Parametric subharmonic instability (PSI) transfers energy from low-mode semidiurnal baroclinic tidal flow to high-mode near-inertial waves at latitudes ∼30°, inducing strong ocean mixing and hence affecting the global ocean circulation. Nevertheless, intuitive descriptions of the physical mechanism for energy transfer by PSI are very sparse. In this study, we reformulate this phenomenon to present a visual image of its mechanism based on a combination of simple classical theories such as beats and parametric excitation without adhering to a strict mathematical formula. It is shown that two small-scale near-inertial waves with slightly different wavenumbers propagating in opposite directions superpose to create beats. When the resulting beats have the peak-to-peak length and the phase velocity equal to the wavelength and the phase velocity of large-scale semidiurnal baroclinic tidal flow, respectively, continuous acceleration of near-inertial motions takes place under the effects of convergence and horizontal shear of the background semidiurnal baroclinic tidal flow. The resonant condition for PSI can thus be easily understood by introducing the well-known concept of beats which also provides a natural explanation for the large difference in spatial scales between the semidiurnal baroclinic tidal flow and the resulting near-inertial waves.
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Bryan F (1987) Parameter sensitivity of primitive equation ocean general circulation models. J Phys Oceanogr 17:970–985
Garrett CJR, Munk WH (1972) Space-time scales of internal waves. Geophys Fluid Dyn 2:225–264
Hasselmann K (1966) Feynman diagrams and interaction rules of wave-wave scattering processes. Rev Geophys Space Phys 4:1–32
Hazewinkel J, Winters KB (2011) PSI of the internal tide on a beta-plane: flux divergence and near-inertial wave propagation. J Phys Oceanogr 41:1673–1682
Hibiya T, Niwa Y, Nakajima K, Suginohara N (1996) Direct numerical simulation of the roll-off range of internal wave shear spectra in the ocean. J Geophys Res 101:14123–14129
Hibiya T, Niwa Y, Fujiwara K (1998) Numerical experiments of nonlinear energy transfer within the oceanic internal wave spectrum. J Geophys Res 103:18715–18722
Hibiya T, Nagasawa M, Niwa Y (2002) Nonlinear energy transfer within the oceanic internal wave spectrum at mid and high latitudes. J Geophys Res 107:3207. doi:10.1029/2001JC001210
Hibiya T, Nagasawa M (2004) Latitudinal dependence of diapycnal diffusivity in the thermocline estimated using a finescale parameterization. Geophys Res Lett 31, L01301. doi:10.1029/2003GL017998
Landau LD, Lifshitz EM (1960) Mechanics. Pergamon Press, Oxford
MacKinnon JA, Winters KB (2005) Subtropical catastrophe: significant loss of low-mode tidal energy at 28.9°. Geophys Res Lett 32, L15605. doi:10.1029/2005GL023376
MacKinnon JA, Alford MH, Sun O, Pinkel R, Zhao Z, Klymak J (2013) Parametric subharmonic instability of the internal tide at 29°N. J Phys Oceanogr 43:17–28
McComas CH, Bretherton FP (1977) Resonant interaction of oceanic internal waves. J Geophys Res 82:1397–1412
Munk WH (1966) Abyssal recipes. Deep-Sea Res 13:707–730
Munk WH, Wunsch C (1998) Abyssal recipes II: energetics of tidal and wind mixing. Deep-Sea Res I 45:1977–2010
Tsujino H, Hasumi H, Suginohara N (2000) Deep Pacific circulation controlled by vertical diffusivity at the lower thermocline depths. J Phys Oceanogr 30:2853–2865
Young WR, Tsang YK, Balmforth NJ (2008) Near-inertial parametric subharmonic instability. J Fluid Mech 607:25–49
The authors express their gratitude to two anonymous reviewers for their invaluable comments on the manuscript.
Responsible Editor: Matthew Robert Palmer
A. Order estimation of the energy production term
In our study, the frequencies of the background and fluctuating components are about 2f and f, respectively, so that, for f < < N, (A1) gives the relation between wavenumbers for the background wave k 0, m 0 and fluctuating waves k ′, m ′ such that
B. Derivation of the growth rate of near-inertial beats
We give the analytical formulation of PSI using the theory of parametric excitation explained in Sect. 2.1.
Since the terms of advection by background flow have nothing to do with energy production, (5) can be rewritten as
Substituting the background field expression and beats solutions
with resonant condition ω 1 + ω 2 = ω 0 and k 1 − k 2 = k 0 into (B1) and neglecting all the terms except those with wavenumbers k 1, k 2, we can get the algebraic equations
In these equations, superscripts “±” represent complex conjugate. In order for (B3) to have nontrivial solutions, ω 1 and ω 2 need to satisfy the following conditions:
This is one of the generalized eigenvalue problems, easily calculated with numerical methods.
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Onuki, Y., Hibiya, T. Excitation mechanism of near-inertial waves in baroclinic tidal flow caused by parametric subharmonic instability. Ocean Dynamics 65, 107–113 (2015). https://doi.org/10.1007/s10236-014-0789-3
- Parametric subharmonic instability
- Resonant triad interaction
- Nonlinear energy cascade
- Semidiurnal tidal flow
- Near-inertial wave
- Beat frequency