  A common situation in galactic and intergalactic gas involves cold dense gas
in motion relative to hot diffuse gas. Kelvin-Helmholtz instability creates a
turbulent mixing layer and populates the intermediate-temperature phase, which
often cools rapidly. The energy lost to cooling is balanced by the advection of
hot high enthalpy gas into the mixing layer, resulting in growth and
acceleration of the cold phase. This process may play a major role in
determining the interstellar medium and circumgalactic medium phase structure,
and accelerating cold gas in galactic winds and cosmic filaments. Cooling in
these mixing layers occurs in a thin corrugated sheet, which we argue has an
area with fractal dimension $D=5/2$ and a thickness that adjusts to match the
hot phase mixing time to the cooling time. These cooling sheet properties form
the basis of a new model for how the cooling rate and hot gas inflow velocity
depend on the size $L$, cooling time $t_{\rm cool}$, relative velocity $v_{\rm
rel}$, and density contrast $\rho_{\rm cold}/\rho_{\rm hot}$ of the system.
Entrainment is expected to be enhanced in environments with short $t_{\rm
cool}$, large $v_{\rm rel}$, and large $\rho_{\rm cold}/\rho_{\rm hot}$. Using
a large suite of three dimensional hydrodynamic simulations, we demonstrate
that this fractal cooling layer model accurately captures the energetics and
evolution of turbulent interfaces and can therefore be used as a foundation for
understanding multiphase mixing with strong radiative cooling.
