Entropy form of dry potential temperature ($ds=c_{p}d\ln \theta$)

For a dry fixed-composition ideal gas with constant specific heats, $ds=c_{p}d\ln T-R_{d}d\ln p$. Defining $\theta =T(p_0/p{)}^{R_d/c_p}$ gives the displayed identity. Reversible adiabatic parcel motion conserves entropy and theta simultaneously. Irreversible entropy production or moist latent heating can change ordinary dry theta even without heat exchanged through the parcel boundary.

Dry potential temperature is conserved for a fixed-composition ideal gas undergoing a reversible adiabatic change. Adiabatic means no heat passes through the boundary; it does not by itself prohibit internal entropy production. With constant specific heat capacity at constant pressure,Positive irreversible entropy production can therefore increase $\theta$ even when the parcel receives no boundary heat.

For example, dissipation of mechanical energy by viscosity heats the gas, and a shock changes its entropy. Mixing also requires care because the resulting parcel is not simply the original unmixed material following one reversible path. This qualification prevents treating all unheated atmospheric motion as exactly conserving dry potential temperature.