Jorgen S. Frederiksen

Nonlinear stability and statistical mechanics of flow over topography

The stability properties and stationary statistics of inviscid barotropic flow over topography are examined. Minimum enstrophy states have potential vorticity proportional to the streamfunction and are nonlinearly stable ; correspondingly, canonical equilibrium based on energy and enstrophy conservation predicts mean potential vorticity is proportional to the mean streamfunction. It is demonstrated that in the limit of infinite resolution the canonical mean state is statistically sharp, that is, without any eddy energy on any scale, and is identical to the nonlinearly stable minimum enstrophy state. Special attention is given to the interaction between small scales and a dynamically evolving large-scale flow. On the b-plane, these stable flows have a westward large-scale component. Possibilities for a general relation between inviscid statistical equilibrium and nonlinear stability theory are examined.

Eddy Viscosity and Stochastic Backscatter Parameterizations on the Sphere for Atmospheric Circulation Models

Abstract Stochastic backscatter, eddy drain viscosity, and net eddy viscosity parameterizations are formulated and calculated for two-dimensional turbulence on the sphere based on eddy damped quasi-normal Markovian and direct interaction approximation closures. The net eddy viscosity is found to be the relatively small difference between the eddy drain viscosity and the backscatter viscosity. These eddy parameterizations have a cusp behavior at the cutoff wavenumber where they have their largest magnitudes. Large-eddy simulations (LES) with the barotropic vorticity equation have been performed incorporating these dynamic subgrid-scale parameterizations and compared with higher-resolution direct numerical simulations (DNS), which are regarded as the benchmark or “truth” for comparisons. Good comparisons are found between kinetic energy spectra for the LES and the DNS at the scales of the LES for both nonrotating two-dimensional turbulence and differentially rotating Rossby wave turbulence. This is contrast...

Subgrid-Scale Parameterizations of Eddy-Topographic Force, Eddy Viscosity, and Stochastic Backscatter for Flow over Topography

Abstract General expressions for the eddy-topographic force, eddy viscosity, and stochastic backscatter, as well as a residual Jacobian term, are derived for barotopic flow over mean (single realization) topography. These subgrid-scale parameterizations are established on the basis of a quasi-diagonal direct interaction closure model, incorporating equations for the mean vorticity, vorticity covariance, and response functions. In general, the subgrid-scale parameterizations have a time–history integral representation, which reflects memory effects associated with turbulent eddies. In the Markov limit, the truncated equations for the ensemble mean and fluctuating parts of the vorticity have the same form as the full resolution equations but with the original “bare” viscosity and bare mean and fluctuating forcings renormalized by eddy drain viscosities, eddy-topographic force, and stochastic backscatter terms. The parameterizations are evaluated at canonical equilibrium states for comparison with G. Hollowa...

Dynamical Subgrid-Scale Parameterizations from Direct Numerical Simulations

Abstract Dynamical subgrid-scale parameterizations of stochastic backscatter, eddy drain viscosity, and net eddy viscosity have been formulated and calculated for two-dimensional turbulent flows on the sphere based on the statistics of direct numerical simulations (DNSs) with the barotropic vorticity equation. A relatively simple methodology based on a stochastic model representation of the subgrid-scale eddies, but which takes into account the memory effects of turbulent eddies, has been employed. The parameterizations have a cusp behavior at the cutoff wavenumber of the retained scales and have closely similar forms to those based on eddy damped quasi-normal Markovian (EDQNM) and direct interaction approximation (DIA) closure models. Large-eddy simulations (LESs) incorporating DNS-based subgrid-scale parameterizations are found to have kinetic energy spectra that compare closely with the results of higher-resolution DNS at the scales of LES for both isotropic turbulence and Rossby wave turbulence. The m...

Singular Vectors, Finite-Time Normal Modes, and Error Growth during Blocking

Abstract The evolution of finite-time singular vectors growing on four-dimensional space–time basic states is studied for cases of block development over the Gulf of Alaska and over the North Atlantic, using a two-level tangent linear model. The initial singular vectors depend quite sensitively on the choice of norm with the streamfunction norm characterized by small-scale baroclinic disturbances, the kinetic energy norm giving intermediate-scale baroclinic disturbances, and the enstophy norm typified by large-scale disturbances with large zonal flow contributions. In all cases, the final evolved singular vectors consist of large-scale equivalent barotropic wave trains across the respective blocking regions. There are close similarities between the evolved singular vectors in each of the norms, particularly for the longer time periods considered, and with corresponding evolved finite-time adjoint modes and evolved maximum sensitivity perturbations. For the longer time periods considered, each of these evo...

Systematic Energy Errors and the Tendency toward Canonical Equilibrium in Atmospheric Circulation Models

Abstract Systematic kinetic energy errors are examined in barotropic and multilevel general circulation models. The dependence of energy spectra on resolution and dissipation and, in addition for the barotropic model, on topography and the beta effect, is studied. We propose explanations for the behavior of simulated kinetic energy spectra by relating them to canonical equilibrium spectra characterized by entropy maximization. Equilibrium spectra at increased resolution tend to have increased large-scale kinetic energy and a drop in amplitude at intermediate and small scales. This qualitative behavior may also be found in forced and/or dissipative simulations if the forcing and dissipation operators acting on the common scales are very similar at different resolutions. An explanation for the tail “wagging the dog” effect is presented. This effect, where scale-selective dissipation operators cause a drop in the tail of the energy spectra and, surprisingly, also an increase in the large-scale energy, is fou...

Southern Hemisphere Storm Tracks, Blocking, and Low-Frequency Anomalies in a Primitive Equation Model

Abstract The results of a study are presented that indicate three-dimensional instability theory is able to generate analogs of a wide variety of Southern Hemisphere observed fluctuations including those associated with cyclogenesis, blocking, and low-frequency teleconnection patterns. This study has been conducted with a two-level primitive equation eigenvalue model and the growing modes for both January and July averaged basic states are examined. In both seasons, the fastest-growing cyclogenesis modes have largest amplitudes in the eastern part of the hemisphere just downstream of the polar jet stream maxima. In July, there is a tendency to form elongated eddies in the region downstream of Australia where growth on both polar and subtropical jet streams occurs. These results are in general agreement with the observational studies of Southern Hemisphere storm tracks. For both January and July, larger-scale slower-propagating dipole or multipole modes are found that are consistent with blocking in some o...

The QDIA and regularized QDIA closures for inhomogeneous turbulence over topography

The dynamics and spectra of the quasi-diagonal direct interaction approximation (QDIA) closure for inhomogeneous two-dimensional turbulence over mean (single realization) topography are compared with results from direct numerical simulations (DNS). A more efficient version of the closure, termed the cumulant update QDIA (CUQDIA), has also been formulated and tested. Studies are performed for a range of resolutions, for large scale Reynolds numbers between very low (

Adjoint Sensitivity and Finite-Time Normal Mode Disturbances during Blocking

R_{L} ) and moderate (

Monsoon disturbances, intraseasonal oscillations, teleconnection patterns, blocking, and storm tracks of the global atmosphere during January 1979: Linear theory

R_{L} approxeq 300