## Dry Convection In A Compressible Atmosphere

Before we can apply the foregoing ideas to atmospheric convection, we must take into account the fact that the atmosphere is a compressible fluid in which p = p(p, T); specifically, since the atmosphere closely obeys the perfect gas law, p = p/RT. For now we will assume a dry atmosphere, deferring consideration of the effects of moisture until Section 4.5. The parcel and environmental pressure, temperature, and density at z = z1 in Fig. 4.5 are p1 =

p(z1), T1 = T(z1), and p1 = p1/RT1. The real difference from the incompressible case comes when we consider the adiabatic displacement of the parcel to z2. As the parcel rises, it moves into an environment of lower pressure. The parcel will adjust to this pressure; in doing so it will expand, doing work on its surroundings, and thus cool. So the parcel temperature is not conserved during displacement, even if that displacement occurs adiabatically. To compute the buoyancy of the parcel in Fig. 4.5 when it arrives at z2, we need to determine what happens to its temperature.

4.3.1. The adiabatic lapse rate (in unsaturated air)

Consider a parcel of ideal gas of unit mass with a volume V, so that pV = 1. If an amount of heat, SQ, is exchanged by the parcel with its surroundings then applying the first law of thermodynamics SQ = dU + dW, where dU is the change in energy and dW is the change in external work done, 4 gives us

where cvdT is the change in internal energy due to a change in parcel temperature of dT and pdV is the work done by the parcel on its surroundings by expanding an amount dV. Here cv is the specific heat at constant volume.

Our goal now is to rearrange Eq. 4-11 to express it in terms of dT and dp so that we can deduce how dT depends on dp. To that end we note that, because pV = 1, dV = d( ^ = -1 dp.

Rudolf Clausius (1822—1888) the Polish physicist, brought the science of thermodynamics into existence. He was the first to precisely formulate the laws of thermodynamics stating that the energy of the universe is constant and that its entropy tends to a maximum. The expression SQ = dU + dW is due to Clausius.

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