Dynamic Transitions and Baroclinic Instability for 3D Continuously Stratified Boussinesq Flows

The main objective of this article is to study the nonlinear stability and dynamic transitions of the basic (zonal) shear flows for the three-dimensional continuously stratified rotating Boussinesq model. The model equations are fundamental equations in geophysical fluid dynamics, and dynamics associated with their basic zonal shear flows play a crucial role in understanding many important geophysical fluid dynamical processes, such as the meridional overturning oceanic circulation and the geophysical baroclinic instability. In this paper, first we derive a threshold for the energy stability of the basic shear flow, and obtain a criteria for nonlinear stability in terms of the critical horizontal wavenumbers and the system parameters such as the Froude number, the Rossby number, the Prandtl number and the strength of the shear flow. Next we demonstrate that the system always undergoes a dynamic transition from the basic shear flow to either a spatiotemporal oscillatory pattern or circle of steady states, as the shear strength $\Lambda$ of the basic flow crosses a critical threshold $\Lambda_c$. Also we show that the dynamic transition can be either continuous or catastrophic, and is dictated by the sign of a transition parameter $A$, fully characterizing the nonlinear interactions of different modes. A systematic numerical method is carried out to explore transition in different flow parameter regimes. We find that the system admits only critical eigenmodes with horizontal wave indices $(0,m_y)$. Such modes, horizontally have the pattern consisting of $m_y$-rolls aligned with the x-axis. Furthermore, numerically we encountered continuous transitions to multiple steady states, continuous and catastrophic transitions to spatiotemporal oscillations.Comment: 20 pages, 7 figure

Türkiye'de İslami Mizah, İktidar Ve Hegemonya: Cafcaf Ve Hacamat Dergileri

This study discusses the hegemonic characteristic of Islamic humour magazine publishing via the examples of Cafcaf magazine which was published between 2008-2015 and Hacamat magazine which was published by a group of people from Cafcaf in 2015. In this direction, the concept of humour, firstly, and relatedly the concept of lughter and caricature, and thereby the meanig of humour is theoretically discussed. Then, by including the debates on the concept of hegemony, a line has been drawn on especially Gramsci‟s theoretical frame. In order to be appropriate for the course of this study and especially for he context of cultural identity debates, the hegemonic feature of humour has been highlighted, on the axis of Laclau and Mouffe‟s contribution to the theory of hegemony after widening the discussion of the concept of hegemony. Then, in the second chapter which is marking the basic moments of hegemonic struggle of Justice and Development Party goverment in the emergence and growing process for the sake of clarifying the political and social context of Cafcaf and Hacamat magazines, the hegemony of Justice and Development Party has been discussed by Gramsci‟s specific notions. In the last chapter, as Islamic humour magazines which follows this aforementioned transformation process, the discourses of Cafcaf and Hacamat magazines have been analyzed and overall critical themes that the magazines mentioned on essays and cartoons have been clarified.Bu çalışma 2008-2015 yılları adasında yayımlanan Cafcaf dergisi ile 2015 yılında Cafcaf kadroları tarafından çıkarılan Hacamat dergi örnekleri üzerinden Türkiye‟de İslami mizah dergiciliğinin hegemonik karakterini ele almaktadır. Bu doğrultuda öncelikle mizah, ilişkili olarak da gülme ve karikatür kavramları ve dolayısıyla mizahın anlamı üzerinde teorik bir tartışma yürütülmüştür. Ardından hegemonya kavramı üzerindeki tartışmalara yer verilerek özellikle Gramsci‟nin teorik çerçevesi üzerinden bir hat çizilmiştir. Çalışmanın izlencesine ve özellikle kültürel kimlik tartışmaları bağlamına uygun olması için de Laclau ve Mouffe‟un hegemonya teorisine olan katkıları ekseninde hegemonya kavramı tartışması genişletildikten sonra mizahın hegemonik niteliğine vurgu yapılmıştır. Ardından, Cafcaf ve Hacamat dergilerinin siyasal ve toplumsal bağlamını netleştirmek adına Adalet ve Kalkınma Partisi iktidarının ortaya çıkış ve gelişim sürecindeki hegemonya mücadelesinin temel uğrakları ve özellikle kültürel dönüşüm süreçlerindeki dönüm noktalarını işaretleyen ikinci bölümde Adalet ve Kalkınma Partisi hegemonyası, Gramsci‟nin özgün kavramları yoluyla ele alınmıştır. Son bölümde ise çalışmanın, mevzubahis dönüşüm sürecini takip ettiği İslami mizah dergileri olarak Cafcaf ve Hacamat dergilerinin söylemleri çözümlenmiş, yazı ve karikatürler üzerinden dergilerin değindiği genel eleştirel temalar netleştirilmiştir

