L

dc z

So, if x = xa + Sx, where 5x is a small increment in x, then, to first order in Sx, we may write:

Taylor expansions were often used in Chapter 4 (in the derivation of vertical stability criteria) and in Chapter 6 (in the derivation of the equations that govern fluid motion).

A.2.2. Vector identities

Cartesian coordinates are best used for getting our ideas straight, but occasionally we also make use of polar (sometimes called cylindrical) and spherical polar coordinates (see Section A.2.3).

The Cartesian coordinate system is defined by three axes at right angles to each other. The horizontal axes are labeled x and y, and the vertical axis is labeled z, with associated unit vectors x, y, z (respectively). To specify a particular point we specify the x coordinate first (abscissa), followed by the y coordinate (ordinate), followed by the z coordinate, to form an ordered triplet (x, y, z).

If $ is a scalar field and a, a 3-dimensional vector field thus:

a = axx + ay y + azv where ax, ay and az are magnitudes of the projections of a along the three axes, then we have the following definitions:

Inspection of I, II, and III shows that we can define the operator V uniquely as:

yd yd yd

dx dy dz

[the scalar product of a and b, a scalar]

VI. a x b = ax ay az bx by bz y (aybz - azby) + y (azbx - axbz) +

[the vector product of a and b, a vector].

Here is the determinant. Thus, for example, in Eq. 7-17, z x Va =

vx yv vz

da da da dx dy dz

= X( - ~dy) + » V dx and so, dug/ dz = jgv x Va yields the component of the thermal wind equation, Eq. 7-16.

Some useful vector identities are:

6. Vx (a x b) = a (V- b) - b (V-a) + (b -V) a - (a -V) b

An important special case of (7) arises when a = b:

These relations can be verified by the arduous procedure of applying the definitions of V, V-, Vx (and a ■ b, a x b as necessary) to all the terms involved.

A.2.3. Polar and spherical coordinates Polar coordinates

Any point is specified by the distance r (radius vector pointing outwards) from the origin and 0 (vectorial angle) measured from a reference line (0 is positive if measured counterclockwise and negative if measured clockwise), as shown in Fig. 6.8.

The velocity vector is u = rvr + 0v0 = (vr,v0) where vr = Dr/Dt and v0 = rD0/Dt are the radial and azimuthal velocities, respectively, and r, 0 are unit vectors.

In polar coordinates:

Spherical polar coordinates

In spherical coordinates a point is specified by coordinates (A, p, z) where, as shown in Fig. 6.19, z is radial distance, p is latitude and A is longitude. Velocity components (u, v, w) are associated with the coordinates (A, p, z), so that u = z cos pDA/Dt is in the direction of increasing A (eastward), v = zDp/Dt is in the direction of increasing p (northward) and w = Dz/Dt is in the direction of increasing z (upward, in the direction opposite to gravity).

In spherical coordinates:

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