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

\[ \bm{f} = f \bm{\hat z}\]

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

\[ \bm{f} = f_y \bm{\hat y} + f_z \bm{\hat z}\]

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

\[ \bm{f} = \left ( f_0 + \beta y \right ) \bm{\hat z} \, ,\]

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.

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

\[ \bm{f} = \left[ 2\Omega\cos\varphi_0 \left( 1 - \frac{z}{R} \right) + \gamma y \right] \bm{\hat y} + \left[ 2\Omega\sin\varphi_0 \left( 1 + 2\frac{z}{R} \right) + \beta y \right] \bm{\hat z} \, ,\]

where $\beta = 2\Omega\cos\varphi_0 / R$ and $\gamma = -4\Omega\varphi_0 / R$.