Dfdy p[dpdxdpdy

where we have substituted Z for the vertical component of the relative vorticity, (dv/dx — du/dy).

Since df/dt = vdf/dy, by re-arranging, we can write (13.7.3) in the form d(Z + f)/dt = —(Z + f)(du/dx + dv/dy) — {(dw/dx dv/dz — dw/dy du/dz)} + [(1/p2)(dp/dxdp/dy — dp/dy dp/dx)] (13.7.4)

The three terms on the right-hand side of (13.7.4) are called the divergence term, the tilting term and the solenoidal term respectively. In general, all the terms contribute to the rate of change of the vertical component of the absolute vorticity (Z + f) following motion in the atmosphere.

The divergence or convergence of air is a powerful mechanism in controlling absolute vorticity. Divergence(convergence) decreases (increases) the absolute vor-ticity, The tilting term represents a contribution from the horizontally-oriented axis of rotation when it is vertically tilted by a non-uniform field of vertical motion, as illustrated in Fig. 13.9, in which the eastward or the u- component of the velocity varies in the vertical, producing a vorticity along the axis of y. Now, if the vertical motion field varies along the y-axis, it will tilt the y-component of the vorticity in the vertical to change the vertical component of the absolute vorticity. The tilting term may be said to be largely responsible for the helical movement of air parcels around the center of a fast-rotating mesoscale convective system, in which the strongest upward velocity is reached at some height above the surface, which separates the regime of low-level cyclonic circulation from that of the anticyclonic circulation above.

The solenoidal term in (13.7.4) represents the contribution of the solenoidal circulation per unit area to the rate of change of absolute vorticity, as already stated in (13.6.2). The following show this relationship:

Using Stokes's theorem, we may write (13.2.4) as

Fig. 13.9 Illustrating how the tilting term contributes to the generation of absolute vorticity

Hence the above expression may be written as dCa/dt = -J J k-(Va x Vp) dA

This may be compared with the solenoidal term in (13.7.4) which can be written

-(I / p2)[dp / dxdp / dy - dp/dy dp/dx] = -k(Va x Vp), where a = 1/p.

It can, therefore, be seen that the solenoidal term truly represents a contribution to the rate of change of the absolute vorticity of the horizontal circulation per unit area.

The same result could be derived if in the solenoidal term we replaced density by temperature using the equation of state. When the circulation is predominantly horizontal, the contribution of this term is negligible.

It is easy to see that if all the three terms on the right-hand side of (13.7.4) are absent as in a barotropic atmosphere, the absolute vorticity following the motion remains conserved.

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