Z dvdx dudy Vdpds dVdn VKs dVdn D dudx dvdy dVds Vdpdn

Fig. 13.5 Distribution of relative vorticity (Z) and divergence (D) in a W'ly jetstream (J) in the northern hemisphere. Lower panel shows how an anticyclonic curvature with negative vorticity has divergence (D) upstream of the jet maximum and convergence (C) downstream

Fig. 13.5 Distribution of relative vorticity (Z) and divergence (D) in a W'ly jetstream (J) in the northern hemisphere. Lower panel shows how an anticyclonic curvature with negative vorticity has divergence (D) upstream of the jet maximum and convergence (C) downstream

13.5 Potential Vorticity

The conservative property of potential temperature in adiabatic flow in a barotropic atmosphere has been used to derive a form of vorticity called potential vorticity which has been found to be extremely useful in tracking changes in atmospheric circulation. We derive an expression for it in this section. It was shown in sect. 13.4 that the absolute circulation remains constant in a barotropic atmosphere. This means that dCa/dt = 0 (13.5.1)

Ca = n A = (Z+f)A, where A is the circulation area.

Substituting for Ca in (13.5.1), we obtain the relation

Equation (13.5.2) is an important relation, since it connects the area of the circulation with the absolute vorticity. It states that for absolute circulation in a barotropic atmosphere to remain constant, the area of the circulation varies inversely as the absolute vorticity.

It can be shown that as in a barotropic atmosphere, the solenoidal term vanishes in frictionless adiabatic flow in which a parcel of air is forced to move along a surface of constant potential temperature. On account of this conservative property, it can be shown that the absolute circulation as given by the relation (13.5.2) remains constant in adiabatic flow. However, in this case, for circulation to remain constant, we can find an expression for change of absolute vorticity as a function of the vertical separation of the potential temperature surfaces.

Since, mass is conserved, we can write

where M is mass, A9 is a small change in potential temperature, and Ap a small change in pressure.

For a constant potential temperature difference, A9, i.e., for an isentropic layer, we may write (13.5.3) as

The relation (13.5.5) assumes a particularly simple form in a homogeneous atmosphere in which the variation of density may be neglected. It then reduces to the form

where Az is the vertical distance between two constant potential temperature surfaces.

The relation (13.5.5) is a mathematical statement of the principle of conservation of potential vorticity in adiabatic, frictionless flow. According to it, the absolute vorticity is directly proportional to the pressure depth of the flow. This means that when a flow enters an area where its vertical depth changes, its absolute vorticity will change so as to conserve potential vorticity. However, this principle works in quite a different way for westerly and easterly flow when the flow approaches a high mountain, such as the north-south-oriented Western Ghats Mountain of peninsular India or the Andes of South America, or the northwest-southeast-oriented lofty Himalayas along the northern boundary of India.

Let us first illustrate the difference by taking the case of a zonal flow without any horizontal shear when Ap remains constant (see Fig. 13.6).

If the flow is westerly, it will develop cyclonic relative vorticity if it turns northward and anticyclonic relative vorticity if it turns southward. But, in turning northward, the westerly flow also acquires anticyclonic relative vorticity because of increase in the value of the Coriolis parameter, whereas a southward-turning flow develops cyclonic relative vorticity because of decrease in the value of the Coriolis parameter. Thus, to conserve absolute vorticity in this case, a westerly flow must remain predominantly zonal. The situation, however, is quite different with an easterly flow when Ap remains constant. It the easterly flow turns northward, it develops enhanced negative relative vorticity on account of both anticyclonic curvature and an increased Coriolis parameter, whereas a southward turn will lead to increased positive relative vorticity on account of both cyclonic curvature and decreased Coriolis parameter.

The situation, however, changes markedly when the flow approaches a mountain where the depth Ap varies. Here, also, the case of a westerly flow differs markedly from that of an easterly flow, as illustrated in Fig. 13.7 and Fig. 13.8 respectively.

The case of the westerly flow is depicted in Fig. 13.7 in two sections: (a) vertical, (b) horizontal. As potential vorticity is to be conserved, the flow will turn

Fig. 13.6 The curvature and the Coriolis effects on a zonal flow for conservation of absolute vorticity

Fig. 13.6 The curvature and the Coriolis effects on a zonal flow for conservation of absolute vorticity

Fig. 13.7 Conservation of potential vorticity in the case of a westerly flow negotiating a mountain (MTN): (a) Vertical section; (b) horizontal section. M-Mass, 6 potential temperature

anticyclonically southward on the windward side where its depth decreases (see lower panel) till it passes the top of the mountain, and then cyclonically on the leeside where the depth increases.

In the case of an easterly flow as depicted in Fig. 13.8, it appears that an air parcel is able to sense the presence of the mountain from a distance and adjust its absolute vorticity by first turning southward in a cyclonic curvature before turning northward in an anticyclonic curvature, so as to partially offset the effect of an enhanced negative relative vorticity (on account of curvature as well as increased value of the Coriolis parameter) at the mountainside. In this way, it can cross the mountain and resume its easterly flow at the original latitude on the leeside.

Fig. 13.8 Conservation of potential vorticity in the case of an easterly flow negotiating a mountain: (a) Vertical section, (b) horizontal section

13.6 The Vorticity Equation in Frictionless Adiabatic Flow

Bjerknes's circulation theorem (13.2.7) may be used to derive an expression for the rate of change of vorticity due to creation of solenoids as well as expansion or contraction of the circulation area over the earth's surface. If we denote the solenoidal term, —<f a dp by N, we can write (13.2.7) in the form d(Z + f)/dt = —(Z + f)(1/A)dA/dt + N/A (13.6.1)

where we have made use of the following relations:

Now, (1/A) dA/dt, which denotes the rate of change of the area A per unit area is called divergence, V-V.

We may, therefore, write (13.6.1) in the form d(Z + f)/dt = —(Z + f)V-V + N/A (13.6.2)

13.7 The Vorticity Equation from the Equations of Motion

We now derive expressions for the rate of change of vorticity in the atmosphere without enforcing adiabatic conditions. For this purpose, we take the more general equations of frictionless motion which were derived in Chaps. 11 and 12 in Cartesian as well as isobaric co-ordinate systems.

13.7.1 Vorticity Equation in Cartesian Co-ordinates (x, y, z)

If we disregard friction, the horizontal momentum equations (11.5.1) may be written in the approximate form du/dt + u du/dx + v du/dy + w du/dz = fv — (1/p)dp/dx (13.7.1)

dv/dt + u dv/dx + v dv/dy + w dv/dz = —fu — (1/p)dp/dy (13.7.2)

To get the vertical component of the vorticity, we differentiate (13.7.1) with respect to y and (13.7.2) with respect to x, and then obtain, by subtracting the former from the latter, the expression dZ/dt + u dZ/dx + v dZ/dy + w dZ/dz = — (Z + f)(du/dx + dv/dy)