Tropical atmospheric circulations with humidity effects.

Tropical atmospheric circulations with humidity effects

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Research Cite this article: Hsia C-H, Lin C-S, Ma T, Wang S. 2015 Tropical atmospheric circulations with humidity effects. Proc. R. Soc. A 471: 20140353. http://dx.doi.org/10.1098/rspa.2014.0353 Received: 1 May 2014 Accepted: 31 October 2014

Subject Areas: mathematical physics, applied mathematics Keywords: tropical atmospheric circulation, humidity, transition Author for correspondence: Chun-Hsiung Hsia e-mail: [email protected]

Chun-Hsiung Hsia1 , Chang-Shou Lin2 , Tian Ma3 and Shouhong Wang4 1 Institute of Applied Mathematical Sciences, and 2 Department of

Mathematics, National Taiwan University, Taipei 10617, Taiwan, Republic of China 3 Department of Mathematics, Sichuan University, Chengdu, People’s Republic of China 4 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA The main objective of this article is to study the effect of the moisture on the planetary scale atmospheric circulation over the tropics. The modelling we adopt is the Boussinesq equations coupled with a diffusive equation of humidity, and the humidity-dependent heat source is modelled by a linear approximation of the humidity. The rigorous mathematical analysis is carried out using the dynamic transition theory. In particular, we obtain mixed transitions, also known as random transitions, as described in Ma & Wang (2010 Discrete Contin. Dyn. Syst. 26, 1399–1417. (doi:10.3934/dcds.2010.26.1399); 2011 Adv. Atmos. Sci. 28, 612–622. (doi:10.1007/s00376-010-9089-0)). The analysis also indicates the need to include turbulent friction terms in the model to obtain correct convection scales for the large-scale tropical atmospheric circulations, leading in particular to the right critical temperature gradient and the length scale for the Walker circulation. In short, the analysis shows that the effect of moisture lowers the magnitude of the critical thermal Rayleigh number and does not change the essential characteristics of dynamical behaviour of the system.

1. Introduction One contribution to a Special feature ’Climate dynamics: multiple scales, memory effects and control perspectives’.

This article is part of a research programme to study low-frequency variability of the atmospheric and oceanic flows. As we know, typical sources of

2014 The Author(s) Published by the Royal Society. All rights reserved.


As in the expression of the thermal critical Rayleigh number Rc , the coefficient of the humidity ˜ is negative; this indicates that the presence of humidity lowers the critical Rayleigh number R temperature difference for the onset of the dynamic transition. Third, we remark that the perturbation analysis in [17] can be applied to the case here to carry out the analysis, and we can show that under the natural boundary condition, the underlying system with humidity effect will undergo a mixed-type transition. In addition, as we argued in [17], it is necessary to include turbulent friction terms in the model to obtain correct convection scales for the large-scale tropical atmospheric circulations, leading in particular to the right critical temperature gradient and the length scale for the Walker circulation. Finally, based on these theoretical results, it is easy then to conclude the same mechanism for ENSO as proposed in Ma & Wang [17,18]. In particular, the random transition behaviour of the system explains general features of the observed abrupt changes between strong El Nino and strong La Nina states. Also, with the deterministic model considered in this article, the randomness is closely related to the uncertainty/fluctuations of the initial data between the narrow basins of attractions of the corresponding metastable events, and the deterministic feature is represented by a deterministic coupled atmospheric and oceanic model predicting the basins

...................................................

Le αk2c αk2c r20 1 α1 αT h2 1 4 2 ˜ +α + 1+ + . R α + Rc = − kc 2 kc Le 24 k Le αq αk2c κT r20 r20

