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Destabilization of mixed Rossby gravity waves and the formation of equatorial zonal jets
- BACH LIEN HUA, MARC D'ORGEVILLE, MARK D. FRUMAN, CLAIRE MENESGUEN, RICHARD SCHOPP, PATRICE KLEIN, HIDEHARU SASAKI
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- Journal:
- Journal of Fluid Mechanics / Volume 610 / 10 September 2008
- Published online by Cambridge University Press:
- 08 August 2008, pp. 311-341
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The stability of mixed Rossby gravity (MRG) waves has been investigated numerically using three-dimensionally consistent high-resolution simulations of the continuously stratified primitive equations. For short enough zonal wavelength, the westward phase propagating MRG wave is strongly destabilized by barotropic shear instability leading to the formation of zonal jets. The large-scale instability of the zonally short wave generates zonal jets because it consists primarily of sheared meridional motions, as shown recently for the short barotropic Rossby wave problem.
Simulations were done in a variety of domain geometries: a periodic re-entrant channel, a basin with a short MRG wave forced in its western part and a very long channel initialized with a zonally localized MRG wave. The characteristics of the zonal jets vary with the geometry. In the periodic re-entrant channel, barotropic zonal jets dominate the total flow response at the equator and its immediate vicinity. In the other cases, the destabilization leads to zonal jets with quite different characteristics, especially in the eastward group propagating part of the signal. The most striking result concerns the formation of zonal jets at the equator, alternating in sign in the vertical, with vertical scale short compared to the scale of the forcing or initial conditions.
A stability analysis of a simplified perturbation vorticity equation is formulated to explain the spatial scale selection and growth rate of the zonal jets as functions of the characteristics of the basic state MRG wave. For both types of zonal jets, the model predicts that their meridional scales are comparable to the zonal scale of the MRG wave basic state, while their growth rates scale as μ ∝ Fr |k|, where Fr is the Froude number of the meridional velocity component of the basic state and k its non-dimensional zonal wavenumber. The vertical scale of the baroclinic zonal jets corresponds to the dominant harmonic ppeak of the basic state in the fastest growing mode, given by ppeak≈0.55k2. Thus, the shorter the zonal wavelength of the basic state MRG wave, the narrower the meridional scale of the zonal jets, both barotropic and baroclinic, with the vertical scale of the baroclinic jets being tied to their meridional scale through the equatorial radius of deformation, which decreases as the square root of the vertical wavenumber. The predictions of the spatial scales are in both qualitative and quantitative agreement with the numerical simulations, where shorter vertical scale baroclinic zonal jets are favoured by shorter-wavelength longer-period MRG wave basic states, with the vertical mode number increasing as the square of the MRG wave period.
An Appendix deals with the case of zonally long and intermediate wavelength MRG waves, where a weak instability regime causes a moderate adjustment involving resonant triad interactions without leading to jet formation. For eastward phase propagating waves, adjustment does not lead to significant angular momentum redistribution.
Equatorial inertial-parametric instability of zonally symmetric oscillating shear flows
- MARC D'ORGEVILLE, BACH LIEN HUA
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- Journal:
- Journal of Fluid Mechanics / Volume 531 / 25 May 2005
- Published online by Cambridge University Press:
- 18 May 2005, pp. 261-291
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This study revisits the problem of the zonally symmetric instability on the equatorial $\beta$-plane. Rather than treating the classical problem of a steady basic flow, it treats a sequence of problems of increasing complexity in which the basic flow is oscillatory in time with a frequency $\omega_0$.
First, for the case of a homogeneous fluid, a time-oscillating barotropic shear forcing may excite a subharmonic parametric resonance of inertial oscillations. Because of the continuous distribution of inertial oscillation frequencies, this resonance occurs at critical inertial latitudes $y_c$ such that $\beta y_c {=}{\pm} {\omega_0}/2$. Next the effects of stratification, characterized by Brunt–Väisälä frequency $N$, are taken into account. It is shown analytically (in the asymptotic limit of a weak shear) that the forced temporal oscillation leads to an inertial-parametric instability, when a resonance condition between the basic flow frequency and the sum of two inertio-gravity free-mode frequencies is met. This inertial-parametric instability has a well-defined inviscid vertical scale selection favouring the high-vertical mode $m_c{\sim}7.45m_0$, where $m_0{=}{\beta N}/{\omega_0^2}$ is the equatorial vertical mode characteristic of frequency $\omega_0$. The viscous critical shear of inertial-parametric instability is lower than the steady inertial instability one.
Finally, this type of setting naturally arises when the basic flow is considered to be an equatorial wave, so the problem is recast with the nonlinear adjustment of the vertically sinusoidal basic state of a zonally symmetric mixed Rossby–gravity (MRG) wave. Initial-value numerical simulations show that the same inertial-parametric instability exists leading to a resonant subharmonic excitation of free modes with vertical scales 7 and 8 times smaller than the basic-state wave. A simplified dynamical model of the instability is introduced, demonstrating that the oscillatory nature of the shear with height for the MRG wave necessarily implies a resonance between distinct vertical modes, the most unstable ones being modes 7 and 8 for a large enough Froude number of the MRG wave. The nonlinear action of the instability is described in terms of angular momentum and potential vorticity changes: a significant mixing due to the breaking of the excited high vertical modes creates a vertically averaged westward flow at the equator and extra-equatorial eastward flows. The ideas exposed may play a part in explaining layering phenomena and the latitudinal structure of the zonal flow in the equatorial oceans below the thermocline.