## Thermodynamic Diagrams

Several thermodynamic diagrams have been devised to study the static stability conditions of the atmosphere. Stability parameters in these diagrams vary but they all seem to have the same common objective: to find out by comparing the environment temperatures at different heights with the dry and moist adiabats at those heights whether stable and unstable conditions exist in any layer, and then, in some diagrams, if the atmosphere is conditionally unstable, to assess the amount of net instability energy that may be available for vertical development. Experience with these different diagrams shows that not all of them are suitable or convenient for practical use. In Appendix-3, we give brief particulars of a few thermodynamic diagrams which are in wide use. The T-S diagram (also known as the T-^ diagram) which is most widely used in meteorology is described in detail, while the main specifications only are given of others.

In the T-S diagram (see Fig. 3.1'), the abscissa is temperature T and the ordinate is entropy S (which is identified with cp ln 9). The dry adiabats (Dry Adiabatic Lapse Rate of temperature, DALR) are horizontal, while the saturated adiabats (Saturated Adiabatic Lapse Rate of temperature, SALR) are curved lines sloping upward from right to left, the slopes varying with temperature, pressure and humidity-mixing ratio. The pressure (p) lines or isobars slope downward from right to left and space out with height, while the humidity-mixing-ratio (x) lines are indicated by broken lines. It is easy to obtain a rough estimate of the instability energy that can be realised from the atmosphere with a given case of radiosonde sounding from this diagram.

None of the thermodynamic diagrams in use, however, take into account the likely effect of water vapour on the vertical stability of the atmosphere. Some ther-modynamic diagrams use the vertical profiles of equivalent potential temperature or wet-bulb potential temperature. The conditions for stability in such diagrams are found by following the standard procedure of comparing the potential temperature of the environment at any level with that of a parcel of air lifted pseudo-adiabatically from a lower level to that level, assuming that both the parcel and the environment possessed the same potential temperature at the lower level. In this procedure, let the potential temperature of the environment at level z0 be 90. Then, at level z0 - 8z', the potential temperature is 9o - (d9/dz) Sz'. Let us now lift a parcel of the environment air from this lower level to the level z0. Then, if we denote the potential temperature of the parcel at level z0 by 91,

where §9 is the difference in the potential temperature of the parcel at the pseudo-adiabatic lapse rate between the two levels.

We can evaluate this difference from Eq. (4.7.2) written in the form

Substitution for §9 from (4.10.2) in (4.10.1) gives the following expression which is proportional to the buoyancy of the parcel at level z0,

(91 - 90)/90 ~ -[(1/90)(99/9z) + (9/90){9(Lxs/Tcp)/9z}j (4.10.3)

If we now visualize a hypothetical fully-saturated atmosphere and write 9e for 9e in Eq. (4.7.3), we obtain dln9e = dln 9 + d(Lxs/Tcp) (4.10.4)

If we assume that the parcel temperature is not too different from that of the environment when it arrives at level z0, we may write (4.10.3) as

(91 -90)90 ^-[(1/9)(99/9z) + {9(Lxs/Tcp)/9z}j (4.10.5)

We use (4.10.4) to obtain the following buoyancy relationship in terms of the equivalent potential temperature of the saturated atmosphere,

From (4.10.6), we observe that the parcel will be positively buoyant at z0 if 9i > 90. So we arrive at the following stability criteria:

(< 0 conditionally unstable (99e/9z) (= 0 Neutral (4.10.7)

(> 0 absolutely stable

Vertical profiles of potential temperature(9) and equivalent potential temperature (9e) in the mean tropical atmosphere in the West Indies area (Jordan, 1958) are shown in Fig. 4.5, along with that of the equivalent potential temperature(9e) that would result if the same atmosphere were hypothetically saturated. While 9 is found simply by reducing the observed temperature dry-adiabatically to 1000 mb, 9e is found by lifting a parcel of air with its existing mixing-ratio x dry-adiabatically to saturation and then lifting it further pseudo-adiabatically to a level till all the water vapour condenses out and then reducing it dry-adiabatically from that level to 1000 mb. 9e is calculated similarly as 9e but with x replaced by xs (the saturation value of the humidity- mixing-ratio at each level).

It is obvious from Fig. 4.5 that the tropical atmosphere is conditionally unstable in the lower and middle troposphere and stable above. However, it does not follow that this latent instability leads to convective overturning automatically. Since the humidity most often is less than 100%, low-level convergence is needed to lift the

Fig. 4.5 Vertical profiles of potential temperature (0) and equivalent potential temperature (0e) in the mean tropical atmosphere in the West Indies area (Jordan, 1958). The third profile gives the equivalent potential temperature (0*) of a hypothetically saturated atmosphere with the same p and T values at each level (Reproduced from Ooyama, 1969, with permission of American Meteorological Society)

unsaturated air to saturation. It is not surprising, therefore, that most of the tropical storms and cyclones originate and develop over the oceans. This is largely due to the fact that most of the moisture evaporated from the oceans remains stored in the boundary layer till a low pressure or depression arrives and produces the required low-level convergence to lift the moist air to higher levels. Experience shows that in the atmosphere, the low-level convergence of moisture must be supported by upper-level divergence for any deep convection to occur for development of severe local storms as well as synoptic-scale deep depressions and cyclones.

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