Turbulence is often subjected to the influence of rotation in many applications such as meteorology, ocean dynamics and astrophysics. Capturing this effect in models for rotating turbulence flows is important for the accurate description of the dynamics related to the presence of the inertial waves, which interact and lead to wave-turbulence. Anisotropy is important feature of these flows. Multiscale models, such as the eddy-damped quasi-normal Markovian one (EDQNM), have been specifically adapted for including rotation, and are consistent with wave-turbulence theory. Being spectral, they assume statistical homogeneity of turbulence. However, turbulence is never really unbounded and homogeneous and the effects of confinement and the consequent lack of homogeneity are usually important. In this study, we propose a numerical implementation for a new wave-turbulence statistical model for confined rotating turbulence (J.F. Scott, JFM 2014).

We study the problem of decaying turbulence confined between two infinite, parallel walls, which rapidly rotate about an axis perpendicular to them. The turbulence is assumed to be statistically homogeneous in unconfined directions, but confinement means that it is inhomogeneous with respect to the third direction. The velocities' and pressure fields expansion is based on inviscid waveguide modes which contain Fourier modes in the unconfined, homogeneous direction and discretized modes in the confined direction. The amplitudes of these modes evolve in time due to viscosity and nonlinearity. These modes are parameterized by the two-dimensional wall-normal wave-vector k and by an integer parameter N due to spectral discretization in the third direction. In order to apply wave-turbulence theory for closing the system, we separated non-dispersive modes N = 0, contributing to the two-dimensional component of the fluid motion, from dispersive ones, contributing to the wave component. The two-dimensional component does not affect the wave component energy distributions and makes non-zero N modes decorrelate in time before there is significant energy transfer between them. The wave-turbulence model provides the time evolution of the two-point statistics related to the amplitudes of the inertial modes.

The numerical solution of this equation is performed by a predictor-corrector scheme that guaranties realizability. At each time step the calculation of the nonlinear terms requires the numerical evaluation of the integral over curve, characterizing resonant interactions between triads of inertial waves. This integral is evaluated using a trapezoidal-style rule, and marching over the resonance curve uses a fourth-order Runge Kutta scheme. However, no resonance curve exists for some range of k. That implies the presence of a critical point, a singularity which is addressed by ad-hoc numerical treatment.

Through a numerical parametric study we have shown the convergence of the numerical scheme with respect to the several numerical parameters involved (number of wave-vectors, number of modes, time discretization, maximum wave number,...). This has permitted to explore the behaviour of the model with respect to the physical parameters (wall-damping, bulk viscosity, initial spectrum). For instance, energy redistribution between modes has been observed as well as the thermalization effect (predicted by theory). An exhaustive exploration of results obtained by our wave-turbulence model will be presented in this conference.