Coriolis

The Coriolis option determines whether the fluid experiences the effect of the Coriolis force, or rotation. Currently three options are available: no rotation, $f$-plane, and $\beta$-plane.

No rotation

By default there is no rotation. This can be made explicit by passing coriolis = nothing to a model constructor.

Traditional $f$-plane

To set up an $f$-plane with, for example, Coriolis parameter $f = 10^{-4} \text{s}^{-1}$

julia> coriolis = FPlane(f=1e-4)
FPlane{Float64}(f=0.0001)

An $f$-plane can also be specified at some latitude on a spherical planet with a planetary rotation rate. For example, to specify an $f$-plane at a latitude of $\varphi = 45°\text{N}$ on Earth which has a rotation rate of $\Omega = 7.292115 \times 10^{-5} \text{s}^{-1}$

julia> coriolis = FPlane(rotation_rate=7.292115e-5, latitude=45)
FPlane{Float64}(f=0.000103126)

in which case the value of $f$ is given by $2\Omega\sin\varphi$.

Coriolis term for constant rotation in a Cartesian coordinate system

One can use ConstantCartesianCoriolis to set up a Coriolis acceleration term where the Coriolis parameter is constant and the rotation axis is arbitrary. For example, with $\boldsymbol{f} = (0, f_y, f_z) = (0, 2, 1) \times 10^{-4} \text{s}^{-1}$,

julia> coriolis = ConstantCartesianCoriolis(fx=0, fy=2e-4, fz=1e-4)
ConstantCartesianCoriolis{Float64}: fx = 0.00e+00, fy = 2.00e-04, fz = 1.00e-04

Or alternatively, the same result can be achieved by specifying the magnitude of the Coriolis frequency f and the rotation_axis. So another way to get a Coriolis acceleration with the same values is:

julia> rotation_axis = (0, 2e-4, 1e-4)./√(2e-4^2 + 1e-4^2) # rotation_axis has to be a unit vector
(0.0, 0.8944271909999159, 0.4472135954999579)

julia> coriolis = ConstantCartesianCoriolis(f=√(2e-4^2+1e-4^2), rotation_axis=rotation_axis)
ConstantCartesianCoriolis{Float64}: fx = 0.00e+00, fy = 2.00e-04, fz = 1.00e-04

An $f$-plane with non-traditional Coriolis terms can also be specified at some latitude on a spherical planet with a planetary rotation rate. For example, to specify an $f$-plane at a latitude of $\varphi = 45°\text{N}$ on Earth which has a rotation rate of $\Omega = 7.292115 \times 10^{-5} \text{s}^{-1}$

julia> coriolis = ConstantCartesianCoriolis(rotation_rate=7.292115e-5, latitude=45)
ConstantCartesianCoriolis{Float64}: fx = 0.00e+00, fy = 1.03e-04, fz = 1.03e-04

in which case $f_z = 2\Omega\sin\varphi$ and $f_y = 2\Omega\cos\varphi$.

Traditional $\beta$-plane

To set up a $\beta$-plane the background rotation rate $f_0$ and the $\beta$ parameter must be specified. For example, a $\beta$-plane with $f_0 = 10^{-4} \text{s}^{-1}$ and $\beta = 1.5 \times 10^{-11} \text{s}^{-1}\text{m}^{-1}$ can be set up with

julia> coriolis = BetaPlane(f₀=1e-4, β=1.5e-11)
BetaPlane{Float64}(f₀=0.0001, β=1.5e-11)

Alternatively, a $\beta$-plane can also be set up at some latitude on a spherical planet with a planetary rotation rate and planetary radius. For example, to specify a $\beta$-plane at a latitude of $\varphi = 10^\circ{S}$ on Earth which has a rotation rate of $\Omega = 7.292115 \times 10^{-5} \text{s}^{-1}$ and a radius of $R = 6,371 \text{km}$

julia> coriolis = BetaPlane(rotation_rate=7.292115e-5, latitude=-10, radius=6371e3)
BetaPlane{Float64}(f₀=-2.53252e-5, β=2.25438e-11)

in which case $f_0 = 2\Omega\sin\varphi$ and $\beta = 2\Omega\cos\varphi / R$.

Non-traditional $\beta$-plane

A non-traditional $\beta$-plane requires either 5 parameters (by default Earth's radius and rotation rate are used):

julia> NonTraditionalBetaPlane(fz=1e-4, fy=2e-4, β=4e-11, γ=-8e-11)
NonTraditionalBetaPlane{Float64}(fz = 1.00e-04, fy = 2.00e-04, β = 4.00e-11, γ = -8.00e-11, R = 6.37e+06)

or the rotation rate, radius, and latitude:

julia> NonTraditionalBetaPlane(rotation_rate=5.31e-5, radius=252.1e3, latitude=10)
NonTraditionalBetaPlane{Float64}(fz = 1.84e-05, fy = 1.05e-04, β = 4.15e-10, γ = -1.46e-10, R = 2.52e+05)