Vortical control of forced two-dimensional turbulence

Abstract

A new numerical technique for the simulation of forced two-dimensional turbulence [D. Dritschel and J. Fontane, “The combined Lagrangian advection method,” J. Comput. Phys. 229, 5408–5417 (2010)10.1016/j.jcp.2010.03.048] is used to examine the validity of Kraichnan-Batchelor scaling laws at higher Reynolds number than previously accessible with classical pseudo-spectral methods, making use of large simulation ensembles to allow a detailed consideration of the inverse cascade in a quasi-steady state. Our results support the recent finding of Scott [R. Scott, “Nonrobustness of the two-dimensional turbulent inverse cascade,” Phys. Rev. E 75, 046301 (2007)10.1103/PhysRevE.75.046301], namely that when a direct enstrophy cascading range is well-represented numerically, a steeper energy spectrum proportional to k−2 is obtained in place of the classical k−5/3 prediction. It is further shown that this steep spectrum is associated with a faster growth of energy at large scales, scaling like t−1 rather than Kraichnan's prediction of t−3/2. The deviation from Kraichnan's theory is related to the emergence of a population of vortices that dominate the distribution of energy across scales, and whose number density and vorticity distribution with respect to vortex area are related to the shape of the enstrophy spectrum. An analytical model is proposed which closely matches the numerical spectra between the large scales and the forcing scale.

Energy and enstrophy spectra and fluxes for the inertial-dissipation range of two-dimensional turbulence

“…In the regime with forward enstrophy transfer, the energy spectrum is typically steeper than k −3 , both in numerical simulations [32] and in experiments [33]. Moreover, Scott [34], Fontane et al [35] report some deviations from the theoretical predictions of Kraichnan [22]. Pandit et al [36] describe properties of 2D flows in the presence of complex forces.…”

Vortex scaling ranges in two-dimensional turbulence

“…In this case strong, nearly-circular coherent vortices can again be unambiguously distinguished from the weak filamentary background flow 6,7,10,[21][22][23][24] . by Kraichnan 9 , and this range extends to larger scales as the minimum allowed distance to vortices increases, as is evident from examining the middle and lower sets of black curves in figure 3b: these spectra correspond, from top to bottom, to increasing minimum distances to vortices.…”

“…by Kraichnan 9 , and this range extends to larger scales as the minimum allowed distance to vortices increases, as is evident from examining the middle and lower sets of black curves in figure 3b: these spectra correspond, from top to bottom, to increasing minimum distances to vortices. Similarly, Fontane et al 6 found E(k) ∼ k −2 for the entire vorticity field, but…”

“…Fontane et al 6 studied the vortex number density in forced two-dimensional turbulence and found a transition of the scaling at k ≈ k f , where k f is the forcing wavenumber. However, there was insufficient range between the forcing and domain scale to draw any firm conclusions.…”

The fate of random initial vorticity distributions for two-dimensional Euler equations on a sphere

“…The fate of random initial vorticity distributions on a sphere complex vorticity distributions and in doubly periodic or spherical domains this clean scaling law has been questioned (Fontane, Dritschel & Scott 2013). When the viscosity is reduced little by little (i.e.…”