Basic Equations of Surface Layer Atmospheric Fluid Mechanics

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

The term in parenthesis is by definition the substantial derivative Dp/Dt and therefore 

To derive the equation of motion, we subtract the reference equilibrium state (16.31) from the equation of motion (16.27) and also use (16.A.5) to obtain 

Let us examine the first and third terms on the RHS of (16.A.6). For a shallow layer the air density p will be approximately equal to the density at the surface. Therefore 

The last term on the RHS of (16.A.6) expresses the vertical acceleration on a fluid element as a result of the density fluctuations. Assuming that the fluctuations in density from the surface value are small, we can expand p in a Taylor series about p0 as follows  

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