# Energy equation

(Redirected from Total energy equation)

## energy equation

1. Thermodynamic energy equation;
same as the first law of thermodynamics.

2. 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.

3. 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 Vn 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.