Tropical Atmospheric Circulations: Dynamic Stability and Transitions

In this article, we present a mathematical theory of the Walker circulation of the large-scale atmosphere over the tropics. This study leads to a new metastable state oscillation theory for the El Nino Southern Oscillation (ENSO), a typical inter-annual climate low frequency oscillation. The mathematical analysis is based on 1) the dynamic transition theory, 2) the geometric theory of incompressible flows, and 3) the scaling law for proper effect of the turbulent friction terms, developed recently by the authors.


Introduction
The atmosphere and ocean around the earth are rotating geophysical fluids, which are also two important components of the climate system. The phenomena of the atmosphere and ocean are extremely rich in their organization and complexity, involving a broad range of temporal and spatial scales. According to J. von Neumann [12], the motion of the atmosphere can be divided into three categories depending on the time scale of the prediction. They are motions corresponding respectively to the short time, medium range and long term behavior of the atmosphere. 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. One typical source of the inter-annual climate low frequency variability is El Niño-Southern Oscillation (ENSO), which is associated with the Walker circulation of the atmosphere over the tropics. The main objective of this article is to introduce a mathematical theory of the general circulation of the atmosphere over the tropics.
The modeling part of the study is based on two important ingredients. First, from the general circulation point of view, there are three invariant regions of the atmosphere: the northern hemisphere, the southern hemisphere and the region over the tropics corresponding to zero latitude. The second is the new scaling law introduced recently by the authors, leading to correct circulation length scales.
The model is then analyzed using the dynamic transition theory and the geometric theory of incompressible flows, both developed recently by the authors. We refer the interested readers to [5] and the references therein for the geometric theory, and to the appendix, [4,10] and the references therein for the dynamic transition The work was supported in part by the Office of Naval Research and by the National Science Foundation.
theory. We mention in particular here that the main philosophy of the dynamic transition theory is to search for the full set of transition states, giving a complete characterization on stability and transition. This complete set of transition states represent the "physical reality" after the transition, and are described by a local attractor, rather than some steady states or periodic solutions or other type of orbits as part of this local attractor. With this theory, many longstanding phase transition problems are either solved or become more accessible leading to a number of physical predictions. For example, the study of phase transitions of helium-3 leads not only to a theoretical understanding of the phase transitions to superfluidity observed by experiments, but also to such physical predictions as the existence of a new superfluid phase C for liquid helium-3 [9].
The main results obtained in this article, on the one hand, verifies the general circulation patterns over the tropics associated with the Walker circulation, and, on the other hand, lead to a new mechanism of El Niño Southern Oscillation (ENSO), which will be published in an accompanying paper [8]. This new mechanism of the ENSO amounts to saying that ENSO, as a self-organizing and self-excitation system, with two highly coupled processes. The first is the oscillation between the two metastable warm (El Niño phase) and cold events (La Niña phase), and the second is the spatiotemporal oscillation of the sea surface temperature (SST) field. The interplay between these two processes gives rises the climate variability associated with the ENSO, leads to both the random and deterministic features of the ENSO, and defines a new natural feedback mechanism, which drives the sporadic oscillation of the ENSO.
The paper is organized as follows. In Section 2, we introduce the atmospheric circulation model over the tropics, which is analyzed in Sections 3 and 4, with concluding remarks given in Section 5.

