Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence

Abstract

We study two-dimensional turbulence in a doubly periodic domain driven by a monoscale-like forcing and damped by various dissipation mechanisms of the form ν µ (−∆) µ . By "monoscale-like" we mean that the forcing is applied over a finite range of wavenumbers k min ≤ k ≤ k max , and that the ratio of enstrophy injection η ≥ 0 to energy injection ε ≥ 0 is bounded by k 2 min ε ≤ η ≤ k 2 max ε. Such a forcing is frequently considered in theoretical and numerical studies of two-dimensional turbulence. It is shown that for µ ≥ 0 the asymptotic behaviour satisfieswhere ||u|| 2 and ||u|| 2 1 are the energy and enstrophy, respectively. If the condition of monoscale-like forcing holds only in a time-mean sense, then the inequality holds in the time mean. It is also shown that for Navier-Stokes turbulence (µ = 1), the timemean enstrophy dissipation rate is bounded from above by 2ν 1 k 2 max . These results place strong constraints on the spectral distribution of energy and enstrophy and of their dissipation, and thereby on the existence of energy and enstrophy cascades, in such systems. In particular, the classical dual cascade picture is shown to be invalid for forced two-dimensional Navier-Stokes turbulence (µ = 1) when it is forced in this manner. Inclusion of Ekman drag (µ = 0) along with molecular viscosity permits a dual cascade, but is incompatible with the log-modified −3 power law for the energy spectrum in the enstrophy-cascading inertial range. In order to achieve the latter, it is necessary to invoke an inverse viscosity (µ < 0). These constraints on permissible power laws apply for any spectrally localized forcing, not just for monoscale-like forcing.

Controlling the dual cascade of two-dimensional turbulence

“…Since energy is transferred to larger scales, a mechanism to dissipate it is necessary in order to reach a statistically stationary state in forced 2-D turbulence. Moreover, Tran & Shepherd (2002) proved that the presence of a large scale sink of energy is necessary in order to obtain the dual cascades with −5/3 and −3 slopes when the forcing is monoscale or band-limited. In the present case where the forcing is full-band the large scale energy dissipation seems to be necessary, and is produced automatically by the optimal control method.…”

“…It has already been proved (Constantin et al 1994;Tran & Shepherd 2002;Tran & Bowman 2003) in the case of monoscale forcing that such an infrared sink is a necessary condition in order to obtain the dual cascades and KLB scaling laws. We have observed a similar sink even when the forcing is full-band.…”

“…Later, Tran & Shepherd (2002) and Tran & Bowman (2003) generalized this result to band-limited forcing and more general types of dissipation. They proved that with monoscale (or band-limited) forcing, the slope of the energy spectrum in the forward cascade cannot be shallower than −5.…”

“…These results show that monoscale and band-limited forcing are actually inconsistent with KLB theory. Tran & Shepherd (2002) show that in the presence of "inverse viscosity" (which removes energy from large scales) the KLB scaling is theoretically possible. However, as was mentioned before, this result suffers from the lack of numerical and experimental evidence.…”

Kolmogorov-Kraichnan Scaling in the Inverse Energy Cascade of Two-Dimensional Plasma Turbulence

“…In the enstrophy cascade, data from experiments [13] and numerical simulations [14,15] showed exponents that vary between ÿ3 and ÿ4. The surface friction forces present in the different experiments are believed to play a role in setting the scaling exponent [16].…”