Water Vapor Mixing Ratio Is Given by Dw = Dws

Home , Adiabatic process, Equivalent temperature, Moist static energy

1. Saturated Adiabatic Processes If vertical ascent continues above the LCL, condensation will occur and the latent heat of phase change will be released. This latent heating will cause the temperature to decrease more slowly with pressure above the LCL than below. This means that the lapse rate above the LCL will be smaller than the lapse rate below the LCL. To describe the behavior of the parcel we use a Pseudoadiabatic process or irreversible saturated adiabatic process where all the condensed water or sublimated ice falls out of the air parcel as soon as it is produced (if we were to include the presence of the condensate it would unnecessarily complicate the process representation). Because the water component accounts for only a small fraction of the system’s mass, the pseudo adiabatic process is nearly identical to a reversible saturated adiabatic process. Through any point of an aerological diagram there is one, and only one, pseudoadiabat. During this process, the change of water vapor mixing ratio is given by dw = dws

1) DERIVATION OF THE EXPRESSIONFOR PSEUDOADIABATIC LAPSE RATE

Our objective is to ﬁnd how the temperature changes as we move up the pseudoadiabat (Γs = −dT/dz). We begin with the First Law of Thermodynamics:

cpdT − αdp = dq = −Ldws (1)

The heat exchange between the air parcel and its environment is zero, but heat is released within the air parcel by the phase change from water vapor to liquid water (ice). As vapor is converted to liquid water, w = ws decreases and dq = −Ldws remember dws < 0

∂w ∂w c dT − αdp = −L s dT + s dp (2) p ∂T ∂p

We can neglect the weak dependence of ws on pressure, and use αdp = −gdz ∂w c + L s dT = −gdz (3) p ∂T L ∂ws 1 + dT = −Γddz (4) cp ∂T dT 1 = −Γd (5) dz 1 + L ∂ws cp ∂T dT − = Γ (6) dz s (7)

This means that the pseudoadiabatic lapse rate (Γs) is not constant, but a function of altitude. Notice that the denominator is larger than unity, so the denominator exceeds the numerator so Γs < Γd. This means that the temperature changes more slowly along a pseudoadiabat than along a dry adiabat. The reason is the release of latent heat by the condensation of vapor to liquid water.

• Because ws increases with increasing T , so Γs decreases with increasing temperature.

1 • The slope of a pseudoadiabat increases and approaches that of a dry adiabat as pressure and temperature decrease. a. Dry Static Energy and Moist Static Energy At this point it is useful to define two concepts. 1) Dry Static Energy. Consider the dry adiabatic lapse rate Γ = g . Since specific enthalpy is d cp dh h = cpT , we can also say that during dry adiabatic ascent dz = −g, which implies that the enthalpy of the parcel decreases during dry adiabatic ascent because of the gravitational work it does. We d can also say that dz (h + gz) = 0. We can define the dry static energy (DSE) as:

DSE = h + gz = cpT + gz (8)

During dry adiabatic ascent/descent: d(DSE) = 0 (9) dz DSE is conserved in the process of dry adiabatic ascent or descent of a parcel.

2) Moist Static Energy: Now consider the equation that governs the ascent of a saturated air parcel: cpdT − αdp + Ldws = 0. We can write this equation as:

cpdT + gdz + Ldws = 0 (10)

So if we deﬁne the moist static energy (MSE) as:

MSE = cpT + gz + Lws (11)

During saturated adiabatic ascent/descent:

d(MSE) = 0 (12) dz This implies that the change in MSE is zero or that MSE is conserved during a saturated adiabatic ascent/descent.

When air is lifted dry adiabatically, enthalpy is converted into potential energy and the latent heat content remains unchanged. In saturated adiabatic ascent, energy is exchanged among all three terms - potential energy increases, while the enthalpy and latent heat content both decrease. However, the sum of the three terms remains constant.