Stability and transitions of the second grade Poiseuille flow

In this study we consider the stability and transitions for the Poiseuille flow of a second grade fluid which is a model for non-Newtonian fluids. We restrict our attention to perturbation flows in an infinite pipe with circular cross section that are independent of the axial coordinate. We show that unlike the Newtonian (epsilon = 0) case, in the second grade model (epsilon > 0 case), the time independent base flow exhibits transitions as the Reynolds number R exceeds the critical threshold R-c = 8.505 epsilon(-1/2) where epsilon is a material constant measuring the relative strength of second order viscous effects compared to inertial effects. At R = R-c, we find that the transition is either continuous or catastrophic and a small amplitude, time periodic flow with 3-fold azimuthal symmetry bifurcates. The time period of the bifurcated solution tends to infinity as R tends to R-c. Our numerical calculations suggest that for low epsilon values, the system prefers a catastrophic transition where the bifurcation is subcritical. We also show that there is a Reynolds number R-E with R-E < R-c such that for R < R-E, the base flow is globally stable and attracts any initial disturbance with at least exponential speed. We show that the gap between R-E and R-c, vanishes quickly as epsilon increases. (C) 2016 Elsevier B.V. All rights reserved

INTERIOR STRUCTURAL BIFURCATION OF 2D SYMMETRIC INCOMPRESSIBLE FLOWS

The structural bifurcation of a 2D divergence free vector field u(., t) when u(., t(0)) has an interior isolated singular point x(0) of zero index has been studied by Ma and Wang [23]. Although in the class of divergence free fields which undergo a local bifurcation around a singular point, the ones with index zero singular points are generic, this class excludes some important families of symmetric flows. In particular, when u(., t(0)) is anti-symmetric with respect to x(0), or symmetric with respect to the axis located on x(0) and normal to the unique eigendirection of the Jacobian Du(., t(0)), the vector field must have index 1 or -1 at the singular point. Thus we study the structural bifurcation when u(., t(0)) has an interior isolated singular point x(0) with index -1, 1. In particular, we show that if such a vector field with its acceleration at t(0) both satisfy the aforementioned symmetries then generically the flow will undergo a local bifurcation. Under these generic conditions, we rigorously prove the existence of flow patterns such as pairs of co-rotating vortices and double saddle connections. We also present numerical evidence of the Stokes flow in a rectangular cavity showing that the bifurcation scenarios we present are indeed realizable

Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

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 l(c), 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 l(c) = 1 and l(c) = 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. In the continuous transition scenario, we rigorously prove that an attractor in the phase space bifurcates which is homeomorphic to the 2l(c) dimensional sphere and consists entirely of degenerate steady state solutions

On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents

This study examines the instability and dynamical transitions of the two-layer western boundary currents represented by the Munk profile in the upper layer and a motionless bottom layer in a closed rectangular domain. First, a bound on the intensity of the Munk profile below which the western boundary currents are locally nonlinearly stable is provided. Second, by reducing the infinite dimensional system to a finite dimensional one via the center manifold reduction, non-dimensional transition numbers are derived, which determine the types of dynamical transitions both from a pair of simple complex eigenvalues as well as from a double pair of complex conjugate eigenvalues as the Reynolds number crosses a critical threshold. We show by careful numerical estimations of the transition numbers that the transitions in both cases are continuous at the critical Reynolds number. After the transition from a pair of simple complex eigenvalue, the western boundary layer currents turn into a periodic circulation, whereas a quasi-periodic or possibly a chaotic circulation emerges after the transition from a pair of double complex eigenvalues. Finally, a comparison between the transitions exhibited in one-layer and two-layer models is provided, which demonstrates the fundamental differences between the two models. (c) 2018 Elsevier B.V. All rights reserved

Dynamic Transitions and Baroclinic Instability for 3D Continuously Stratified Boussinesq Flows

The main objective of this article is to study the nonlinear stability and dynamic transitions of the basic (zonal) shear flows for the three-dimensional continuously stratified rotating Boussinesq model. The model equations are fundamental equations in geophysical fluid dynamics, and dynamics associated with their basic zonal shear flows play a crucial role in understanding many important geophysical fluid dynamical processes, such as the meridional overturning oceanic circulation and the geophysical baroclinic instability. In this paper, first we derive a threshold for the energy stability of the basic shear flow, and obtain a criterion for local nonlinear stability in terms of the critical horizontal wavenumbers and the system parameters such as the Froude number, the Rossby number, the Prandtl number and the strength of the shear flow. Next, we demonstrate that the system always undergoes a dynamic transition from the basic shear flow to either a spatiotemporal oscillatory pattern or circle of steady states, as the shear strength of the basic flow crosses a critical threshold. Also, we show that the dynamic transition can be either continuous or catastrophic, and is dictated by the sign of a transition number, fully characterizing the nonlinear interactions of different modes. Both the critical shear strength and the transition number are functions of the system parameters. A systematic numerical method is carried out to explore transition in different flow parameter regimes. In particular, our numerical investigations show the existence of a hypersurface which separates the parameter space into regions where the basic shear flow is stable and unstable. Numerical investigations also yield that the selection of horizontal wave indices is determined only by the aspect ratio of the box. We find that the system admits only critical eigenmodes with roll patterns aligned with the x-axis. Furthermore, numerically we encountered continuous transitions to multiple steady states, as well as continuous and catastrophic transitions to spatiotemporal oscillations