## Hwn FIGURE 4.11. Dry convection viewed from the perspective of temperature (left) and potential temperature (right). Air parcels rise from the surface and follow a dry adiabat until their temperature matches that of the environment, when they will become neutrally buoyant. If the surface temperature is T (or T2), convection will extend to an altitude Z1 (or z2). The same process viewed in terms of potential temperature is simpler. The stable layer above has d6/dz > 0; convection returns the overturning layer to a state of uniform 6 corresponding to neutral stability, just as observed in the laboratory experiment (cf. the right frame of Fig. 4.6). Note that by definition, 6 = T at p = 1000 hPa.

FIGURE 4.11. Dry convection viewed from the perspective of temperature (left) and potential temperature (right). Air parcels rise from the surface and follow a dry adiabat until their temperature matches that of the environment, when they will become neutrally buoyant. If the surface temperature is T (or T2), convection will extend to an altitude Z1 (or z2). The same process viewed in terms of potential temperature is simpler. The stable layer above has d6/dz > 0; convection returns the overturning layer to a state of uniform 6 corresponding to neutral stability, just as observed in the laboratory experiment (cf. the right frame of Fig. 4.6). Note that by definition, 6 = T at p = 1000 hPa. FIGURE 4.12. A parcel displaced a distance A from height Z1 to height zp. The density of the parcel is pp, and that of the environment, pE.

air parcel. Consider Fig. 4.12. An air parcel has been displaced upward adiabatically a distance A from level z1 to level zP = z1 + A. The environment has density profile pE (z), and a corresponding pressure field pE(z) in hydrostatic balance with the density. The parcel's pressure must equal the environmental pressure; the parcel's density is pP = pP/RTP = pE(zP)/RTP. We suppose the parcel has height Sz and cross-sectional area SA.

Now the forces acting on the parcel are (following the arguments and notation given in Section 3.2):

-g pP SASz (downward), ii) net pressure force FT + FB = SpE SA

where we have used hydrostatic balance. Hence the net force on the parcel is

The parcel's mass is PpSzSA, and the equation of motion for the parcel is therefore d2A

so that d2A dt2

The quantity b = -g (pP - pE) /pP is, of course, the buoyancy of the parcel as defined in Eq. 4-3: if pP > pE the parcel will be negatively buoyant. Now, because the parcel always has the same pressure as the environment, we may write its buoyancy, using the ideal gas law, Eq. 1-1, and the definition of potential temperature, Eq. 4-17, as b = gPE-P = g (Tp - Te) = g (6p - 6e) .

PP Te 6e

For small A, 6e (z + A) = 6e (z\) + Ad6E/ dz. Moreover, since the potential temperature of the parcel is conserved, 6P = 6e(zi), and so 6P - 6E = -Ad6E/dz, enabling Eq. 4-19 to be written:

which depends only on the vertical variation of 6e. Note that, under the stable conditions assumed here, N2 > 0, so N is real. Then Eq. 4-21 has oscillatory solutions of the form

A = Ai cos Nt + A2 sin Nt, where A1 and A2 are constants set by initial conditions, and N, defined by Eq. 4-22, is the frequency of the oscillation. It is for this reason that the quantity N (with units of s-1) is known as the buoyancy frequency.

Thus in the stable case, the restoring force associated with stratification allows the existence of waves, which are known as internal gravity waves, and which are in fact analogous to those commonly seen on water surfaces. The latter, known as surface gravity waves, owe their existence to the stable, "bottom heavy,'' density difference at the water-air interface. Internal gravity waves owe their existence to a continuous, internal, stable stratification.

Under typical tropospheric conditions (see the 6 profile in Fig. 4.9), we estimate

9.81ms 