In the past the methods of statistical mechanics had been very successfully applied to the microscopic motion of molecules. In extending their application to the realm of turbulent motion, one encounters the difficulty that the fluid dynamical equations are continuous. By using a representation of the flow fields in terms of Fourier components this problem can be solved to the extent that a phase space can still be defined although it is infinite dimensional. Another difficulty is that, in turbulent motion, energy is not conserved but flows through the system. Burgers therefore proposed that the statistics of a turbulent system is controlled by an average balance between input and output of energy and not, as is appropriate to assume in the realm of molecular motion, by the conservation of energy. Taking the dissipation to be quadratic, when expressed in terms of the Fourier coefficients, and constraining the statistics to respect an average balance between forcing and dissipation, he applied the techniques of statistical mechanics and concluded that the dissipation is equally partitioned among the Fourier components.
This conclusion was both interesting and problematic. Equipartition of dissipation leads to an unphysical infinite total dissipation if the phase space of the system is infinite dimensional. Quantum mechanics does not come to the rescue here as it had done earlier when an analogous problem arose in the statistical mechanics of electromagnetic radiation. Despite a series of publications , many of which are reprinted in the memorial volume by Nieuwstadt and Steketee (2), a completely satisfying solution did not emerge and Burgers finally abandoned the subject. Several years later Onsager (3) took it up again but decided to pursue a course that is more in line with equilibrium statistical mechanics, as detailed in the review article by Eyink and Sreenivasan (4).
The work described below (5) can be considered as an attempt to revisit Burgers’ approach. It will thus be investigated whether statistical mechanics can be used to deal with forced-dissipative turbulent systems, using as a basic assumption that the statistics is controlled by an average balance between forcing and dissipation. The problem of the infinite dissipation is not resolved but moderated by limiting ourselves to finitely truncated spectral representations of fluid flows. We will phrase the theory in the language of probability theory and the principle of maximum entropy, as advocated by Jaynes (6).
The method will be applied to a simple one-layer model of the large-scale atmospheric circulation. In this context the model’s statistics can be identified with the model’s climate. From the perspective of this model our aim is to deduce the model’s climate from its basic equations, as an alternative to averaging over long numerical time-integrations.
The model to be considered describes the motion of a single layer of incompressible fluid on the surface of a rotating sphere. Orography is taken into account and the flow is assumed to be geostrophically balanced and thus approximately governed by the horizontal advection of quasigeostrophic potential vorticity. The equation that is used, is a somewhat simplified version of an equation discussed by the author (7). The system is forced by relaxation towards a zonally symmetric circulation that consists of jet-streams in both hemispheres, and is damped by a term that has the same structure as the viscosity term in fluid dynamics. The two-dimensional streamfunction, in terms of which the horizontal velocity is expressed, is the basic field of the model and is represented by a finite set of spherical harmonics, indexed by the integers m and n, where n runs from 1 to N=42 and m runs from -n to +n. The variables of the model are expressed in units formed by appropriate combinations of the earth’s radius and its angular velocity of rotation.
The phase space of the model consists of the Fourier coefficients ψ_mn of the streamfunction. By projecting the advection equation of potential vorticity onto the finite set of spherical harmonics, one obtains a dynamical system of quadratically non-linear equations in the Fourier coefficients. When integrated numerically, this finite-dimensional dynamical system displays chaotic turbulent motion, not unlike what is seen in large-scale atmospheric flow. To demonstrate this, we show in Figure 1 two snapshots of the vorticity and the zonally averaged zonal velocity, separated by 10 days in time, at the end of an integration of 2000 days.
The central concept in a statistical mechanical theory is the probability density function, denoted by P, which in our case gives the probability to find the system in a state with Fourier coefficients ψ_mn. According to Jaynes’ principle of maximum entropy, the probability density function should have a maximum value of its information entropy S_I. All available information on the system is used to constrain the maximization of S_I, such as the normalization condition on P. Additional information consists of fixed averages of certain functions Q, denoted by 〈Q〉. Without these averages as constraints, the maximization of the information entropy S_I would result in a probability density function in which all Fourier coefficients would have equal probability.
In the equilibrium statistical mechanical theory that emerged from Onsager’s approach, the information entropy is maximized with fixed values of the average energy 〈E〉 and the average enstrophy 〈Z〉. This has been shown (5) to work rather well if the statistics is controlled by conservation of energy and enstrophy, i.e., in the unforced-undamped case - which is not very realistic. If forcing and dissipation are present then, in line with Burgers’ approach, it is more appropriate to maximize the information entropy with fixed (zero) values of 〈F-D〉 and 〈G-H〉. Here dE/dt=F-D and dZ/dt=G-H and in this case we thus require that, on average, forcing and dissipation of energy and enstrophy balance. Fortunately, the mathematics is similar in both cases because all constraints are quadratic and leads to a probability density function that is a product of normal distributions. Once the probability density function is known, all relevant statistics can be calculated, such as spectra of energy and enstropy and average vorticity fields.