Coriolis forces
The Coriolis model controls the manifestation of the term $\bm{f} \times \bm{u}$ in the momentum equation.
The $f$-plane approximation
Under an $f$-plane approximation[3] the reference frame in which the momentum and tracer equations are are solved rotates at a constant rate around a vertical axis, such that
\[ \bm{f} = f \bm{\hat z}\]
where $f$ is constant and determined by the user.
The $\beta$-plane approximation
Under the $\beta$-plane approximation, the rotation axis is vertical as for the $f$-plane approximation, but $f$ is expanded in a Taylor series around a central latitude such that
\[ \bm{f} = \left ( f_0 + \beta y \right ) \bm{\hat z} \, ,\]
where $f_0$ is the planetary vorticity at some central latitude, and $\beta$ is the planetary vorticity gradient. The $\beta$-plane model is not periodic in $y$ and thus can be used only in domains that are bounded in the $y$-direction.
- 3The $f$-plane approximation is used to model the effects of Earth's rotation on anisotropic fluid motion in a plane tangent to the Earth's surface. In this case, $\bm{f}$ is
math \bm{f} \approx \frac{4 \pi}{\text{day}} \sin \varphi \bm{\hat z} \, , $where $\phi$ is latitude and the Earth's rotation rate is approximately $2 \pi / \text{day}$. This approximation neglects the vertical component of Earth's rotation vector at $\varphi$.