Effects of friction on two-dimensional Navier-Stokes turbulence

Large-scale dissipation mechanisms have been routinely employed in
numerical simulations of two-dimensional turbulence to absorb energy
at large scales, presumably mimicking the quasi-steady picture of
Kraichnan in an unbounded fluid. Here, ``side effects'' of such a
mechanism---mechanical friction---on the small-scale dynamics of
forced two-dimensional Navier--Stokes turbulence are elaborated by
both theoretical and numerical analysis. Given a positive friction
coefficient $\alpha$, viscous dissipation of enstrophy has been known
to vanish in the inviscid limit $\nu\to0$. This effectively renders
the scale-neutral friction the only mechanism responsible for
enstrophy dissipation in that limit. The resulting dynamical picture
is that the classical enstrophy inertial range becomes a dissipation
range, in which the dissipation of enstrophy by friction mainly occurs.
For each $\alpha>0$, there exists a critical viscosity $\nu_c$, which
depends on physical parameters, separating the regimes of predominant
viscous and frictional dissipation of enstrophy. It is found that
$\nu_c=[{\eta'}^{1/3}/(Ck_f^2)]\exp\{-{\eta'}^{1/3}/(C\alpha)\}$, where
$\eta'$ is half the enstrophy injection rate, $k_f$ is the forcing wave
number, and $C$ is a nondimensional constant (the Kraichnan--Batchelor
constant). The implication of the present results is that mechanical
friction is a poor choice in numerical attempts to address fundamental
issues concerning the direct enstrophy transfer.