2013/01/31 by Eleftherios Gkioulekas
Chemistry · Earth and Planetary Sciences · Engineering · Mathematics · Physics and Astronomy · #Chemistry #Classical mechanics #Dissipation #Energy flux #Enstrophy #Fluid Dynamics and Turbulent Flows #Flux (metallurgy) #Geometry #Mathematics #Mechanics #Navier-Stokes equation solutions #Oceanographic and Atmospheric Processes #Physics #Quantum mechanics #Scaling #Statistical physics #Vorticity #Wavenumber #nlin.CD
paper · pdf · doi:10.1016/j.physd.2014.06.002
23 pages, resubmitted to Physica D. arXiv admin note: substantial text overlap with arXiv:1206.0315, arXiv:1201.0567
arxiv created 2013/12/22 · openalex publication_date 2014/06/13 · arxiv updated 2015/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We investigate an inequality constraining the energy and potential enstrophy flux spectra in two-layer and multi-layer quasi-geostrophic models. Its physical significance is that it can diagnose whether any given multi-layer model that allows co-existing downscale cascades of energy and potential enstrophy can allow the downscale energy flux to become large enough to yield a mixed energy spectrum where the dominant k-3 scaling is overtaken by a subdominant k-5/3 contribution beyond a transition wavenumber kt situated in the inertial range. The validity of the flux inequality implies that this scaling transition cannot occur within the inertial range, whereas a violation of the flux inequality beyond some wavenumber kt implies the existence of a scaling transition near that wavenumber. This flux inequality holds unconditionally in two-dimensional Navier-Stokes turbulence, however, it is far from obvious that it continues to hold in multi-layer quasi-geostrophic models, because the dissipation rate spectra for energy and potential enstrophy no longer relate in a trivial way, as in two-dimensional Navier-Stokes. We derive the general form of the energy and potential enstrophy dissipation rate spectra for a generalized symmetrically coupled multi-layer model. From this result, we prove that in a symmetrically coupled multi-layer quasi-geostrophic model, where the dissipation terms for each layer consist of the same Fourier-diagonal linear operator applied on the streamfunction field of only the same layer, the flux inequality continues to hold. It follows that a necessary condition to violate the flux inequality is the use of asymmetric dissipation where different operators are used on different layers. etc.