On final states of two-dimensional decaying turbulence

Abstract

Numerical and analytical studies of final states of two-dimensional (2D) decaying turbulence are carried out. The first part of this work is trying to give a definition for final states of 2D decaying turbulence. The functional relation of -, which is frequently adopted as the characterization of those final states, is merely a sufficient but not necessary condition; moreover, it is not proper to use it as the definition. It is found that the method through the value of the effective area S covered by the scatter -plot, initially suggested by Read, Rhines, and White ["Geostrophic scatter diagrams and potential vorticity dynamics," J. Atmos. Sci. 43, 3226 (1986)] is more general and suitable for the definition. Based on this concept, a definition is presented, which covers all existing results in late states of decaying 2D flows (including some previous unexplainable weird double-valued -scatter plots). The remaining part of the paper is trying to further study 2D decaying turbulence with the assistance of this definition. Some numerical results, leading to "bar" final states and further verifying the predictive ability of statistical mechanics [Yin, Montgomery, and Clercx, "Alternative statistical-mechanical descriptions of decaying two-dimensional turbulence in terms of patches and points," Phys. Fluids 15, 1937 (2003)], are reported. It is realized that some simulations with narrow-band energy spectral initial conditions result in some final states that cannot be very well interpreted by the statistical theory (meanwhile, those final states are still in the scope of the definition).

A new parallel strategy for two-dimensional incompressible flow simulations using pseudo-spectral methods

“…Although the physical mechanism of the multi-valued structure is still unclear, our simulations reveal some limitations of the statistical mechanics [21], and it is worthwhile to carry on further investigations in this direction. Of course, the PTF scheme may play an important role in these longtime computations because of its high parallel efficiency.…”

“…18 and 19 of Yin et al [20] gives a counter example: the double-valued xw structure (see Yin [21] for a detailed discussion about this kind of structure). We will seek some other simulations to validate the generality of multi-valued structures, and there are mainly two choices to do this: (1) make changes in the initial conditions; (2) test different values of m for certain kind of initial condition.…”

“…The simulation lasted for 17 h, and reached the quasi-steady state at t % 200 with a simple function relation of x-w ( Fig. 7(b) in [21]). The same run lasts for 29 h if we use the PFFT scheme (32 CPUs), and 82 h for the PTD scheme (4 CPUs).…”

“…see [20,21]). Those simulations normally need to last as long as 100-1000 eddy turnover times before final states of the 2D decaying turbulence are reached.…”

“…In Section 2, the PTF scheme is applied to solve the 2D NS equation; the benchmark of the parallel code on SGI 3800 is presented. We also show several long-term numerical simulations with high resolutions, which reveal some interesting physical phenomena of 2D decaying turbulence [20,21]. The adoption of the PTF scheme saves about 20-40% of cpu time in those longtime runs (the largest number of time steps in those simulations is up to 1.6 · 10 6 ).…”