Non-equilibrium Closure Research Articles

A Stochastic Framework for Modeling the Population Dynamics of Convective Clouds

AbstractA stochastic prognostic framework for modeling the population dynamics of convective clouds and representing them in climate models is proposed. The framework follows the nonequilibrium statistical mechanical approach to constructing a master equation for representing the evolution of the number of convective cells of a specific size and their associated cloud‐base mass flux, given a large‐scale forcing. In this framework, referred to as STOchastic framework for Modeling Population dynamics of convective clouds (STOMP), the evolution of convective cell size is predicted from three key characteristics of convective cells: (i) the probability of growth, (ii) the probability of decay, and (iii) the cloud‐base mass flux. STOMP models are constructed and evaluated against CPOL radar observations at Darwin and convection permitting model (CPM) simulations. Multiple models are constructed under various assumptions regarding these three key parameters and the realisms of these models are evaluated. It is shown that in a model where convective plumes prefer to aggregate spatially and the cloud‐base mass flux is a nonlinear function of convective cell area, the mass flux manifests a recharge‐discharge behavior under steady forcing. Such a model also produces observed behavior of convective cell populations and CPM simulated cloud‐base mass flux variability under diurnally varying forcing. In addition to its use in developing understanding of convection processes and the controls on convective cell size distributions, this modeling framework is also designed to serve as a nonequilibrium closure formulations for spectral mass flux parameterizations.

A Nonequilibrium Statistical Model of Spectrally Truncated Burgers‐Hopf Dynamics

Exact spectral truncations of the unforced, inviscid Burgers‐Hopf equation are Hamiltonian systems with many degrees of freedom that exhibit intrinsic stochasticity and coherent scaling behavior. For this reason recent studies have employed these systems as prototypes to test stochastic mode reduction strategies. In the present paper the Burgers‐Hopf dynamics truncated to n Fourier modes is treated by a new statistical model reduction technique, and a closed system of evolution equations for the mean values of the m lowest modes is derived for m ≪ n. In the reduced model the m‐mode macrostates are associated with trial probability densities on the phase space of the n‐mode microstates, and a cost functional is introduced to quantify the lack of fit of paths of these densities to the Liouville equation. The best‐fit macrodynamics is obtained by minimizing the cost functional over paths, and the equations governing the closure are then derived from Hamilton‐Jacobi theory. The resulting reduced equations have a fractional diffusion and modified nonlinear interactions, and the explicit form of both are determined up to a single closure parameter. The accuracy and range of validity of this nonequilibrium closure is assessed by comparison against direct numerical simulations of statistical ensembles, and the predicted behavior is found to be well represented by the reduced equations. © 2014 Wiley Periodicals, Inc.

Reynolds stress closure for nonequilibrium effects in turbulent flows

From consideration of turbulence anisotropy dynamics due to spatial or temporal variations in the mean strain rate, a new Reynolds stress closure for nonequilibrium effects in turbulent flows has been developed. This closure, formally derived from the Reynolds stress anisotropy transport equation, results in an effective strain rate tensor that accounts for the strain rate history to which the turbulence has been subjected. In contrast to prior nonequilibrium models that have sought to address nonequilibrium effects via changes in the eddy viscosity, the present approach accounts for nonequilibrium effects in the fundamental relation between the anisotropy tensor and the strain rate tensor. The time-local form of the nonequilibrium closure can be readily implemented in place of the classical equilibrium Boussinesq closure on which most existing computational frameworks are currently based. This new closure is applied here to four substantially different classes of nonequilibrium test problems. Results show dramatically improved agreement with experimental and computational data, without the need to vary any model parameters, when compared with the standard equilibrium closure and with various prior nonequilibrium closures.

A thermodynamic model of turbulent motions in a granular material

This paper is devoted to a thermodynamic theory of granular materials subjected to slow frictional as well as rapid flows with strong collisional interactions. The microstructure of the material is taken into account by considering the solid volume fraction as a basic field. This variable is of a kinematic nature and enters the formulation via the balance law of the configurational momentum, including corresponding contributions to the energy balance, as originally proposed by Goodman and Cowin [1], but modified here. Complemented by constitutive equations, the emerging field equations are postulated to be adequate for motions, be they laminar or turbulent, if the resolved length scales are sufficiently small. On large length scales the sub-grid motion may be interpreted as fluctuations, which manifest themselves in correspondingly filtered equations as correlation products, like in the turbulence theory. We apply an ergodic (Reynolds) filter to these equations and thus deduce averaged equations for the mean motions. The averaged equations comprise balances of mass, linear and configurational momenta, energy, and turbulent kinetic energy as well as turbulent configurational kinetic energy. They are complemented by balance laws for two internal fields, the dissipation rates of the turbulent kinetic energy and of the turbulent configurational kinetic energy. We formulate closure relations for the averages of the laminar constitutive quantities and for the correlation terms by using the rules of material and turbulent objectivity, including equipresence. Many versions of the second law of thermodynamics are known in the literature. We follow the Muller-Liu theory and extend Muller’s entropy principle to allow the satisfaction of the second law of thermodynamics for both laminar and turbulent motions. Its exploitation, performed in the spirit of the Muller-Liu theory, delivers restrictions on the dependent constitutive quantities (through the Liu equations) and a residual inequality, from which thermodynamic equilibrium properties are deduced. Finally, linear relationships are proposed for the nonequilibrium closure relations.

Stability of algebraic non-equilibrium second-order closure models

Stability problems of algebraic non-equilibrium second-moment closure models have given rise to the so-called quasi-equilibrium versions in which turbulence equilibrium is used as an additional constraint. In this paper, we investigate reasons for the failure of the G.L. Mellor, T. Yamada [Reviews of Geophysics 20 (1982) 851] level 2.5 closure model and suggest a remedy for this. We further discuss a new non-equilibrium closure model by V.M. Canuto, A. Howard, Y. Cheng, M.S. Dubovikov (Journal of Physical Oceanography, 2000, accepted for publication) which has proven to allow for stable calculations. All models are then numerically tested with a simple wind entrainment experiment motivated by the H. Kato, O.M. Phillips [Journal of Fluid Mechanics 37 (1969) 643] laboratory experiment, with the aid of which the instability of the Mellor and Yamada (1982) and the stability of the Canuto et al. (2000) model are confirmed. The Canuto et al. (2000) model has three advantages compared to the Mellor and Yamada (1982) which are (i) the symmetry of stability functions, (ii) a higher critical Richardson number, and (iii) that the normalised shear stress increases with normalised shear for turbulence equilibrium. The latter advantage of the new model causes its high physical and numerical stability.

On the density dependence of the self-diffusion coefficient of interacting Brownian particles

A brief summary is given of our theoretical approach to the statistical description of interacting Brownian particles. For the discussion of particle transport processes on the Smoluchowski level a hierarchy of coupled diffusion-like equations in configuration space for the reduced particle density functions and the corresponding particle flows is derived. Non-equilibrium processes are taken into account by means of linear perturbation theory. Applying the direct correlation force formalism, the hierarchy is cut off by replacing the three-particle density function nabc ( r 1, r 2, r 3, t). Here closure relations between the direct correlation force and nab ( r 1, r 2, t) have to be added. Specializing the brownons to hard spheres and neglecting the hydrodynamic interaction, the two-particle density function is evaluated from the well-known equilibrium PY approximation and (as a first step) from the low-density limit of the non-equilibrium closure. Using a linear response theory of particle transport, the self-diffusion coefficient in a one-component hard sphere system of Brownian particles is calculated through a formal non-equilibrium picture based on tracer diffusion.