<p>This equation is a form of the <a href="https://en.wikipedia.org/wiki/Shallow_water_equations" rel="nofollow noreferrer">Shallow-water equations</a>, which are derived from Navier-Stokes in the incompressible limit and the vertical direction is integrated out.</p>

<p>One then takes the equation for the vorticity (which has only one component then) from the Shallow-water equations. The vorticity is split into planetary vorticity, as is the Coriolis term with the <span class="math-container">$\beta$</span>-sheet-approximation.</p>

<p>Additionally it looks like they're using a velocity streamfunction.</p>

CC BY-SA 4.0
2019-05-09T16:49:54.273Z
2019-05-09T16:49:54.273Z
AtmosphericPrisonEscape
<p>I came across the barotropic vorticity equation (below) in <a href="https://doi.org/10.1080/03091920290020977" rel="nofollow noreferrer">Sakamoto (2002)</a> and I cannot figure it out. The notation is not clear to me. How can it be derived from the Navier-Stokes equation?</p>

<p><a href="https://i.sstatic.net/HyaHJ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/HyaHJ.png" alt="excerpt from paper"></a></p>

<p><span class="math-container">$$ \beta\psi_x=\frac{\tau_x^y-\tau_y^x}{\rho_0H}-r\zeta+A_H\nabla^2\zeta-\frac{f_0}{H}w_D $$</span></p>

<p>where</p>

<p><span class="math-container">$\psi$</span> is the streamfunction</p>

<p><span class="math-container">$\nabla$</span> is the horizontal gradient operator</p>

<p><span class="math-container">$\zeta=\nabla^2\psi$</span> is the vertical component of relative vorticity</p>

<p><span class="math-container">$(\tau_x,\tau_y)$</span> is the zonal wind stress</p>

<p><span class="math-container">$f=f_0+\beta y$</span> is the Coriolis parameter</p>

<p><span class="math-container">$\rho_0$</span> is the mean density</p>

<p><span class="math-container">$H$</span> is the depth of the model ocean</p>

<p><span class="math-container">$r$</span> is the inverse time scale for vorticity decay by bottom friction </p>

<p><span class="math-container">$A_H$</span> is a horizontal eddy viscosity</p>

<p><span class="math-container">$w_D$</span> is not explained in this paper.</p>

<geophysics><fluid-dynamics>
Barotropic vorticity equation
Cong Gao
# Barotropic vorticity equation

(Asked by: username_0 on 2019-05-09)

I came across the barotropic vorticity equation (below) in [Sakamoto (2002)](https://doi.org/10.1080/03091920290020977) and I cannot figure it out. The notation is not clear to me. How can it be derived from the Navier-Stokes equation?

[![excerpt from paper](https://i.sstatic.net/HyaHJ.png)](https://i.sstatic.net/HyaHJ.png)

$$ \beta\psi\_x=\frac{\tau\_x^y-\tau\_y^x}{\rho\_0H}-r\zeta+A\_H\nabla^2\zeta-\frac{f\_0}{H}w\_D $$

where

$\psi$ is the streamfunction

$\nabla$ is the horizontal gradient operator

$\zeta=\nabla^2\psi$ is the vertical component of relative vorticity

$(\tau\_x,\tau\_y)$ is the zonal wind stress

$f=f\_0+\beta y$ is the Coriolis parameter

$\rho\_0$ is the mean density

$H$ is the depth of the model ocean

$r$ is the inverse time scale for vorticity decay by bottom friction

$A\_H$ is a horizontal eddy viscosity

$w\_D$ is not explained in this paper.

**Tags:** geophysics, fluid-dynamics


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## 1 answers

### Answer by username_1 on 2019-05-09. Score: 1

This equation is a form of the [Shallow-water equations](https://en.wikipedia.org/wiki/Shallow_water_equations), which are derived from Navier-Stokes in the incompressible limit and the vertical direction is integrated out.

One then takes the equation for the vorticity (which has only one component then) from the Shallow-water equations. The vorticity is split into planetary vorticity, as is the Coriolis term with the $\beta$-sheet-approximation.

Additionally it looks like they're using a velocity streamfunction.


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