Energy equation

energy equation

Thermodynamic energy equation;
same as the first law of thermodynamics.
Mechanical energy equation (
or kinetic energy equation): an expression for the rate of change of kinetic energy, which is obtained by scalar multiplication of the three-dimensional vector equation of motion by the vector velocity u; it may be written in the form
where φ = gz is the geopotential energy, ρ is density, p is pressure, F is the vector frictional force per unit volume, and ∇ is the del operator.
Total energy equation: An expression relating all forms of energy obtained by combining the thermodynamic energy equation with the mechanical energy equation. When integrated over a fixed volume of the atmosphere, this equation takes the form
where dV is the volume element, ds is the element of the surface of the volume, and V_n is the inwardly directed velocity normal to the surface of the volume.

This equation expresses the fact that the combined internal, kinetic, and potential energy in a given volume can vary only as a result of 1) the transport of these forms of energy across the boundaries of the volume; 2) the work done by pressure forces on the boundary; 3) the addition or removal of heat; and 4) the dissipational effect of friction.

Gill, A. E. 1982. Atmosphere–Ocean Dynamics. Academic Press, . 76–82.