Atmospheric Model over the Tropics
Physical laws governing the motion and states of the atmosphere and ocean can be described by the general equations of hydrodynamics and thermodynamics. As discussed in the Introduction, the atmospheric motion equations over the tropics are the Boussinesq equations restricted on θ = 0, where the meridional velocity component u θ is set to zero, and the effect of the turbulent friction is taking into considering using the scaling law derived in [10]: Here σ i = C i h 2 (i = 0, 1) represent the turbulent friction, a is the radius of the earth, the space domain is taken as M = S 1 a × (a, a + h) with S 1 a being the onedimensional circle with radius a, and For simplicity, we denote In atmospheric physics, the temperature T 1 at the tropopause z = a + h is a constant. We take T 0 as the average on the lower surface z = a. To make the nondimensional form, let Also, we define the Rayleigh number, the Prandtl number and the scaling laws by Omitting the primes, the nondimensional form of (2.1) reads where (x 1 , x 2 ) ∈ M = (0, 2πr 0 ) × (r 0 , r 0 + 1), δ 0 and δ 1 are as in (2.2), (u · ∇) and ∆ as usual differential operators, and The problem is supplemented with the natural periodic boundary condition in the x 1 -direction, and the free-slip boundary condition on the top and bottom boundary: (2.6) Here ϕ(x 1 ) is the temperature deviation from the average T 0 on the equatorial surface and is periodic, i.e., 2πr0 0 ϕ(x 1 )dx 1 = 0 and ϕ(x 1 ) = ϕ(x 1 + 2πr 0 ).
The deviation ϕ(x 1 ) is mainly caused by a difference in the specific heat capacities between the sea water and land.

Walker Circulation under the Idealized Conditions
In an idealized case, the temperature deviation ϕ vanishes. In this case, the study of transition of (2.3) is of special importance to understand the longitudinal circulation. Here, we are devoted to discuss the dynamic bifurcation of (2.3), the Walker cell structure of bifurcated solutions, and the convection scale under the idealized boundary condition (3.1) ϕ(x 1 ) = 0 for any 0 ≤ x 1 ≤ 2πr 0 , For the problem (2.3) with (2.5) and (2.6), let T ) satisfies one of (2.5) and (2.6)}.
Then, define the operators L R = A + B R and G : where ψ = (u, T ) ∈ H 1 , P : L 2 (M ) 3 → H is the Leray Projection, and δ i = 2 r 2 0 + δ i (i = 0, 1). Under the definitions (3.2), the problem (2.3) with one of (2.5) and (2.6) is equivalent to the following abstract equation Consider the eigenvalue problem which is equivalent to with the boundary conditions (2.5) and (2.6).

TROPICAL ATMOSPHERIC CIRCULATIONS: DYNAMIC STABILITY AND TRANSITIONS 5
We shall see later that these equations (3.5) are symmetric, which implies, in particular, that all eigenvalues β j (R) are real, and there exists a number R c , called the first critical Rayleigh number, such that The following theorem provides a theoretical basis to understand the equatorial Walker circulation. (1) When the Rayleigh number R ≤ R c , the equilibrium solution (u, T ) = 0 is globally stable in the phase space H. Proof of Theorem 3.1. We proceed in the following several steps.
Step 1. We shows that equations (2.3) have an equivalent form as the classical 2D Bénard problem.
Since the velocity field u defined on M = S 1 × (r 0 , r 0 + 1) is divergence-free, there exists a stream function ϕ such that satisfying the given boundary conditions. Therefore, the following two vector fields are gradient fields, which can be balanced by ∇p in (2.3). Hence, (2.3) are equivalent to (3.9) Therefore, equation (3.3) is also an abstract form of (3.9). Thus, equations (3.4) and (3.5) are symmetric. It is clear that the operator G defined in (3.2) is orthogonal. Hence, the attractor bifurcation theorem for the Rayleigh-Bénard convection proved in [3] is also valid for the problem (2.3) with (2.5), (2.6). Thus, this transition of the problem is Type-I (i.e., continuous), and Assertion (1) is proved.
Step 2. Proof of Assertion (2). We only need to verify that the attractor A R is homeomorphic to a circle, and consisting of singular points of (3.3). We consider the first eigenvectors of (3.5) at R = R c . By (3.8), equations (3.5) at R = R c are equivalent to the form (3.10) with the boundary conditions (2.5) and (2.6) As in [3,6], we see that the multiplicity of R c in (3.10) is m = 2, and the corresponding eigenvectors are given by where the functions H(x 2 ) and Φ(x 2 ) satisfy that . Since (3.13) with (3.14) are symmetric, we define a number α by where α is the number as in (A.6), and Ψ is the center manifold function and by [4] it satisfies that and L R is defined by (3.2), and G(ψ, ψ) defined by for anyψ = (ũ,T ), ψ = (u, T ) ∈ H 1 . Obviously, for any ψ,ψ ∈ H 1 , we have Hence, we derive from (3.15) and (3.16) that Due to (3.8), −L R is a symmetric sectorial operator. Thus, we have Then, as in [6], Assertion (2) follows.
By the structural stability theorem in [5], the vector field e 0 in (3.19) is not structurally stable in H 1 , because the boundary saddle points on x 2 = r 0 are connected to saddle points on x 2 = r 0 + 1, a different connected component. However, e 0 is structurally stable in the spacẽ To see this, we know that if u ∈H, then u = (u 1 , u 2 ) =constant, and u 1 has the Fourier expansion It follows that If we can prove that the vector field e given by (3.18) is inH, then, as R c < R < R c + δ, e is topologically equivalent to e 0 . Hence, to prove Assertion (3), it suffices to verify that e ∈H.
Obviously,H is an invariant space for the operator L R + G defined by (3.2), and the orthogonal complementaryH ⊥ ofH in H is It is readily to prove that all steady state solutions of (3.3) are inH. Thus, Assertion (3) is proved.
Then, for any ψ 0 ∈ H \ (H ∪ Γ), it follows that there exists a time t 0 > 0 such that the velocity field u(t, ψ 0 ) of ψ(t 0 , ψ 0 ) is topologically equivalent to the following vector field for any t > t 0 where e is as in (3.18). The vector field e has the roll structure as shown in Figure 3.1, and c 0 e −δ 0 t is a parallel channel flow. Hence, it is easy to show thatũ = e + c 0 e −δ 0 t is topologically equivalent to the structure as shown in Figure 3.2 (a) as α 0 < 0, and to the structure as shown in Figure 3.2 (b) as α 0 > 0, for t > t 0 large. Assertion (4) is proved.

