In a pseudo-adiabatic process, the lapse rate of temperature of a saturated parcel of air with height may be found from the entropy form of the First law of thermodynamics (3.2.1) as follows:
d (ln T)/dz - k d(lnp)/dz = -(L/cp T)dxs/dz (4.6.1)
where k = R/cp, and xs is the saturation mixing-ratio.
With the aid of the hydrostatic approximation and the equation of state, (4.6.1) may be written as dT/dz + g/cp = -(L/cp)d xs/dz = -(L/cp)(dxs/dT)(dT/dz) (4.6.2)
Since, by (3.4.8), the dry adiabatic lapse rate, g/cp = -dT/dz = rd, we may write (4.6.2) as rs = -dT/dz = rd /[1 + (L/cp)(dxs/dT)] (4.6.3)
where rs is the saturated adiabatic lapse rate.
Since dxs/dT is always positive, rs < rd. In the tropics, rs has a mean value of about 6 °C per km as against about 10 °C per km for rd.
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