A Few Examples of Novel Behavior of Turbulence in the Presence of Waves: Intermittency, Helicity and Dual Energy Cascades in Geophysical Flows

The ocean and the atmosphere, and hence the climate, are governed by multi-scale interactions between turbulent eddies and waves. Turbulence in the presence of such waves proves to be richer and more complex than for homogeneous isotropic (HIT) flows, both for fluids and in magnetohydrodynamic (MHD). Several examples will be reviewed in this talk and some cases will be illustrated with concepts using phenomenology, and with results stemming from recent high-resolution direct numerical simulations using the Navier-Stokes or Boussinesq equations. The role of anisotropy will be emphasized, and the context is that of geophysical or astrophysical fluid dynamics.

For example, turbulence has been shown to display strong intermittency, with fat wings in the probability distribution function (PDF) of the vertical velocity for stratified flows, when for HIT the PDFs of the velocity are Gaussian. This takes place in the so-called saturated regime where a balance between nonlinearities and waves occur. These localized patches of turbulence lead to enhanced mixing in the atmosphere and the ocean, and such a bursty behavior has also been observed in the planetary boundary layer and in the stratosphere.

Furthermore, helicity (velocity-vorticity correlation), which is an ideal invariant in HIT, can be generated in rotating stratified flows in which it can lead to the creation of large-scale magnetic fields. Helicity is also used as an indicator of hurricane activity, and it develops at river confluences, inducing mixing and erosion.

Finally, turbulence allows for an inverse cascade of energy to large scales, and a direct cascade to small scales, either of enstrophy in quasi-two-dimensional fluids, or of helicity in three dimensions, because of the anisotropy which is imposed by the boundaries or due to external agents. It has also been shown to lead to a dual transfer of energy to both large and small scales, each with a constant flux, e.g. for rotating stratified flows, reinforcing large-scale geostrophic balance and small-scale transport and dissipation. This dual cascade, which is also observed in MHD, can be justified on the basis of statistical mechanics of simplified truncated systems and thus it can be seen as corresponding to a broad class of nonlinear phenomena.