[M-IS21] Geophysical fluid dynamics-Transfield approach to geoscience

[MIS21-04] New numerical methods for calculating statistical equilibria of two-dimensional turbulent flows

*Koki Ryono¹, Keiichi Ishioka¹ (1.Graduate School of Science, Kyoto University)

Keywords:two-dimensional turbulence, Miller-Robert-Sommeria theory, statistical equilibrium, mixing entropy, Casimir invariants

We propose two types of methods to numerically calculate the equilibrium state of the vorticity field on a sphere, in the framework of the statistical mechanics theory of two-dimensional turbulence proposed by Miller, Robert, and Sommeria. Here, the equilibrium state of the vorticity field is defined as the state that maximizes the mixing entropy among the states obtained by mixing the initial vorticity field under the constraints of conservation laws (energy, three components of angular momentum, and all Casimir invariants). From the viewpoint of computational complexity or the nature of the algorithm, it was difficult to calculate non-zonal statistical equilibrium states for a general initial vorticity field with the previously proposed methods. Our new methods make it possible to calculate the statistical equilibria with a realistic computational complexity even when the initial vorticity field consists of multi-level vorticity patches. Furthermore, it is possible to calculate multiple equilibrium states with our new methods, and the relationship between the states can be made clearer than with the previous methods. The proposed new methods include the bridge-building method, in which different branches of equilibrium solutions are connected by a path that can be followed numerically, and the gradient method, in which the spectrum of the macroscopic vorticity field is treated as variables and the mixing entropy is maximized. The former method is for the canonical problem, in which the inverse temperature beta is introduced and the free energy is maximized. The latter method can be applied to the microcanonical problem, in which the mixing entropy is maximized under the energy constraint, by using the gradient projection method. In the microcanonical problem, it is necessary to pay attention to the feasible region of mixing entropy maximization, and the latter method incorporates the geometric properties of the feasible region and the mixing entropy.
To test the proposed methods, statistical equilibria of turbulent flow arising from a barotropically-unstable zonal initial vorticity field on a sphere are calculated. As a result, we obtained an equilibrium state of the zonal wavenumber 2 type, which is very similar to the result of time integration of the initial vorticity field. On the other hand, the equilibrium state with a wavenumber 1-type structure, which does not appear in the time integration, is also calculated and has a higher mixing entropy than the wavenumber 2-type equilibrium state. By calculating the eigenvalues of the Hessian matrix of the mixing entropy at each equilibrium state, it is found that the equilibrium state of wavenumber 1 type is a local maximum of the mixing entropy and the equilibrium state of wavenumber 2 type is a saddle point of the mixing entropy. The reason that a wavenumber 2-type state appears in the end-state of time integration is considered to be that the initial zonal vorticity field is linearly stable to wavenumber 1-type disturbances.