Note the fact that the tidal potential is a function primarily of the horizontal position, with a small percentage change in the vertical direction, so the volume integration of the time derivative of the tidal potential is equal to po dtjjdv ~ po hdtjjdxdy + po / Zdtjrdxdy (3.97)
Integrating Eqn. (3.97) over a tidal cycle, the first term associated with the time-invariant depth of the ocean (h) vanishes, so that only the free surface elevation term remains. After integration by parts, the corresponding tidal contribution averaged over a tidal cycle is —po dt ZJt .As discussed above, to maintain the tidal circulation, the tidal potential and the vertical velocity should be positively correlated, so this term is negative. As a result, the tidal contribution in this formulation turns out to be negative.
This formulation is thus incapable of providing a clear explanation for the role of tidal energy in the global circulation. This failure is not a surprise: and a closer examination reveals the problem associated with this formulation. In Eqn. (3.91), tidal potential is included as a part of the total energy of the oceanic circulation system. It is well known that the total tidal energy declines due to tidal dissipation; thus the negative sink term on the right-hand side of Eqn. (3.91) reflects the decline of tidal potential over longer time scales. The exact formulation of tidal potential energy involves complicated dynamics of the gravitational field and orbital motions of the solar system: this is beyond the scope of our discussion.
It is interesting to note that the first term on the right-hand side of Eqn. (3.83) does not explicitly appear on the right-hand side of the energy balance equations in this formulation. To dig for this term, we reformulate the problem as follows. First, we notice that, in the previous formulation, the GPE includes contributions associated with the tidal potential and non-tidal motions. It is well known that tidal dissipation is associated with loss of GPE in the Earth-Moon-Sun system. In the following analysis we will divide both GPE and KE into two parts.
The time-invariant part of GPE 0o and the tidal part of GPE satisfy the following equation d (p&o) -> ->
Changes in the total GPE can be rewritten as
TiJJJv pH v = TtJJJv p*od v + TtJJJv ^(t) d v = Tt $o + Tt (3.99)
Note that the second term is associated with tidal potential , which should decline with time. If the tidal potential term is moved from the right-hand side of Eqn. (3.99) to its left-hand side, we can see that for a balance of GPE of the non-tidal motions, tidal dissipation should appear as a source of energy for the non-tidal motions.
Similarly, if we assume that the tidal velocity and non-tidal velocity are uncorrelated, then we have
It is worth noting that the postulation of a complete uncorrelation between the tidal and non-tidal motions is only conceptual, but not strictly rigorous. In fact, one of the most important issues in energetic theory of the oceanic general circulation is that tidal dissipation can affect the non-tidal motions through mixing sustained by tidal dissipation.
Nevertheless, the separation of tidal and non-tidal motions can help us to understand the contribution of tidal dissipation to the non-tidal motions; thus, we adopt this postulation in the following discussion.
The corresponding energy equations for the non-tidal motions are
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