Finite-Amplitude Baroclinic Instability of Time-Varying Abyssal Currents

2006 ◽  
Vol 36 (1) ◽  
pp. 122-139 ◽  
Author(s):  
Seung-Ji Ha ◽  
Gordon E. Swaters
Keyword(s):  
Baroclinic Instability ◽  
Soliton Solutions ◽  
Zonal Flow ◽  
Finite Amplitude ◽  
Marginal Stability ◽  
Time Variability ◽  
Time Varying ◽  
Weakly Nonlinear ◽  
Abyssal Current ◽  
The Stability

Abstract The weakly nonlinear baroclinic instability characteristics of time-varying grounded abyssal flow on sloping topography with dissipation are described. Specifically, the finite-amplitude evolution of marginally unstable or stable abyssal flow both at and removed from the point of marginal stability (i.e., the minimum shear required for instability) is determined. The equations governing the evolution of time-varying dissipative abyssal flow not at the point of marginal stability are identical to those previously obtained for the Phillips model for zonal flow on a β plane. The stability problem at the point of marginally stability is fully nonlinear at leading order. A wave packet model is introduced to examine the role of dissipation and time variability in the background abyssal current. This model is a generalization of one introduced for the baroclinic instability of zonal flow on a β plane. A spectral decomposition and truncation leads, in the absence of time variability in the background flow and dissipation, to the sine–Gordon solitary wave equation that has grounded abyssal soliton solutions. The modulation characteristics of the soliton are determined when the underlying abyssal current is marginally stable or unstable and possesses time variability and/or dissipation. The theory is illustrated with examples.

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°.

Evolution of solitary marginal disturbances in baroclinic frontal geostrophic dynamics with dissipation and time-varying background flow

2007 ◽  
Vol 463 (2083) ◽  
pp. 1749-1769 ◽  
Author(s):  
Mattea R Turnbull ◽  
Gordon E Swaters
Keyword(s):  
Lower Layer ◽  
Baroclinic Instability ◽  
Variable Coefficients ◽  
Time Variability ◽  
Perturbation Approach ◽  
Dynamical Regime ◽  
Time Varying ◽  
Mean Flow ◽  
Background Flow ◽  
Frontal Geostrophic Dynamics

A two-layer frontal geostrophic flow corresponds to a dynamical regime that describes the low-frequency evolution of baroclinic ocean currents with large amplitude deflections of the interface between the layers on length-scales longer than the internal deformation radius within the context of a thin upper layer overlying a dynamically active lower layer. The finite-amplitude evolution of solitary disturbances in baroclinic frontal geostrophic dynamics in the presence of time-varying background flow and dissipation is shown to be governed by a two-equation extension of the unstable nonlinear Schrödinger (UNS) equation with variable coefficients and forcing. The soliton solution of the unperturbed UNS equation corresponds to a saturated isolated coherent anomaly in the baroclinic instability of surface-intensified oceanographic fronts and currents. The adiabatic evolution of the propagating soliton and the uniformly valid first-order perturbation fields are determined using a direct perturbation approach together with phase-averaged conservation relations when both dissipation and time variability are present. It is shown that the soliton amplitude parameter decays exponentially due to the presence of the dissipation but is unaffected by the time variability in the background flow. On the other hand, the soliton translation velocity is unaffected by the dissipation and evolves only in response to the time variability in the background flow. The adiabatic solution for the induced mean flow exhibits a dissipation-generated ‘shelf region’ in the far field behind the soliton, which is removed by solving the initial-value problem.

On the dynamics of finite-amplitude baroclinic waves as a function of supercriticality

1976 ◽  
Vol 78 (3) ◽  
pp. 621-637 ◽  
Author(s):  
Joseph Pedlosky
Keyword(s):  
Steady State ◽  
Baroclinic Instability ◽  
Finite Amplitude ◽  
Steady State Mode ◽  
Final State ◽  
Nonlinear Modes ◽  
Stream Structure ◽  
The Stability ◽  
Amplitude Mode ◽  
Stream Mode

A finite-amplitude model of baroclinic instability is studied in the case where the cross-stream scale is large compared with the Rossby deformation radius and the dissipative and advective time scales are of the same order. A theory is developed that describes the nature of the wave field as the shear supercriticality increases beyond the stability threshold of the most unstable cross-stream mode and penetrates regions of higher supercriticality. The set of possible steady nonlinear modes is found analytically. It is shown that the steady cross-stream structure of each finite-amplitude mode is a function of the supercriticality.Integrations of initial-value problems show, in each case, that the final state realized is the state characterized by the finite-amplitude mode with the largest equilibrium amplitude. The approach to this steady state is oscillatory (nonmonotonic). Further, each steady-state mode is a well-defined mixture of linear cross-stream modes.

