## The Thermodynamic Energy Equation

In the isobaric co-ordinate system, the thermodynamic energy equation (11.9.1) may be written in the form dT/dt = (am + 8Q/dt)/cp (12.2.7)

Expanding the left side of (12.2.7) with the aid of (12.2.2) and re-arranging, we obtain dT/dt = -V • V T + am+(1/cp) 5Q/dt (12.2.8)

where a (= KT/p - dT/dp) is the static stability parameter, m (= dp/dt) is the vertical p-velocity, and k = R/cp.

Since T = (-p/R)dO/dp, the Eq. (12.2.8) may also be expressed in terms of dO/dp.

As already mentioned in Chap. 10, diabatic heating or cooling occurs due to convergence of net radiative as well as sensible and latent heat fluxes, and it is possible to get an approximate measure of it in a steady state (dT/dt = 0) by evaluating the adiabatic effects it produces in the form of horizontal and vertical thermal advections of heat as given by the first and the second terms on the right-hand side of equation (12.2.8). Usually, in the tropics the second term approximately balances the third term, but in midlatitudes the first and second terms together balance the third term.

### 12.3 Balanced Flow in Natural Co-ordinates

It is convenient to deal with horizontal motion depicted on weather maps in Natural co-ordinates. Depending upon the terms in balance, we may define some idealized balanced flows. This was realized more than a century ago.

12.3.1 Velocity and Acceleration in Natural Co-ordinate System

The natural co-ordinate system is characterized by a system of unit vectors, t, n, k, in which t is in the direction of the wind, n is in a direction normal to the wind (positive to the left of the wind), and k is in the vertical direction. This means that k = t x n.

Let the horizontal velocity along a curve s = s (t) in this co-ordinate system be denoted by V = Vt, where V = ds/dt, and t is a unit vector along the curve. Then, the acceleration a is given by

[Note that t is a unit vector, while t denotes time]

To evaluate dt/dt, we assume that the wind changes direction and that after time St it turns through an angle in an anticlockwise direction (Fig. 12.1).

Fig. 12.1 Evaluation of dt/dt Let R be the radius of curvature of the path of the parcel. Then it can be shown from Fig. 12.1 that dt/dt = dy/dt n = V/R n (12.3.2)

since dy/dt, the angular velocity, is equal to V/R.

Substituting for dt/dt from (12.3.2) in (12.3.1), we obtain a = (dV/dt) t +(V2/R) n (12.3.3)