We present new results for the time reversal of weakly nonlinear pulses traveling in a random dissipative environment. Also we describe a new theory for calculating the eddy viscosity for weakly nonlinear waves propagating over a random surface. The turbulent viscosity is calculated from first principles, namely, without imposing any stress-strain hypothesis. A viscous shallow water model is considered and its effective viscosity characterized. We also show that weakly nonlinear waves can still be time reversed under weak dissipation. Incoherently scattered signals are recompressed, both for time reversal in transmission as well as in reflection. Under the weakly nonlinear, weakly dissipative regime, dissipation only affects the refocused pulse profile regarding its amplitude, but its shape is not corrupted. Numerical experiments are presented.