RESEARCH STATEMENT. 1. dynamic bifurcation theories in fluid dynamics and geophysical fluid dynamics,

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1 1 OBJECTIVES AND SCIENTIFIC TASKS One of the primary goals in climate dynamics is to document through careful theoretical and numerical studies the presence of climate low frequency variability, to verify the robustness of this variability s characteristics to changes in model parameters and to help explain its physical mechanisms. The thorough understanding of this variability is a challenging problem with important practical implications for geophysical efforts to quantify predictability, analyze error growth in dynamical models and develop efficient forecast methods. This study is part of long term research project to study low frequency variability of the atmospheric and oceanic flows. I propose to study the structure, stability and transition of these incompressible fluid flows and geophysical fluid flows. I focus on two main physical problems; (i)classical problems arising in fluid/geophysical fluid dynamics including the Complex Ginzburg-Landau equation, the Kuramoto- Sivashinsky equation, the Rayleigh-Bénard convection, the Swift-Hohenberg equation and the thermosolutal convection, and (ii)long time period proximity of solutions in geophysical fluid dynamics including the thermosolutal convection and the wind-driven ocean circulation. Their variability, independently and interactively, may play a significant role on climate change, past and future. It is hoped that the study of these fluid or geophysical fluid problems can involve in part the application of existing mathematical and computational theories that can further the understanding of underlying physical problems, as well as advance the development of new mathematical theories. When we study weather and climate, we want to know how they change. Scientists have to understand the basic principles for how any complicated system can change. Studies, using highly complicated physical models, could see predictable behaviors such as periodic, quasi-periodic turbulent characteristics. Bifurcation theory enables one to determine how qualitatively different flow regimes appear and disappear as control parameters change; it provides us with an important method to explore the theoretical limits of predicting these flow regimes and consists of 1) complete bifurcation when the system parameter crosses some critical values, 2) asymptotic stability of bifurcated solutions and 3) the structure/pattern of the bifurcated solutions and phase transitions in physical spaces. The study in the first two directions is related to application of a new bifurcation theory, called attractor bifurcation, which was developed by T. Ma and S. Wang [17]. The third direction of the study is related to geometric study of fluid flows and includes structural stability theory. I propose to study specific topics in the following areas: 1. dynamic bifurcation theories in fluid dynamics and geophysical fluid dynamics, 2. stability issues in both phase and physical spaces, and 3. long time orbital stability of solutions of the Boussinesq system in geophysical fluid dynamics. In the remaining part of this section, I will state these three specific topics in details. 1.1 Dynamic bifurcation theories in fluid dynamics and geophysical fluid dynamics. Recently, the ideas of dynamical systems theory and nonlinear functional analysis have been applied so far to climate dynamics by careful numerical studies [15, 33, 35]. In particular, pseudo-arclength continuation methods have been applied to atmospheric and oceanic models increasing horizontal resolution [13, 32]. These numerical bifurcation studies have produced so far fairly reliable results for two classes of geophysical flows: atmospheric flows in a periodic mid latitude channel and oceanic flows in a rectangular mid latitude basin, subject to wind stress on its upper surface. In these studies, the symmetry properties of the forcing Page 1

