Calculation of Moments of Dynamical Systems Describing Turbulence

Abstract

This Letter presents a general method for calculation of moments of invariant measure of multidimensional continuous-time dissipative dynamical systems with noise. In particular, moments of the Lorenz model are calculated.PACS 03.40.Gc This Letter addresses the problem of finding the invariant measure (IM) of continuous-time multidimensional dissipative dynamical systems describing hydrodynamical turbulence and some other chaotic phenomena. The dynamical system includes whitenoise terms that correspond to the thermal fluctuations existing in any real fluid; it is shown that without this noise the problem has many solutions and leads to solving a singular system of linear algebraic equations for the moments of the IM. With nonzero noise the system is nonsingular and here it is described how to solve it by a convergent series. The general method is applied to the well-known representative of the dynamical systems describing turbulence, the Lorenz model; this is probably the first time that moments of the Lorenz (or any similar continuous-time multidimensional) system are calculated by a method other than direct numerical integration. Finally, the convergence of the series is discussed.We start with the system of stochastic differential equations dXJdt ~2 Jk <iijkXjX k + 2jb i jX J~-/j,/c = 1, . . . , N, which is known to describe hydrodynamical (HD) turbulence and similar phenomena; the variables X t (t) usually represent Fourier coefficients of the velocity field, and £, are N independent components of Gaussian white noise. The coefficients of (1) satisfy the following conditions:2)y byXfXj = 0 for all x,gfllb^e independent of /.Here (2) expresses energy conservation by inertial forces, (3) energy conservation by Coriolis or, in magnetic fluids, Lorentz force, (4) positive energy dissipation, and (5) the fluctuation-dissipation theorem. The quantity €, proportional to the fluid temperature, is the mean square of thermal fluctuations of the variables X t and normally it is very small; typically e =^= 10~~2 0 m 2 /s 2 in the HD systems. We shall see that despite its smallness it cannot be neglected. The best-known representative of the general system (1) is perhaps the Lorenz model; we write it here with the origin shifted by the transformation X 3 -> X 3 4-r 4-a to accomodate it to the general form (1) dX l /dt=-cr(X l -X 2 )+g l t lf dX 2 /dt--o-X x -X 2 -X x X 3 + g 2 t 2 ,dX 3 /dt = -bX 3 -b(r + a-) + X X X 2 + ftf 3 .From (1) or (6) we can easily obtain a system of linear algebraic equations for the moments (X^1), . . . , (X^N) of the IM (stationary moments). Suppose first that € = 0; hence g^O and the system is deterministic. Since (dX x /dt) = 0, we have from (6) the relation -or (X x ) + a (X 2 ) = 0, and similarly for {dX 2 /dt)~0 and (dX 3 /dt)=0. Also, since (d(X x X 2 )/dt)=0, we have -cr(<X x X 2 ) -(X$) + (X?))-(X x X 2 )-(X?X 3 )~0, and so on. The linear system we get is infinite dimensional; there have been numerous attempts to make it finite dimensional by an appropriate closure hypothesis and t...

Ab initio method of calculating invariant measure for turbulent flow

“…The time evolution of v(t) is given by the N-dimensional system of stochastic differential equations (34), where the indices I, J and K assume the finite number N of values.…”

“…In sections 3.1 and 3.2 we examine the deterministic part of the evolution equation (34), in particular we prove the existence of global solution of the system without noise. In subsection 3.3 we prove that in the stochastic system (34) the moments E |v(t)| n remain bounded, hence the global solution of this system exists and is finite with probability 1.…”

“…Nonzero noise makes the representative point of the system in the phase space 'jump' from one otherwise deterministic and separated trajectory to another, in particular away from the fixed point corresponding to the laminar flow. Therefore there is a single IM (whose density, a smooth function if ε > 0, will be denoted ψ from now on) and the ensemble average with respect to this distribution is equal to the long-time average of any solution of the stochastic differential equation (34):…”

“…Although the physical problem he is solving in these papers is that of climate change, still the mathematical methods he uses are very inspiring also for the different problem of stationary NSE turbulence. Macháček [34] uses the method described here to calculate long time mean values of the first and second moments of the Lorenz dynamical system, the well-known model of turbulent hydrodynamic systems [35].…”