and so the kinetic energy equation can be combined with eq.(58) to give:
Using the fact that to good approximation, this may be written in the form:
Both terms on the RHS of eq.(59) can be neglected : the pressure tendency term is only important for high frequency acoustic waves and the second term is smaller than the corresponding term involving on the LHS by a factor . Therefore, to a good approximation :
which expresses the conservation of a quantity related to the Bernoulli energy. For ascending or descending air parcels the change of kinetic energy following the motion is much less than the change in gravitational
potential energy and eq.(60) is commonly simplified to:
which is conservation of moist static energy for saturated air (after approximating with ).
Conservation of virtual moist static energy ( ) can be used to define a liquid water conservation variable similar to , i.e.
where we have defined virtual liquid water temperature according to:
Since this definition does not require the approximation of an exponential as in the case of , it is more accurate for air parcels making large vertical excursions.