2 term played an important role in determining the onset of bifurcations such as the Hopf bifurcation [13, 12], saddle node bifurcation [2, 30] and pitchfork or Hopf [1, 32]. In addition to the numerical studies, rigorous bifurcation and stability analysis for the original partial differential equation models should be conducted. However, most studies on bifurcation issues on geophysical fluid dynamics and climate dynamics have only considered systems of ordinary differential equations that are obtained by projecting the partial differential equations onto a finite dimensional solution space, either by finite differencing or by truncating a Fourier expansion; see [5] and references there. Challenging mathematical problem is to conduct rigorous bifurcation and stability analysis in both phase and physical spaces for the original partial differential equations that govern the geophysical flows. Some progresses have been made in this direction in various problems including the Kuramoto-Sivashinsky equation, the Cahn-Hillard equation, the complex Ginzburg-Landau equation, the Rayleigh-Bénard convection and the Thermosolutal convection in fluid or geophysical fluid dynamics; see among others [3, 11, 18, 21, 22, 24, 25, 27]. These progresses in this area allow us to overcome some of the inherent limitations of the numerical bifurcation results that dominate the climate dynamics literature up to this point, and to capture the essential dynamics of the governing partial differential equation systems. The proposed work in this study concentrates on application of dynamic bifurcation theory to partial differential equation models including the Complex-Ginzburg Landau equation, Rayleigh-Bénard convection, the thermosolutal convection and wind-driven ocean circulation, as well as some other convection related geophysical fluid problems. The structure of bifurcated solutions is also explored. The main ingredient of this study is the center manifold reduction theorem. The analysis is the reduction of the problem to the center manifold in the first unstable eigendirections, based on the approximation formula for the center manifold function. The key idea is to find the approximation of the reduction to certain order, leading to a nondegenerate system with higher order perturbations Stability issues in both phase and physical spaces. The geometric theory of incompressible flows is initiated by many researchers including Ma and Wang [18] to study the stability and transition of 2-D incompressible flows in physical spaces. It is worthwhile to mention that structural stability is different from dynamical stability such as the asymptotic stability. A solution of partial differential equations is called asymptotically stable if any solution with perturbed initial data converges to the given solution. Structural stability, used in this proposal, refers to the stability of vector fields in physical space, and the idea of structural stability can be extended to vector fields on higher or infinite dimensional manifolds. A vector field is said to be structurally stable if, for any small perturbation, the perturbed vector field is topologically equivalent to the original vector field. More precisely, we are led to study the set of certain vector fields at once by defining a topology over it and hence an equivalence relation between arbitrarily close vector fields (connected for instance to the existence of some homeomorphism identifying different orbits). At this point, structural stability is obviously a global requirement, implying that the perturbation of the whole vector field leaves its qualitative properties preserved no matter what the initial conditions are. There are examples of steady states which are structurally stable but dynamically unstable and vice versa. For this study it is essential to study the topological structure of the divergence-free vector fields for 2D-incompressible flows governed by the Navier-Stokes equations or Euler equations. Such a result would involve a specific connection between solutions of the equations and flow structure in physical space, i.e., this area of research links the kinematics to the dynamics of fluid flows Long time orbital stability of solutions of Boussinesq system in geophysical fluid dynamics. Since the velocity equation in the Boussinesq system which governs the Rayleigh-Bénard convection Page 2

