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Frederiksen, JS, Carnevale GF.  1986.  Stability properties of exact nonzonal solutions for flow over topography. Geophysical and Astrophysical Fluid Dynamics. 35:173-207.   10.1080/03091928608245892   AbstractWebsite

The nonlinear stability properties of stationary exact nonzonal solutions for inviscid flow over topography are examined within a barotropic model in spherical geometry. For stationary solutions, such that the potential vorticity is proportional to the streamfunction, necessary and sufficient conditions for nonlinear stability are established. For a truncated system with rhomboidal truncation wave number J these are that the solid body rotation component of the zonal wind u(i) be negative, corresponding to westward flow, as J ->infinity. These results are established by using the methods of statistical mechanics. The sufficient condition for stability is also established by applying Arnol'd's method. The results are illustrated by numerical calculations. The stationary solutions are perturbed by disturbances in the streamfunction fields or by small changes in the topographic height; the climatic states for the system are obtained directly using statistical mechanics methods. The nonlinear stability properties of the stationary solutions are obtained by comparing the stationary solution with the climate, which for inviscid flow is shown to be unique. Stationary flows for which u(i) is eastward, are found to be unstable even in the limit as the streamfunction perturbation or change in the topographic height vanish. Large amplitude transient waves are generated which break the time invariance symmetry of the initial stationary flows. In contrast, for stationary flows with westward u(i), the climate is identical to the initial flow in the limit as the initial streamfuncton perturbation or the change in the topographic height vanishes. The linear instability characteristics of the stationary solutions are also obtained by solving a linear eigenvalue problem. The difficulties in establishing the stability properties of more general exact solutions, where the streamfunction is a general differentiable function of the potential vorticity, within numerical spectral models are discussed.

Carnevale, GF, Frederiksen JS.  1983.  Viscosity renormalization based on direct-interaction closure. Journal of Fluid Mechanics. 131:289-303.   10.1017/s0022112083001330   AbstractWebsite

Approximations in statistical turbulence theory often rely on modelling the decay in time of velocity correlations with a simple exponential decay. The decay rate is viewed as a renormalized viscosity. The three simplest implementations of this approximation scheme were originally given independently by Kraichnan, Edwards and Leslie. Each of these investigators used a different formalism and each achieved different renormalization prescriptions. These three different results are reexamined here entirely in terms of direct-interaction theory. The difference in the prescriptions of Kraichnan and Leslie is shown to be the product of different definitions of renormalized viscosity. Edwards’ prescription is shown to result from an inconsistent identification of the non-stationary energy-spectrum relaxation rate with the viscosity. An assessment of the validity of the Markovian closure approximation, and a prescription for non-stationary renormalized viscosity are provided.

Weiss, J, Tabor M, Carnevale G.  1983.  The Painlevé property for partial differential equations. Journal of Mathematical Physics. 24:522-526.   10.1063/1.525721   AbstractWebsite

In this paper we define the Painlevé property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Bäcklund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painlevé property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.

Carnevale, GF, Frederiksen JS.  1983.  A statistical dynamical theory of strongly nonlinear internal gravity waves. Geophysical and Astrophysical Fluid Dynamics. 23:175-207.   10.1080/03091928308209042   AbstractWebsite

A statistical dynamical closure theory describing the interaction of strongly (and weakly) nonlinear two-dimensional internal waves in the presence of viscous dissipation and thermal conduction is derived. By applying renormalization methods originally formulated for quantum and classical statistical field theory, closures similar to the Direct Interaction and eddy-damped quasi-normal procedures of turbulence are derived. These methods are applied directly to the strongly nonlinear primitive field equations in Eulerian variables, thus avoiding the small amplitude assumptions inherent in the resonant interaction formalism. Propagator renormalization techniques provide formulas for the nonlinear internal wave frequency and spectral width in terms of the energy spectrum. The commonly used multiple time and space scale analysis is replaced by an analysis of the two-point correlation functions in terms of sum and difference variables. This permits the systematic development of a Landau equation. This generalization of the Boltzmann equation incorporates spatial variation of the group velocity and scattering due to spatial inhomogeneity. In the limit of weakly interacting waves and zero viscosity, the closures reduce to the resonant interaction approximation formalism. It is shown that the inviscid resonant interaction limit is singular in the sense that the quilibrium spectrum differs from that of the general inviscid nonlinear off-resonant case. This is due to the fact that in the resonant interaction limit there is an additional constant of motion, viz. “z-momentum”. The implications of these results are discussed.

Carnevale, GF, Martin PC.  1982.  Field theoretical techniques in statistical fluid dynamics: With application to nonlinear wave dynamics. Geophysical and Astrophysical Fluid Dynamics. 20:131-164.   10.1080/03091928208209002   AbstractWebsite

A derivation of two-point Markovian closure is presented in classical statistical field theory formalism. It is emphasized that the procedures used in this derivation are equivalent to those employed in the quantum statistical field theory derivation of the Boltzmann equation. Application of these techniques to the study of two-dimensional flow on a β-plane yields a quasi-homogeneous, quasi-stationary transport equation and a renormalized dispersion relation for Rossby waves

Carnevale, GF, Holloway G.  1982.  Information decay and the predictability of turbulent flows. Journal of Fluid Mechanics. 116:115-121.   10.1017/s0022112082000391   AbstractWebsite

A measure of predictability that has many superior features compared to currently used measures is introduced. Through statistical theory it is demonstrated that in inviscid truncated flow this new predictability measure increases monotonically in time while all initial information about the system decays. Under the influence of forcing and viscosity the behaviour of this measure is shown always to satisfy intuitive expectations.

Carnevale, GF.  1982.  A nonstationary solution to Liouville’s equation for a randomly forced two‐dimensional flow. Physics of Fluids. 25:1547-1549.   10.1063/1.863942   AbstractWebsite

Liouville’s equation for randomly forced two‐dimensional flow with Rayleigh friction is examined. An exact nonstationary solution is presented for a special form of the forcing and zero energy initial condition. This solution is such that the fluctuation‐dissipation relation is valid at all times.

Carnevale, GF.  1982.  Statistical features of the evolution of two-dimensional turbulence. Journal of Fluid Mechanics. 122:143-153.   10.1017/s0022112082002134   AbstractWebsite

Statistical fluid dynamics identifies a functional of the fluid energy spectrum that plays the role of Boltzmann's entropy for fluids. Through a series of two-dimensional flow simulations we confirm the theoretical predictions for the behaviour of this entropy functional. This includes a demonstration of Loschmidt's paradox and an examination of the effects of Rossby waves and viscosity on the behaviour of the entropy.

Carnevale, GF, Frisch U, Salmon R.  1981.  H theorems in statistical fluid dynamics. Journal of Physics a-Mathematical and General. 14:1701-1718.   10.1088/0305-4470/14/7/026   AbstractWebsite

It is demonstrated that the second-order Markovian closures frequently used in turbulence theory imply an H theorem for inviscid flow with an ultraviolet spectral cut-off. That is, from the inviscid closure equations, it follows that a certain functional of the energy spectrum (namely entropy) increases monotonically in time to a maximum value at absolute equilibrium. This is shown explicitly for isotropic homogeneous flow in dimensions d>or=2, and then a generalised theorem which covers a wide class of systems of current interest is presented. It is shown that the H theorem for closure can be derived from a Gibbs-type H theorem for the exact non-dissipative dynamics.