A THEORY OF NONLOCAL MIXING-LENGTH CONVECTION .1. THE MOMENT FORMALISM

Nonlocal theories of convection that have been developed for the study of convective overshooting often make unwarranted assumptions which p reordain the conclusions. We develop a flexible and potentially powerf ul theory of convection, based on the mixing length picture, which is designed to make unbiased, self-consistent predictions about overshoot ing and other complicated phenomena in convection. In this paper we se t up the basic formalism and demonstrate the power of the method by sh owing that a simplified version of the theory reproduces all the stand ard 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 th e time, z is the vertical position of a blob, v is its vertical veloci ty, and T is its temperature. The distribution function satisfies a Bo ltzmann equation, where the physics of the interactions of blobs is in troduced through dynamical equations for v and T. We assume horizontal pressure equilibrium. The equation for v includes terms due to buoyan cy, microscopic viscosity, and turbulent viscosity. The latter effect is modeled with a turbulent viscosity coefficient, v(turb) = sigmal(w) , where sigma is the local velocity dispersion of the blobs and l(w) i s the mixing length corresponding to momentum exchange between blobs. Similarly, the equation for T includes adiabatic heating, radiative di ffusion of heat, and turbulent diffusion of heat, which is modeled thr ough a diffusion coefficient, chi(turb) = sigmal(theta), where l(theta ) is the thermal mixing length. By taking various moments of the Boltz mann equation, we generate a series of equations which describe the ev olution of the mean fluid and various moments of the turbulent fluctua tions. The equations, taken to various orders, are useful for describi ng turbulent fluids at corresponding levels of complexity. In this pic ture, fluid blobs define the background, and the background tells the blobs how to move through v-T phase space. We consider the second-orde r 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, DELTAdelT(crit), below which the only possible steady state of a fluid is nonconvecting. Above this cr itical value, a fluid is convectively unstable. We identify two distin ct regimes of convection which we identify as efficient and inefficien t convection. The equations we derive for convection in these regimes are very similar to the standard equations employed in stellar astroph ysics. We also develop the theory of local convection in a composition -stratified fluid. We reproduce the various known regimes of convectio n in this problem, including semiconvection and the ''salt fingers'' p henomena. Surprisingly, we find that the well-known Ledoux criterion h as no bearing at all on the physics of convection in a stratified medi um, except in certain limits that are not of interest in astrophysics. To investigate nonlocal effects like convective overshooting, it is n ecessary to consider third-order equations. We write down the appropri ate 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.