Split energy cascade in turbulent thin fluid layers

Abstract

We discuss the phenomenology of the split energy cascade in a three-dimensional thin fluid layer by mean of high resolution numerical simulations of the Navier-Stokes equations. We observe the presence of both an inverse energy cascade at large scales, as predicted for two-dimensional turbulence, and of a direct energy cascade at small scales, as in three-dimensional turbulence. The inverse energy cascade is associated with a direct cascade of enstrophy in the intermediate range of scales. Notably, we find that the inverse cascade of energy in this system is not a pure 2D phenomenon, as the coupling with the 3D velocity field is necessary to guarantee the constancy of fluxes.

When Gas Dynamics Decouples from Galactic Rotation: Characterizing ISM Circulation in Disk Galaxies

“…A turbulent velocity field δ v is characterized by its kinetic energy spectrum E(k) such that the mean turbulent kinetic energy per unit mass is 1 2 δv 2 = ∞ 0 E(k)dk. The power spectrum E(k) of two dimensional turbulence is characterized by the existence of two inertial regimes: (i) an inverse energy cascade E(k) ∝ k −5/3 for k < k f and (ii) a direct enstrophy cascade with E(k) ∝ k −3 for k > k f , where k f is the wavenumber of the forcing scale (Kraichnan 1967;Wada et al 2002;Bournaud et al 2010;Musacchio & Boffetta 2017). In our model, the random component of the velocity v nc field is characterized by V(k).…”

Condensate in quasi-two-dimensional turbulence

“…In the early stage in which the condensate grows, (t = 25t f in Fig. 6) the energy flux shows clearly the splitting of the energy cascade (as in [2,3]). A fraction of the energy is transported toward small wavenumbers k < k f with a negative flux ε inv , while the remnant energy perform a direct cascade toward large k with flux ε dir In the late stage (t = 2650t f in Fig.6) when the condensate has reached a steady state, the average flux is zero for k < k f and is equal to the energy input ε for k > k f .…”

“…Turbulent flows in such quasi two-dimensional (2D) geometries display an interesting phenomenology with both 2D and threedimensional (3D) features. Numerical [1][2][3][4] and experimental [5][6][7][8] works have demonstrated the emergence of a split energy cascade in which a fraction of the energy flow to large scales (as in a pure 2D flow) and the remaining part goes to small scales producing the 3D direct cascade. The key parameter which controls the relative flux of the two energy cascades is the geometric ratio S = L z /L f between the confining scale L z and the forcing scale L f [1,2].…”

“…Nonetheless, the saturation of the energy at long times can be achieved only if all the energy injected is dissipated at small scales, which is the only dissipative term in the equations. In Within the scenario of the split-energy cascade [3], the dissipation rate is equal to the flux of the direct energy cascade ε dir = ε − ε inv . The measured values of ε dis are consistent with this picture.…”

“…According to the phenomenology described in [3], three cascade processes take place in the turbulent the enstrophy cascade at the scale L z , that is, ε dir ∝ βL 2 z [3]. Recalling that ε inv = ε−ε dir , and that ε inv = 0 for S larger than the critical aspect ratio S * = L * z /L f [2], one gets ε inv /ε ∝ 1 − (S/S * ) 2 .…”