pydune.physics.dune.bedinstability_1D.spatial_growth_rate#

spatial_growth_rate(w, A0, B0, mu, r)[source]#

Dune instability spatial growth rate - imaginary part of the complex wavenumber \(k_{+}\). Note that here, \(\mathcal{A} = \mathcal{A}_{0}\) where \(\mathcal{B} = \mathcal{B}_{0} - 1/(r^{2}\mu)\), taking into account slope effects.

Parameters:

  • w (scalar, numpy array) – non dimensional pulsation \(\omega\).

  • A0 (scalar, numpy array) – hydrodynamic coefficient \(\mathcal{A}_{0}\) (in-phase).

  • B0 (scalar, numpy array) – hydrodynamic coefficient \(\mathcal{B}_{0}\) (in-quadrature).

  • mu (scalar, numpy array) – friction coefficient \(\mu\).

  • r (scalar, numpy array) – velocity ratio \(u_{*}/u_{\rm d} = \sqrt{\theta/\theta_{d}}\)

Returns:

sigma_s – spatial dune growth rate \(\sigma_{\rm s}\).

Return type:

scalar, numpy array

Notes

Note that all quantities are made non dimensional:

  • length scales by the saturation length \(L_{\rm sat}\).

  • time scales by \(L_{\rm sat}^{2}/Q_{*}\).

Examples

>>> import numpy as np
>>> w = np.linspace(0.001, 1, 1000)
>>> A0, B0, mu, r = 3.5, 2, 0.63, 2
>>> sigma_s = spatial_growth_rate(w, A0, B0, mu, r)