Amplitude equations at the critical points of unstable dispersive physical systems

1981 ◽  
Vol 377 (1769) ◽  
pp. 185-219 ◽  
Keyword(s):  
Critical Point ◽  
Pulse Propagation ◽  
Baroclinic Instability ◽  
Soliton Solutions ◽  
Complex Conjugate ◽  
Amplitude Equations ◽  
Weakly Nonlinear ◽  
Physical Systems ◽  
Scattering Transform ◽  
Induced Transparency

The amplitude equations that govern the motion of wavetrains near the critical point of unstable dispersive, weakly nonlinear physical systems are considered on slow time and space scales T m ═ ε m t ; X m ═ ε m x ( m ═ 1, 2,...). Such systems arise when the dispersion relation for the harmonic wavetrain is purely real and complex conjugate roots appear when a control parameter ( μ ) is varied. At the critical point, when the critical wavevector k c is non-zero, a general result for this general class of unstable systems is that the typical amplitude equations are either of the form ( ∂/∂ T 1 + c 1 ∂/∂ X 1 ) (∂/∂ T 1 + c 2 ∂/∂ X 1 ) A ═ ±α A ─ β AB , ( ∂/∂ T 1 + c 2 ∂/∂ X 1 ) B ═ (∂/∂ T 1 + c 1 ∂/∂ X 1 ) | A | 2 , or of the form ( ∂/∂ T 1 + c 1 ∂/∂ X 1 ) (∂/∂ T 1 + c 2 ∂/∂ X 1 ) A ═ ±α A - β A | A | 2 . The equations with the AB -nonlinearity govern for example the two-layer model for baroclinic instability and self-induced transparency (s. i. t.) in ultra-short optical pulse propagation in laser physics. The second equation occurs for the two-layer Kelvin-Helmholtz instability and a problem in the buckling of elastic shells. This second type of equation has been considered in detail by Weissman. The AB -equations are particularly important in that they are integrable by the inverse scattering transform and have a variety of multi-soliton solutions. They are also reducible to the sine-Gordon equation ϕ ξƬ ═ ± sin ϕ when A is real. We prove some general results for this type of instability and discuss briefly their applications to various other examples such as the two-stream instability. Examples in which dissipation is the dominant mechanism of the instability are also briefly considered. In contrast to the dispersive type which operates on the T 1 -time scale, this type operates on the T 2 -scale.

scholarly journals Mega riverbed-patterns: linear and weakly nonlinear perspectives

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210331
Author(s):  
Sk Zeeshan Ali ◽  
Subhasish Dey ◽  
Rajesh K. Mahato
Keyword(s):  
Pattern Formation ◽  
Transport Model ◽  
Landau Equation ◽  
Linear Perspective ◽  
Marginal Stability ◽  
Mathematical Framework ◽  
Weakly Nonlinear ◽  
Sediment Size ◽  
Channel Aspect Ratio ◽  
The Stability

In this paper, we explore the mega riverbed-patterns, whose longitudinal and vertical length dimensions scale with a few channel widths and the flow depth, respectively. We perform the stability analyses from both linear and weakly nonlinear perspectives by considering a steady-uniform flow in an erodible straight channel comprising a uniform sediment size. The mathematical framework stands on the dynamic coupling between the depth-averaged flow model and the particle transport model including both bedload and suspended load via the Exner equation, which drives the pattern formation. From the linear perspective, we employ the standard linearization technique by superimposing the periodic perturbations on the undisturbed system to find the dispersion relationship. From the weakly nonlinear perspective, we apply the centre–manifold-projection technique, where the fast dynamics of stable modes is projected on the slow dynamics of weakly unstable modes to obtain the Stuart–Landau equation for the amplitude dynamics. We examine the marginal stability, growth rate and amplitude of patterns for a given quintet formed by the channel aspect ratio, wavenumber of patterns, shear Reynolds number, Shields number and relative roughness number. This study highlights the sensitivity of pattern formation to the key parameters and shows how the classical results can be reconstructed on the parameter space.

