## energy equation

- 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 formwhere 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 formwhere
*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.