A similar expression can be written for the y-component of the gradient.

The transformation of the gradient of any arbitrary function from z to c coordinate is then given by the vector operator

Vc = Vz + (Vc z)d/dz i.e., Vz = Vc - (Vcz)d/dz (11.6.3)

Equation (11.6.3) is quite general and may be used for transformation to any vertical co-ordinate 'c' which is a function of z or p, such as an isobaric surface, an isentropic surface or a sigma (a) surface (where a = p/p0).

As examples of the use of (11.6.3), let us consider the following transformations:

(a) From height (z=constant) to isobaric (p=constant) surface

In this case, we put c = p, and since we wish to compute the gradient of z along the isobaric (p = constant) surface, we apply the operator (11.6.3) to write

where the left-hand side disappears, and, since dp/dz = —pg (the hydrostatic relation), the right-hand side reduces to the required relation

Thus, on the isobaric surface, density does not appear explicitly and the horizontal pressure gradient is given by the gradient of the geopotential at constant pressure surface.

(b) From height (z=constant) to an isentropic (9=constant) surface

In this case, we put c = 9, and apply the operator (11.6.3) to p which varies along the z = constant surface and write

With the aid of the hydrostatic relation and re-arranging, (11.6.5) may be written

Since 9 = T (1000/p)K, dp/dz = —pg, and p = pRT (equation of state), it can be shown that the right-hand side of (11.6.6) is equal to V9 (cp T + gz). Thus, along with (11.6.4), we have the relationships

where (cpT + gz) is called the dry static energy. It is also called the Montgomery potential or isentropic stream function.

(c) From isobaric (p=constant) to sigma (o-constant) surface

Here we put c = o, and, since o = p/ps, we obtain, using (11.6.3), the transformation equation

11.7 The Equations of Motion in Spherical Co-ordinate System

It is well-known that the surface of the earth may be treated as spherical for most meteorological purposes. So a curvilinear co-ordinate system in which the horizontal co-ordinate axes follow the curvature of the earth is more appropriate for

resolving the relative motion into components than any other co-ordinate system. In the spherical co-ordinate system (shown in Fig. 11.9), the horizontal co-ordinates are replaced by longitude X (to the east) and latitude ^ (to the north) and the earth's radius vector r is taken as the vertical co-ordinate. The relationships with the corresponding Cartesian co-ordinates in measurement of distances and velocity components are as follows:

Sx = a cos ^ SX; Sy = a S^; Sz = Sr u = (a cos 4>)dX/dt; v = ad^/dt; w = dr/dt (11.7.1)

In the spherical co-ordinate system, the velocity vector changes direction as a parcel moves along the earth's surface. So, we compute the velocity and acceleration in this co-ordinate system as follows: Let i, j, k be the unit vectors in the eastward, northward and vertical direction respectively. Then, we write for the velocity vector

V = u i + v j + w k, and, for the acceleration following motion, dV/dt = u di/dt + v dj/dt + w dk/dt

In (11.7.2), the rate of change of the unit vectors may be evaluated as follows:

For d i/dt, we note that the unit vector i varies only along the x-direction, so we may write, di/dt = udi/dx (11.7.3)

Since the vector di lies in a plane perpendicular to the axis of the earth and always directed towards the same axis, it has components in the northward and the upward directions, as shown in Fig. 11.10.

With the aid of Fig. 11.10, we, therefore, write di/dx = (j sin ^ — k cos \$)/(a cos (11.7.4)

The unit vector j varies with longitude and latitude but not with k. So we write dj/dt = u d j/dx + v dj/dy

Fig. 11.10 The components of the vector di along j and k axes

The components of the variations in these directions are dj/dx = —i tan^/a, dj/dy = —k/a

Fig. 11.11 shows how the unit vector j varies with (a) longitude, and (b) latitude. Similarly, we see that the unit vector k does not vary in the vertical but has variations along (a) longitude and (b) latitude. So we write dk/dt = u dk/dx + v dk/dy

The variations of k along longitude and latitude are shown in Fig. 11.12 (a, b) respectively:

Substituting from (11.7.3) to (11.7.6) in (11.7.2), we get for the acceleration of motion in spherical co-ordinates the expression

(a) Along longitude

Fig. 11.11 Components of the dj vector along (a) longitude and (b) latitude

(a) Along longitude (b) Along latitude

(a) Along longitude (b) Along latitude

Fig. 11.12 Components of the dk vector along (a) longitude and (b) latitude

dV/dt =[du/dt - uv tan ^/a + uw/a]i +[dv/dt + (u2 tan ^)/a + vw/a]j

Finally, substituting (11.7.7) in (11.5.1), we get the components of the equations of motion in spherical co-ordinates du/dt - uv tan^/a + uw/a = -adp/dx + 2 Q v sin^ - 2 Q w cos ^ + v d2u/dz2

dv/dt + u2 tan ^/a + vw/a = -adp/dy - 2 Q u sin^ + v d2 v/dz2 (11.7.9) dw/dt - (u2 + v2)/a = -adp/d z + 2 Q u cos ^ - g (11.7.10)

The momentum equations (11.7.8-11.7.10) are non-linear, since they contain terms which are quadratic in the dependent variables. Even the time derivatives of the dependent variables u, v, w, following motion are non-linear partial differential equations, as given by the relation for du/dt du/dt = du/dt + u du/dx + vdu/dy + w dw/dz and similar ones for dv/dt and dw/dt. The terms involving 'a' on the left-side of the momentum equations (11.7.8-11.7.10) are called the curvature terms, since they are due to the curvature of the earth.

