Nonlinear transfer and spectral distribution of energy in $\alpha$ turbulence

Abstract

Two-dimensional turbulence governed by the so-called $\alpha$ turbulence equations, which include the surface quasi-geostrophic equation ($\alpha =1$), the Navier--Stokes system ($\alpha =2$), and the governing equation for a shallow flow on a rotating domain driven by a uniform internal heating ($\alpha =3$), is studied here in both the unbounded and doubly periodic domains. This family of equations conserves two inviscid invariants (energy and enstrophy in the Navier--Stokes case), the dynamics of which are believed to undergo a dual cascade. It is shown that an inverse cascade can exist in the absence of a direct cascade and that the latter is possible only when the inverse transfer rate of the inverse-cascading quantity approaches its own injection rate. Constraints on the spectral exponents in the wavenumber ranges lower and higher than the injection range are derived. For Navier--Stokes turbulence with moderate Reynolds numbers, the realization of an inverse energy cascade in the complete absence of a direct enstrophy cascade is confirmed by numerical simulations.

arXiv comment5 figures, 28 pages

References in corpus (5)

• Inverse cascade in two-dimensional turbulence: deviations from Gaussianity
• Energy Spectrum of Quasi-Geostrophic Turbulence
• Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence
• On the dual cascade in two-dimensional turbulence
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Cited by in corpus (14)

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• Velocity statistics for point vortices in the local α-models of turbulence
• Number of degrees of freedom of two-dimensional turbulence
• Enstrophy dissipation in freely evolving two-dimensional turbulence
• Thermalization and free decay in Surface Quasi-Geostrophic flows
• Statistical Measures and Selective Decay Principle for Generalized Euler Dynamics: Formulation and Application to the Formation of Strong Fronts
• Constraints on scalar diffusion anomaly in three-dimensional flows having bounded velocity gradients
• Local and Nonlocal Dispersive Turbulence
• Remarks on the KLB theory of two-dimensional turbulence
• Cascades transition in generalised two-dimensional turbulence
• Is the subdominant part of the energy spectrum due to downscale energy cascade hidden in quasi-geostrophic turbulence?
• Diminishing inverse transfer and non-cascading dynamics in surface quasi-geostrophic turbulence