Nonlinear effects in two-layer large-amplitude geostrophic dynamics. Part 2. The weak-beta case

2000 ◽  
Vol 412 ◽  
pp. 161-196 ◽  
Author(s):  
RICHARD H. KARSTEN ◽  
GORDON E. SWATERS
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Stability ◽  
Large Amplitude ◽  
Baroclinic Instability ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Steady Flows ◽  
Leading Order ◽  
Unstable Flow ◽  
Amplitude Analysis

This paper is a continuation of our study on nonlinear processes in large-amplitude geostrophic (LAG) dynamics. Here, we examine the so-called weak-β models. These models arise when the intrinsic length scale is large enough so that the dynamics is geostrophic to leading order but not so large that the β-effect enters into the dynamics at leading order (but remains, nevertheless, dynamically non-negligible). In contrast to our previous analysis of strong-β LAG models in Part 1, we show that the weak-β models allow for vigorous linear baroclinic instability.For two-layer weak-β LAG models in which the mean depths of both layers are approximately equal, the linear instability problem can exhibit an ultraviolet catastrophe. We argue that it is not possible to establish conditions for the nonlinear stability in the sense of Liapunov for a steady flow. We also show that the finite-amplitude evolution of a marginally unstable flow possesses explosively unstable modes, i.e. modes for which the amplitude becomes unbounded in finite time. Numerical simulations suggest that the development of large-amplitude meanders, squirts and eddies is correlated with the presence of these explosively unstable modes.For two-layer weak-β LAG models in which one of the two layers is substantially thinner than the other, the linear stability problem does not exhibit an ultraviolet catastrophe and it is possible to establish conditions for the nonlinear stability in the sense of Liapunov for steady flows. A finite-amplitude analysis for a marginally unstable flow suggests that nonlinearities act to stabilize eastward and enhance the instability of westward flows. Numerical simulations are presented to illustrate these processes.

Finite-amplitude internal wavepacket dispersion and breaking

2001 ◽  
Vol 429 ◽  
pp. 343-380 ◽  
Author(s):  
BRUCE R. SUTHERLAND
Keyword(s):  
Numerical Simulations ◽  
Internal Waves ◽  
Large Amplitude ◽  
Finite Amplitude ◽  
Critical Amplitude ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
The Stability ◽  
The Waves ◽  
Horizontal Wavelength

The evolution and stability of two-dimensional, large-amplitude, non-hydrostatic internal wavepackets are examined analytically and by numerical simulations. The weakly nonlinear dispersion relation for horizontally periodic, vertically compact internal waves is derived and the results are applied to assess the stability of weakly nonlinear wavepackets to vertical modulations. In terms of Θ, the angle that lines of constant phase make with the vertical, the wavepackets are predicted to be unstable if [mid ]Θ[mid ] < Θc, where Θc = cos−1 (2/3)1/2 ≃ 35.3° is the angle corresponding to internal waves with the fastest vertical group velocity. Fully nonlinear numerical simulations of finite-amplitude wavepackets confirm this prediction: the amplitude of wavepackets with [mid ]Θ[mid ] > Θc decreases over time; the amplitude of wavepackets with [mid ]Θ[mid ] < Θc increases initially, but then decreases as the wavepacket subdivides into a wave train, following the well-known Fermi–Pasta–Ulam recurrence phenomenon.If the initial wavepacket is of sufficiently large amplitude, it becomes unstable in the sense that eventually it convectively overturns. Two new analytic conditions for the stability of quasi-plane large-amplitude internal waves are proposed. These are qualitatively and quantitatively different from the parametric instability of plane periodic internal waves. The ‘breaking condition’ requires not only that the wave is statically unstable but that the convective instability growth rate is greater than the frequency of the waves. The critical amplitude for breaking to occur is found to be ACV = cot Θ (1 + cos2 Θ)/2π, where ACV is the ratio of the maximum vertical displacement of the wave to its horizontal wavelength. A second instability condition proposes that a statically stable wavepacket may evolve so that it becomes convectively unstable due to resonant interactions between the waves and the wave-induced mean flow. This hypothesis is based on the assumption that the resonant long wave–short wave interaction, which Grimshaw (1977) has shown amplifies the waves linearly in time, continues to amplify the waves in the fully nonlinear regime. Using linear theory estimates, the critical amplitude for instability is ASA = sin 2Θ/(8π2)1/2. The results of numerical simulations of horizontally periodic, vertically compact wavepackets show excellent agreement with this latter stability condition. However, for wavepackets with horizontal extent comparable with the horizontal wavelength, the wavepacket is found to be stable at larger amplitudes than predicted if Θ [lsim ] 45°. It is proposed that these results may explain why internal waves generated by turbulence in laboratory experiments are often observed to be excited within a narrow frequency band corresponding to Θ less than approximately 45°.

