Weiss, J.B., and J.C. McWilliams, 1993: Temporal scaling behavior of decaying two-dimensional turbulence. Phys. Fluids A 5, 608-621.
Abstract: Decaying two-dimensional turbulence is characterized by the emergence of coherent vortices, which subsequently dominate the evolution. The temporal scaling behavior of the flow is analyzed using a scaling theory, a long-time integration of the fluid equations, and a dissipative, modified point-vortex model that represents the turbulence as a system of interacting coherent structures. Good agreement is found in the behavior of average vortex properties, low-order moments of the flow fields, and the form of self-similar evolution.
Abstract: Results are presented for initial-value problems in two-dimensional and three-dimensional, rotating, stably stratified (i.e., planetary-scale) flows with small diffusivities. The domains and initial conditions are spatially homogeneous and isotropic (though only in a particular sense for the latter problem). In each problem the evolution is dominated by coherent vortices that spontaneously emerge around the time of maximum dissipation. The vortex populations evolve by a lengthy sequence of combinatorial interactions until non-chaotic end-state configurations are achieved. In the three-dimensional flow, the vortices effect a significant departure from the isotropy prediction of Charney (1971).
Abstract: High-resolution numerical simulations were made of unforced, planetary-scale fluid dynamics. In particular, the simulation was based on the quasi-geostrophic equations for a Boussinesq fluid in a uniformly rotating and stably stratified environment, which is an idealization for large regions of either the atmosphere or ocean. The solutions show significant discrepancies from the long-standing theoretical prediction of isotropy. The discrepancies are associated with the self-organization of the flow into a large population of coherent vortices. Their chaotic interactions govern the subsequent evolution of the flow towards a final configuration that is non-turbulent.
Abstract: A survey is made of many types of coherent vortices in the Earth's ocean and atmosphere. These vortices often occur with strong, environmentally induced anisotropy in their velocity and vorticity fields. We propose a definition of the essential characteristics of coherent vortices and formulate hypotheses concerning their dynamical role in complex, anisotropic fluid motions. Finally, we analyze numerical solutions both for uniformly rotating, stably stratified three-dimensional flow and for two-dimensional flow for the phenomena of enstrophy cascade and dissipation, intermittency, isotropy in the appropriate coordinate frame, coherent vortex emergence, vortex population dynamics, and approach to a nonturbulent end state.
Abstract: Balance Equation models describing accurate, gravity-wave-free states on the so-called "slow manifold" of the Primitive Equations are of wide and growing interest, both theoretical and practical, for geophysical fluid dynamics. As a particular example with only two spatial dimensions, the Shallow-Water Balance Equations are a coupled, highly nonlinear, nonsymmetric system of partial differential equations, for which only ad hoc solvers of limited robustness have previously been developed. Two multigrid algorithms are presented, one explicit and one implicit in time, which are shown by analysis and numerical examples to be efficient and robust solution techniques for this system. These examples include modons and Shallow-Water turbulence at finite Rossby number. It is found that, in some regimes of physical parameters, quite large time steps can be taken with the implicit solver, with little loss of accuracy or efficiency. This is interpreted as due to significantly slower evolutionary rates of the dominant patterns compared to parcel trajectory rates.
Abstract: Numerical solutions are obtained by implicit multigrid solvers for initial-value problems in the rotating Shallow-Water Equations (SWE) with spatially complex initial conditions. Companion solutions are also obtained with the Shallow-Water Balance Equations (SWBE), both to determine the initial conditions for the SWE and to provide a comparison solution that lies entirely on the slow, advective manifold. We make use of a control parameter (here the Rossby number, R) to regulate the degree of slowness and balance. While there are measurable discrepancies between the evolving SWE and SWBE solutions for all R, there is a distinct, spatially local breakdown both of the slow manifold in the SWE solution and in the closeness of correspondence between the SWE and SWBE solutions. This critical value for breakdown is only slightly smaller than the R values at which, first, the SWE evolution becomes singular (i.e., the fluid depth vanishes), or, second, a consistent initial condition for the SWBE cannot be defined. This breakdown is most clearly evident in a sudden increase in vertical velocity near the center of a strong, cyclonic vortex; its behavior is primarily associated with an enhanced dissipation rather than an initiation of gravity-wave propagation. The numerical performance of the multigrid solvers is satisfactory even in the difficult circumstances near solution breakdown or singularity.
Abstract: Two approaches are investigated to the multigrid solution of the quasigeostrophic equations---a fundamental nonlinear system of partial differential equations that models large--scale planetary flows. One approach employs standard coarsening with pointwise SOR, and the other line relaxation with partial coarsening. The latter solver is implemented in turbulent--flow simulations on the CRAY C--90 supercomputer. This solver is robust with respect to anisotropy of the operator due to stratification, and it efficiently exploits the vectorization and parallelization capabilities of the machine. The approach taken is applicable to more complex related systems.