11.8 The Equation of Continuity

The law of conservation of mass requires that the rate of change of mass following motion should be zero. That is, d(p 8V)/dt = 0

In the above expression, the right-hand side gives the rate of expansion of the volume element §V per unit time, which is equivalent to the three-dimensional divergence of the wind, V • V.

So the above equation can be written in the Lagrangian form dp/dt = —pV • V = —p(du/dx + dv/dy + dw/dz) (11.8.1)

This is the continuity equation for change of density and volume of an individual parcel of the fluid following motion. It states that the density of the parcel will decrease if there is divergence of air from the volume and increase if there is convergence into it.

Equation (11.8.1) may also be written in the Eulerian form dp/dt = —V • pV (11.8.2)

The Eq. (11.8.2) states that the rate of change of mass in a unit volume of a fluid is equal to the convergence or divergence of mass through the boundaries of the volume. Equations (11.8.1) and (11.8.2) are only the different expressions of the principle of conservation of mass.

If we assume that the air is incompressible, the left-hand side of (11.8.2) vanishes and we write

V • V = du/dx + dv/dy + dw/dz = 0 Or, dw/dz = —(du/dx + dv/dy) = —Vh • V (11.8.3)

where VH • V denotes horizontal divergence.

11.9 The Thermodynamic Energy Equation

The thermodynamic energy equation is virtually a restatement of the First law of thermodynamics (3.2) in the form

Or, using the equation of state for an ideal gas, and the relation, cp - cv = R, the above equation may be written as

where the left-side denotes the rate of diabatic heating, and the right-side constitutes the adiabatic response of the atmosphere in the form of an increase of its internal energy and doing work against external pressure, and the symbols have their usual meanings (see Chap. 3).

11.10 Scale Analysis and Simplification of the Equations of Motion

Motions in the atmosphere occur on different space and time scales. From molecular vibrations to very large planetary-scale flow, there is a vast range of motions and each type of motion has its own characteristic scale of length, velocity and time. Although the equations of motion (11.7.8-11.7.10) are quite general, it may bepos-sible for any particular type of motion to simplify them by eliminating terms which may be irrelevant or vanishingly small, and retaining those which more closely represent the motion being considered. This process of simplification and approximation is called scale analysis. It is like weighing the terms with an atmospheric scale and finding out which is heavy and which is light. We present the average length-scale of a few familiar types of motion systems in Table 11.1.

The scale is based on commonly-observed values of the basic variables and their observed variations in time and space for the particular type of motion system. For example, for large-scale tropical or midlatitude motion systems, we adopt the following typical values of the scaling parameters.

Horizontal pressure fluctuation Ap

It should be pointed out that the time scale't' refers to the advective time scale for the motion system and is usually the time taken for the system to move through one wavelength at approximately the velocity of the wind. The vertical velocity 'w' is usually not directly measured but can be computed in principle from the

Table 11.1 Length scale of some types of atmospheric motions

Type of motion

Horizontal scale (m)

Molecular vibration (Mean free path)

Small-scale eddy

Dust devil

Cumulus cloud

Cloud clusters

Cyclone/Hurricane/Typhoon

Synoptic-scale wave

Planetary-scale wave

104 105

divergence of the horizontal wind. Computations show that w is about two orders of magnitude smaller than the horizontal velocity. The frictional terms in Eqs. (11.7.8) and (11.7.9) are not considered, since it is held that except in the narrow boundary layer close to the earth's surface, friction does not play an important role.

We now consider a synoptic-scale wave disturbance centered at latitude, say ^ = 30°, where each of the components of the earth's angular velocity vector, viz., the vertical, 2Q sin and the horizontal, 2Q cos has magnitude of the order of 10-4s-1. The vertical component is usually denoted by 'f' and called the Coriolis parameter in meteorology. The scale analysis of the various terms of the horizontal momentum equations (11.7.8) and (11.7.9) for this wave system is shown in Table 11.2 which shows that the magnitudes of the terms involving the curvature of the earth as well as the horizontal component of earth's angular velocity vector are relatively unimportant for the type of motion under consideration. The most important terms which appear to be in approximate geostrophic balance are the pressure gradient force and the Coriolis force, since they are approximately of the same order of magnitude (10-3). This balance was first noted by Buys Ballot on weather charts in 1857.

11.10.1 The Geostrophic Approximation and the Geostrophic Wind

The approximate balance between the pressure gradient force and the Coriolis force gives us the geostrophic relationships:

ug = -(1/pf) dp/dy vg = (1/pf) dp/dx Or, in vector notation, Vg = -(1/pf) kx Vp (11.10.1)

where Vg(= ug i + vgj) is called the geostrophic wind.

Table 11.2 Scale analysis of the horizontal momentum equations

Terms

Table 11.2 Scale analysis of the horizontal momentum equations

Terms

 1 2 3 4 5 6 (x-comp) du/dt —2Qv sin ^ +2Qwcos ^ +u w/a — (uv ■ tan ^)/a = —adp/dx (y-comp) dv/dt +2Qu sin ^ +v w/a + (u2 tan /a = —adp/dx Scales V2 /L V x 10—4 W x 10—4 VW/a V2/a ((Ap)/p)/L Order of Magnitudes of the terms (ms —2) 10—4 10—3 10—6 10—8 10—5 10—3