Linear model of infinite dimensional viscous Mathieu system

The linear stability analysis of the motion of a fluid inside a stably stratified square cavity under vertical harmonic oscillation leads to a damped Mathieu equation whose coefficients are spatial operators. We describe an ansatz for the modal effect of viscosity which still provides a correct estimate of critical parameter values from a superposition of standard modal damped Mathieu oscillators, despite the non-commutativity of the operators due to confinement.