A theory of nonlocal mixing-length convection. I. The moment formalism

Nonlocal theories of convection that have been developed for the study of convective overshooting often make unwarranted assumptions which preordain the conclusions. We develop a flexible and potentially powerful theory of convection, based on the mixing length picture, which is designed to make unbiased, self-consistent predictions about overshooting and other complicated phenomena in convection. In this paper we set up the basic formalism and demonstrate the power of the method by showing that a simplified version of the theory reproduces all the standard results of local convection. The mathematical technique we employ is a moment method, where we develop a Boltzmann transport theory for turbulent fluid elements. We imagine that a convecting fluid consists of a large number of independent fluid blobs. The ensemble of blobs is described by a distribution function, f_A(t, z, v, T), where t is the time, z is the vertical position of a blob, v is its vertical velocity, and T is its temperature. The distribution function satisfies a Boltzmann equation, where the physics of the interactions of blobs is introduced through dynamical equations for v and T. We assume horizontal pressure equilibrium. The equation for v̇ includes terms due to buoyancy, microscopic viscosity, and turbulent viscosity. The latter effect is modeled with a turbulent viscosity coefficient, v_turb = σℓ_w, where σ is the local velocity dispersion of the blobs and ℓ_w is the mixing length corresponding to momentum exchange between blobs. Similarly, the equation for T includes adiabatic heating, radiative diffusion of heat, and turbulent diffusion of heat, which is modeled through a diffusion coefficient, χ_turb = σℓ_θ, where ℓ_θ is the thermal mixing length. By taking various moments of the Boltzmann equation, we generate a series of equations which describe the evolution of the mean fluid and various moments of the turbulent fluctuations. The equations, taken to various orders, are useful for describing turbulent fluids at corresponding levels of complexity. In this picture, fluid blobs define the background, and the background tells the blobs how to move through v-T phase space. We consider the second-order equations of our theory in the limit of a steady state and vanishing third moments, and show that they reproduce all the standard results of local mixing-length convection. We find that there is a particular value of the superadiabatic gradient, Δ∇T_crit, below which the only possible steady state of a fluid is nonconvecting. Above this critical value, a fluid is convectively unstable. We identify two distinct regimes of convection which we identify as efficient and inefficient convection. The equations we derive for convection in these regimes are very similar to the standard equations employed in stellar astrophysics. We also develop the theory of local convection in a composition-stratified fluid. We reproduce the various known regimes of convection in this problem, including semiconvection and the "salt fingers" phenomena. Surprisingly, we find that the well-known Ledoux criterion has no bearing at all on the physics of convection in a stratified medium, except in certain limits that are not of interest in astrophysics. To investigate nonlocal effects like convective overshooting, it is necessary to consider third-order equations. We write down the appropriate equations and see that they involve fourth moments in a nontrivial way. Closure relations for the fourth moments and the solution of the full nonlocal equations are the topics of future papers.