The Vorticity Equation in Isobaric Coordinates

A somewhat simpler form of the vorticity equation may be derived by using the equations of motion in isobaric co-ordinates.

Combining (12.2.3) and (12.2.4) and using vector notations, we can write the equations of motion in isobaric co-ordinates as dV/dt +(V-V)pV + mdV/dp = -VO - k x f V Or, dV/dt = -V [O + (V-V)/2] - kx V(Z + f) - mdV/dp (13.7.5)

where we have used the vector identity, (V-V)V =(V2) (V • V)+ k x ZV, and put Z = k-V x V.

We now operate on the vector equation (13.7.5) with the Del operator Vx, and obtain, after some vector operations, the vorticity equation dZ/dt = -V-Vp(Z + f) - mdZ/dp - (Z + f)Vp-V-k-VmxdV/dp (13.7.6)

Or, since df/dt and df/dp are both 0, we can write vorticity equation (13.7.6) as d(Z + f)/dt = -(Z + f)Vp-V - k-Vm x dV/dp (13.7.7)

where the divergence of the velocity is along the isobaric surface.

13.8 Circulation and Vorticity in the Real Atmosphere (In Three Dimensions)

We have in the foregoing sections considered for simplicity mostly the vorticity of a circulation in the horizontal plane about a vertical axis. However, in the earth-atmosphere system, circulation depends upon the actual locations of heat sources and sinks, and, as such, may occur in any plane in space with vorticity in a direction normal to the plane of the circulation, according to Stokes's theorem. In a rectangular system of co-ordinates, a given circulation can, therefore, be resolved into three components so as to have the vorticity of each component along the co-ordinate axes, x (eastward),y (northward) and z (upward). The component circulation may be one or more of the following:

(a) Horizontal circulation about the vertical z-axis, involving wind components u and v along x and y axes respectively.

The Stokes's relation for this circulation may be written as

^"(udx + v dy) = J/(dv/dx - du/dy)dxdy (13.8.1)

A purely horizontal circulation of this type is schematically shown in Fig. 13.10a.

(b) Zonal- vertical circulation about the y-axis, involving wind components u and w along the x and z axes respectively.

The Stokes's relation in this case is

^"(udx + w dz) = J f(du/dz - dw/dx)dxdz (13.8.2)

The zonal-vertical circulation of this type, shown in Fig. 13.10b, is usually described as a Walker circulation.

(c) Meridional-vertical circulation about the x-axis, involving the v and w components of the wind along y and z axes respectively.

The Stokes's relation in this case is

^"(vdy + wdz)=J J (dw/dy - dv/dz) dydz (13.8.3)

The meridional-vertical circulation of this type, usually called a Hadley-type circulation, is shown in Fig. 13.10c.

The component circulations depicted in Fig. 13.10 (a, b, c) are, however, highly idealized. In nature, they always get superimposed on each other and what we actually observe is a resultant circulation which could be in any plane with vorticity about a direction normal to it.

Fig. 13.10 Resolution of a circulation in space into component circulations with vorticity about coordinate axes: (a) Horizontal circulation in the x-y plane about a vertical z-axis, (b) Zonal-vertical Walker-type circulation in the x-z plane about the y-axis, and (c) Meridional- vertical Hadley-type circulation in the y-z plane about the x-axis

Fig. 13.10 Resolution of a circulation in space into component circulations with vorticity about coordinate axes: (a) Horizontal circulation in the x-y plane about a vertical z-axis, (b) Zonal-vertical Walker-type circulation in the x-z plane about the y-axis, and (c) Meridional- vertical Hadley-type circulation in the y-z plane about the x-axis

13.9 Vertical Motion in the Atmosphere

In the foregoing sections, we assumed air motion to be either along a horizontal or an isobaric, isentropic or a spherical surface. Over most parts of the globe, especially over high latitudes, this assumption is well borne out by observations and the wind is found to be largely horizontal. But there is also a vertical component of the air motion, the magnitude of which is ordinarily found to be about two orders of magnitude smaller than the horizontal component in large-scale motion systems. No matter how small this vertical motion may be, it is always significant. Measurements reveal that in some sub-synoptic and meso-scale circulation systems, such as cyclones and tornadoes, the vertical motion can be as large and important as the horizontal motion. Vertical motion plays a key role in the formation of such weather phenomena as cloud, precipitation, thunderstorms, etc. Because of its small magnitude, vertical velocity is difficult to measure directly. Several indirect methods have been devised to infer it from the observed winds and other atmospheric parameters. Three methods which are in common use for the purpose are: (1) the kinematic method which uses the equation of continuity in isobaric co-ordinates (12.2.6); (2) The adiabatic method which uses the thermodynamic energy equation (12.2.8); and (3) The vorticity method which uses different forms of the vorticity equation (13.7.6).

13.9.1 The kinematic Method

If we assume the atmosphere to be incompressible, the right-hand side of the continuity equation (12.2.6) may be put equal to zero. Thus, we get