Basic Equations of Atmospheric Fluid Mechanics

We will derive first the equations that govern the fluid density, temperature, and velocities in the lowest layer of the atmosphere. These equations will form the basis from which we can subsequently explore the processes that influence atmospheric turbulence. In our discussion we shall consider only a shallow layer adjacent to the surface, in which case we can make some rather important simplifications in the equations of continuity, motion, and energy. The equation of continuity for a compressible fluid is 

Equation (16.25) can be simplified further if we use the summation convention, that is, if we omit the summation symbol from the equation, assuming that repeated symbols are summed from 1 to 3. As a result, the continuity equation becomes 

Because we are interested only in processes taking place on limited spatial and temporal scales over which the air motion is not influenced by the rotation of the Earth, we will neglect the Coriolis acceleration (see Chapter 21) and write the equation of motion of a compressible, Newtonian fluid in a gravitational field as 

Finally the energy equation, assuming that the contribution of viscous dissipation to the energy balance of the atmosphere is negligible, is 

These six equations can therefore be solved, in principle, subject to appropriate boundary and initial conditions to yield velocity, pressure, density, and temperature profiles in the atmosphere. 

We wish to derive the equations that govern the fluid density, temperature, and velocities in the lowest layers of the atmosphere. These equations will form the basics from which we can subsequently explore the processes that influence atmospheric turbulence. 

The equations of continuity and motion for a compressible, Newtonian fluid in a gravitational field are 

Equations (16.1) to (16.4) represent six equations for the six unknowns i, U2, M3, p, p, and r. These equations can therefore be solved, in principle, subject to appropriate boundary and initial conditions to yield velocity, pressure, density, and temperature profiles in an ideal gas. Because of the highly coupled nature of (16.1) to (16.4), these equations are virtually impossible to solve analytically. However, we can exploit certain aspects characteristic of the lower atmosphere to simplify them. 

We can express the equilibrium profiles of pressure, density, and temperature in terms of functions of X3 only as follows  

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APPENDIX 16 DERIVATION OF THE BASIC EQUATIONS OF SURFACE LAYER ATMOSPHERIC FLUID MECHANICS... 

To describe the theoretical dynamical and thermal behavior of the atmosphere, the fundamental equations of fluid mechanics must be employed. In this section these equations are presented in a relatively simple form. A more conceptual view will be presented in Section 3.6. The circulation of the Earth s atmosphere is governed by three basic principles Newton s laws of motion, the conservation of energy, and the conservation of mass. Newton s second law describes the response of a fluid to external forces. In a frame of reference which rotates with the Earth, the first fundamental equation, called the momentum equation, is given by ... 

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