3 and the thermosolutal convection (2.3) and (2.6) is exactly the Navier-Stokes equations(forced by a buoyancy term) whose regularity of solutions is one of the million dollar mathematical problems of the new millennium. For such a complex system, simplification is highly desirable and the infinite Prandtl number model is one of the ways. In the infinite Prandtl number regime, one can prove global existence and uniqueness of smooth solutions [9]. The infinite Prandtl number limit of the Boussinesq equations has been used as the standard model for the convection of earth s mantle, where it is argued that P r could be of the order 10 24, as well as for many gases under high pressure. These features make this problem worthy of study and allows us to focus on the velocity field which will behave much more regularly than the finite Prandtl number convection. This topic will be addressed in Section 2.5 and Section RESULTS FROM PRIOR STUDIES 2.1. Complex Ginzburg Landau Equation [16, 23, 29]. The complex Ginzburg Landau (GL) equation reads u t (α + iβ) u + (σ + iρ) u 2 u λu = 0, (2.1) where the unknown function u : Ω( R n ) [0, ) C is a complex-valued function. The main results obtained in [16] can be summarized as follows. (See Figure 1.) First, for the GL equation with the Dirichlet boundary condition, let λ 1 be the first eigenvalue of the elliptic operator. 1. If λ αλ 1, the trivial solution u = 0 is globally asymptotically stable. The global attractor of the GL equation consists of exactly the trivial steady state solution u = As λ crosses αλ 1, i.e., there exists an ϵ > 0 such that for any αλ 1 < λ < αλ 1 + ϵ, (2.1) bifurcates from the trivial solution an attractor Σ λ. The bifurcated attractor Σ λ attracts the open set L 2 (Ω, C)/Γ, where Γ is the stable manifold of u = 0 having codimension two in L 2 (Ω, C). Second, bifurcation from any eigenvalue of the Laplacian can also be obtained as for the first eigenvalue. It is worth mentioning that the complete structure of the global attractor for the bifurcations from the first eigenvalue is obtained, while no such information is available for bifurcations from the rest of the eigenvalues. H Σ 0 Σ 2 Γ Γ αλ λ 1 Σ λ λ C Σ 1 ρ λ 1 ρ Σ 1 Σ 0 Σ 2 Figure 1: (Left)Bifurcation diagram for the GL equation with the Dirichlet boundary condition: (1) u=0 bifurcates to an attractor Σ λ = S 1 for λ = αλ 1, which is the boundary of the shaded region, and (2) the global attractor A λ is the 2D disk. (Middle)Phase space structure in L 2 (Ω, C) {λ} in the case where β + ρ = 0. Here the bifurcated attractor Σ λ = S 1 is a stable limiting cycle. (Right)Subcritical bifurcation diagram for the cubic-quintic CGLE. For λ c < ρ < λ 1, Σ 1 represents the inner circle and Σ 2 represents the outer circle, both of which are bifurcated from nonzero point (u, λ c ) = (e iθ u, λ c ). Page 3

4 I also studied so called the generalized complex Ginzburg-Landau equation [23, 29] which involves an unstable cubic and a stable quintic term which saturates the expected blowup of the solutions. u t = ρu + (1 + iρ 0) u + (1 + iρ 1 ) u 2 u (1 + iρ 2 ) u 4 u. (2.2) In [23], if we let λ 1 be the first eigenvalue of under the Dirichlet boundary condition, then the equation (2.2) bifurcates subcritically from u = 0 for ρ < λ 1. Moreover, with the help of the global attractor near ρ = 0, we obtain a saddle node bifurcation point (u, ρ) = (u, λ c ) (0 < λ c < λ 1 ) from which two branches of solutions Σ 1 and Σ 2 bifurcate. We can see that Σ 1 is a repeller and convergent to (0, λ 1 ) as ρ λ 1 and Σ 2 is an attractor and extends to ρ > λ 1. (see Figure 1.) 2.2. Rayleigh Bénard Convection. Rayleigh-Bénard convection, that is, a buoyancy-driven convection in a fluid layer heated from below and cooled from above, is one of the prime examples of bifurcating high-dimensional systems. It has long been a subject of intense theoretical and experimental study and has been applied to many different areas of study such as meteorology, geophysics, and astrophysics. The governing equations are the following Boussinesq equations: [ ] 1 u + (u )u + p u RT k = 0, P r t T + (u )T T = 0, u = 0, t where u = (u 1, u 2 ) is the velocity field, p is the pressure function, T is the temperature field and k = (0, 1) is the unit vector in x 2 -direction. In the Boussinesq equations, we have two important numbers: the Rayleigh number, R which measures the ratio of overall buoyancy force to the damping