  This paper presents a theory describing the energy budget of a fluid under
the Boussinesq approximation: the theory is developed in a manner consistent
with the conservation law of mass. It shows that no potential energy is
available under the Boussinesq approximation, and also reveals that the work
done by the buoyancy force due to changes in temperature corresponds to the
conversion between kinetic and internal energy. This energy conversion,
however, makes only an ignorable contribution to the distribution of
temperature under the approximation.
  The Boussinesq approximation is, in physical oceanography, extended so that
the motion of seawater can be studied. This paper considers this extended
approximation as well. Under the extended approximation, the work done by the
buoyancy force due to changes in salinity corresponds to the conversion between
kinetic and potential energy. It also turns out that the conservation law of
mass does not allow the condition $\nabla\cdot\boldsymbol{u}=0$ on the fluid
velocity $\boldsymbol{u}$ to be imposed under the extended approximation; the
condition to be imposed instead is presented.