2

rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20140353

climate low-frequency variability include the wind-driven (horizontal) and thermohaline (vertical) circulations (THC) of the ocean, and the El Niño Southern Oscillation (ENSO). Their variability, independently and interactively, may play a significant role in climate change, past and future. The primary goal of our study 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 the variability is a challenging problem with important practical implications for geophysical efforts to quantify predictability, analyse error growth in dynamical models and develop efficient forecast methods. ENSO is one of the strongest interannual climate variabilities associated with strong atmosphere–ocean coupling, with significant impacts on global climate. ENSO consists of warm events (El Niño phase) and cold events (La Niña phase) as observed by the equatorial eastern Pacific sea-surface temperature (SST) anomalies, which are associated with persistent weakening or strengthening in the trade winds; see among others [1–16]. An interesting current debate is whether ENSO is best modelled as a stochastic or chaotic system—linear and noise-forced, or nonlinear oscillatory and unstable system [14]. It is obvious that a careful fundamental level examination of the problem is crucial. For this purpose, Ma & Wang [17] initiated a study of ENSO from the dynamical transition point of view and derived in particular a new oscillation mechanism of ENSO. Namely, ENSO is a self-organizing and self-excitation system, with two highly coupled oscillation processes–the oscillation between metastable El Nino and La Nina and normal states, and the spatio-temporal oscillation of the SST. The main objective of this article is to address the moisture effect on the low-frequency variability associated with ENSO. First, as our main purpose is to capture the patterns and general features of the large-scale atmospheric circulation over the tropics, it is appropriate to use the Boussinesq equations coupled with a diffusive equation of humidity. In addition, the humidity effect is also taken into consideration by treating the heating source as a linear approximation of the humidity function. Second, although the introduction of the humidity effect leads to substantial difficulty from the mathematical point of view, we have shown in theorem 4.1 that the humidity does not affect the type of dynamic transition the system undergoes. Namely, we show that under the idealized boundary conditions, only continuous transition (Type I) occurs. However, the critical thermal Rayleigh number is slightly smaller than that in the case without moisture factor. To see this effect of humidity, we refer to formula (3.31), which reads


(a) Atmospheric circulation model The hydrodynamical equations governing the atmospheric circulation is the Navier–Stokes equations with the Coriolis force generated by the Earth’s rotation, coupled with the first law of thermodynamics. Let (ϕ, θ , r) be the spheric coordinates, where ϕ represents the longitude, θ the latitude and r the radial coordinate. The unknown functions include the velocity field u = (uϕ , uθ , ur ), the temperature function T, the humidity function q, the pressure p and the density function ρ. Then the equations governing the motion and states of the atmosphere consist of the momentum equation, the continuity equation, the first law of thermodynamics, the diffusion equation for humidity and the equation of state (for ideal gas), which read ∂u (2.1) + ∇u u + 2Ω × u + ∇p + kρg = μu, ρ ∂t

and

∂ρ + div(ρu) = 0, ∂t ∂T ˜ + κ˜ T T, + u · ∇T + p div u = Q ρcv ∂t ∂q + u · ∇q = S˜ + κ˜ q q ρ ∂t

(2.3)

p = RρT.

(2.5)

(2.2)

(2.4)

Here, 0 ≤ ϕ ≤ 2π , −π/2 ≤ θ ≤ π/2, a < r < a + h, a is the radius of the Earth, h is the height of the troposphere, Ω is the Earth’s rotating angular velocity, g is the gravitational constant, μ, ˜ and S˜ are heat and humidity sources, and k = (0, 0, 1). The differential κ˜ T , κ˜ q , cv , R are constants, Q operators used are as follows: (1) The gradient and divergence operators are given by ∂ 1 ∂ ∂ 1 , , ∇= r cos θ ∂ϕ r ∂θ ∂r and

div u =

1 ∂uϕ 1 ∂ 2 1 ∂(uθ cos θ) (r ur ) + + 2 r cos θ ∂θ r cos θ ∂ϕ r ∂r

(2) In the spherical geometry, although the Laplacian for a scalar is different from the Laplacian for a vectorial function, we use the same notation for both of them uϕ 2 sin θ ∂uθ ∂ur 2 + 2 − 2 ,

u = uϕ + 2 r cos θ ∂ϕ r cos2 θ ∂ϕ r cos2 θ

uθ +

uθ 2 sin θ ∂uϕ 2 ∂ur − 2 , − 2 2 2 ∂θ r r cos θ r cos2 θ ∂ϕ

...................................................

