41.1.1 The kinetic energy density

For the scalings relevant for a hydrostatic and Boussinesq fluid (i.e., the ocean primitive equations using the ``Traditional Approximation''), the kinetic energy per unit volume (kinetic energy density) is determined just by the energy in the horizontal currents

$$e \equiv {\rho_{o} \over 2}(u^{2} + v^{2}) = {\rho_{o} \over 2} \, \vec{u}_{h} \cdot \vec{u}_{h}.$$ (41.1)

Hence, to develop an equation for the kinetic energy, it is necessary to consider the horizontal momentum equations

  

$$- \nabla \cdot (\vec{u} \, u) + v \left(f + { u \tan \phi \over a... ...t) - { p_{\lambda} \over a \rho_{o} \cos \phi} + (\kappa_{m} u_{z})_{z} + F^{u}$$ u_t = (41.2)

$$- \nabla \cdot (\vec{u} \, v) - u \left(f + { u \tan \phi \over a} \right) - { p_{\phi} \over a \rho_{o} } + (\kappa_{m} v_{z})_{z} + F^{v},$$ v_t = (41.3)

where the horizontal frictional terms

$$F^{\vec{u}} = (F^{u},F^{v},0)$$ (41.4)

were defined in Equations (9.187) and (9.193), and

$$\vec{u} = (u,v,w) = (\vec{u}_{h},w)$$ (41.5)

is the velocity field.