Before discussing the general properties of (18.3.13), we consider two special cases:

Special case 1

In this case, we put UT = 0 in (18.3.13), and obtain two values of c given by:

The phase speeds c1 and c2 are real quantities which correspond to free stable (normal mode) oscillations of the two-level model with a vertically-averaged barotropic basic state zonal current,Um.

It will be seen from (18.3.14) that c1 is simply the phase speed of the barotropic Rossby wave moving westward relative to the barotropic basic state zonal wind, which was derived in (15.6.26) (l, the meridional wave number, being 0 in the present case). c2 given by (18.3.15), however, is in a different category. It may be interpreted as the phase speed of an internal baroclinic Rossby wave mode.

However, Lindzen et al. (1968) have shown that this baroclinic mode is a spurious one, since it does not arise in the real atmosphere but appears in the two-level quasi-geostrophic model because of the top boundary condition, m = 0 at p = 0. This boundary condition is equivalent to putting a rigid lid at the top, whereas in the real atmosphere there is no such barrier.

Holton (1979) has shown how an analogous baroclinic mode is excited in a homogeneous incompressible ocean with a free surface and a motionless basic state when a rigid-lid boundary condition with no vertical motion at the top is specified. It is shown that a small perturbation defined by u = u'(x, t), v = v/(x, t) at the surface of an ocean of depth h = H+h', where H is mean depth of the ocean, will move westward with a phase speed given by c = -p/(k2 + f02/gH) (18.3.16)

It is obvious that the phase speed given by (18.3.16) is that of a mixed Rossby-gravity mode, i.e., a westward-moving Rossby wave under the stabilizing influence of gravity. The two expressions (18.3.15) and (18.3.16) are analogous, since in the oceanic case, Um = o and in the denominator fo2/gH replaces 2X2{ = 2fo2/o(Ap)2}. For details of the derivation of (18.3.16), the original reference may be consulted.

Special case 2

The expression (18.3.13) then reduces to c = Um ±Ur{(k2 - 2X2)/(k2 + 2X2)}1/2 (18.3.17)

It is easy to see that when k2 < 2X2, c in (18.3.17) has an imaginary part. This means that all waves longer than a certain critical length given by,

Lc = ^2(n/X), will amplify. Since X = fo/(o1/2Ap), Lc = (Ap)n(2 o)1/2/fo.

For a value of o normally obtaining in midlatitudes, say at latitude of 45 °N, Lc turns out to be about 3ooo km. However, it is clear from the formula above that the critical wavelength Lc for baroclinic instability increases with the static stability o. Further, with P = o, although Lc does not depend upon the thermal wind, the exponential growth rate a of the baroclinic wave, given by a = kci, where ci denotes the imaginary part of the phase speed c, depends upon it as given by the expression a = kci = kUx{(2X2 - k2)/(2X2 + k2)}1/2 (18.3.18)

If all terms are retained in (18.3.13), a stability criterion called a neutral curve which connects all values of UT and k for the case in which 5 is just about zero and for which the flow may be treated as marginally stable may be worked out. Thus, from (18.3.13), for 5 = o, we have p2X4/{k4(k2 + 2X2)} = Ut2(2X2 - k2) (18.3.19)

or, solving for k4/2X4, we obtain the relation k4/2X4 = 1 ±{1 - P2/(4X4Ut2 )}1/2 (18.3.2o)

between the thermal wind and the wavelength in the case of the marginally stable flow. In Fig. 18.1, the nondimensional quantity k2/2X2, which gives a measure of the marginally stable zonal wavelength, is plotted against the nondimensional parameter 2X2UT/P, which is proportional to the thermal wind.

Fig. 18.1 Neutral stability curve for the two-level baroclinic model (After Holton, 1979, published by Academic Press, Inc)

As shown in the diagram, the neutral curve separates the unstable region of the (UT, k) plane from the stable region. It is evident that the inclusion of the p-effect stabilizes the flow, for now unstable waves can occur only when |UT | > p/2X2. Further, the minimum value of UT required for unstable growth depends strongly on k. This means that the p-effect stabilizes the long wave end of the spectrum. The flow is also stable for waves shorter than the critical wavelength Lc. The long wave stabilization associated with the p-effect is caused by the westward propagation of long waves which occurs only when the p-effect is included in the model. It can be shown that the baroclinically unstable waves propagate westward at a speed which lies between the maximum and the minimum mean zonal wind speeds.

To find the minimum value of UT for which unstable baroclinic waves may exist, we differentiate (18.3.19) with respect to k and find that the required condition dUT/dk = 0

This wave number corresponds to the wave of maximum baroclinic instability. When UT increases from zero and reaches this minimum value, the perturbation which first becomes unstable and amplifies has a wave number, k = 21/4 X, which corresponds to a wavelength of about 4000 km under normal conditions of static stability.

Observations show that the average wavelength of midlatitude synoptic-scale wave disturbances which amplify and decay is also close to this value. This may, perhaps, explain why baroclinically unstable waves are so common in midlatitude westerly currents. During amplification, the wave extracts energy from the mean thermal wind thereby decreasing the thermal wind and stabilizing the flow till the thermal wind builds up again to start another cycle of energy transformations. It is also observed that the average thermal wind in midlatitude westerlies exceeds that required for maximum baroclinic instability at wavelength near 4000 km which works out to be about 4ms-1 which arises from a wind shear of 8 m s-1 between 250 and 750 hPa in the two-level model. We, therefore, find that the observed behaviour of the midlatitude synoptic systems is consistent with the hypothesis that the disturbances develop from infinitesimally small perturbations of a baroclinically unstable basic westerly current by deriving energy for growth from the thermal wind.

Of course, in the real atmosphere, many other sources of energy, such as those due to lateral shear of an unstable zonal current, nonlinear interaction of finite amplitude perturbations, release of latent heat due to condensation, etc., may also influence the development of synoptic-scale disturbances. However, the findings of observational studies, laboratory experiments, and theoretical models all suggest that baroclinic instability is the primary mechanism for development of synoptic-scale wave disturbances in the midlatitudes.

18.4 Vertical Motion in Baroclinically Unstable Waves

The quasi-geostrophic baroclinic model that we discussed in the foregoing sections has to satisfy two important constraints at the same time. These are: that (1) the vorticity changes are geostrophic, and (2) the temperature changes are hydrostatic. In order that both the constraints be satisfied, the vertical motion field at every instant must be adjusted so that the divergent motions keep the vorticity changes geostrophic and the vertical motions keep the temperature changes hydrostatic. These properties of the quasi-geostrophic model are clearly revealed when we use the linearized equations (18.3.5-18.3.7) to compute the vertical motion field. An omega equation for the linearized model can be obtained by taking the second derivative of (18.3.7) and eliminating the time derivatives with the aid of (18.3.5) and (18.3.6). If we neglect the P-effect for simplicity, we obtain the following omega equation

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