The Meridional Flow of Source-Driven Abyssal Currents in a Stratified Basin with Topography. Part I: Model Development and Dynamical Properties

2006 ◽  
Vol 36 (3) ◽  
pp. 335-355 ◽  
Author(s):  
Gordon E. Swaters
Keyword(s):  
Baroclinic Instability ◽  
Model Development ◽  
Finite Amplitude ◽  
Meridional Flow ◽  
Gravity Models ◽  
Western Boundary ◽  
Reduced Gravity ◽  
The Stability ◽  
Abyssal Currents ◽  
Variable Topography

Abstract The equatorward flow of source-driven grounded deep western boundary currents within a stratified basin with variable topography is examined. The model is the two-layer quasigeostrophic (QG) equations, describing the overlying ocean, coupled to the finite-amplitude planetary geostrophic (PG) equations, describing the abyssal layer, on a midlatitude β plane. The model retains subapproximations such as classical Stommel–Arons theory, the Nof abyssal dynamical balance, the so-called planetary shock wave balance (describing the finite-amplitude β-induced westward propagation of abyssal anomalies), and baroclinic instability. The abyssal height field can possess groundings. In the reduced gravity limit, a new nonlinear steady-state balance is identified that connects source-driven equatorward abyssal flow (as predicted by Stommel–Arons theory) and the inertial topographically steered deep flow described by Nof dynamics. This model is solved explicitly, and the meridional structure of the predicted grounded abyssal flow is described. In the fully baroclinic limit, a variational principle is established and is exploited to obtain general stability conditions for meridional abyssal flow over variable topography on a β plane. The baroclinic coupling of the PG abyssal layer with the QG overlying ocean eliminates the ultraviolet catastrophe known to occur in inviscid PG reduced gravity models. The baroclinic instability problem for a constant-velocity meridional abyssal current flowing over sloping topography with β present is solved and the stability characteristics are described.

Rolls versus squares in thermal convection of fluids with temperature-dependent viscosity

1987 ◽  
Vol 178 ◽  
pp. 491-506 ◽  
Author(s):  
D. R. Jenkins
Keyword(s):  
Thermal Convection ◽  
Fluid Layer ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Temperature Dependent Viscosity ◽  
Weakly Nonlinear ◽  
Expansion Procedure ◽  
The Stability ◽  
Dependent Viscosity

We consider finite-amplitude thermal convection, in a horizontal fluid layer. The viscosity of the fluid is dependent upon its temperature. Using a weakly nonlinear expansion procedure, we examine the stability of two-dimensional roll and three-dimensional square planforms, in order to determine which should be preferred in convection experiments. The analysis shows that the roll planform is preferred for low values of the ratio of the viscosities at the top and bottom boundaries, but the square planform is preferred for larger values of the ratio. At still larger values, subcritical convection is predicted. We also include the effects of boundaries having finite thermal conductivity, which enables favourable comparison to be made with experimental studies. A discrepancy between the present work and a previous study of this problem (Busse & Frick 1985) is discussed.

Investigation on the behaviors of the soliton solutions for a variable-coefficient generalized AB system in the geophysical flows

2017 ◽  
Vol 31 (28) ◽  
pp. 1750254 ◽  
Author(s):  
Xiao-Hang Jiang ◽  
Yi-Tian Gao ◽  
Xin-Yi Gao
Keyword(s):  
Variable Coefficient ◽  
Baroclinic Instability ◽  
Soliton Solutions ◽  
Geophysical Flows ◽  
Bilinear Forms ◽  
Bilinear Method ◽  
Small Perturbations ◽  
Asymptotic Reduction ◽  
The Stability ◽  
Hirota Bilinear

Under investigation in this paper is a variable-coefficient generalized AB system, which is used for modeling the baroclinic instability in the asymptotic reduction of certain classes of geophysical flows. Bilinear forms are obtained, and one-, two- and three-soliton solutions are derived via the Hirota bilinear method. Interaction and propagation of the solitons are discussed graphically. Numerical investigation on the stability of the solitons indicates that the solitons could resist the disturbance of small perturbations and propagate steadily.

