Instability of quasi-geostrophic vortices in a two-layer ocean with a thin upper layer

2003 ◽  
Vol 475 ◽  
pp. 303-331 ◽  
Author(s):  
E. S. BENILOV
Keyword(s):  
Boundary Value Problem ◽  
Lower Layer ◽  
Boundary Value ◽  
Stability Boundary ◽  
Normal Modes ◽  
Layer Depth ◽  
Asymptotic Results ◽  
Gaussian Profile ◽  
Swirl Velocity ◽  
The Stability

We examine the stability of a quasi-geostrophic vortex in a two-layer ocean with a thin upper layer on the f-plane. It is assumed that the vortex has a sign-definite swirl velocity and is localized in the upper layer, whereas the disturbance is present in both layers. The stability boundary-value problem admits three types of normal modes: fast (upper-layer-dominated) modes, responsible for equivalent-barotropic instability, and two slow baroclinic types (mixed- and lower-layer-dominated modes). Fast modes exist only for unrealistically small vortices (with a radius smaller than half of the deformation radius), and this paper is mainly focused on the slow modes. They are examined by expanding the stability boundary-value problem in powers of the ratio of the upper-layer depth to the lower-layer depth. It is demonstrated that the instability of slow modes, if any, is associated with critical levels, which are located at the periphery of the vortex. The complete (sufficient and necessary) stability criterion with respect to slow modes is derived: the vortex is stable if and only if the potential-vorticity gradient at the critical level and swirl velocity are of the same sign. Several vortex profiles are examined, and it is shown that vortices with a slowly decaying periphery are more unstable baroclinically and less barotropically than those with a fast-decaying periphery, with the Gaussian profile being the most stable overall. The asymptotic results are verified by numerical integration of the exact boundary-value problem, and interpreted using oceanic observations.

On the stability of large-amplitude geostrophic flows in a two-layer fluid: the case of ‘strong’ beta-effect

1995 ◽  
Vol 284 ◽  
pp. 137-158 ◽  
Author(s):  
E. S. Benilov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Stability Boundary ◽  
Phase Speed ◽  
Stability Theorems ◽  
Beta Effect ◽  
Zonal Velocity ◽  
Special Cases ◽  
Geostrophic Flows ◽  
The Stability

This paper examines the stability of two-layer geostrophic flows with large displacement of the interface and strong β-effect. Attention is focused on flows with non-monotonic interface profiles which are not covered by the Rayleigh-style stability theorems proved by Benilov (1992a, b) and Benilov & Cushman-Roisin (1994). For such flows the coefficient of the highest derivative in the corresponding boundary-value problem vanishes at the point where the depth profile has an extremum. Although this singularity is similar to a critical level, it cannot be regularized by the simplistic introduction of infinitesimal viscosity through the assumption that the phase speed of the disturbance is complex. In order to regularize the singularity properly, one should consider the problem within the framework of the original ageostrophic viscous equations and, having obtained the boundary-value problem for harmonic disturbance, take the limit Rossby number → 0, viscosity → 0.The results obtained analytically and (for special cases) numerically indicate that the stability of flows with non-monotonic profiles strongly depends on the depth of the upper layer. If the upper layer is ‘thick’ (i.e. if the average depth H1 of the upper layer is of the order of the total depth of the fluid H0), the stability boundary-value problem does not have any solutions at all, which means stability (however, this stability is structurally unstable, and the flow, generally speaking, can be made weakly unstable by any small effect such as external forcing, viscosity, or ageostrophic corrections). In the case of ‘thin’ upper layer (H1/H0 [lsim ] Ro), the order of the singularity changes and all non-monotonic flows are unstable regardless of their profiles. It is also demonstrated that thin-upper-layer flows do not have to be non-monotonic to be unstable: if u–βR20 (where u is the zonal velocity, β is the β-parameter, and R0 is the deformation radius) changes sign somewhere in the flow, the stability boundary-value problem has another singular point which leads to instability.

On the stability of large-amplitude vortices in a continuously stratified fluid on the f-plane

1998 ◽  
Vol 355 ◽  
pp. 139-162 ◽  
Author(s):  
E. S. BENILOV ◽  
D. BROUTMAN ◽  
E. P. KUZNETSOVA
Keyword(s):  
Growth Rate ◽  
Boundary Value Problem ◽  
Large Amplitude ◽  
Boundary Value ◽  
Normal Modes ◽  
Large Displacement ◽  
Primitive Equations ◽  
Rossby Number ◽  
Spatial Coordinates ◽  
The Stability

