Statistical mechanics of Arakawa’s discretizations

Abstract

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a Modelling, Analysis and Simulation Modelling, Analysis and SimulationStatistical mechanics of Arakawa's discretizations S.B. Dubinkina, J.E. Frank Statistical mechanics of Arakawa's discretizations ABSTRACT The results of statistical analysis of simulation data obtained from long-time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization chosen. Statistical mechanical theories are constructed for three discretizations of the quasi-geostrophic model due to Arakawa (1966), each having different conservation properties. It is shown that the three statistical theories accurately explain the differences observed in statistics derived from the discretizations. The effect of the time discretization is also considered, and it is shown that projection is insufficient if the underlying spatial discretization is not conservative. REPORT MAS-E0712 JULY 2007 ABSTRACTThe results of statistical analysis of simulation data obtained from long-time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization chosen. Statistical mechanical theories are constructed for three discretizations of the quasi-geostrophic model due to Arakawa (1966), each having different conservation properties. It is shown that the three statistical theories accurately explain the differences observed in statistics derived from the discretizations. The effect of the time discretization is also considered, and it is shown that projection is insufficient if the underlying spatial discretization is not conservative.

Statistical relevance of vorticity conservation in the Hamiltonian particle-mesh method

“…In [8] we analyzed energy and enstrophy conserving finite difference methods for the QG model under topographic forcing, and observed that the discrete time-averaged mean fields q and ψ obtained depend heavily on the conservation properties of the discretizations used. For a discretization that conserves energy only, the predicted mean field is uniformly zero velocity Ψ = 0.…”

Equilibrium states of finite-dimensional approximations of a two-dimensional incompressible inviscid fluid

“…Равнораспределение энергии по модам наблюдается и в численных экспериментах, однако мы не будем здесь приводить их результаты по причине тривиальности полученных решений. Аналогичный результат был получен в работе [13]. Равновесный спектр энстрофии для дискретизации с двумя инвариантами (2.38) имеет вид…”

“…В обеих работах аналогичная нашей задача ре-шалась спектральными методами. В работе [13] относительная точность составляла 3·10 −11 для дискретизации Аракавы, однако при значительно более низком разрешении 22 2 .…”

“…Гауссово распределение по площадям для аппроксимаций Аракавы наблюдается в ши-роких пределах разрешений от 22 2 в работе [13] до 8192 2 , а для галёркинских приближений от 11 2 в работе [4] до 1024 2 в работе [40]. Судя по всему, гауссов вид распределения является универсальным свойством этих двух конечномерных аппроксимаций, чем они сильно отли-чаются от идеальной жидкости, которая сохраняет распределение по площадям.…”

A new high order energy and enstrophy conserving Arakawa-like Jacobian differential operator

“…Arakawa, in [3], proposed a particular second-order mimetic finite-difference Jacobian operator and showed the superiority of his solution with respect to other non-mimetic schemes [4]. His solution have been widely used [25], [12], studied [18], [8], and generalized [7], [24], [19].…”