2. Model for atmospheric motion with humidity

3


of attraction and the SST. In addition, from the predictability and prediction point of view, it is crucial to capture more detailed information on the delay feedback mechanism of the SST. For this purpose, the study of an explicit multi-scale coupling mechanism to the ocean is inevitable. In fact, the new mechanism strongly suggests the need and importance of the coupled ocean–atmosphere models for ENSO predication in [1,3–11]. This article is organized as follows. Section 2 gives the objective Boussinesq model with humidity. The eigenvalue problem is analysed in §3. The transition theorem is stated and proved in §4. In §5, the turbulence friction factors are considered and the corresponding critical temperature difference and the wavenumbers of Walker circulation are checked.


(3) The convection terms are given by ∇u u = u · ∇uϕ + u · ∇uθ +

u ϕ uθ u ϕ ur − tan θ, r r

u2ϕ u2ϕ + u2θ u θ ur + tan θ, u · ∇ur − . r r r

(4) The Coriolis term 2Ω × u is given by 2Ω × u = 2Ω(cos θ ur − sin θuθ , sin θuϕ , − cos θuϕ ). Here, Ω is the angular velocity vector of the Earth, and Ω is the magnitude of the angular velocity. The above system of equations is basically the equations used by L. F. Richardson in his pioneering work [19]. However, they are in general too complicated to conduct theoretical analysis. As practised by earlier researchers such as Charney [20], and from the lessons learned by the failure of Richardson’s pioneering work, one tries to be satisfied with simplified models approximating the actual motions to a greater or lesser degree instead of attempting to deal with the atmosphere in all its complexity. By starting with models incorporating only what are thought to be the most important of atmospheric influences, and by gradually bringing in others, one is able to proceed inductively and thereby to avoid the pitfalls inevitably encountered when a great many poorly understood factors are introduced all at once. The simplifications are usually done by taking into consideration some of the main characterizations of the large-scale atmosphere. One such characterization is the small aspect ratio between the vertical and horizontal scales, leading to a hydrostatic equation replacing the vertical momentum equation. The resulting system of equations is called the primitive equations (see among others [21]). Another characterization of large-scale motion is the fast rotation of the Earth, leading to the celebrated quasi-geostrophic equations [22].

(b) Tropical atmospheric circulation model In this article, our main focus is on formation and transitions of the general circulation patterns. For this purpose, the approximations we adopt involve the following components. First, we often use the Boussinesq assumption, where the density is treated as a constant except in the buoyancy term and in the equation of state. Second, because the air is generally not incompressible, we do not use the equation of state for ideal gas; rather, we use the following empirical formula, which can be regarded as the linear approximation of (2.5): (2.6) ρ = ρ0 [1 − αT (T − T0 ) − αq (q − q0 )], where ρ0 is the density at T = T0 and q = q0 , and αT and αq are the coefficients of thermal and humidity expansion. Third, as the aspect ratio between the vertical scale and the horizontal scale is small, the spheric shell, which the air occupies, is treated as a product space S2a × (a, a + h) with the product metric ds2 = a2 dθ 2 + a2 sin2 θ dφ 2 + dz2 . This approximation is extensively adopted in geophysical fluid dynamics. Fourth, the hydrodynamic equations governing the atmospheric circulation over the tropical zone are the Navier–Stokes equations coupled with the first law of thermodynamics and the

...................................................

and

4


∂uϕ ∂(uθ cos θ) 2ur 2 2 − − ∂θ r2 r2 cos θ r2 cos θ ∂ϕ ∂ 1 ∂f ∂ 2f 1 ∂f 2 ∂ 1 cos θ + 2 r . +

f = 2 ∂θ ∂r r cos θ ∂θ r cos2 θ ∂ϕ 2 r2 ∂r

ur −


and

(u · ∇) =

uφ ∂ ∂ + uz , a ∂φ ∂z

f =

∂2 1 ∂2 f + f a2 ∂φ 2 ∂z2

u = (uφ eφ + uz ez ) uφ 2uz 2 ∂uz 2 ∂uφ − 2 eθ + uz − 2 − 2 ez , = uφ + 2 a ∂φ a a ∂φ a

where f is the Laplacian for scalar functions, and u is the Laplace–Beltrami operator on (0, 2π ) × (a, a + h) with the product metric. Based on the thermodynamics, the heat source Q is a function of the humidity q. For simplicity, we take the linear approximation Q = α0 + α1 q,

(2.8)

S = 0.

