Type 
Article 
Date 
200504 
Language 
English 
Author(s) 
D'Orgeville Marc^{2}, Hua BachLien^{1} 
Affiliation(s) 
1 : IFREMER, Lab Phys Oceans, F29280 Plouzane, France. 
Source 
Journal of Fluid Mechanics (00221120) (Cambridge University Press), 200504 , Vol. 531 , P. 261291 
DOI 
10.1017/S0022112005004015 
WOS© Times Cited 
16 
Keyword(s) 
Equatorial wave, Dynamical model, Oscillating shear flows 
Abstract 
This study revisits the problem of the zonally symmetric instability on the equatorial betaplane. 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 timeoscillating 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) = +/omega(0)/2. Next the effects of stratification, characterized by BruntVaisala 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 inertialparametric instability, when a resonance condition between the basic flow frequency and the sum of two inertiogravity freemode frequencies is met. This inertialparametric instability has a welldefined inviscid vertical scale selection favouring the highvertical mode m(c) similar to 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 inertialparametric 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 Rossbygravity (MRG) wave. Initialvalue numerical simulations show that the same inertialpara metric instability exists leading to a resonant subharmonic excitation of free modes with vertical scales 7 and 8 times smaller than the basicstate 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 Fronde 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 extraequatorial 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. 
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