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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)

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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].

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.

R/Cp= 2/7 for an ideal, diatomic gas

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:

T

Z

TAtm

Γadiabatic

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.

θ Atm Γadiabatic

θ

Z

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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.

T

Z TATM

Γadiabatic

Stable air and inertial gravity waves

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Thermal Inversions – The solid earth is a great black-body

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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

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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

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The connection with radiation

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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.

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Hartmann

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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.