Dynamic Transitions of Quasi-geostrophic Channel Flow

Dijkstra, Henk A.; Şengül, Taylan; Shen, Jie; Wang, Shouhong

doi:10.1137/15m1008166

Cited by 28 publications

(24 citation statements)

References 10 publications

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“…In particular, Chen et al first proved the existence of a Hopf bifurcation in as the Reynolds number crosses a critical value. Dijkstra et al extend the results in Chen et al by addressing the stability problem of the bifurcated periodic solutions. Inspired by the importance of NS‐ α model ( γ ≠0) in geophysical fluid dynamics and understanding turbulent flow, we try to extend the results of QG model in Chen et al to NS‐ α model ( γ ≠0).…”

Section: Introductionmentioning

confidence: 85%

“…When γ =0, reduces to the following idealized two‐dimensional (2‐D) QG flow problem:

\partial_{t} false(normalΔ ψ false) = E Δ^{2} ψ - ϵ J false(ψ, normalΔ ψ false) - \partial_{x} ψ - \sin π y .

The QG flow problem has been intensively studied (for instance, Chen et al and Dijkstra et al ). In particular, Chen et al first proved the existence of a Hopf bifurcation in as the Reynolds number crosses a critical value.…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by the importance of NS‐ α model ( γ ≠0) in geophysical fluid dynamics and understanding turbulent flow, we try to extend the results of QG model in Chen et al to NS‐ α model ( γ ≠0). Also, we will use an approach different from that in Dijkstra et al to study the corresponding transition, which is simpler and easily understood.…”

Section: Introductionmentioning

confidence: 99%

“…The QG flow problem (5) has been intensively studied (for instance, Chen et al 8 ( ≠ 0). Also, we will use an approach different from that in Dijkstra et al 9 to study the corresponding transition, which is simpler and easily understood. The theoretical foundation of our present study is the phase transition dynamics theory recently developed by Ma and Wang.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the stability and transition for the Navier‐Stokes‐alpha model

Wang

Yan

2019

Math Methods in App Sciences

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The main aim of this paper is to investigate the stability and transition of the Navier‐Stokes‐alpha model. By using the continued‐fraction method, combining with the dynamic transition theory, we show the existence of a Hopf bifurcation in this model as Reynolds number crosses a critical value. Upon deriving the explicit expression of a non‐dimensional number P called transition number, which is a function of the critical Reynolds number and the aspect ratio, we further analyze the transition associated with the Hopf bifurcation. More precisely, it is shown that the modeled flow exhibits either a continuous or catastrophic transition at the critical Reynolds number, whose specific type of the transition is determined by the sign of the real part of P at the critical Reynolds number, and the spatio‐temporal structure of the limit cycle bifurcated that corresponds to a wave that propagates slowly westward and is symmetric about the mid‐axis of the channel.

show abstract

Section: Introductionmentioning

confidence: 85%

“…When γ =0, reduces to the following idealized two‐dimensional (2‐D) QG flow problem:

\partial_{t} false(normalΔ ψ false) = E Δ^{2} ψ - ϵ J false(ψ, normalΔ ψ false) - \partial_{x} ψ - \sin π y .

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the stability and transition for the Navier‐Stokes‐alpha model

Wang

Yan

2019

Math Methods in App Sciences

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“…This paper arises out of a research program to generate rigorous mathematical results on climate variability developed from the viewpoint of dynamical transitions [10, 5,6]. The basic philosophy of dynamic transition theory is to search for the full set of transition states, giving a complete characterization of stability and transition.…”

Section: Introductionmentioning

confidence: 99%

Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

Özer

Şengül

2017

J. Math. Fluid Mech.

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Abstract. The main aim of the paper is to investigate the transitions of the thermohaline circulation in a spherical shell in a parameter regime which only allows transitions to multiple equilibria. We find that the first transition is either continuous (Type-I) or drastic (Type-II) depending on the sign of the transition number. The transition number depends on the system parameters and lc, which is the common degree of spherical harmonics of the first critical eigenmodes, and it can be written as a sum of terms describing the nonlinear interactions of various modes with the critical modes. We obtain the exact formulas of this transition number for lc = 1 and lc = 2 cases. Numerically, we find that the main contribution to the transition number is due to nonlinear interactions with modes having zero wave number and the contribution from the nonlinear interactions with higher frequency modes is negligible. In our numerical experiments we encountered both types of transition for Le < 1 but only continuous transition for Le > 1. In the continuous transition scenario, we rigorously prove that an attractor in the phase space bifurcates which is homeomorphic to the 2lc dimensional sphere and consists entirely of degenerate steady state solutions.

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On the nonlinear stability and the existence of selective decay states of 3D quasi‐geostrophic potential vorticity equation

Esen

Han

Şengül

et al. 2019

Math Methods in App Sciences

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In this article, we study the dynamics of large scale motion in atmosphere and ocean governed by the 3D-quasi-geostrophic potential vorticity (QGPV) equation with a constant stratification. It is shown that for a Kolmogorov forcing on the first energy shell, there exist a family of exact solutions which are dissipative Rossby waves. The nonlinear stability of these exact solutions are analyzed based on the assumptions on the growth rate of the forcing. In the absence of forcing, we show the existence of selective decay states for the 3D-QGPV equation. The selective decay states are the 3D-Rossby waves traveling horizontally at a constant speed. All these results can be regarded as the expansion of that of the 2D-QGPV system and in the case of 3D-QGPV system with isotropic viscosity. Finally, we present a geometric foundation for the model as a general equation for non-equilibrium reversible-irreversible coupling.

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Dynamic Transitions of Quasi-geostrophic Channel Flow

Cited by 28 publications

References 10 publications

On the stability and transition for the Navier‐Stokes‐alpha model

On the stability and transition for the Navier‐Stokes‐alpha model

Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

On the nonlinear stability and the existence of selective decay states of 3D quasi‐geostrophic potential vorticity equation

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