Winds on a Rotating Earth The Dynamical Equations and the Conservation Laws

11.1 Introduction

Nature as a whole tends to remain in a quasi-balanced state and nullify any imbalance that may arise or is created between its different parts or regions on account of differences of pressure, temperature or density at any time. The principle of heat balance requires that any imbalance in the distribution of heat in the earth- atmosphere system, such as that between a heat source over a warm low pressure area and a heat sink over a neighboring relatively cold high pressure area will force the atmosphere to circulate so as to remove the imbalance and restore the original heat balance. In the balancing process, the oceans also take part, but because of much lower density and faster movement of air, the atmosphere plays a much more active and effective role in the required heat transfer process than the oceans in shorter time scales. On longer time scales, say months or seasons, however, the contributions of oceans in this regard may become just as important, if not greater.

The forces that cause air motion are basically three: (i) the pressure gradient force, (ii) the force of gravity, and (iii) the force of friction or viscosity. In deriving the equations of motion involving these forces, we shall make use of the Newton's second law of motion which relates the motion to the forces acting on the moving body. The law of conservation of mass will be used in deriving the equation of continuity and the first law of thermodynamics which is the law of conservation of energy will be used in deriving the thermodynamic energy equation. Further, while applying the equations of motion to the real atmosphere, it is important to recognize that motions occur on varieties of space and time scales ranging from the smallest and fastest molecular vibrations to the largest and slowest-moving planetary-scale waves and that it is sometimes realistic to simplify the equations to adapt them to the scale of motion under consideration.

11.2 Forces Acting on a Parcel of Air

As already mentioned, the forces acting on a parcel of air in the atmosphere are:

(i) Pressure gradient force,

(ii) Gravity force, and

(iii) Frictional force

11.2.1 Pressure Gradient Force

Since pressure on a surface is defined as the force exerted on unit area of the surface and it is a continuous function of space variables, we take p0 as pressure at a point P (x0, y0, zo ) which lies at the center of an infinitesimally small rectangular volume with sides, Sx, Sy, Sz in a rectangular co-ordinate system (x, y, z) (see Fig. 11.1). It is assumed that the pressure increases eastward towards the positive x-direction, northward towards the positive y-direction, and upward towards the positive z-direction.

Taking the x-direction first, the force of the outside air on face ABCD of the volume is {p0 — (dp/dx) (Sx/2)} Sy Sz, whereas that on the opposite face PQRS is -{p0 + (dp/dx) (Sx/2)} Sy Sz.

The net force in the x-direction is then given by the vector addition of the forces on the two faces, i.e., by

— (dp/dx) Sx Sy Sz Similarly, the net forces in the y and z direction are respectively,

The total net force per unit mass on the volume element is then obtained by adding the above three forces and dividing the sum by the mass of the volume which is given by p Sx Sy Sz. Thus we get

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