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 solved rotates at a constant rate.
The traditional $f$-plane approximation
In the traditional $f$-plane approximation, the coordinate system rotates around a vertical axis such that
where $f$ is constant and determined by the user.
The non-traditional $f$-plane approximation
In the non-traditional $f$-plane approximation, the coordinate system rotates around an axis in the $y,z$-plane, such that
where $f_y$ and $f_z$ are constant and determined by the user.
The $\beta$-plane approximation
The traditional $\beta$-plane approximation
Under the traditional $\beta$-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
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.
The non-traditional $f$-plane approximation
The non-traditional $\beta$-plane approximation accounts for the latitudinal variation of both the locally vertical and the locally horizontal components of the rotation vector
where $\beta = 2\Omega\cos\varphi_0 / R$ and $\gamma = -4\Omega\varphi_0 / R$.
- 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, the projection of Earth's rotation vector at latitude $\varphi$ and onto a coordinate system in which $x, y, z$ correspond to the directions east, north, and up is
math \bm{f} \approx \frac{4 \pi}{\text{day}} \left ( \cos \varphi \bm{\hat y} + \sin \varphi \bm{\hat z} \right ) \, , $where the Earth's rotation rate is approximately $2 \pi / \text{day}$. The traditional $f$-plane approximation neglects the $y$-component of this projection, which is appropriate for fluid motions with large horizontal-to-vertical aspect ratios.