METR/OCN 465: Computer Programming with Applications in Meteorology and Oceanography Potential Temperature Potential temperature is defined as the temperature that a parcel of air would have if it were brought adiabatically to some reference pressure. For adiabatic processes, potential temperature is a conserved property of air, which makes it worth keeping track of for many theoretical and some practical purposes. To find an expression for potential temperature in terms of other quantities, we note first that for an air parcel subject only to adiabatic temperature changes (that is, changes associated only with expansion and compression in response to decreases and increases in pressure, respectively), the law of conservation of energy can be written as: DT
1 Dp
=
Dt ρC p Dt
(where T is the parcel temperature, p is the parcel pressure, ρ is the parcel density, Cp is the specific heat of air at constant pressure, and D( )/Dt (1) signifies the rate of change of some property of the parcel, “( )”, with respect to time). The ideal gas law applies in its usual form as well: (2) p = ρRT Solving Eq. (2) for ρ, substituting into Eq. (1), and rearranging terms gives: or (3) 1 DT ⎛ R ⎞ 1 Dp
=
T Dt ⎜⎝ C p ⎟⎠ p Dt
D ln(T ) ⎛ R ⎞ D ln( p)
=⎜ ⎟
Dt
⎝ C p ⎠ Dt
Now imagine bringing the parcel adiabatically from its starting location at time t1, where it has temperature T(t1) and pressure p(t1), to a reference pressure at time t2, pref (t2), at which it’s temperature will be T(t2) (possibly different from T(t1) because it will have warmed or cooled adiabatically as a result of the change in pressure).
To see what the final temperature will be, integrate Eq. (3) between the two times and solve for T(t2): ⎛ pref ( t 2 ) ⎞
(4) T ( t 2 ) = T ( t1 )⎜
⎟
⎝ p( t1 ) ⎠
R
Cp
The final temperature is, by definition, the potential temperature. It is commonly represented using the Greek letter “theta” (θ). Dropping the explicit time references from Eq. (4), the equation relating potential temperature to temperature and pressure can be written as the Poisson Equation: ⎛ pref ⎞
(5) θ = T ⎜
⎟
⎝ p ⎠
R
Cp
In practice, the reference pressure is almost always chosen to be 1000 millibars. Also, the gas constant and specific heat of air at constant pressure are assumed to be those for dry air. (For most practical purposes, adding water vapor doesn’t change the gas constant or the specific heat at constant pressure enough to worry about when calculating potential temperature.) From Eq. (5), we can see that: (a) for a pressure lower than the reference pressure, the parcel’s potential temperature is higher than the parcel’s temperature; (b) for a given temperature, air with a lower pressure has a higher potential temperature; (c) at a given pressure, warmer air has a higher potential temperature than cooler air; and (d) for a parcel that conserves its potential temperature (as it will if it undergoes only adiabatic processes), as its pressure increases then so does its temperature (that is, it warms as the pressure on it increases), while it cools as its pressure drops. For these reasons, air the rises in the atmosphere (where the pressure is lower) cools as it expands, while air that subsides (sinks), warms as it is compressed.