White-noise and geometrical optics limits of Wigner-Moyal equation for beam waves in turbulent media II: Two-frequency formulation

We introduce two-frequency Wigner distribution in the setting of parabolic approximation to study the scaling limits of the wave propagation in a turbulent medium at two different frequencies. We show that the two-frequency Wigner distribution satisfies a closed-form equation (the two-frequency Wigner-Moyal equation). In the white-noise limit we show the convergence of weak solutions of the two-frequency Wigner-Moyal equation to a Markovian model and thus prove rigorously the Markovian approximation with power-spectral densities widely used in the physics literature. We also prove the convergence of the simultaneous geometrical optics limit whose mean field equation has a simple, universal form and is exactly solvable.