Anisotropic phase-mixing in homogeneous turbulence in a rapidly rotating or in a strongly stratified fluid: An analytical study

Abdelaziz Salhi (1) , Claude Cambon (2)

1 département de physique de la faculté des sciences de Tunis
2 LMFA - Laboratoire de Mecanique des Fluides et d'Acoustique

Angular phase mixing in rapidly rotating or in strongly stratified flows is quantified for single-time single-point energy components, using linear theory. In addition to potential energy, turbulent kinetic energy is more easily analyzed in terms of its toroidal and poloidal components, and then in terms of vertical and horizontal components. Since the axial symmetry around the direction n (which bears both the system angular velocity and the mean density gradient) is consistent with basic dynamical equations, the input of initial anisotropy is investigated in the axisymmetric case. A general way to construct axisymmetric initial data is used, with a classical expansion in terms of scalar spherical harmonics for the 3D spectral density of kinetic energy e, and a modified expansion for the polarization anisotropy Z, which reflects the unbalance in terms of poloidal and toroidal energy components. The expansion involves Legendre polynomials of arbitrary order, P2n0(cos θ), (n =0,1,2,...,N0), in which the term [cos θ = (k∙n)/∣k∣] characterizes the anisotropy in k-wavespace; two sets of parameters, β2n(e) and β2n(z), separately generate the directional anisotropy and the polarization anisotropy. In the rotating case, the phase mixing results in damping the polarization anisotropy, so that toroidal and poloidal energy components asymptotically equilibrate after transient oscillations. Complete analytical solutions are found in terms of Bessel functions. The envelope of these oscillations decay with time like (ft)−2 (f being the Coriolis parameter), whereas those for the vertical and horizontal components decay like (ft)−3. The long-time limit of the ratio of horizontal component to vertical one depends only on β2(e), which is eventually related to a classical component in structure-based modeling, independently of the degree of the expansion of the initial data. For the stratified case, both the degree of initial anisotropy and the initial unbalance in terms of potential and poloidal (or kinetic gravity wave) energy are investigated. The latter unbalance is characterized by a ratio χ/2, assuming initial proportionality between the kinetic energy spectrum and the potential energy one. The phase mixing yields asymptotic equipartition in terms of poloidal and potential energy components, and analytical solutions are found in terms of Weber functions. At large time, the damped oscillations for poloidal, potential and vertical components decay with time like (Nt)−1/2 (N is the buoyancy frequency), while the oscillations for the horizontal component decay with time like (Nt)−3/2. The long-time limit of the ratio of horizontal component to vertical one depends only on the parameters χ, β2(e), β0(z), β2(z), and β4(z).

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]

Abdelaziz Salhi, Claude Cambon. Anisotropic phase-mixing in homogeneous turbulence in a rapidly rotating or in a strongly stratified fluid: An analytical study. Physics of Fluids, 2007, 19, pp.055102. ⟨10.1063/1.2728934⟩. ⟨hal-00272172⟩