Critical layer and radiative instabilities in shallow-water shear flows

Riedinger, Xavier; Gilbert, Andrew D.

doi:10.1017/jfm.2014.303

Cited by 6 publications

(20 citation statements)

References 39 publications

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“…The energy of normal modes can concentrate within the critical level, as further described in Nguyen et al (2012) or Riedinger & Gilbert (2014). In our study, we show that the presence of the critical level for mode 2 induces an intense stretching of the flow S. The latter is defined as the sum of the strain and the shear:…”

Section: Critical Level Computationsupporting

confidence: 60%

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Study of the stability of a large realistic cyclonic eddy

et al. 2020

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We investigate the stability of a composite cyclone representative of Arabian Sea eddies using a high resolution primitive equation model. We observe that the eddy is unstable with respect to a mixed barotropic/baroclinic instability, leading to the growth of an azimuthal mode 2 perturbation. The latter deforms the eddy, which eventually evolves into a tripole after about 4 months of simulation. The presence of a critical level for the most unstable mode generates sharp fronts in the surface mixed layer where the Rossby number is large. These fronts then become unstable, and this generates submesoscale cyclones and filaments. Near these fronts, diapycnal mixing occurs, causing the potential vorticity to change sign locally, and symmetric instability to develop in the core of the cyclonic eddy. Despite the instabilities, the eddy is not destroyed and remains a large-scale coherent structure for the last 6 months of the simulation. Looking at Sea Surface Height only, the composite eddy evolves little, and fairly represents the eddy observed in the altimetry which can live for several months. The study of this simulation thus illustrates the numerous kinds of instabilities which may occur in large cyclonic eddies but can not be captured directly by altimetric data. Highlights ► A composite cyclone representative of the Arabian Sea eddies is unstable with respect to a mixed barotropic/baroclinic instability. ► Submesoscale features are generated at the surface via barotropic, baroclinic, and symmetric instabilities. ► Currently available altimetric and floats data cannot capture the instability occurring in the eddy.

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Section: Critical Level Computationsupporting

confidence: 60%

“…At the beginning of the simulation, transient perturbations occur in the core of the eddy, mostly stemming from the initial adjustment of the eddy. They are evanescent and rapidly decay (Riedinger & Gilbert, 2014). However, during the first 20 days, their energy has similar magnitude to that of the unstable mode 2.…”

Section: Journal Pre-proofmentioning

confidence: 94%

Study of the stability of a large realistic cyclonic eddy

et al. 2020

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“…This can be understood easily for the hydrodynamic case : determines the strength of the singularity at , and the singularity becomes removable if is absent. Similar arguments have also been given by Riedinger & Gilbert (2014).…”

Section: The Asymptotic Analysissupporting

confidence: 86%

Critical-layer instability of shallow-water magnetohydrodynamic shear flows

2022

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In this paper, the instability of shallow-water shear flow with a sheared parallel magnetic field is studied. Waves propagating in such magnetic shear flows encounter critical levels where the phase velocity relative to the basic flow, $c-U(y)$ , matches the Alfvén wave velocities $\pm B(y)/\sqrt {\mu \rho }$ , based on the local magnetic field $B(y)$ , the magnetic permeability $\mu$ , and the mass density of the fluid $\rho$ . It is shown that when the two critical levels are close to each other, the critical layer can generate an instability. The instability problem is solved, combining asymptotic solutions at large wavenumbers and numerical solutions, and the mechanism of instability explained using the conservation of momentum. For the shallow-water magnetohydrodynamic system, the paper gives the general form of the local differential equation governing such coalescing critical layers for any generic field and flow profiles, and determines precisely how the magnetic field modifies the purely hydrodynamic stability criterion based on the potential vorticity gradient in the critical layer. The curvature of the magnetic field profile, or equivalently the electric current gradient $J' = - B''/\mu$ in the critical layer, is found to play a complementary role in the instability.

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“…Candelier, Le Dizès & Millet (2012) showed that an inflection-free boundary layer profile becomes unstable with respect to an inviscid 'radiative instability' as soon as there is an angle between the directions of shear and stratification, the instability being the strongest for an angle of π/2, that is for a vertical wall. This instability, which results from the coupling between shear and internal waves, has been obtained in other contexts: shallow water flows (Satomura 1981;Balmforth 1999;Riedinger & Gilbert 2014), compressible flows (Mack 1990;Parras & Le Dizès 2010) and rotating flows (Riedinger, Le Dizès & Meunier 2010. It has often been associated with a phenomenon of resonant over reflection (McIntyre & Weissman 1978;Grimshaw 1979;Lindzen & Barker 1985), negative energy waves (Kopev & Leontev 1983) or spontaneous wave emission (Plougonven & Zeitlin 2002;Le Dizès & Billant 2009).…”

mentioning

confidence: 79%

Instability of a boundary layer flow on a vertical wall in a stably stratified fluid

Bai

Dizès

2016

J. Fluid Mech.

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The stability of a horizontal boundary layer flow on a vertical wall in a viscous stably stratified fluid is considered in this work. A temporal stability analysis is performed for a tanh velocity profile as a function of the Reynolds number Re = UL/ν and the Froude number F = U/(LN) where U is the main stream velocity, L the boundary layer thickness, N the buoyancy frequency and ν the kinematic viscosity. The diffusion of density is neglected. The boundary layer flow is found to be unstable with respect to two instabilities. The first one is the classical viscous instability which gives rise to Tollmien-Schlichting (TS) waves. We demonstrate that, even in the presence of stratification, the most unstable TS wave remains two-dimensional and therefore independent of the Froude number. The other instability is three-dimensional, inviscid in nature and associated with the stratification. It corresponds to the so-called radiative instability. We show that this instability appears first for Re Re (r) c ≈ 1995 for a Froude number close to 1.5 whereas the viscous instability develops for Re Re (v) c ≈ 3980. For large Reynolds numbers, the radiative instability is also shown to exhibit a much larger growth rate than the viscous instability in a large Froude number interval. We argue that this instability could develop in experimental facilities as well as in geophysical situations encountered in ocean and atmosphere.

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Critical layer and radiative instabilities in shallow-water shear flows

Cited by 6 publications

References 39 publications

Study of the stability of a large realistic cyclonic eddy

Study of the stability of a large realistic cyclonic eddy

Critical-layer instability of shallow-water magnetohydrodynamic shear flows

Instability of a boundary layer flow on a vertical wall in a stably stratified fluid

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