Nonlinear effects in two-layer large-amplitude geostrophic dynamics. Part 1. The strong-beta case

2000 ◽  
Vol 412 ◽  
pp. 125-160 ◽  
Author(s):  
RICHARD H. KARSTEN ◽  
GORDON E. SWATERS
Keyword(s):  
Large Amplitude ◽  
Nonlinear Effects ◽  
Breaking Waves ◽  
Short Wave ◽  
Finite Amplitude ◽  
Length Scale ◽  
Leading Order ◽  
Length Scales ◽  
Geostrophic Balance ◽  
Suitable Framework

Baroclinic large-amplitude geostrophic (LAG) models, which assume a leading-order geostrophic balance but allow for large-amplitude isopycnal deflections, provide a suitable framework to model the large-amplitude motions exhibited in frontal regions. The qualitative dynamical characterization of LAG models depends critically on the underlying length scale. If the length scale is sufficiently large, the effect of differential rotation, i.e. the β-effect, enters the dynamics at leading order. For smaller length scales, the β-effect, while non-negligible, does not enter the dynamics at leading order. These two dynamical limits are referred to as strong-β and weak-β models, respectively.A comprehensive description of the nonlinear dynamics associated with the strong- β models is given. In addition to establishing two new nonlinear stability theorems, we extend previous linear stability analyses to account for the finite-amplitude development of perturbed fronts. We determine whether the linear solutions are subject to nonlinear secondary instabilities and, in particular, a new long-wave–short-wave (LWSW) resonance, which is a possible source of rapid unstable growth at long length scales, is identified. The theoretical analyses are tested against numerical simulations. The simulations confirm the importance of the LWSW resonance in the development of the flow. Simulations show that instabilities associated with vanishing potential- vorticity gradients can develop into stable meanders, eddies or breaking waves. By examining models with different layer depths, we reveal how the dynamics associated with strong-β models qualitatively changes as the strength of the dynamic coupling between the barotropic and baroclinic motions varies.

scholarly journals On the Nonlinear Stability of Inviscid Homogeneous Shear Flows in Sea Straits of Arbitrary Cross Sections

2012 ◽  
Vol 11 (3) ◽  
pp. 31-50
Author(s):  
V Ramakrishnareddy ◽  
M Subbiah
Keyword(s):  
Nonlinear Stability ◽  
Cross Sections ◽  
Shear Flows ◽  
Stability Result ◽  
Finite Amplitude ◽  
Steady Flows ◽  
Stability Theorems ◽  
Sea Straits ◽  
Parallel Shear Flows ◽  
Special Case

In this paper we study the nonlinear stability of steady flows of inviscid homogeneous fluids in sea straits of arbitrary cross sections. We use the method of Arnol'd [1] to obtain two general stability theorems for steady basic flows with respect to finite amplitude disturbances. For the special case of plane parallel shear flows we find a finite amplitude extension of the linear stability result of Deng et al [2]. We also present some examples of basic flows which are stable to finite amplitude disturbances.

scholarly journals On the baroclinic instability of axisymmetric rotating gravity currents with bottom slope

2000 ◽  
Vol 408 ◽  
pp. 149-177 ◽  
Author(s):  
PAUL F. CHOBOTER ◽  
GORDON E. SWATERS
Keyword(s):  
Numerical Simulations ◽  
Linear Stability ◽  
Lower Layer ◽  
Baroclinic Instability ◽  
Laboratory Data ◽  
Gravity Current ◽  
Gravity Currents ◽  
Finite Amplitude ◽  
Rotating System ◽  
A Current

