Statistical Dynamics of Internal Gravity Waves-Turbulence

Numerical simulations of internal gravity waves-turbulence are carried out for the inviscid, viscous and forceddissipative two-dimensional primitive equations using the spectral method. Some of the results are compared with the predictions of the eddy damped quasi-normal Markovian (EDQNM) closure for internal waves of Carnevale and Frederiksen, generalized for periodic boundary conditions and possible random forcing and dissipation. The EDQNM reduces to the Boltzman equation of resonant interaction theory in the continuum space limit and as the forcing and dissipation vanish. However, the limit is singular in the sense that as well as conserving total energy, E, and total cross-correlation between the vorticity and buoyancy fields, C, an additional conservation law, viz. z-momentum, P,, occurs in the limit. This means that the resonant interaction equilibrium (RIE) solution of the Boltzmann equation differs from the statistical mechanical equilibrium (SME) solution of the EDQNM closure. The statistical stability of the SME and RIE spectra for the primitive equations is tested by integrating the inviscid equations using initial realizations of these spectra with random phases. It is found that E and C are accurately conserved while P, undergoes large amplitude variations. The approach to equilibrium of initial disequilibrium spectra is monitored by examining the evolution of the entropy. The increase and asymptotic approach to a constant value corresponding to complete chaos is consistent with the behaviour predicted by the EDQNM closure. For the viscous decay and forceddissipative experiments, the behaviour of the entropy is also consistent with that predicted by the EDQNM closure. There is approximate equipartition of potential and total kinetic energies throughout the integrations from initial conditions having equal potential and total kinetic energies and as well equal vertical and horizontal energies, but as expected, the ratio of horizontal to vertical kinetic energy increases with time to a value greater than unity.