math lessons - Vorticity equation

The vorticity equation is an important prognostic equation in the atmospheric sciences. It describes the total derivative (that is, the local change due to local change with time and advection) of vorticity, and thus can be stated in either relative or absolute form.

The more compact version is that for absolute vorticity, $η$ , using the pressure system:

$\frac{d \eta}{d t} = -\eta \cdot \nabla_h \mathbf{v}_h - \left( \frac{\partial \omega}{\partial x} \frac{\partial v}{\partial z} - \frac{\partial \omega}{\partial y} \frac{\partial u}{\partial z} \right) - \frac{1}{\rho^2} \mathbf{k} \cdot ( \nabla_h p \times \nabla_h \rho )$

Here, $ρ$ is density, u, v, and $ω$ are the components of wind velocity, and $\nabla_h$ is the 2-dimensional (i.e. horizontal-component-only) del.

The terms on the RHS denote the positive or negative generation of absolute vorticity by divergence of air, twisting of the axis of rotation, and baroclinity, respectively.

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Vorticity equation

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