fridom.nonhydro.initial_conditions.geostrophic_spectra.geostrophic_energy_spectrum

fridom.nonhydro.initial_conditions.geostrophic_spectra.geostrophic_energy_spectrum(kx, ky, kz, d=7, k0=6, c=2)

Geostrophic energy spectrum.

Description

The geostrophic energy spectrum is separated into horizontal and vertical components. Following the work of Masur & Oliver [2020], the horizontal energy spectrum \(S_h\) is given by:

\[S_h = \frac{k^7}{\left(k^2 + a k_0^2\right)^{2b}}\]

where \(k = \sqrt{k_x^2 + k_y^2}\) is the horizontal wavenumber, \(a\) and \(b\) are constants:

\[a = \frac{4}{7}b - 1, \quad b = \frac{7+d}{4}\]

where \(d\) is the power law exponent for large horizontal wavenumbers (\(S_h(k) \sim k^{-d}\) for \(k \to \infty\)). The parameter \(k_0\) is the wavenumber with the maximum energy.

The vertical component \(S_v\) is given by:

\[S_v = \exp(-c|k_z|)\]

where \(k_z\) is the vertical wavenumber and \(c\) is a constant. The total energy spectrum is given by \(S = S_h S_v\).

Parameters

kxfloat

The horizontal wavenumber in the x-direction.

kyfloat

The horizontal wavenumber in the y-direction.

kzfloat

The vertical wavenumber.

dfloat, optional (default=7)

The power law exponent for large horizontal wavenumbers (\(S_h(k) \sim k^{-d}\) for \(k \to \infty\)).

k0float, optional (default=6)

The wavenumber with the maximum energy.

cfloat, optional (default=2)

The decay rate of the vertical energy spectrum.