Minimum Enstrophy Research Articles

Coarse‐grain entropy in two‐dimensional turbulence

A generalized coarse‐grain entropy for a simple two‐dimensional system is derived and used to explain the emergence of coherent structures, which are the most probable states, and their equilibrium with random fluctuations. The relationship between the canonical distribution function and the most probable states, which appear at critical values of the inverse temperature β, is elucidated. Under some conditions, the coherent structures are localized and intermittent, i.e., occupy only a small fraction of the system. For small fluctuation amplitudes, the macrostate entropy is equal to minus the coarse‐grain enstrophy, in agreement with the selective decay and minimum enstrophy theories. Maximizing the macrostate entropy leads to vortex coalescing, and in some cases, a process of concentration occurs in which vorticity and energy are concentrated into vortices of decreasing area.

Peeling of convection cells and the generation of sheared flow as relaxation to minimum enstrophy

Recent work by Drake et al. [Phys. Fluids B 4, 488 (1992)] has found that convection cells can generate large-scale shear flows by a dynamical process called ‘‘peeling.’’ Here, it is shown that this process can be viewed as a relaxation to a state where enstrophy is minimized relative to a fixed energy.

Selective Decay of Enstrophy and the Excitation of Barotropic Waves in a Channel

Abstract If the momentum, energy and circulation of a fluid in a periodic, quasi-geostrophic, β-plane channel are specified, then there is a minimum enstrophy implied. This minimum enstrophy flow is obtained using the calculus of variations and is found to be also a solution of the quasi-geostrophic equations. It is either a parallel flow or a finite-amplitude Rossby wave, depending on the aspect ratio of the channel and the amount of energy and momentum within it. The most geophysically relevant case is a channel whose zonal length is substantially greater than its meridional breadth. In this instance the form of the minimum enstrophy solution is decided by the ratio of the energy to the squared momentum. When this parameter is below a Critical value one has a parallel flow, while if this value is exceeded, the minimum enstrophy, solution is a Rossby wave. Heuristic arguments based on the enstrophy cascade in two-dimensional turbulence suggest a “selective decay hypothesis”. This is that scale-selective ...

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 β-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.

Minimum enstrophy vortices

Two kinds of minimum enstrophy vortices in two-dimensional flows are found by variational analysis. Each represents the ideal limit of a selective decay of enstrophy with energy and angular momentum or circulation, respectively, remaining fixed. For each, the vorticity is confined to a disk whose radius is determined by the fixed integrals. The definition of the vortices assures a large degree of stability, but many questions about their formation and destruction have not yet been answered.

Two-dimensional turbulence above topography

In a turbulent two-dimensional flow enstrophy systematically cascades to very small scales, at which it is dissipated. The kinetic energy, on the other hand, remains at large scales and the total kinetic energy is constant. Above random topography an initially turbulent flow tends to a steady state with streamlines parallel to contours of constant depth, anticyclonic around a bump. A numerical experiment verifies this prediction. In a closed basin on a beta-plane the solution with minimum enstrophy implies a westward flow in the interior, returning in narrow boundary layers to the north and south. This result is interpreted using a parameterization of the effects of the eddies on the large-scale flow. The numerical solution is in qualitative agreement, but corresponds to a minimum of a more complex measure of the total enstrophy than the usual quadratic integral.

Energy Streamfunctions in the Vertical-Meridional Plane for the Northern Atmosphere

The zonal mean flux vector of atmospheric heat transports is very closely nondivergent in the vertical-meridional plane. This is demonstrated for potential (cpT + gz) and latent heat. Thus the heat flux vector fields can be represented by streamfunctions. The top of the atmosphere is a streamline for latent heat. For potential heat, the radiation flux across the top determines the upper boundary condition. The conservation equations are invariant with respect to arbitrarily choosing a constant reference heat but the streamfunctions are not. The impact on the streamfunctions of shifting the reference heat is equivalent to subtracting the mass transport, scaled with that constant, from the heat flux. To remove this ambiguity it is postulated that the curt of the heat flux vector in the vertical-meridional plane be minimized in the least-squares sense. This principle of minimum mean enstrophy is rationalized by analogy to electro-dynamics. It yields a formula for the reference heat in terms of hemispheric integrals of heat and mass flux curl. The formula is applied to the circulation statistics of the MIT-Library. The reference constants turn out to be numerically identical to the observed hemispheric annual mean of the respective heat form (∼324 J g−1 for potential heat and ∼6 J g−1, corresponding to 2.6 g kg−1, for latent heat). Streamfunctions reduced in this way are presented for the seasons. The potential heat circulation is highly variable. It changes sign from summer to winter over the entire northern atmosphere. The water circulation is less variable; it changes sign only in the tropics and fluctuates in intensity in the extratropics. It is shown that there is no further ambiguity in the streamfunction concept. The interrelationships between kinetic energy and potential, latent and static heat flux are discussed with the main result that the potential heat circulation is largely governed by radiation and precipitation flux but very little by the kinetic energy flux and not at all by the water vapor, and further that the streamfunction concept is not applicable to the kinetic energy flux. The main virtue of the streamfunctions is that they quantitatively represent the net heat flux on all scales and phases.