The baroclinic stability characteristics of axisymmetric gravity currents in a rotating system with a sloping bottom are determined. Laboratory studies have shown that a relatively dense fluid released under an ambient fluid in a rotating system will quickly respond to Coriolis effects and settle to a state of geostrophic balance. Here we employ a subinertial two-layer model derived from the shallow-water equations to study the stability characteristics of such a current after the stage at which geostrophy is attained. In the model, the dynamics of the lower layer are geostrophic to leading order, but not quasi-geostrophic, since the height deflections of that layer are not small with respect to its scale height. The upper-layer dynamics are quasi-geostrophic, with the Eulerian velocity field principally driven by baroclinic stretching and a background topographic vorticity gradient.Necessary conditions for instability, a semicircle-like theorem for unstable modes, bounds on the growth rate and phase velocity, and a sufficient condition for the existence of a high-wavenumber cutoff are presented. The linear stability equations are solved exactly for the case where the gravity current initially corresponds to an annulus flow with parabolic height profile with two incroppings, i.e. a coupled front. The dispersion relation for such a current is solved numerically, and the characteristics of the unstable modes are described. A distinguishing feature of the spatial structure of the perturbations is that the perturbations to the downslope incropping are preferentially amplified compared to the upslope incropping. Predictions of the model are compared with recent laboratory data, and good agreement is seen in the parameter regime for which the model is valid. Direct numerical simulations of the full model are employed to investigate the nonlinear regime. In the initial stage, the numerical simulations agree closely with the linear stability characteristics. As the instability develops into the finite-amplitude regime, the perturbations to the downslope incropping continue to preferentially amplify and eventually evolve into downslope propagating plumes. These finally reach the deepest part of the topography, at which point no more potential energy can be released.

scholarly journals Baroclinic instability of a symmetric, rotating, stratified flow: a study of the nonlinear stabilisation mechanisms in the presence of viscosity

2002 ◽  
Vol 9 (5/6) ◽  
pp. 487-496 ◽  
Author(s):  
R. Mantovani ◽  
A. Speranza
Keyword(s):  
Parameter Space ◽  
Baroclinic Instability ◽  
Vertical Wall ◽  
Time Integration ◽  
Stable State ◽  
Physical Nature ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Vertical Shear ◽  
Direct Role

Abstract. This paper presents the analysis of symmetric circulations of a rotating baroclinic flow, forced by a steady thermal wind and dissipated by Laplacian friction. The analysis is performed with numerical time-integration. Symmetric flows, vertically bound by horizontal walls and subject to either periodic or vertical wall lateral boundary conditions, are investigated in the region of parameter-space where unstable small amplitude modes evolve into stable stationary nonlinear solutions. The distribution of solutions in parameter-space is analysed up to the threshold of chaotic behaviour and the physical nature of the nonlinear interaction operating on the finite amplitude unstable modes is investigated. In particular, analysis of time-dependent energy-conversions allows understanding of the physical mechanisms operating from the initial phase of linear instability to the finite amplitude stable state. Vertical shear of the basic flow is shown to play a direct role in injecting energy into symmetric flow since the stage of linear growth. Dissipation proves essential not only in limiting the energy of linearly unstable modes, but also in selecting their dominant space-scales in the finite amplitude stage.

scholarly journals Optimal Stability Thresholds in Rotating Fully Anisotropic Porous Medium with LTNE

2021 ◽  
Author(s):  
Florinda Capone ◽  
Maurizio Gentile ◽  
Jacopo A. Gianfrani
Keyword(s):  
Porous Layer ◽  
Nonlinear Stability ◽  
Steady Motion ◽  
Vertical Axis ◽  
Linear Instability ◽  
List Type ◽  
Nonlinear Stability Analysis ◽  
Onset Of Convection ◽  
Anisotropic Porous Medium ◽  
Necessary And Sufficient

Abstract The onset of thermal convection in an anisotropic horizontal porous layer heated from below and rotating about vertical axis, under local thermal non-equilibrium hypothesis is studied. Linear and nonlinear stability analysis of the conduction solution is performed. Coincidence between the linear instability and the global nonlinear stability thresholds with respect to the L2—norm is proved. Article Highlights A necessary and sufficient condition for the onset of convection in a rotating anisotropic porous layer has been obtained. It has been proved that convection can occur only through a steady motion. A detailed proof is reported thoroughly. Numerical analysis shows that permeability promotes convection, while thermal conductivities and rotation stabilize conduction.