1) DERIVATION OF EQUIVALENT POTENTIAL TEMPERATURE Analogous to the potential temperature, which is conserved along the dry adiabat, the quantity that is conserved along a pseudoadiabat is the equivalent potential temperature. To obtain an expression for θe, we begin, again, with the First Law (Equation 1), use pα = RT and divide by T.

2 dT dp −Ldw c − R = s (13) p T d p T

We can also take the ln of the potential temperature ln θ = ln T +κ ln p0 −κ ln p, differentiating and multyplying by cp we have: dθ dT dp c = c − R (14) p θ p T d p which means that −L dθ dws = (15) cpT θ we can demonstrate that L dw ≈ d Lws , so we can integrate to obtain: cpT s cpT

Lw Lw d s + d ln θ = d ln θexp s = 0 (16) cpT cpT Integrated to obtain Lws θe = θexp = constant (17) cpT

The equivalent potential temperature is conserved along a pseudoadiabat. Θe is the potential temperature Θ of a parcel of air when all the water vapor has condensed so that its saturation mixing ratio ws is zero. In many applications it is necessary to determine the temperature of a parcel as it ascends (or descends) moist adiabatically over some depth, which for cumulus convection can extend from near the surface to the tropopause. Althouogh the expression for the moist adiabatic lapse rate could be integrated to obtain the temperature of a parcel as it ascends, it was derived using the hydrostatic approximation which would not be valid for vigorous cumulus convection. We will derive a relation that doesn’t rely on this approximation: equivalent potential temperature. Θe can be found as follows: the air is lifted pseudoadiabatically until all the vapor has condensed, released its latent heat, and fallen out. The air is then compressed dry adiabatically to the standard pressure of 1000hPa, at which point it will attain the temperature Θe. If the air is initially unsaturated, ws and T are the saturation mixing ratio and temperature at the point where the air ﬁrst becomes saturated after being lifted dry adiabatically.

2) ADIABATIC WET BULB TEMPERATURE Taw In the derivation of equivalent potential temperature we considered moist ascent of a parcel to a height at which all its water vapor condenses. Another process that can be considered is the descent of a saturated parcel moist adiabatically to its initial pressure p to a temperature Taw. This may require that water be added to the parcel as it descends in order to maintain saturation. If the parcel under consideration is initially unsaturated, it ﬁrst would have to be lifted to the LCL and then taken moist adiabatically o pressure p with water vapor added to maintain saturation. Point (Taw, p) on the θe(Ts, ps) pseudoadiabat which, is given by:

3 κ 1000 Lws(Taw, p) Taw exp = θe(Ts, ps) (18) p cpTaw

On a skew T, it is given by the isotherm passing through the intersection of the θe(Ts, ps) pseudoadiabat with the isobar p.

3) ADIABATIC WET-BULB POTENTIAL TEMPERATURE θaw

Point (θaw, 1000) on the θe(Ts, ps) pseudoadiabat given by Lws(θaw, 1000) θawexp = θe(Ts, ps) (19) cpθaw

4) ADIABATIC EQUIVALENT TEMPERATURE Tae

If we follow the pseudoadiabat (θe(Ts, ps)) to the top of the atmosphere, all the water is condensed out and r = rs = 0. Because the air is completely dry, the pseudoadiabat is equal to the dry adiabat given by:

1000κ T = θ (T , p ) (20) ae p e s s p κ T = θ (T , p ) (21) ae 1000 e s s On a SkewT, follow the pseudoadiabat up to the lowest pressure, then down to the dry adiabat until it intersects the p isobar.

5) ADIABATIC EQUIVALENT POTENTIAL TEMPERATURE θae

The temperature that crosses the dry adiabat passing through Tae at p=1000.

θae = θe(Ts, ps) (22) Obtained by following the pseudoadiabat to the lowest pressure on the chart and down the dry adiabat through this point to the 1000 mb isobar. This is the way to determine the value of θe(Ts, ps) on a skew T, log p graph.