(2.9)

and the humidity source is taken as zero

The boundary conditions are periodic in the φ-direction (u, T, q)(φ + 2π , z) = (u, T, q)(φ, z),

(2.10)

and free-slip on the Earth surface z = a and the tropopause z = a + h, uz = 0, and

uz = 0,

∂uφ = 0, ∂z ∂uφ = 0, ∂z

T = T0 , T = T1 ,

q = q0 , q = 0,

at z = a

⎫ ⎪ ⎪ ⎬

⎪ ⎪ at z = a + h,⎭

(2.11)

where T0 , T1 , q0 and q1 are constants satisfying T0 > T1

and q0 > 0.

(2.12)

...................................................

where Ω is the Earth’s rotating angular velocity, g the gravitational constant, ρ0 the density ˜ 0 cν , S = S/ρ ˜ 0 and αT and αq are the of air at T = T0 , ν = μ/ρ0 , κT = κ˜ T /ρ0 cν , κq = κ˜ q /ρ0 , Q = Q/ρ coefficients of thermal and humidity expansion. However, the differential operators used in (2.7) are as follows:

5


diffusion equation of the humidity. These equations are restricted on the lower latitude region where the meridional velocity component uθ is zero. Let (φ, z) ∈ M = (0, 2π ) × (a, a + h) be the coordinate, where φ is the longitude, a the radius of the Earth and h the height of the troposphere. The unknown functions include the velocity field u = (uφ , uz ), the temperature function T, the humidity function q and the pressure p. Then, the equations governing the motion and states of the atmosphere read ⎫ uφ u φ uz ∂uφ 2 ∂uz 1 ∂p ⎪ + (u · ∇)uφ + = ν uφ + 2 − 2 − 2Ωuz − ,⎪ ⎪ ⎪ ∂t a ρ0 a ∂φ ⎪ a ∂φ a ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ uφ ∂uz 2uz 2 ∂uφ 1 ∂p ⎪ ⎪ ⎪ + (u · ∇)uz − = ν uz − 2 − 2 + 2Ωuφ − ⎪ ⎪ ∂t a ρ0 ∂z a ∂φ a ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ − g(1 − αT (T − T0 ) − αq (q − q0 )), (2.7) ⎪ ∂T ⎪ ⎪ + (u · ∇)T = κT T + Q, ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ∂q ⎪ ⎪ + (u · ∇)q = κq q + S ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ∂uz 1 ∂uφ ⎪ ⎪ ⎭ + = 0, and a ∂φ ∂z


The problem (2.7)–(2.11) possesses a steady-state solution

6

γ0 2 γ1 3 z − z + c1 z + c0 , T˜ = 6κT 2κT q0 q0 ⎪ q˜ = − z + (h + a) ⎪ ⎪ h h ⎪ ⎪ ⎪ z ⎪ ⎪ ⎪ ˜ ⎭ p˜ = − ρ0 g(1 − αT (T − T0 ) − αq (˜q − q0 )) dz,⎪

and

(2.13)

0

where α1 q0 (h + a) α1 q0 , γ1 = , h h γ1 3 γ1 a a+h γ0 2 γ0 − − c0 = a + a + T0 − (a + h)3 + (a + h)2 + T1 h 6κT 2κT h 6κT 2κT 1 1 γ1 2 γ0 c1 = − (T0 − T1 ) − (h + 3 ha + 3a2 ) − (h + 2a) . h κT 6 2

γ0 = α0 +

and

To obtain the non-dimensional form, let x = hx , t=

a = hr0 ,

h2 t , κT

u=

κT u , h

˜ T = (T0 − T1 )T + T, p=

and

q = q0 q + q˜

ρ0 νκT p + p˜ , h2

and consider (x1 , x2 ) = (r0 φ, z)

and

(u1 , u2 ) = (uφ , uz ).

