By Buehler O.
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5) yields ∂2 ∂t2 H|ξ |2 2 + ∇ · (gh Hξ ) = H|u |2 − gh 2 . 7) In a bounded (or periodic) domain the second term vanishes after integrating over the domain. Moreover, under the assumption that ξ · u is ﬁnite, it also follows that the time-average of the domain-integrated left-hand side is zero for large averaging time, and therefore the same must be true for the timeaveraged domain-integrated diﬀerence between kinetic and potential energy on the right-hand side. 1 Linear dynamics 25 linear waves in shallow water.
The physical conditions of validity for the circulation theorem remain unchanged. 46) is simply modiﬁed to q= (∇ × u + f ) · ∇s . 1. In a rotating frame the integral conservation of momentum is replaced by the conservation of so-called absolute momentum, whose density per unit mass is u + f × r. This follows from the identity f ×u= D (f × r). 52) Interestingly, this conserved momentum is not the absolute momentum as observed in an inertial frame, which would involve Ω = 12 f instead of f . The reason for this discrepancy can be traced back to the earlier tacit assumption 18 Elements of ﬂuid dynamics about the pressure ﬁeld, which was assumed to balance the centrifugal force in a state of rest relative to the rotating frame.
By construction, the aspect ratio of the three-dimensional ﬂow is very small, with horizontal scales greatly exceeding the vertical scale. Speciﬁcally, if the typical layer depth is H then kH 1 where k is a typical horizontal wavenumber of the ﬂow ﬁeld. In the shallow-water regime this small parameter is exploited asymptotically and it can be shown that the leading-order ﬂow consists of horizontal velocities u = (u, v) that are constant across the layer and a much smaller vertical velocity w that varies linearly across the layer.