The instability of an axisymmetric vortex with monotonic potential vorticity in rotating shallow water

Ford, Rupert

doi:10.1017/s0022112094002946

Cited by 82 publications

(112 citation statements)

References 13 publications

(11 reference statements)

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“…This approximation is shown in figure 3(a) (dot) and shows good agreement with increasing k. These limit modes have a peak localised close to y = 1 for large k, as illustrated in figure 4 (solid), and so can be thought of as driven by the jump in potential vorticity Q at y = 1, much like a normal mode on a Rankine vortex in the analogous problem in plane polar geometry (Ford, 1994).…”

Section: Asymptotic Theory For Limit Modesmentioning

confidence: 60%

“…Recent work by Yim & Billant (2013) shows that a bending, non-radiative instability can also exist for an isolated vortex in a stratified anticyclonic fluid, and that this instability is due to a critical layer. Finally, it would be interesting to investigate the existence of critical layer instablility for other types of flows, in particular for coherent vortices in shallow water, extending the study of Ford (1994) to smooth profiles with critical layers.…”

Section: Resultsmentioning

confidence: 99%

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Critical layer and radiative instabilities in shallow-water shear flows

Riedinger

Gilbert

2014

J. Fluid Mech.

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In this study a linear stability analysis of shallow water flows is undertaken for a representative Froude number F = 3.5. The focus is on monotonic base flow profiles U without an inflexion point, in order to study critical layer instability (CLI) and its interaction with radiative instability (RI). First the dispersion relation is presented for the piecewise linear profile studied numerically by Satomura (1981) and using WKBJ analysis an interpretation given of mode branches, resonances and radiative instability. In particular surface gravity waves can resonate with a limit mode (or Rayleigh wave), localised near the discontinuity in shear in the flow; in this piecewise profile there is no critical layer. The piecewise linear profile is then continuously modified in a family of nonlinear profiles, to show the effect of the vorticity gradient Q = −U on the nature of the modes. Some modes remain as modes and others turn into quasi-modes, linked to Landau damping of disturbances to the flow, depending on the sign of the vorticity gradient at the critical point. Thus an interpretation of CLI for continuous profiles is given, as the remnant of the resonance with the limit mode. Numerical results and WKBJ analysis of CLI and RI for more general smooth profiles are provided. A link is made between growth rate formulae obtained by considering wave momentum and those found via the WKBJ approximation. Finally the competition between the stabilising effect of vorticity gradients in a critical layer and the destabilising effect of radiation (RI) is studied.

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Section: Asymptotic Theory For Limit Modesmentioning

confidence: 60%

Section: Resultsmentioning

confidence: 99%

Critical layer and radiative instabilities in shallow-water shear flows

Riedinger

Gilbert

2014

J. Fluid Mech.

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“…Note that a similar reasoning applied for the shallow-water system leads to the conclusion that phase locking is impossible at small , so that unbalanced instabilities require finite (see Ford (1994) and Dritschel & Vanneste (2006)). Note also that the condition means that the instability processes are restricted to the models governed by PDEs, since ODE models typically involve motion at fixed spatial scales which are independent of the Rossby number.…”

Section: Unbalanced Instabilitiesmentioning

confidence: 86%

“…It is probably worth emphasising the difference between the small-Rossby-number regime considered here, and the small-Froude number regime treated in great detail by Ford (1994) and Ford et al (2000) in the context of the shallow-water model (see Saujani & Shepherd 2002, Ford et al 2002 for a discussion). In the latter regime, the spectrum of the linear operator in (2.3) is not bounded from below by a large parameter.…”

Section: Lighthill Radiationmentioning

confidence: 96%

Exponential Smallness of Inertia–Gravity Wave Generation at Small Rossby Number

Vanneste

2008

Journal of the Atmospheric Sciences

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This paper discusses some of the mechanisms whereby fast inertia-gravity waves can be generated spontaneously by slow, balanced atmospheric and oceanic flows. In the small-Rossby-number regime relevant to mid-latitude dynamics, high-accuracy balanced models, which filter out inertia-gravity waves completely, can in principle describe the evolution of suitably initialised flows up to terms that are exponentially small in the Rossby number , i.e, of the form exp(−α/ ) for some α > 0.This suggests that the mechanisms of inertia-gravity-wave generation, which are not captured by these balanced models, are also exponentially weak. This has been confirmed by explicit analytical results obtained for a few highly-simplified models. We review these results and present some of the exponential-asymptotic techniques that have been used in their derivation. We examine both spontaneous-generation mechanisms which generate exponentially small waves from perfectly balanced initial conditions, and unbalanced instability mechanisms which amplify unbalanced initial perturbations of steady flows. The relevance of the results to realistic flows is discussed.2