Omitting the primes, equations (2.7) become 2 ∂u2 1 ∂p ∂u1 = Pr u1 + − 2 u1 − ∂t r0 ∂x1 ∂x1 r0

∂u2 ∂t

∂T ∂t

and

∂q ∂t ∂u2 ∂u1 + ∂x1 ∂x2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ − ωu2 − (u · ∇)u1 − u1 u2 , ⎪ ⎪ r0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ∂u1 2 ∂p ⎪ ⎪ ˜ ⎪ = Pr u2 − − 2 u2 + RT + Rq − ⎪ r0 ∂x1 ∂x2 ⎪ r0 ⎪ ⎪ ⎬ 1 2 ⎪ + ωu1 − (u · ∇)u2 + u1 , ⎪ ⎪ r0 ⎪ ⎪ ⎪ ⎪ ⎪ ˜ ⎪ dT(hx2 ) 1 ⎪ ⎪ u2 + αq − (u · ∇)T, = T − ⎪ ⎪ ⎪ T0 − T1 dx2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = Le q + u2 − (u · ∇)q ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ = 0,

(2.14)

...................................................

u˜ = 0,


⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬


where the non-dimensional physical parameters are

7

and

T

the Lewis number,

ω=

2Ωh2 κT

α=

α1 q0 h2 , κT (T0 − T1 )

the Earth rotation

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(2.15)

and −

˜ 2) 1 dT(hx γ0 h2 1 x2 − r0 − =1+ T0 − T1 dx2 κT (T0 − T1 ) 2 2 1 α1 q0 h x22 − r20 − r0 − , − 2κT (T0 − T1 ) 3

(2.16)

where r0 < x2 < r0 + 1, and r0 is the non-dimensional radius of the Earth. For simplicity, we take the average approximation x2 = r0 + 12 . In this case, (2.16) is given by dT˜ 1 α1 q0 h2 . − =1+ T0 − T1 dx2 24κT (T0 − T1 ) x2 =r0 +1/2

Thus, equation (2.14) is approximated by ∂u1 2 ∂u2 1 ∂p − 2 u1 − = Pr u1 + ∂t r0 ∂x1 ∂x1 r0

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ − ωu2 − (u · ∇)u1 − u1 u2 , ⎪ ⎪ r0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ∂u1 2 ∂p ∂u2 ⎪ ˜ ⎪ = Pr u2 − − 2 u2 + RT + Rq − ⎪ ⎪ ∂t r0 ∂x1 ∂x r0 2 ⎬ + ωu1 − (u · ∇)u2 +

1 2 u , r0 1

∂T = T + γ u2 + αq − (u · ∇)T, ∂t ∂q = Le q + u2 − (u · ∇)q ∂t and

div u = 0,

where γ =1+

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

α . 24

The domain is M = [0, 2π r0 ] × (r0 , r0 + 1), and the boundary conditions are given by ⎫ ⎪ (u, T, q)(x1 + 2π r0 , x2 ) = (u, T, q)(x1 , x2 ) ⎬ ∂u1 ⎭ and u2 = 0, = 0, T = 0, q = 0, at x2 = r0 , r0 + 1.⎪ ∂x2

(2.17)

(2.18)

(2.19)

...................................................

κq Le = κT


⎫ ⎪ the Prandtl number, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ αT g(T0 − T1 )h ⎪ ⎪ the thermal Rayleigh number,⎪ R= ⎪ ⎪ κT ν ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ αq gq0 h ⎪ ⎪ ˜ the humidity Rayleigh number, R= ⎪ ⎬ κ ν

ν Pr = κT


For the problem (2.17)–(2.19), we set the spaces ⎫ H = {(u, T, q) ∈ L (M, R ) | div u = 0, u2 = 0 at r0 , r0 + 1}⎬ 2

(2.20)

3. Eigenvalue problem and principle of exchange of stability (a) Eigenvalue problem To study the transition of (2.17)–(2.19) from the basic state, we need to consider the following eigenvalue problem: ⎫ ⎪ 2 ∂u2 1 ∂p ⎪ ⎪ − 2 u1 − − ωu2 = βu1 , Pr u1 + ⎪ ⎪ r0 ∂x1 ∂x1 ⎪ r0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ∂u1 2 ∂p ˜ − ⎪ − 2 u2 + RT + Rq Pr u2 − + ωu1 = βu2 ,⎪ ⎪ ⎬ r0 ∂x1 ∂x2 r0 (3.1) ⎪ ⎪ ⎪

