Coriolis forces

The Coriolis model controls the manifestation of the term   in the momentum equation.

-plane approximation

Under an -plane approximation^[1] the reference frame in which the momentum and tracer equations are solved rotates at a constant rate.

The traditional -plane approximation

In the traditional -plane approximation, the coordinate system rotates around a vertical axis such that

where 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 -plane, such that

where , , and are constants determined by the user.

-plane approximation {#\beta-plane-approximation}

The traditional -plane approximation {#The-traditional-\beta-plane-approximation}

Under the traditional -plane approximation, the rotation axis is vertical as for the -plane approximation, but is expanded in a Taylor series around a central latitude such that

where is the planetary vorticity at some central latitude, and is the planetary vorticity gradient. The -plane model is not periodic in and thus can be used only in domains that are bounded in the -direction.

The non-traditional -plane approximation {#The-non-traditional-\beta-plane-approximation}

The non-traditional -plane approximation accounts for the latitudinal variation of both the locally vertical and the locally horizontal components of the rotation vector

as can be found in the paper by Dellar (2011), where   and   .

Spherical Coriolis

On curvilinear grids on the sphere, the Coriolis parameter varies with latitude according to

where is the planetary rotation rate and is latitude.

For hydrostatic models, only the vertical component of the Coriolis force is retained (the traditional approximation), contributing and to the zonal and meridional momentum equations respectively.

For nonhydrostatic models, the full Coriolis force includes additional terms involving the horizontal component   that couples the horizontal and vertical momentum equations.

Discretization of the Coriolis term

On the Arakawa C-grid, the two velocity components and are staggered: is defined at the west faces of each cell, while is defined at the south faces. Computing the Coriolis acceleration (e.g., in the -equation) therefore requires interpolating to the -point, and vice versa.

The choice of interpolation scheme affects two important properties:

1. Conservation: whether the scheme conserves kinetic energy, potential enstrophy, or both.

2. Boundary accuracy: whether the scheme correctly handles masked (land) points near immersed boundaries.

Enstrophy-conserving scheme

The enstrophy-conserving scheme (Sadourny, 1975) evaluates at cell centers and interpolates velocity directly:

where denotes the interpolation of from cell centers to the -point, and is the 4-point average of to the -point.

This scheme conserves potential enstrophy ( where   ) for horizontally non-divergent flow, but does not conserve kinetic energy.

Energy-conserving scheme

The energy-conserving scheme (Sadourny, 1975) evaluates at vorticity (corner) points and interpolates transport ( ) rather than velocity:

where   is the volume transport per unit depth. The product   is computed at each vorticity point before the spatial averaging, which ensures that the Coriolis terms cancel when forming the kinetic energy equation (Dobricic, 2006).

This scheme conserves kinetic energy but not potential enstrophy.

Triad (Energy- and Enstrophy-Conserving) scheme

The Triad scheme (Arakawa and Lamb, 1981) uses triads to achieve simultaneous conservation of both kinetic energy and potential enstrophy. Each triad at a cell center sums 3 of the 4 surrounding vorticity values, paired with transports at diagonally adjacent velocity points. The four triads at cell center are:

where is the potential vorticity at corner (vorticity) points. The Coriolis tendency is then:

where the sum is over the four triads and is the transport at the diagonally paired velocity point. This 12-point stencil conserves both kinetic energy and potential enstrophy in the limit of horizontally non-divergent flow.

Active-weighted (wet-points-only) correction

Near immersed boundaries on a C grid, the conventional averaging of velocities in the Coriolis term includes masked (land) points where velocity is zero. As shown by Jamart and Ozer (1986), this underestimates the Coriolis force along solid boundaries:

"The calculation of the term in the momentum equation is usually performed by averaging the values of the four closest neighbors of the point under consideration. [...] In cases where the interior solution is uniform, the procedure amounts to reducing the Coriolis parameter by a factor of 2 along such a wall."

This creates a spurious numerical boundary layer with artificial residual currents that are entirely an artifact of the discretization.

The wet-points-only correction eliminates this artifact by dividing the interpolated Coriolis term by the number of active (non-masked) nodes in the stencil, rather than the full stencil size:

where is the count of non-peripheral (wet) velocity nodes in the 4-point interpolation stencil. When all nodes are active, this reduces to the standard scheme.

When to use the wet-points-only correction

The active-weighted correction can reduce spurious numerical boundary layers along simple, flat immersed boundaries (Jamart and Ozer, 1986). However, for complex topography (narrow passages, sharp capes, jagged coastlines) or large values, the amplification factor can inject energy and produce grid-scale checkerboard artifacts along coastlines. For this reason, the standard (non-active-weighted) schemes are the default. Users should test the active-weightedschemes carefully and verify the possible benefits before adopting them in production simulations.

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1. The -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 and onto a coordinate system in which correspond to the directions east, north, and up is    where the Earth's rotation rate is approximately . The traditional -plane approximation neglects the -component of this projection, which is appropriate for fluid motions with large horizontal-to-vertical aspect ratios. ↩︎