Lapse Rate and Static Stability

Convection, Clouds and Radiation
Chapter 4 – Marshall & Plumb
Lapse Rate and Static Stability
-Γ≡ dT/dz = -g/Cp = 9.8 oC / km for dry air. (Marshall&Plumb 39-41)
This expression tells us that for adiabatic motion in the vertical, the temperature
will fall by ~10 oC/km in a dry atmosphere. Throughout the troposphere, the
average lapse rate is about -6.5 oC reflecting both the importance of H2O vapor
condensation (latent heat) and dynamical constraints (small static stability).
Why -g/Cp?
1st Law:
dQ = dU + dW = cv dT + p dV
Substituting (we want to express the 1st law in terms of p and T):
for the work term, we have:
p dV = p d(1/ρ) = - (p/ρ2)dρ (ρV=1 by mass conservation)
Ideal Gas:
p = ρRT
dp = ρRdT+RTdρ
Substitution: p dV = - (p/ρ2)dρ = -(p/(ρ2RT))dp + p/(-ρT)dT = -dp/ρ + R dT
dQ = (R + cv)dT – dp/ρ = cpdT – dp/ρ
(cp = R + cv for ideal gas)
For dQ = 0, cpdT = dp/ρ
Finally, from hydrostatic balance, dp = -gρdz, so dT/dz = -g/cp (text 4-13)
1
Potential Temperature
It is useful at this time to define the potential temperature, θ. θ is the temperature that a parcel of dry
air would have if it were compressed adiabatically to the surface (1000 mb). [see derivation from
4.3 on page 41].
R/Cp= 2/7 for an ideal, diatomic gas
Air moved adiabatically conserves its potential temperature.
We can now diagnose whether a thermal profile is stable against (dry) convection:
If -dTatm/dz > Γ (dθ/dz<0), the atmosphere is unstable.
If -dTatm/dz < Γ (dθ/dz>0), the atmosphere is stable.
At times, the atmospheric temperature can actually increase with altitude,
dTatm/dz > 0. This thermal profile is characteristic of the much of the
stratosphere and is extremely stable.
To understand the stability requirement, consider the following thermal
structure:
Z
Z
Γadiabatic
Γadiabatic
θ Atm
TAtm
T
θ
Imagine the parcel of air marked with the
is pushed upward and then
released. If the upward motion is adiabatic, the temperature of the parcel
will fall as it expands (along the adiabatic lapse rate). As a result of the
vertical displacement, the parcel will be warmer than its surroundings and
thus less dense (lighter) and the parcel is buoyant and its vertical velocity
will accelerate. Alternatively if the parcel is pushed downward its
temperature will be less than its surroundings and therefore more dense.
The parcel will be accelerated downward.
2
Z
TATM
Γadiabatic
T
In this case, if the parcel of air is pushed upward it will be colder than its
surroundings and thus more dense and the parcel will oscillate about its
initial position.
As we have seen earlier, the simple radiative model can not adequately
describe the energy transfer from the surface to the atmosphere because it
produces a discontinuity. In fact, only ~40% of the energy transfer from
the surface to the atmosphere occurs via longwave radiation. The other
60% is transferred by latent and sensible heat.
Stable air and inertial gravity waves
3
Thermal Inversions – The solid earth is a great black-body
4
Moisture
• 
Some terms:
q = specific humidity is ratio of the mass of water to the mass of air (ρv / ρ)
q*= saturated-specific humidity
= (R/Rv)(es/p) [note R here is the
mass weighed gas constant, so
depends on molecular weight; Rv is ~1/2 R).
Empirically, es = AeβT A = β = 0.067oC-1
U= Relative humidity = q/q* x 100%
es
5
6
7
Saturated adiabatic lapse rate
Returning to the 1st law ….
dQ = (R + cv)dT – dp/ρ = cpdT – dp/ρ = -Ldq for a condensing
atmosphere, where L = heat of condensation.
Or,
d(cpT + gz + Lq) = 0
in words, “the change in moist static energy is zero”
Following the math on pg 49, we derive the saturated adiabatic lapse
rate, Γs:
-dT/dz = Γs = Γd { (1+Lq*/RT)/(1 + β Lq*/cp)}
Always < 1
8
The connection with radiation
9
Convection redistributes the thermal energy yielding (globally-averaged), a mean
lapse rate of ~ -6.5 oC/km. Radiative processes tend to produce a more negative
temperature gradient (cooling). Nonradiative upward heat transfer opposes such
cooling. The smaller lapse rate, -6.5 oC/km, reflects both the importance of
convection (and water condensation in particular) and the fact that the atmosphere
is, on mean, stable.
10
Hartmann
11
Despite its simplicity, a radiative-convective model is useful for
understanding the relative importance of various processes for
controlling the surface temperature. In Fig. 3.17 from Hartmann, for
example, the importance of CO2, H2O, and O3, are illustrated for a
clear sky calculation.
12