coefficients, and the Prandtl number, P r which measures the relative importance of kinematic viscosity over thermal diffusivity. We impose the periodic boundary condition with spatial periods L in the horizontal direction x 1 and the top and bottom plates are assumed to be stress free: T = 0, u 2 = 0, or no-slip (Dirichlet): u = 0, T = 0 at x 2 = 0, 1. (2.3) u 1 x 2 = 0 at x 2 = 0, 1, (2.4) 2.3. Thermosolutal Convection. We take into consideration the situation with warm salty fluid below cold fluid. As warm fluid tends to rise, salty fluid being heavier than fresh fluid, tends to fall. In this case either an oscillatory motion or a steady convective motion may be seen. In contrast to the Rayleigh-Bénard convection, the linear problem of the thermosolutal convection near the basic solution is nonsymmetric. This requires much more work in order to analyze the eigenvalue problem. The system reads transport equations for the temperature and solute concentration, the Navier-Stokes equation with the Boussinesq approximation, and incompressibility as follows: [ ] 1 u + (u )u = u p + Ta k u + (R T T R S S)k, P r t T t = T + u S 3 (u )T, t = τ S + u 3 (u )S u = 0, in the dimensionless domain D = R 2 (0, 1). Here, u = (u 1, u 2, u 3 ) is the velocity field, T is the temperature field and S is the solute concentration. (2.5) (2.6) In the Boussinesq equations, we have four important positive Page 4

5 numbers: the thermal Rayleigh number, R T which measures the ratio of overall buoyancy force to the damping coefficients, the salinity Rayleigh number, R S which measures the ratio of the coefficients of salt diffusion and temperature, the Prandtl number P r which measures the relative importance of kinematic viscosity over thermal diffusivity, the Taylor number, Ta which measures the rotation, and the Lewis number τ which characterizes fluid flows where there is simultaneous heat and mass transfer by convection. Double diffusive convection is a fundamental fluid dynamical phenomenon that can arise when a layer of fluid with a dissolved solute, such as salt, is heated from below. This involves the difference between the diffusivities of thermal and chemical components in releasing gravitational potential energy. The subject also covers a more general study of fluids in which there are gradients of two or more properties with different molecular diffusivities Infinite Prandtl Number Convection [21, 22, 24, 25, 27]. The two dimensional infinite Prandtl number model of (2.3) is described as follows: p u RT k = 0, T t + (u )T T Ru 2 = 0, u = 0. The linear eigenvalue problem for the Rayleigh number R is symmetric, all eigenvalues R k are real numbers such that 0 < R 1 < < R k < R k+1 <. The first eigenvalue R 1 is a function of the period L. The critical Rayleigh number R c is given by R c = min L>0 R 1(L) and the minimum can be achieved. For simplicity, we take L such that R c = R 1 (L). The bifurcation and stability analysis of this problem is then carried out by (i) explicitly reducing the problem to the center manifold generated by the first eigenfunctions, (ii) classifying the structure of the bifurcated attractor, and (iii) using the geometric theory of incompressible flows to classify the structure of the solutions in the physical space. This analysis leads to the following. First, for the problem (2.7) with (2.4), the equation bifurcates from the trivial solution (T, R) = (0, R c ) to an attractor Σ R for R > R c such that Σ R consists of exactly one cycle of steady state solutions and homeomrophic to S 1. Second, we can classify the structure of the solutions of the infinite Prandtl number convection problem in a physical space leading to the justification of the roll structure. More precisely, for any T 0 H\Γ, there exists a time t 0 0 such that for any t t 0, the associated vector field u(t, T ) = u(t, T, Ψ 0 ) is structurally stable and is topologically equivalent to the structure as shown in Figure 2(left), where Ψ = (u(t, Ψ 0 ), T (t, Ψ 0 )) is the solution of (2.7), Ψ 0 = (u 0 (T 0 ), T 0 ) ( H H)\ Γ and Γ(resp. Γ) is the stable manifold of T = 0(resp. of (u, T ) = 0) with co-dimension 2 in H(resp. in H H). Here, H = L 2 (Ω), and H = {u L 2 (Ω) 2 u = 0, u dx = 0, u 1 is periodic in x 1 direction, and u 2 x2 =0,1 = 0}. Ω