## Static Stability of Dry Air Buoyancy Oscillations

where we write re for -dT/dz which is the actual lapse rate of temperature with height in the environment.

From (3.4.9), it follows that if the actual lapse rate re is less than the dry adiabatic lapse rate rd, d9/dz is positive, which means that the potential temperature increases with height. This makes the atmosphere statically stable. Thus, the atmosphere is vertically stable, neutral or unstable, according as re is less than, equal to or greater than rd. In a stable atmosphere, any displacement (upward or downward) will restore the parcel to its original position through a series of vertical oscillations which are known as buoyancy oscillations. The frequency of such oscillations, which is known as Brunt-Vaisalla frequency, may be derived as follows:

Let a parcel with pressure p, density p and potential temperature 9 be displaced upward from its equilibrium level z through a small distance 5z without disturbing the environment. If the corresponding variables in the environment be pe, pe and 9e, dpe/dz = -pe g, and the acceleration of the parcel is given by the relation d2(§z)/dt2 = -g - (1/p)(dp/dz) = -g(9e - 9)/9e (3.4.10)

where it has been assumed that since the environment is not disturbed, p = pe, and dp/dz = dpe/dz = -peg.

Now, since the parcel moves adiabatically, its potential temperature does not change with height. But, in the environment, the potential temperature varies with height, so

Equation (3.4.10) may, therefore, be written as d2(§z)/dt2 = -(g/9e)(d9e/dz) 5z = -N2 5z (3.4.12)

where N = {(g/9e) (d9e/dz)}1/2 is called the Brunt-Vaisalla frequency.

The general solution of (3.4.12) is of the form

Therefore, if N > 0, the parcel will oscillate about its equilibrium level with a frequency N and period 2n/N.

The value of N computed from vertical soundings of the mean tropical atmosphere is about 0.012s-1. This gives for the period of the oscillations about 8.8 min.

In the case where N = 0, there is no acceleration and the parcel will be in equilibrium with its environment at the end of the motion. However, if N < 0, the parcel will continue to rise, if displaced upward, and there is no equilibrium. Thus, the stability conditions are as follows:

d9/dz > 0 Stable d9/dz = 0 Neutral d9/dz < 0 Unstable

The atmosphere in general remains in static equilibrium most of the time. However, if static instability develops in any part of the atmosphere at any time, rapid vertical motion occurs so as to restore stability over the region again The situation, however, changes in cases of moist convection due to release of latent heat of condensation and this will be discussed in the next chapter.