Homogeneous, isotropic turbulence and its collapse in stratified and rotating fluids

We derive properties of turbulence in a homogeneous, incompressible, Newtonian fluid, The discussion features an invariant N (characteristic circulation in material circuits containing vorticity but imbedded in nearly irrotational masses of fluid). The key to closure is to see that the turbulence is immature for a while after creation of the vorticity in that friction, important in the creation of the vorticity, is subsequently unimportant for a period tau such that N tau/d(2) = gamma(o)R(1/4) (d is an initial length, R = N/nu and gamma(o) is a constant). After t = tau, any statistical property can be expressed to within certain undetermined constants by a known function of N, t and R. Thus u(2) = A(u)(N/t)R-1/4, where t is time, u is r.m.s. (root-mean-square) velocity and A(u) is a constant. The theory predicts that the dissipation is weakly R-dependent as R-1/4, a behavior conjectured long ago by Saffman as both theoretically possible and not inconsistent with the data. The theory may be applied to find the collapse time T-s when rotation and stratification are present but at lesser times are unimportant. We find gamma T-s = beta R-2/5 where beta is a constant and gamma is the Brunt-Vaisala frequency. (C) 1997 Elsevier Science B.V.