41.1.2 External and internal mode kinetic energies

In MOM, the splitting of the flow into a vertically averaged velocity and a deviation from that average prompts an analysis of the kinetic energy which takes such a split into account. For this purpose, it is useful to introduce a depth averaging operator

$$\overline{\alpha} \equiv { 1 \over H + \eta} \int_{-H}^{\eta} dz \; \alpha.$$ (41.6)

The symbol for deviations from the depth average is given by a hat

$$\widehat{\alpha} = \alpha - \overline{\alpha}.$$ (41.7)

Using this notation, the horizontal velocity components can be split into the external (depth averaged) and internal modes

$$(u,v) = (\overline{u},\overline{v}) + (\widehat{u},\widehat{v}).$$ (41.8)

Substituting these velocities into the kinetic energy density yields

$$e = \frac{\rho_{o}}{2} (\overline{u} \, \overline{u} + \overline{... ... ) + \rho_{o} \, ( \overline{u} \, \widehat{u} + \overline{v} \, \widehat{v} ).$$ (41.9)

The depth averaged kinetic energy density is given by

$$\overline{e} = \frac{\rho_{o}}{2} (\overline{u} \, \overline{u} +... ...overline{\widehat{u} \, \widehat{u}} + \overline{\widehat{v} \, \widehat{v}} ).$$ (41.10)

Note the uncoupling of the external and internal modes in the depth averaged kinetic energy density. Hence, the depth averaged kinetic energy density can be thought of as a contribution from the external mode kinetic energy density

$$e_{ext} = \frac{\rho_{o}}{2} (\overline{u} \, \overline{u} + \overline{v} \, \overline{v} ),$$ (41.11)

and the depth averaged internal mode kinetic energy density

$$\overline{e_{int}} = \frac{\rho_{o}}{2}( \overline{\widehat{u} \, \widehat{u}} + \overline{\widehat{v} \, \widehat{v}} ).$$ (41.12)

Of central interest is how the budget for the volume averaged kinetic energy density, derived in equation (A.37), breaks up into external and internal mode components. Namely, with

$$\langle e \rangle V^{-1} \int d\Omega \int^{\eta}_{-H} dz \; e$$ =

$$V^{-1} \int d\Omega \; (H+\eta) \, e_{ext} + V^{-1} \int d\Omega \int^{\eta}_{-H} dz \; e_{int}$$ =

$$\langle e_{ext} \rangle + \langle e_{int} \rangle,$$ = (41.13)

what are the terms determining the individual time evolution of $\langle e_{ext} \rangle$ and $\langle e_{int} \rangle$? That question is answered for the external mode in Section A.1.5, and the internal mode in the Section A.1.6.