T + γ u2 + αq = βT, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Le q + u2 = βq ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂u2 ∂u1 ⎪ ⎭ + =0 and ∂x1 ∂x2 supplemented with the boundary conditions (2.19). By the boundary conditions (2.19), the eigenvalues of (3.1) can be solved by separation of variables as follows: ⎧ kx1 ⎪ ⎪ u11 = −uk (x2 ) sin , ⎪ ⎪ r0 ⎪ ⎪ ⎪ kx ⎪ 1 ⎪ ⎪ , u12 = vk (x2 ) cos ⎪ ⎪ r0 ⎪ ⎨ kx 1 ψ 1 = T1 = Tk (x2 ) cos (3.2) , ⎪ r0 ⎪ ⎪ ⎪ kx ⎪ ⎪q1 = qk (x2 ) cos 1 , ⎪ ⎪ r0 ⎪ ⎪ ⎪ ⎪ 1 kx kx ⎪ ⎩p = Ak (x2 ) cos 1 − Bk (x2 ) sin 1 r0 r0 and

⎧ kx1 ⎪ ⎪ u21 = uk (x2 ) cos , ⎪ ⎪ r0 ⎪ ⎪ ⎪ kx1 ⎪ 2 ⎪ ⎪ , u2 = vk (x2 ) sin ⎪ ⎪ r0 ⎪ ⎨ kx1 ψ 2 = T2 = Tk (x2 ) sin , ⎪ r0 ⎪ ⎪ ⎪ kx1 ⎪ ⎪ , q2 = qk (x2 ) sin ⎪ ⎪ r0 ⎪ ⎪ ⎪ ⎪ kx kx ⎪ ⎩p2 = Ak (x2 ) sin 1 + Bk (x2 ) cos 1 , r0 r0

(3.3)

for k = 1, 2, 3, . . .. Based on the continuity equation in (3.1), we have uk =

r0 dvk . k dx2

(3.4)

Let Bk =

ωr0 vk (x2 ) kPr

and

Ak = pk +

2 vk (x2 ). r0

(3.5)

...................................................

H1 = {(u, T, q) ∈ H2 (M, R4 ) ∩ H | (u, T, q) satisfies (2.12)}. ⎭


and

8

4


+ γ vk + αqk = βTk ,

Le D2k qk + vk = βqk Duk = 0,

and

vk = 0,

Tk = 0,

qk = 0,

where D=

d dx2

and D2k =

d2 dx22

at x2 = r0 , r0 + 1,

−

k2 r20

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(3.6)

.

Plugging vk = sin jπ (x2 − r0 ) into (3.6), we see that the eigenvalue β satisfies the cubic equation Pr k2 Pr 2 2 2 2 2 2 (αkj + β)(Le αkj + β) Pr αkj + 2 + β αkj + 4 (αkj + β)(Le αkj + β) r0 r0 −

k2 Pr R r20

2 (γ Le αkj + γβ + α) −

where

2 2

αkj = j π +

k2

k2 Pr R˜ r20

2 (αkj + β) = 0,

(3.7)

(j ≥ 1, k ≥ 1).

(3.8)

1/2

r20

(b) Principle of exchange of stabilities The linear stability of the problem (2.17)–(2.19) is dictated precisely by the eigenvalues of (3.1), which are determined by (3.7). The expansion of (3.7) is Pr k2 Pr 3 2 β + (1 + Le + Pr)αkj + 2 + 4 2 β 2 r0 r0 αkj Pr k2 Pr k2 4 2 ˜ β + (Pr + Le Pr + Le)αkj + 2 (1 + Le) αkj + 2 − 2 2 (γ R + R) r0 r0 r0 αkj 4 Le αkj Pr Pr k2 4 2 2 ˜ 2 + Le αkj Pr αkj + 2 + 2 2 −αkj R − αR − γ Le αkj R + r0 r0 αkj r20 = 0.

(3.9)

We see that β = 0 is a solution of (3.9) if and only if 4 Le αkj Pr Pr k2 4 2 2 ˜ 2 Le αkj Pr αkj + 2 + 2 2 −αkj R − αR − γ Le αkj R + = 0. r0 r0 αkj r20

(3.10)

Inferring from (2.15) and (2.18), αR = and

α1 αT gq0 h5 κT2 ν

γR=R +

α1 αT h2 ˜ = R αq κT

α1 αT gq0 h5 24κT2 ν

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ α1 αT h2 ˜ ⎪ ⎪ =R+ R.⎪ ⎭ 24αq κT

(3.11)

9 ...................................................

