Effective viscosity due to local turbulence interactions near the cutoff wavenumber in a constrained numerical simulation

Stochastic subgrid-scale modelling for non-equilibrium geophysical flows

Methods motivated by non-equilibrium statistical mechanics of turbulence are applied to solve an important practical problem in geophysical fluid dynamics, namely the parametrization of subgrid-scale eddies needed in large-eddy simulations (LESs). A direct stochastic modelling scheme that is closely related to techniques based on statistical closure theories, but which is more generally applicable to complex models, is employed. Here, we parametrize the effects of baroclinically unstable subgrid-scale eddies in idealized flows with broad similarities to the Antarctic Circumpolar Current of the Southern Ocean. The subgrid model represents the effects of the unresolved eddies through a generalized Langevin equation. The subgrid dissipation and stochastic forcing covariance matrices as well as the mean subgrid forcing required by the LES model are obtained from the statistics of a high resolution direct numerical simulation (DNS). We show that employing these parametrizations leads to LES in close agreement with DNS.

Spectral intermode coupling in a model of isotropic turbulence

We investigate the nonlinear coupling between the so-called explicit modes, identified with wave numbers k such that 0< or =k< or =k(c), and implicit modes, defined such that k(c)< or =k< or =k(max). Here k(c) is an arbitrarily chosen cutoff wave number and k(max) is the ultraviolet cutoff as determined by viscous damping. The stresses arising from the nonlinearity in the Navier-Stokes equations are categorized as "implicit-implicit" (or "Reynolds") and "explicit-implicit" (or "cross"). These arise from dynamic coupling between different regions of wave number space. Their respective effects on momentum, kinetic energy, and energy flux are assessed. The analysis is based on a model system comprising the Navier-Stokes equations and the Edwards-Fokker-Planck energy equation [S. F. Edwards, J. Fluid Mech. 18, 239 (1964)] which is known to retain all the symmetries of homogeneous, isotropic turbulence. The Reynolds stress is found to be responsible for long-range energy transfers. It can be represented by an effective viscosity and is mainly determined by dynamical friction. The cross term is more complicated, involving both diffusive and frictional effects. For long-range coupling it can be expressed as a modification of the effective viscosity, while for short-range coupling it may be modeled on the assumption that implicit scales are slaved to explicit scales. Thus, both the random and coherent aspects of intermode coupling in turbulent flows are relevant in the cross term. The imposition of a continuity requirement on energy transfer leads to a new parametrization that represents the effect of absent modes in a truncated spectral simulation, and takes into account the phase-coupling (coherent) effects, as well as the usual viscositylike (random) effects.