Atmosphere-altitude relationship

Fig. I. Relationship between temperature and altitude (a) cold and te) hot are the composites of extremes of cold and hot atmospheres (b) arctic and (d) tropical are llie composites of the arctic and tropical regions (e l is the standard almusphere Upon which altimetry is based...

The first question that we may ask is the form of the relationship between D/Dt and the ordinary partial time derivative 3/31. The so-called sky-diver problem illustrated in Fig. 2-4 provides a simple physical example that may serve to clarify the nature of this relationship without the need for notational complexity. A sky diver leaps from an airplane at high altitude and begins to record the temperature T of the atmosphere at regular intervals of time as he falls toward the Earth. We denote his velocity as —Uc/iveri-, where izis a unit vector in the vertical direction, and the time derivative of the temperature he records as D T/Dt. Here, D /Dt represents the time rate of change (of 7) measured in a reference frame that moves with the velocity of the diver. Evidently there is a close relationship between this derivative and the convected derivative that was introduced in the preceding paragraph. Let us now suppose, for simplicity, that the temperature of the atmosphere varies with the distance above the Earth s surface but is independent of time at any fixed point, say, z = z. In this case, the partial time derivative 3 T/dt is identically equal to zero. Nevertheless, in the frame of reference of the sky diver, D T/Dt is not zero. Instead,... 

As can be seen from the hydrostatic equation (3.12), there exists a single-valued monotonic relationship between pressure and geometric height in a static atmosphere. Therefore, meteorologists often use the pressure (rather than the altitude) as the independent vertical coordinate. It is also convenient to use the log-pressure altitude... 

The exponential relationship between pressure and altitude arises from the compressibility of air. Note how this contrasts with the essentially linear relationship between pressure and depth in water, which results from the very low compressibility of water (Section 2.2.2, Hydrostatic Pressure). The exponential relationship for the atmosphere can be derived by referring to Fig. 4.3, which shows a sketch of a volume of air located directly above sea level. Pressure at the bottom of the volume, Pq, is approximately 1 atm, but can vary somewhat in response to meteorological conditions. For simplicity, vertical temperature variations are neglected, and a constant temperature of 288 K (15 °C) is assumed. 

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