scholarly journals Propagation regimes of interfacial solitary waves in a three-layer fluid

2015 ◽  
Vol 2 (1) ◽  
pp. 1-41
Author(s):  
O. E. Kurkina ◽  
A. A. Kurkin ◽  
E. A. Rouvinskaya ◽  
T. Soomere
Keyword(s):  
Internal Waves ◽  
Fluid Layer ◽  
Soliton Solutions ◽  
Boussinesq Approximation ◽  
Finite Amplitude ◽  
Stable Configuration ◽  
Weakly Nonlinear ◽  
Small Parameters ◽  
Specific Equation ◽  
Kdv Type Equations

Abstract. Long weakly nonlinear finite-amplitude internal waves in a fluid consisting of three inviscid immiscible layers of arbitrary thickness and constant densities (stable configuration, Boussinesq approximation) bounded by a horizontal rigid bottom from below and by a rigid lid at the surface are described up to the second order of perturbation theory in small parameters of nonlinearity and dispersion. First, a pair of alternatives of appropriate KdV-type equations with the coefficients depending on the parameters of the fluid (layer positions and thickness, density jumps) are derived for the displacements of both modes of internal waves and for each interface between the layers. These equations are integrable for a very limited set of coefficients and do not allow for proper description of several near-critical cases when certain coefficients vanish. A more specific equation allowing for a variety of solitonic solutions and capable of resolving most of near-critical situations is derived by means of the introduction of another small parameter that describes the properties of the medium and rescaling of the ratio of small parameters. This procedure leads to a pair of implicitly interrelated alternatives of Gardner equation (KdV-type equations with combined nonlinearity) for the two interfaces. We present a detailed analysis of the relationships for the solutions for the disturbances at both interfaces and various regimes of the appearance and propagation properties of soliton solutions to these equations depending on the combinations of the parameters of the fluid. It is shown both the quadratic and the cubic nonlinear terms vanish for several realistic configurations of such a fluid.

Meridional dynamics of grounded abyssal water masses on a sloping bottom in a mid-latitude -plane

2018 ◽  
Vol 841 ◽  
pp. 674-701 ◽  
Author(s):  
Gordon E. Swaters
Keyword(s):  
Baroclinic Instability ◽  
Volume Transport ◽  
Finite Amplitude ◽  
Meridional Flow ◽  
Water Masses ◽  
Time Dependent ◽  
Meridional Transport ◽  
Initial Current ◽  
Abyssal Current ◽  
Parallel Shear Flow

Observations, numerical simulations and theoretical scaling arguments suggest that in mid-latitudes, away from the source regions and the equator, the meridional transport of abyssal water masses along a continental slope corresponds to planetary geostrophic flows that are gravity- or density-driven and topographically steered. We investigate these dynamics using a nonlinear reduced-gravity model that can describe grounded abyssal meridional flow over sloping topography that crosses the planetary vorticity gradient. Exact nonlinear steady and time-dependent solutions are obtained. The general steady theory is illustrated for a non-parallel equatorward flow that possesses a single along-slope grounding along the upslope flank of the current (complementing previous work). Four specific nonlinear time-dependent solutions are described. Two initial-value problems are solved exactly. The first initial configuration corresponds to an equatorward abyssal flow that has no cross-slope shear in the along-slope velocity and possesses a single grounding along the upslope flank of the current. The nonlinear time-dependent evolution of this initial current into a non-parallel shear flow is described. The second initial condition corresponds to an isolated radially symmetric grounded abyssal pool or dome. The nonlinear time-dependent evolution of this abyssal dome, which propagates equatorward with unsteady along- and cross-slope velocities while deforming into an elliptically shaped abyssal dome with $\unicode[STIX]{x1D6FD}$-induced diminishing height, is described. Finally, the nonlinear time-dependent boundary-value problem can be solved exactly in which the in-flow boundary condition on the poleward boundary of the mid-latitude domain corresponds to a time-dependent abyssal current with both an upslope and downslope grounding. Two specific time-dependent boundary conditions are examined. The first corresponds to a time-limited surge in the equatorward volume transport in the abyssal current along the poleward boundary. The second configuration corresponds to the nonlinear evolution of a finite-amplitude downslope plume or loop that forms in the abyssal current that is reminiscent of those seen in baroclinic instability simulations.

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