Information theory and the spectrum of isotropic turbulence

The problem of determining the spectrum of isotropic turbulence can be thought of as one of finding the most appropriate joint probability distribution for the flow taken as a whole. From the point of view of information theory, what one means by the most appropriate distribution is clearly defined and easily justified; it is the probability distribution that maximises the information theory entropy, subject to whatever constraints one can impose on the flow. In this work, the relevant constraints are taken to be the Reynolds number and energy dissipation rate of the flow, energy balance (on average) at every point in wavenumber space, and adherence to the Navier-Stokes equations. Using these constraints, it is shown that the maximum entropy formalism leads to a pair of coupled equations describing the distribution of energy in the turbulent spectrum, and the correlations between the amplitudes of velocity components with nearly identical wavenumbers. Although solutions to these equations are not presented, it develops that if a power-law solution exists, it can only be the Kolmogorov law E(k) varies as k-53/. In arriving at this result, a useful concept is that of the 'turbulent temperature', defined as the reciprocal of the derivative of the entropy with respect to the local energy dissipation rate. This quantity plays a role directly analogous to the thermodynamic temperature, governing the rate of energy exchange between different wavenumbers. It is found that, within the spectrum's inertial subrange, the turbulent temperature is virtually constant, with only a minute temperature gradient required to drive the energy cascade.