ZonalFlow

Generalized Quasilinear approximation and Generalized Cumulant Expansion

• nl(): Numerically solving the fully non-linear (NL) dynamics of dissipative and driven rotational flows can be a computational expensive task if one is interested in the statistical behaviour of turbulent zonal jets. In a spectral code, the computational bottleneck is computing the sum of all interactions arising from the non-linear terms in the dynamical field equations.

• gql(): The Generalized Quasilinear (GQL) approximation [1] simplifies these non-linear terms by inducing rules for interaction between low (L) and high (H) wavenumber zonal projections of the dynamical field obtained using a spectral filter with cutoff Λ. Specifically, the HH→H (eddy-eddy non-linearity interactions as well as the HL→L and LL→H interactions are eliminated. In this manner, the field equations can be extrapolated between Quasilinear (QL) dynamics for Λ = 0, and NL dynamics for Λ = M (the maximum zonal wavenumber). Therefore, a GQL system obtained for Λ < M, and that suffices to simulate zonal jet statistics for given friction and driving parameters, is by nature a reduced model of the underlying dynamics. However, obtaining statistics of the flow may still require performing simulations with large spin-up times to arrive at a statistical significant sample.

• gce2(): The Generalized Cumulant Expansion (GCE2) circumvents this last problem, by posing equations in terms of the statistics -- the first and second cumulant -- derived from the GQL equations for a given cutoff Λ. This allows the required cumulants (or equaivalently the moments) to be obtained directly, precluding the need for large spin-up dynamical calculations. In practice, the cumulant sizes for low Λ are large and can be computationally more expensive per timestep; however, this can be partly offset by the fact that fewer timesteps need to be solved. Use of dimensional reduction techniques can help speed this up further.

ZonalFlow allows you to solve all three sets of equations: NL, GQL and GCE2 for dynamics on the β-plane. It uses the OrdinaryDiffEq package for numerically time-integrating the spectral ODE problem, giving access to a range of integration algorithms. A matrix-free representation is used to minimize computational cost. Currently, only fixed timestep algorithms are recommended and computations are serial for now.

Functions nl(params...), gql(params...) and gce2(params...) solver functions are exported by the package together with a number of solution analysis functions.

References

[1] Marston, J. B. and Chini, G. P. and Tobias, S. M. (2016) Physical Review Letters 116 214501

Use

The current release solves for equations on the β-plane with forcing by a deterministic point-jet and relaxation; the choice of hyperviscosity is available. Set the parameters for a given solution as follows:

Domain:

• lx: axial domain length
• ly: transverse domain length
• nx: axial resolution
• ny: transverse resolution
• Λ (for GQL and GCE2)

Problem:

• Ω: Rotational rate
• θ: Latitude
• β: Coriolis parameter
• Ξ: jet strength
• τ: relaxation time

Timestepping and solution save:

• jw: Jet width; defaults to 0.2
• ic: Initial Condition; defaults to point jet with random seed.
• dt: Timestep size; defaults to 0.001.
• t_end: Final solution time
• savefreq: Data saving frequency

Each solver function returns the solution variable to which the following analysis algorithms may be applied.

• zonalenergy(params...)
• fourierenergy(params...)
• meanvorticity(params...)
• zonalvelocity(params...)

License

MIT