The stability of continuously stratified vortices with large displacement of isopycnal surfaces on the f-plane is examined both analytically and numerically. Using an appropriate asymptotic set of equations, we demonstrated that sufficiently large vortices (i.e. those with small values of the Rossby number) are unstable. Remarkably, the growth rate of the unstable disturbance is a function of the spatial coordinates. At the same time, the corresponding boundary-value problem for normal modes has no smooth square-integrable solutions, which would normally be regarded as stability.We conclude that (potentially) stable vortices can be found only among ageostrophic vortices. Since this assumption cannot be verified analytically due to complexity of the primitive equations, we verify it numerically for the particular case of two-layer stratification.

scholarly journals Implicit nonlinear fractional differential equations of variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Mohammed Said Souid ◽  
Kanokwan Sitthithakerngkiet ◽  
Ali Hakem
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Variable Order ◽  
Stability Of Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

Stability and constant boundary-value problems of harmonic maps with potential

2000 ◽  
Vol 68 (2) ◽  
pp. 145-154 ◽  
Author(s):  
Qun Chen
Keyword(s):  
Riemannian Manifold ◽  
Energy Density ◽  
Boundary Value Problem ◽  
Riemannian Manifolds ◽  
General Class ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Boundary Value ◽  
The Stability ◽  
Harmonic Maps With Potential

AbstractLet M, N be Riemannian manifolds, f: M → N a harmonic map with potential H, namely, a smooth critical point of the functional EH(f) = ∫M[e(f)−H(f)], where e(f) is the energy density of f. Some results concerning the stability of these maps between spheres and any Riemannian manifold are given. For a general class of M, this paper also gives a result on the constant boundary-value problem which generalizes the result of Karcher-Wood even in the case of the usual harmonic maps. It can also be applied to the static Landau-Lifshitz equations.

scholarly journals Boundary value problem for nonlinear fractional differential equations of variable order via Kuratowski MNC technique

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Dumitru Baleanu ◽  
Mohammed Said Souid ◽  
Ali Hakem ◽  
Mustafa Inc
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Measure Of Noncompactness ◽  
Research Study ◽  
Boundary Value ◽  
Existence Of Solution ◽  
Variable Order ◽  
Kuratowski Measure Of Noncompactness ◽  
Fractional Differential ◽  
The Stability

AbstractIn the present research study, for a given multiterm boundary value problem (BVP) involving the Riemann-Liouville fractional differential equation of variable order, the existence properties are analyzed. To achieve this aim, we firstly investigate some specifications of this kind of variable-order operators, and then we derive the required criteria to confirm the existence of solution and study the stability of the obtained solution in the sense of Ulam-Hyers-Rassias (UHR). All results in this study are established with the help of the Darbo’s fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of our observed results.

scholarly journals Stability and Convergence of Two-Step Obrechkoff Scheme For Second-Order Two-Point Boundary Value Problem

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
Olufemi Bosede ◽  
Ashiribo Wusu ◽  
Moses Akanbi
Keyword(s):  
Mathematical Modeling ◽  
Differential Equation ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
Stability And Convergence ◽  
Point Boundary ◽  
Difference Methods ◽  
The Stability

Mathematical modeling of scientific and engineering processes often yield Boundary Value Problems (BVPs). One of the broad categories of numerical methods for solving BVPs is the finite difference methods, in which the differential equation is replaced by a set of difference equations which are solved by direct or iterative methods. In this paper, we use some properties of matrices to analyze the stability and convergence of the prominent finite difference methods - two-step Obrechkoff method - for solving the boundary value problem $u^{\prime \prime} = f(t,u)$, $a < x < b$, $u(a) = \eta_1$, $u(b) = \eta_2$. Conditions for the stability and convergence of the two-step Obrechkoff method method were established.

Stable Running with Asymmetric Legs: A Bifurcation Approach

2015 ◽  
Vol 25 (11) ◽  
pp. 1550152 ◽  
Author(s):  
Andreas Merker ◽  
Dieter Kaiser ◽  
Andre Seyfarth ◽  
Martin Hermann
Keyword(s):  
Boundary Value Problem ◽  
System Dynamics ◽  
Boundary Value ◽  
Mass Model ◽  
Animal Locomotion ◽  
Bifurcation Points ◽  
Stable Solutions ◽  
The Stability

The spring-mass model is a frequently used gait template to describe human and animal locomotion. In this study, we transform the spring-mass model for running into a boundary value problem and use it for the computation of bifurcation points. We show that the analysis of the region of stable solutions can be reduced to the calculation of its boundaries. Using the new bifurcation approach, we investigate the influence of asymmetric leg parameters on the stability of running. Like previously found in walking, leg asymmetry does not necessarily restrict the range of stable running and may even provide benefits for system dynamics.

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