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“…By setting the determinant of B equal to zero, two cubic equations can be obtained that can be solved for the real and imaginary portions (ν r and ν i ) of the complex Doppler-shifted frequency ν = ν r + iν i −ωm. The real portion encapsulates lower frequency vortex Rossby waves, higher frequency inertia-gravity waves, and mixed inertia-gravity-vortex Rossby waves (with inseparable dispersion characteristics) and when combined with a nonzero imaginary portion, exponentially growing and decaying modes exist -the former of which is the Rossby-inertia gravity wave instability (Ford, 1994;Schecter & Montgomery, 2004;Hodyss & Nolan, 2008;. It is not our intent in here to describe this more complex dispersion relationship, but rather to isolate the primary modes of variability on shallow water vortices by making certain simplifying assumptions on (13)-(15), foremost of which is neglecting the unstable modes.…”

Section: The Decomposition For All Variables Is As Follows: H(r)=h(r)mentioning

confidence: 99%

Barotropic Aspects of Hurricane Structural and Intensity Variability

Hendricks¹,

Kuo²,

Peng³

et al. 2011

Recent Hurricane Research - Climate, Dynamics, and Societal Impacts

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and Intensity Variability 3 Diagnostic balance equationBy setting ∂δ/∂t = 0 in (4), and neglecting terms involving the velocity potential χ,t h e relationship between ψ and h is obtained. This is the nonlinear balance equation in polar coordinates. Equation (9) indicates that given ψ or h with appropriate boundary conditions, the other field which is in quasi-balance with that field may be obtained. Since P =(∇ 2 ψ + f )/h, equations (8) and (9) are sufficient to describe the quasi-balanced evolution of a shallow water vortex. By substituting h =( ∇ 2 ψ + f )/P into the right hand side of (9), an invertibility relationship is obtained, where ψ, h, ζ, u and v may be obtained diagnostically, given P. This invertibility principle is the reason P is such a useful dynamical quantity. The shallow water dynamics can be succintly speficied by one prognostic PV equation and a diagnostic relationship for the other quasi-balanced fields. For the special case of axisymmetric flow in which v = ∂ψ/∂r and u = −∂ψ/r∂φ = 0, equation (9) reduces toimplying that, when enforcing the boundary conditions,which is the gradient wind balance equation. In this context, (9) is a more general formulation of the gradient wind balance equation when the flow is both asymmetric and balanced. LinearizationEquations (3)-(5) cannot be solved analytically due to the nonlinear terms. However, under the assumption that perturbations are small in comparison to their means, a modified set of equations can be obtained that can be solved analytically. This process is referred to as linearization. In order to linearize (3)-(5), a basic state chosen here is that of an arbitrary axisymmetric vortexω(r)=v(r)/r that is in gradient balance with the heighth(r). While our basic state vortex is prescribed to be axisymmetric, the basic state could similarly be obtained by taking an azimuthal mean, i.e.,vand similar definitions hold for the other variables. ′ (r, φ, t), χ(r, φ, t)=χ ′ (r, φ, t),andψ(r, φ, t)=ψ ′ (r, φ, t). The perturbations u ′ , v ′ , ζ ′ ,andδ ′ are related to ψ ′ and χ ′ as above. Upon substituting the decomposed variables into (3)-(5), enforcing the gradient balance constraint, and neglecting higher order terms involving multiplication of perturbation quantities, we obtain the linearized versions of (3)-(5) which govern small amplitude disturbances about the basic state vortex, The decomposition for all variables is as follows: h(r)=h(r)+h ′ (r, φ, t), v(r)=v(r)+v ′ (r, φ, t), u(r)=u ′ (r, φ, t), ζ(r)=ζ(r)+ ζ ′ (r, φ, t), δ(r)=δwhereη = f +ζ is the basic state absolute vertical vorticity. We wish to understand the nature and behavior of small disturbances (i.e., waves) on this basic state vortex. Using the Wentzel-Kramers-Brillouin (WKB) approximation, we assume normal mode solutions of the form ψ ′ (r, φ, t)=ψ exp[i(kr + mφ − νt)], χ ′ (r, φ, t)=χ exp[i(kr + mφ − νt)],a n d h ′ (r, φ, t)=ĥ exp[i(kr + mφ − νt)],w h e r ek is the radial wavenumber (i.e., the number of wave crests per unit radial distance), m is the azimuthal wave number (i.e.,...

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The instability of an axisymmetric vortex with monotonic potential vorticity in rotating shallow water

Cited by 82 publications

References 13 publications

Critical layer and radiative instabilities in shallow-water shear flows

Critical layer and radiative instabilities in shallow-water shear flows

Exponential Smallness of Inertia–Gravity Wave Generation at Small Rossby Number

Barotropic Aspects of Hurricane Structural and Intensity Variability

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