# Coriolis forces

The Coriolis model controls the manifestation of the term $\boldsymbol{f} \times \boldsymbol{v}$ in the momentum equation.

## $f$-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

\[ \boldsymbol{f} = f \boldsymbol{\hat z} \, ,\]

where $f$ is constant and determined by the user.

## The arbitrary-axis constant-Coriolis approximation

In this approximation, the coordinate system rotates around an axis in the $x,y,z$-plane, such that

\[ \boldsymbol{f} = f_x \boldsymbol{\hat x} + f_y \boldsymbol{\hat y} + f_z \boldsymbol{\hat z} \, ,\]

where $f_x$, $f_y$, and $f_z$ are constants determined by the user.

## $\beta$-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

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

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 $\beta$-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

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

as can be found in the paper by Paul J Dellar (2011) where $\beta = 2 \Omega \cos \varphi_0 / R$ and $\gamma = -4 \Omega \sin \varphi_0 / R$.

- 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 the 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 $\boldsymbol{f} \approx \frac{4 \pi}{\text{day}} \left ( \cos \varphi \boldsymbol{\hat y} + \sin \varphi \boldsymbol{\hat z} \right ) \, ,$ 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.