Walker circulation under natural conditions
We now return the natural boundary condition ϕ(x 1 ) ≡ 0 and Q = 0.
In this case, equations (2.3) admits a steady state solution Consider the deviation from this basic state:

Then (2.3) becomes (4.2)
The boundary conditions are the free-free boundary conditions given by Let the operators L R = A+B R and G : H 1 → H be as in (3.2), and L ε : H 1 → H be defined by Then, the problem (4.2) and (4.3) is equivalent to the abstract form Consider the eigenvalue problem It is known that |ϕ(x 1 )| and Q = 0 are small, with ∆T = T 0 − T 1 ∼ = 100 • C as unit. Hence, the steady state solution (V, J) is also small: Thus, (4.2) is a perturbation equation of (2.3).
Since perturbation terms involving (V, J) are not invariant under the zonal translation (in the x 1 -direction), for general small functions ϕ(x 1 ) = 0, the first eigenvalues of (4.2) are (real or complex) simple, and by the perturbation theorems in [4], all eigenvalues of linearized equation of (4.2) satisfy the following principle of exchange of stability (PES): Reβ e j (R ε c ) < 0 for any j ≥ m + 1, where m = 1 as β ε 1 (R) is real, m = 2 as β ε 1 (R) is complex near R ε c , and R ε c is the critical Rayleigh number of perturbed system (4.2).
The following two theorems follow directly from Theorem 3.1 and the perturbation theorems in the appendix or in [4].
Theorem 4.1. Let β ε 1 (R) near R = R ε c be a real eigenvalue. Then the system (4.2) has a transition at R = R ε c , which is either mixed (Type-III) or continuous (Type-I), depending on the temperature deviation ϕ(x 1 ). Moreover, we have the following assertions: (1) If the transition is Type-I, then as R ε c < R < R ε c + δ for some δ > 0, the system bifurcates at R ε c to exactly two steady state solutions ψ 1 and ψ 2 in H, which are attractors. In particular, space H can be decomposed into two open sets U 1 , U 2 : such that ψ i ∈ U i (i = 1, 2), and ψ i attracts U i . (2) If the transition is Type-III, then there is a saddle-node bifurcation at R = R * with R * < R ε c such that the following statements hold true: (a) if R * < R < R ε c + δ with R = R ε c , the system has two steady state solutions ψ + R and ψ − R which are attractors, as shown in Figure 4.1, such that for R > R ε c , there exists a time t 0 ≥ 0 such that for any t > t 0 the velocity field u(t, ψ 0 ) is topologically equivalent to the structure as shown in Figure  3  Theorem 4.2. Let β ε 1 (R) be complex near R = R ε c . Then the system (4.2) bifurcates from (ψ, R) = (0, R ε c ) to a periodic solution ψ R (t) on R ε c < R, which is an attractor, and ψ R (t) can be expressed as , α and ρ are constants depending on ϕ(x 1 ), and ψ 1 ,ψ 1 are first eigenfunctions of linearized equations of (2.3).