Parameterization of Mixed Layer Eddies. Part I: Theory and Diagnosis

2008 ◽  
Vol 38 (6) ◽  
pp. 1145-1165 ◽  
Author(s):  
Baylor Fox-Kemper ◽  
Raffaele Ferrari ◽  
Robert Hallberg
Keyword(s):  
Mixed Layer ◽  
Mixed Layer Depth ◽  
Baroclinic Instability ◽  
Finite Amplitude ◽  
Climate Impacts ◽  
Surface Mixed Layer ◽  
Rotation Rates ◽  
Wide Range ◽  
Horizontal Density Gradient ◽  
Mixed Layers

Abstract Ageostrophic baroclinic instabilities develop within the surface mixed layer of the ocean at horizontal fronts and efficiently restratify the upper ocean. In this paper a parameterization for the restratification driven by finite-amplitude baroclinic instabilities of the mixed layer is proposed in terms of an overturning streamfunction that tilts isopycnals from the vertical to the horizontal. The streamfunction is proportional to the product of the horizontal density gradient, the mixed layer depth squared, and the inertial period. Hence restratification proceeds faster at strong fronts in deep mixed layers with a weak latitude dependence. In this paper the parameterization is theoretically motivated, confirmed to perform well for a wide range of mixed layer depths, rotation rates, and vertical and horizontal stratifications. It is shown to be superior to alternative extant parameterizations of baroclinic instability for the problem of mixed layer restratification. Two companion papers discuss the numerical implementation and the climate impacts of this parameterization.

Solitary Filaments of Coupled Electromagnetic Wave and Finite Amplitude Ion Fluctuations

1976 ◽  
Vol 31 (12) ◽  
pp. 1517-1519 ◽  
Author(s):  
P. K. Shukla ◽  
M. Y. Yu ◽  
S. G. Tagare
Keyword(s):  
Electromagnetic Wave ◽  
Electromagnetic Radiation ◽  
Plasma Density ◽  
Large Amplitude ◽  
Small Amplitude ◽  
Finite Amplitude ◽  
Nonlinear Coupling ◽  
Incoming Radiation

Abstract We show analytically that the nonlinear coupling of a large amplitude electromagnetic wave with finite amplitude ion fluctuations leads to filamentation. The latter consists of striations of the electromagnetic radiation trapped in depressions of the plasma density. The filamentation is found to be either standing or moving normal to the direction of the incoming radiation. Criteria for the existence of localized filaments are obtained. Small amplitude results are discussed.

scholarly journals Finite-amplitude steady waves in plane viscous shear flows

1985 ◽  
Vol 160 ◽  
pp. 281-295 ◽  
Author(s):  
F. A. Milinazzo ◽  
P. G. Saffman
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Navier Stokes ◽  
Unidirectional Flow ◽  
Viscous Shear ◽  
Finite Amplitude Waves

Computations of two-dimensional solutions of the Navier–Stokes equations are carried out for finite-amplitude waves on steady unidirectional flow. Several cases are considered. The numerical method employs pseudospectral techniques in the streamwise direction and finite differences on a stretched grid in the transverse direction, with matching to asymptotic solutions when unbounded. Earlier results for Poiseuille flow in a channel are re-obtained, except that attention is drawn to the dependence of the minimum Reynolds number on the physical constraint of constant flux or constant pressure gradient. Attempts to calculate waves in Couette flow by continuation in the velocity of a channel wall fail. The asymptotic suction boundary layer is shown to possess finite-amplitude waves at Reynolds numbers orders of magnitude less than the critical Reynolds number for linear instability. Waves in the Blasius boundary layer and unsteady Rayleigh profile are calculated by employing the artifice of adding a body force to cancel the spatial or temporal growth. The results are verified by comparison with perturbation analysis in the vicinity of the linear-instability critical Reynolds numbers.

The Linear and Weakly-Nonlinear Stability of a Core Annular Flow in a Corrugated Tube

2000 ◽  
Author(s):  
Hsien-Hung Wei ◽  
David S. Rumschitzki
Keyword(s):  
Thin Film ◽  
Nonlinear Stability ◽  
Annular Flow ◽  
Linear Instability ◽  
Base Flow ◽  
Nonlinear Regime ◽  
Weakly Nonlinear ◽  
Corrugated Tube

Abstract Both linear and weakly nonlinear stability of a core annular flow in a corrugated tube in the limit of thin film and small corrugation are examined. Asymptotic techniques are used to derive the corrugated base flow and corresponding linear and weakly nonlinear stability equations. Interesting features show that the corrugation interaction can excite linear instability, but the nonlinearity still can suppress such instability in the weakly nonlinear regime.

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