Atmospheric Thermodynamics and Humidity

Moist thermodynamics and humidity variables

Humidity in the atmosphere is described by several variables that differ mainly in their reference mass. Vapour pressure e is the partial pressure of water vapour, and saturation vapour pressure e_s(T) is its maximum value at temperature T, which rises steeply with T. The mixing ratio r = \varepsilon e/(p-e), with \varepsilon \approx 0.622, is the mass of vapour per unit mass of dry air. The specific humidity q = r/(1+r) is the mass of vapour per unit mass of moist air. At small vapour contents the two are nearly equal, so the difference matters only in humid air. The psychrometric equation e = e_s(T_w) - \gamma P (T - T_w) relates vapour pressure to the dry-bulb temperature T and wet-bulb temperature T_w of a ventilated psychrometer, where \gamma is the psychrometric constant. Its derivation assumes that the process is isobaric and adiabatic, with evaporation taking place without net heat exchange beyond the air-to-bulb exchange. The process is not isothermal, because T \neq T_w, and not isentropic, because evaporation is irreversible. Virtual temperature T_v = T(1+0.61q) lets moist air be treated as dry air of the same density. Since water vapour is lighter than dry air, T_v exceeds T, and at fixed p and T an increase in mixing ratio raises T_v and lowers the density. For humid tropical air the change in T_v from a moderate change in mixing ratio is of the order of a few kelvin. For an unsaturated parcel rising adiabatically without mixing, the mixing ratio is conserved. Vapour pressure then falls roughly in proportion to total pressure, whereas e_s falls much faster because temperature drops by about 10 K per km. The two curves therefore converge and cross, so the parcel becomes supersaturated. In very clean air with no condensation nuclei, heterogeneous nucleation is absent and the parcel can remain supersaturated without condensing.

Dry adiabatic processes

A dry adiabatic process is one in which unsaturated air expands or compresses without heat exchange, so its entropy is conserved. The first law for a parcel is dq = c_p\,dT - \alpha\,dp with \alpha = 1/\rho. Setting dq = 0 and combining with hydrostatic balance gives the dry adiabatic lapse rate \Gamma_d = g/c_p \approx 9.8 K km^{-1}. The same process obeys Poisson's relation, so T_2 = T_1 (p_2/p_1)^{R/c_p}, with R/c_p \approx 0.286 for dry air. Temperatures must be in kelvin, and the height of the parcel is not needed when the pressures are known. A parcel that is lifted from a pressure of about 900 hPa to 700 hPa at 290 K cools to about 270 K by this relation. For an adiabatic process the density and pressure are linked by \rho \propto p^{1/\gamma}, where \gamma = c_p/c_v. Halving the density therefore requires the pressure to fall to (1/2)^{\gamma} of its initial value. The result depends only on the pressure ratio and \gamma, so the initial temperature does not enter the calculation. If a rising parcel is to keep a constant temperature, the first law requires an external heating rate dq/dt = -\alpha\, dp/dt. Under hydrostatic balance this equals g w per unit mass, where w is the vertical velocity, so a parcel rising slowly still needs a small but nonzero heat input.

Temperature advection by wind

The local temperature change at a fixed site is governed by horizontal advection, \partial T/\partial t = -\mathbf{V}\cdot\nabla T. The sign is essential: wind blowing from a warmer region warms the site, and wind from a colder region cools it. With the y-axis pointing north, the relation becomes \partial T/\partial t = -v\,\partial T/\partial y. A wind from the north has v < 0, so it carries air from the north toward the site, and a warmer region there produces warming. For a steady wind the total change over a time t is \Delta T = -v\,(\partial T/\partial y)\,t. Distances must be converted to metres and times to seconds before the product is evaluated.