However, under the Dirichlet boundary conditions [27], we have a different result. Third, for any (u 0 (T 0 ), T 0 ) H H\(Γ E), there exists a time t 0 0 such that for any t t 0, the associated vector field u(t, Ψ 0 ) is structurally stable and is topologically equivalent to the structure as shown in Figure 2(right), where Ψ = (u(t, Ψ 0 ), T (t, Ψ 0 )) is the solution of (2.7) with initial data Ψ 0 = (u 0 (T 0 ), T 0 ) and Γ is the stable manifold of (u(t ), T ) = 0 with co-dimension 2 in H H. Here, H = L 2 (Ω), E = {(u, T ) H H 1 0 u 1 dx 2 = 0}, and H = {u L 2 (Ω) 2 u = 0, u 1 is periodic in x 1 direction, and u 2 x2 =0,1 = 0}, (2.7) Page 5

6 Figure 2: (Left)The roll structure of solutions of the infinite Prandtl number convection under free boundary conditions. (Right)Schematic Flow Structure of u with no slip boundary conditions which means that the flow does not move on the top and bottom boundaries. The infinite Prandtl number model for the 3-D rotating thermosolutal convection (2.6) is u p + Ta k u + ( λt ηs)k = 0, T t = T + λ u 3 (u )T, u = 0. S t = τ S + η u 3 (u )S, (2.8) We impose the periodic boundary condition in the horizontal directions x 1 and x 2, and stress-free on the top and bottom plates. Unlike the spectrum analysis of linear theory for finite Prandtl numbers, the infinite Prandtl number convection has only two eigenvalues. This analysis leads us to: Let λ = λ c. If η > η, then the linear eigenvalue problem of the rotating thermosolutal convection (2.8) has two pure imaginary eigenvalues for all k 1, k 2 0 and j 1. However, the problem has two real eigenvalues if η < η, where, λ c = η + (τ + 1)c π2 (τ + 1) ( c Ta) a 2, η = τ 2 a 2 11 c π2 (c Ta) 11 1 τ a 2, 11 λ = 1 a 2 11 c π2 (c Ta) ( ) 2π 2 ( ) 2π 2 1 τ a 2, a 2 11 = +, c = (a π 2 ) 2, 11 L 1 L 2 The bifurcation and the structure of the bifurcated solution can be summarized as follows: Assume η < η. for the problem (2.8) with boundary conditions, the steady state (T, S) = 0 is locally asymptotically stable if λ λ c and the equations bifurcate from (T, S, λ) = (0, 0, R c ) to an attractor Σ λ for λ > λ c, such that Σ λ consists of exactly one cycle of steady state solutions and is homeomorphic to S 1. The effect of rotation on convective motion is an important issue in astrophysical and geophysical applications. The presence of the rotation, due to the earth s spin about its axis, introduces in the equations of motion in the rotating framework that can be interpreted as forces. It is the Coriolis force whose major effect is to impart a certain vertical rigidity to the fluid, and introduces the Taylor number, Ta, which is proportional to the square of the rotation rate. I showed that the first critical number of the rotating thermosolutal convection depends not only on the aspect of ratio but also on the Taylor number. (see Figure 3.) Therefore, it can be seen that the first critical thermal Rayleigh number in the fast rotating flows becomes significantly larger than that in the non-rotating flows, which means that the rotation inhibits the onset of instability. flows. Therefore, the rotating flows are much more stable than non-rotating 2.5. Long time orbital stability of solutions of Boussinesq system in geophysical fluid dynamics [26]. A Broader rationale for investigating the infinite Prandtl number convection is based on the observation of both the linear and weakly nonlinear theories; that fluids with P r > 1 convect in a Page 6

7 log Rc log Ta Figure 3: The values of R c T = λ c determined by Ta. similar fashion. However, no fluid has the infinite Prandtl number. Thus, it is of interest to investigate long time properties of the solutions of the Boussinesq system at large Prandtl number and that becomes an interesting initial layer problem. Let (u, T, S) be the solution of (2.6) with initial conditions (u 0, T 0, S 0 ), and (u, T, S ) be the solution of (2.8) with the boundary conditions and the initial conditions as u x3 =0,1 = 0, T, S x3 =0 = 1, T, S x3 =1 = 0, (u, T, S ) t=0 = (u 0, T 0, S 0 ). (2.9) Let P be the Leray projector and E be an elliptic operator. Due to the coercivity of E, the infinite Prandtl number convection with the operator E is u = R T E 1 (P (T k)) R S E 1 (P (S k)), T t = T (u )T, S t T, S x3 =0 = 1, T, S x3 =1 = 0, = τ S (u )S, (2.10) This is natural to ask a question if (u, T, S) (u, T, S ) and (u 0, T 0, S 0 ) (u 0, T 0, S 0 ) as P r? Due to the singularity of perturbation, lim u 0 u 0 in the limit system. We then observe P r that the infinite Prandtl number limit is a singular problem involving an initial layer. I recognized that this problem has two time scales that are O(1) and O(1/P r) and this requires us to use the method of multiple scales [10, 19]. I derived systematically an effective approximating system for the limiting dynamics of thermosolutal convection (2.6): u 0 (t) = e Eη u 0 e Eη E 1 [R T P (T 0 k) R S P (S 0 k)] + E 1 [ R T P (T 0 (t)k) R S P (S 0 (t)k) ], T 0 t = T 0 (u 0 )T 0, S 0 t T 0, S 0 x3 =0 = 1, T 0, S 0 x3 =1 = 0. = τ S 0 (u 0 )S 0, (2.11) And I compared (2.11) with (2.10) to see how close the effective approximating system and the infinite Prandtl number system remain for a finite time interval and then, the showed that the solutions of the Boussinesq system (2.6) converge to those of the effective approximating in the sense of L 2 -norm: For any finite time interval [0, T ] with T fixed, and any initial data Ψ 0 = (u 0, T 0, S 0 ), we have: First, 1. T 0 T L (0,T ;L 2 ) Cε(R T + R S )e c(r T +R S )t, 2. S 0 S L (0,T ;L 2 ) Dε(R T + R S )e d(r T +R S )t, 3. u 0 u e Eη u 0 + R T e Eη E 1 (P (T 0 k)) R S e Eη E 1 (P (S 0 k)) L (0,T ;L 2 ) KεR T R S (R T + R S )e k(r T +R S )t. Second, 1. u u 0 L (0,T ;L 2 ) C 1 εe (8C 2R 2 T +8D 1R 2 S )t (C 1 R T + D 2 R S ) 2 2. T T 0 L (0,T ;L 2 ) C 4 εe (8C 2R 2 T +8D 1R 2 S )t (C 5 R T + D 3 R S ) Page 7

8 3. S S 0 L (0,T ;L 2 ) D 4 εe (8C 2RT 2 +8D 1RS 2 )t (C 5 R T + D 3 R S ) where, C, D, K, c, d, C i, D i and k are constants which are independent on ε = 1 P r. The above results lead us to believe that the infinite Prandtl number model of thermosolutal convection can also be justified as the limit of the Boussinesq approximation to the thermosolutal convection as the Prandtl number approaches infinity in any finite time interval and the convergence rate is O(ε). This means that we can examine the nature of thermosolutal convection in the infinite Prandtl number regime admitting initial layers. 3 CURRENT AND FUTURE WORK 3.1. Dynamic Bifurcation Theories in fluid dynamics and geophysical fluid dynamics. My current work on this direction started as joint work with Yuncherl Choi(Kwangwoon Univ., KOREA) and Jongmin Han(Kyung Hee Univ., KOREA) for Dynamical bifurcation of the generalized Swift-Hohenberg equation [4] which reads u t = αu (1 + xx ) 2 u + µu 2 u 3. (3.1) In this collaboration, I study the primary instability of the generalized Swift-Hohenberg equation under a periodic boundary condition. I prove that it bifurcates from the trivial solution to an attractor which determines the long time dynamics of the system. Using the attractor bifurcation theorem and the center manifold theory, I also describe the bifurcated attractor in detail. The next problem will be about the 2-dimensional GL equation defined on the periodic spatial domain Ω = ( l, l) ( l, l) with odd periodic boundary condition on the domain: { u(x1, x 2 ) = u(x 1 + 2l, x 2, t) = u(x 1, x 2 + 2l, t), (x 1, x 2 ) R 2, t 0 u( x 1, x 2, t) = u(x 1, x 2, t), (x 1, x 2 ) Ω, t 0 This problem is worth of study because the multiplicity of the first eigenvalue determines the dimension of the center manifold and has a variety range according to the periodicity 2l. This phenomenon is not common for other equations. There are many interesting and long standing problems in this area which are parts of my long term project; the low frequency of variability of the atmospheric and oceanic flows such as 1. The wind driven circulation which is a horizontal circulation. The wind driven circulation plays a role mostly in the ocean s subannual to interannual variability while the thermosolutal convection is most important in decadal to millennial variability. Some numerical results were reported but not much rigorous analysis is published yet. 2. The Hopf bifurcation of the thermosolutal convection which enables us to understand the transitions of different flow regimes. Since Hopf bifurcation to a stable periodic solution is associated with continuous transitions and bifurcation to unstable periodic solutions is related to discontinuous transition, often leading to transitions to more complex flow regimes, this study will help to understand the variability of the atmospheric and ocean flows. 