)}\]

is the magnitude of the total velocity at the bottom boundary, both nonhydrostatic and hydrostatic transforms linear bottom drag have the form

\[\begin{align} \mathcal{S}_u &= -\frac{C_d}{dz} |\mathbf{u}(x,y,-D)| u(x,y,-D) \\ \mathcal{S}_v &= -\frac{C_d}{dz} |\mathbf{u}(x,y,-D)| v(x,y,-D) \\ \mathcal{S}_w &= 0 \\ \mathcal{S}_\eta &= 0 \end{align}\]

and for quasigeostrophic transforms,

\[\begin{align} \mathcal{S}_\textrm{qgpv} &= -\frac{C_dd}{dz} \left( \frac{\partial}{\partial x} \left( |\mathbf{u}(x,y,-D)| v(x,y,-D) \right) - \frac{\partial}{\partial y} \left( |\mathbf{u}(x,y,-D)| u(x,y,-D) \right) \right) \end{align}\]

Usage

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

wvt.addForcing(WVBottomFrictionQuadratic(Cd=0.001));

Topics

• Initialization
  • WVBottomFrictionQuadratic initialize the WVBottomFrictionQuadratic
• Properties
  • Cd non-dimensional quadratic drag coefficient
  • cd \(\frac{Cd}{dz}\) scaled quadratic drag coefficient with units \(m^{-1}\)
• CAAnnotatedClass requirement
  • classRequiredPropertyNames Returns the required property names for the class

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