Concluding Remarks
In this article, a careful examination of the dynamic transitions and stability of the large-scale atmospheric flows over the tropics, associated with the Walker circulation and the ENSO, is given. The analysis and the results obtained show the following from the physical point of view. Third, Theorem 4.1 does give a characterization of the ENSO, leading a correct oscillation mechanism of the ENSO between two metastable El Niño and La Niña events, which is further studied in [8].

Appendix A. Dynamic Transition Theory for Nonlinear Systems
In this appendix we recall some basic elements of the dynamic transition theory developed by the authors [4,7], which are used to carry out the dynamic transition analysis for the binary systems in this article.
A.1. New classification scheme. Let X and X 1 be two Banach spaces, and X 1 ⊂ X a compact and dense inclusion. In this chapter, we always consider the following nonlinear evolution equations where u : [0, ∞) → X is unknown function, and λ ∈ R 1 is the system parameter. Assume that L λ : X 1 → X is a parameterized linear completely continuous field depending continuously on λ ∈ R 1 , which satisfies In this case, we can define the fractional order spaces X σ for σ ∈ R 1 . Then we also assume that G(·, λ) : X α → X is C r (r ≥ 1) bounded mapping for some 0 ≤ α < 1, depending continuously on λ ∈ R 1 , and Hereafter we always assume the conditions (A.2) and (A.3), which represent that the system (A.1) has a dissipative structure.
Let the eigenvalues (counting multiplicity) of L λ be given by The following theorem is a basic principle of transitions from equilibrium states, which provides sufficient conditions and a basic classification for transitions of nonlinear dissipative systems. This theorem is a direct consequence of the center manifold theorems and the stable manifold theorems; we omit the proof.
Theorem A.1. Let the conditions (A.4) and (A.5) hold true. Then, the system (A.1) must have a transition from (u, λ) = (0, λ 0 ), and there is a neighborhood U ⊂ X of u = 0 such that the transition is one of the following three types: (1) Continuous Transition: there exists an open and dense set U λ ⊂ U such that for any ϕ ∈ U λ , the solution u λ (t, ϕ) of (A.1) satisfies lim λ→λ0 lim sup t→∞ u λ (t, ϕ) X = 0.
(2) Jump Transition: for any λ 0 < λ < λ 0 + ε with some ε > 0, there is an open and dense set U λ ⊂ U such that for any ϕ ∈ U λ , where δ > 0 is independent of λ. This type of transition is also called the discontinuous transition. (3) Mixed Transition: for any λ 0 < λ < λ 0 + ε with some ε > 0, U can be decomposed into two open sets U λ 1 and U λ 2 (U λ i not necessarily connected): With this theorem in our disposal, we are in position to give a new dynamic classification scheme for dynamic phase transitions.
Definition A.1 (Dynamic Classification of Phase Transition). The phase transitions for (A.1) at λ = λ 0 is classified using their dynamic properties: continuous, jump, and mixed as given in Theorem A.1, which are called Type-I, Type-II and Type-III respectively.
An important aspect of the transition theory is to determine which of the three types of transitions given by Theorem A.1 occurs in a specific problem. Hereafter we present a few theorems in this theory to be used in this article, and we refer interested readers to [10] for a complete description of the theory.
A.3. Singular Separation. In this section, we study an important problem associated with the discontinuous transition of (A.1), which we call the singular separation.
(1) An invariant set Σ of (A.1) is called a singular element if Σ is either a singular point or a periodic orbit.