Near-inertial waves in the ocean: beyond the ‘traditional approximation’

Abstract

The dynamics of linear internal waves in the ocean is analysed without adopting the ‘traditional approximation’, i.e. the horizontal component of the Earth's rotation is taken into account. It is shown that non-traditional effects profoundly change the dynamics of near-inertial waves in a vertically confined ocean. The partial differential equation describing linear internal-wave propagation can no longer be solved by separation of spatial variables; it was however pointed out earlier in the literature that a reduction to a Sturm–Liouville problem is still possible, a line that is pursued here. In its formal structure the Sturm–Liouville problem is the same as under the traditional approximation, but its eigenfunctions are no longer normal vertical modes of the full problem. The question is addressed of whether the solution found through this reduction is the general one: a set of eigenfunctions to the full problem is constructed, which depend in a non-separable way on the two spatial variables; these functions are orthogonal and form, under mild assumptions, a complete basis.In the near-inertial range, non-traditional effects act as a singular perturbation; this is seen from the sub-inertial short-wave limit, which is present whenever the ‘non-traditional’ terms are there, but disappears under the traditional approximation. In the dispersion relation the sub-inertial modes represent a smooth continuation of the super-inertial ones. The combined effect of the horizontal component of rotation and a vertical inhomogeneity in the stratification is found to play a crucial role in the dynamics of sub-inertial waves. They are trapped in waveguides localized around minima of the buoyancy frequency. The presence of horizontal inhomogeneities in the effective Coriolis parameter (such as shear currents or beta effect) are shown to enable a transition from super-inertial to sub-inertial waves (and thus effectively an irreversible transformation of large-scale into small-scale motions). It is suggested that this transformation provides a mechanism for mixing in the deep ocean.The notion of critical reflection of internal waves at a sloping bottom is also modified by non-traditional effects, and they strongly increase the probability of critical reflection in the near-inertial to tidal range.

Studies of inertio‐gravity waves on midlatitude beta‐plane without the traditional approximation

“…The exponential form of N(z) given by (4.19) was used for the experiments. It turns out that the non-TA effects have a significant role in producing the sub-inertial mode of IG waves near the bottom of the ocean where the value of N(z) is small, as predicted by Gerkema and Shrira (2005a). Moreover, because the vertical variability of the sub-inertial mode of IG waves is very large, the near-inertial waves thus generated likely contribute to a mechanism of deep-ocean mixing.…”

“…It turns out that the new mode which they obtained is identical in physics to the sub-inertial wave mode that was found during the 1970s mentioned earlier. More detailed properties of the non-traditional wave mode are now clarified by Kasahara (2003aKasahara ( , 2003bKasahara ( , 2004 and Durran and Bretherton (2004) in the fluid of constant buoyancy frequency N. Gerkema and Shrira (2005a) analysed the general solutions of linear Boussinesq equations without the TA in the fluid of variable N(z) as a function of vertical coordinate z and pointed out a possibility that 'the non-traditional effects profoundly change the dynamics of near-inertial waves in a vertically confined ocean'.…”

“…Moreover, it is well-known that the properties of near-inertial oscillations are rather unique because not only are they observed ubiquitously, though intermittently, even in deep seas as well as in the upper part of ocean, but also the vertical correlation of wave motions are very weak (Webster, 1968;Fu, 1981). Through an analytical consideration, Gerkema and Shrira (2005a) suggested that the non-traditional Coriolis effects appear as sub-inertial waves in the region of weakest stratification, i.e. where the value of N 2 (z) becomes minimal.…”

Low-frequency internal waves in magnetized rotating stellar radiation zones

“…This corresponds to the elliptic modes family (the E 1 modes in Dintrans et al 1999), which propagate in the whole sphere. In the other regime where |ν M;m | ≥ 1, which corresponds to equatorially trapped hyperbolic modes (the H 1 modes in Dintrans et al 1999), the MHD TA fails to reproduce the wave behaviour rigourously and the full dynamical equation has to be solved (a detailed discussion is given in Gerkema & Shrira 2005a;Gerkema et al 2007). For a strong stratification, which is considered here, Miles (1974) proposed an asymptotic solution to the problem, for which it is necessary to construct a boundary layer about the trapping latitude,…”

Low-frequency internal waves in magnetized rotating stellar radiation zones

“…This approximation has to be carefully used however, because it changes the nature of the Poincaré operator, and removes the singularities and associated shear layers that appear. Then, in the subinertial regime, where σ ≤ 2Ω s , which corresponds to the equatorially trapped hyperbolic modes (the H 2 modes in Dintrans et al 1999;and Dintrans & Rieutord 2000), the TA fails to reproduce the waves behaviour and the complete momentum equation has to be solved (see detailed examples in Gerkema & Shrira 2005;and Gerkema et al 2008). Note also the work by Fruman (2009), who examines the validity of the TA on the equatorial β-plane in function of the 2Ω s /N value.…”