We present a generic mechanism for the thermal damping of compressive waves in the interstellar medium (ISM), occurring due to radiative cooling. We solve for the dispersion relation of magnetosonic waves in a two-fluid (ion-neutral) system in which density- and temperature-dependent heating and cooling mechanisms are present. We use this dispersion relation, in addition to an analytic approximation for the nonlinear turbulent cascade, to model dissipation of weak magnetosonic turbulence. We show that in some ISM conditions, the cutoff wavelength for magnetosonic turbulence becomes tens to hundreds of times larger when the thermal damping is added to the regular ion-neutral damping. We also run numerical simulations which confirm that this effect has a dramatic impact on cascade of compressive wave modes.
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