3. The thermosolutal convection is highly nonlinear due to the combined effects of the temperature and the salinity on density and it causes the existence of multiple equilibria and thresholds in regimes in the thermosolutal convection. two equilibria flow regimes in the thermosolutal convection. The sudden climate change is related to the shift between any convection to anthropogenic climate forcing is still an open question; see [31]. The sensitivity of the thermosolutal Page 8

9 4. The other fluid or geophysical fluid problems including atmospheric circulation motion with humidity effect, surface tension driven convection, Ericksen-Leslie equations for the flow of nematic liquid crystal flows which is well suited for describing many special flows for the materials with small molecules, and some coupled atmospheric and oceanic models Stability Issues in both phase and physical spaces. Whenever we study fluid problems, we can find that there are two different but connected stability and transitions associated with the problems. These are stability and transitions of the solutions of the original partial differential equation in phase space and the structure and transition of solutions in physical space. This problem has been initiated by many authors including [6, 7]. One of the example issues related to this direction is the western boundary current separation which is a long standing problem in physical oceanography. The boundary layer separation problem is key to understand the transition to turbulence and stability of flows and linked to spatial and phase space structure changes of solutions of the original partial differential equations Long time orbital stability of solutions of Boussinesq system in geophysical fluid dynamics. Inspired by the successful analysis in the finite time interval I want to answer the question: Do solutions of the Boussinesq system and solutions of the infinite Prandtl number model remain close on a large time interval for large Prandtl number? In general we should not expect long time proximity of each individual orbit due to their turbulent and chaotic behavior. Instead, the statistical properties for such systems are much more important and physically relevant[19, 20] and hence it is natural to ask if the statistical properties as well as global attractors remain close. I am trying to establish the proximity of the solutions to the thermosolutal convection and those of the infinite Prandtl number model over a long time period. For that purpose I try to establish in [28], the regularity of suitable weak solutions to the Boussinesq system at large Prandtl number and the existence of a global attractor as well; There exists a constant κ 0 (D) depending only on the domain D such that 1. For ε(r T T R S S) κ 0 and any initial data Ψ 0, there exists a constant T so that all the suitable weak solutions to the Boussinesq systems become regular for t T. 2. All suitable weak solutions converge to a global attractor of the Boussinesq system for ε(r T T R S S) κ 0. The topics in this direction include but are not limited to; 1. To improve the convergence rate: I am recently trying to improve the convergence rate from O(ε) to O(ε 2 ) using the asymptotic expansion methods of the singular perturbation theory [36, 14]. 2. Comparison of global attractors: due to the discrepancy in natural phase spaces of Boussinesq system involving the velocity, the temperature and the salinity, and the infinite Prandtl number system involving only the temperature and the salinity, we need to project the phase space of the Boussinewq system down to the temperature and salinity filed only, and shift the phase space of the infinite Prandtl number system to the product space of velocity and temperature-salinity. We can compare the global attractor s after projection which is easier than the comparison after shifting the phase space of the infinite Prandtl number system because it requires similar analysis to the upper semicontinuity of global attractors for dynamical systems [8, 34]. Page 9

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