41.1.7.1 Splitting of the energy density

In the following, brackets will always denote a global volume average

$$\langle \alpha \rangle = V_{globe}^{-1} \int^{globe} d\Omega \int^{\eta}_{-H} dz \; \alpha,$$ (41.60)

where V_globe is the constant global volume of the ocean water. The ``globe'' subscript is dropped in further equations in this section. Recall that V_globe is constant, even with the free surface, since the ocean is incompressible and it is assumed that the budget for the surface fresh water forcing is closed.

Because

$$\langle \widehat{u} \, \overline{u} \rangle = \langle \widehat{v} \, \overline{v} \rangle = 0,$$ (41.61)

the volume averaged kinetic energy density splits into two terms

$$\langle e \rangle = \langle e_{ext} \rangle + \langle e_{int} \rangle,$$ (41.62)

where

$$\langle e_{ext} \rangle = (\rho_{o}/2) \langle \overline{u} \, \overline{u} + \overline{v} \, \overline{v} \rangle$$ (41.63)

and

$$\langle e_{int} \rangle = (\rho_{o}/2) \langle \widehat{u} \, \widehat{u} + \widehat{v} \, \widehat{v} \rangle.$$ (41.64)

Now the time tendency of the global averaged external mode kinetic energy density is given by

$$\partial_{t} \, \langle e_{ext} \rangle \rho_{o} \, \langle \overline{u} \, \partial_{t} \, \overline{u} + \overline{v} \, \partial_{t} \, \overline{v} \rangle$$ =

$$\rho_{o} \, \langle \overline{u} \, \partial_{t} \, (u -\widehat{u}) + \overline{v} \, \partial_{t} \, (v-\widehat{v}) \rangle.$$ = (41.65)

Similarily, the time tendency of the global averaged internal mode kinetic energy density is given by

$$\partial_{t} \, \langle e_{int} \rangle \rho_{o} \, \langle \widehat{u} \, \partial_{t} \, \widehat{u} + \widehat{v} \, \partial_{t} \, \widehat{v} \rangle$$ =

$$\rho_{o} \, \langle \widehat{u} \, \partial_{t} \, (u -\overline{u}) + \widehat{v} \, \partial_{t} \, (v-\overline{v}) \rangle.$$ = (41.66)