D2k Tk

system of ordinary


Owing to (2.19), plugging (3.2) or (3.3) into (3.1), we obtain the following differential equations: 1 k Pr D2k uk − 2 uk − pk = βuk , r0 r0 2 2 ˜ Pr Dk vk − 2 vk + RTk + Rqk − Dpk = βvk , r0


Hence, we rewrite (3.10) as

α1 αT h2 2κ Le αq αkj T

4 R˜ + αkj

r20 k2

2 + αkj

1

10

r20

+

2 αkj

r20

.

(3.12)

We define the critical Rayleigh number Rc as 2 2 Le αkj αkj r20 1 α1 αT h2 1 4 2 ˜ +α + 1+ Rc = min − R kj 2 αkj + 2 + 2 2κ Le 24 k k,j≥1 Le αq αkj r0 r0 T 2 2 2 Le αk1 αk1 1 α1 αT h2 1 4 r0 2 ˜ + 1+ R + αk1 2 αk1 + 2 + 2 , = min − 2 κ Le 24 k k≥1 Le αq αk1 r0 r0 T

(3.13)

where 2 = π2 + αk1

k2 r20

.

Based on (3.9)–(3.12), by definition (3.13), we shall prove the following principle of exchange of stabilities later. Lemma 3.1. Let βjk be the eigenvalues of (3.1) that satisfy (3.9). Let kc ≥ 1 be integer minimizing (3.13), and βki c (1 ≤ i ≤ 3) be solution of (3.9) with (k, j) = (kc , 1), and Reβk3c ≤ Reβk2c ≤ Reβk1c . Then βk1c must be real near R = Rc , and ⎧ ⎪ ⎪ ⎨< 0, 1 βkc (R) = 0, ⎪ ⎪ ⎩> 0, j Reβkc (Rc ) < 0,

R < Rc , R = Rc , R > Rc , for j = 2, 3.

Moreover, we have βjk (Rc ) < 0

for βjk being real and βjk = βk1c .

Remark 3.2. The value Rc defined by (3.13) is called the critical Rayleigh number at which the principle of exchange of stability (PES) holds. It provides the critical temperature difference

Tc = T0 − T1 given by αT gh3 Tc = Rc . κT ν Equivalently, we have

Tc =

Le αk2c 1 1 α1 αT h2 κT ν + 1 + − R˜ Le 24 αT gh3 Le αq αk2c 1 κT 2 αk2c 1 1 2 4 r0 + αkc 1 2 αkc 1 + 2 + 2 , kc r0 r0

where kc ≥ 1 is the integer which satisfies (3.13).

(3.14)

...................................................

2 Le αkj 1 + 1+ Le 24


R=−


Next, we consider the PES for the complex eigenvalues of (3.1). Let β = iρ0 (ρ0 = 0) be a zero of (3.9). Then

ρ02

and

= Le

4 αkj

2 Pr αkj

+

2 (1 + Le + Pr)αkj

Pr

+

Pr

r20

+

r20 +

Pr

(1 + Le) Pr k2

2 αkj

+

k2

r20

−

Pr k2 2 r20 αkj

2 ˜ R − αR − γ Le −αkj

2 r20 αkj

˜ (γ R + R)

2 R+ αkj

4 Le αkj

r20

.

r20

Hence, equation (3.9) has a pair of purely imaginary solutions ±iρ0 if and only if the following condition holds true: 2 + (1 + Le + Pr)αkj

+

Pr r20

= Le

(1 + Le)

Pr

2 Pr αkj

4 (Pr + Le + Le Pr)αkj

r20 2 αkj

4 αkj

+

+

k2

r20 2Pr r20

−

+

Pr k2

˜ (γ R + R)

2 r20 αkj Pr k2 2 r20 αkj


2 R+ αkj

4 Le αkj

r20

> 0.

(3.15)

It follows from (3.11) and (3.15) that

Pr

2 + (Pr + 1)αkj r20

α1 αT h2 Pr 1 Pr 2 2 + (Pr + 1)αk∗ + (Pr + Le)αk∗ R= 1− − R˜ c c αq κT 24 r20 r20 2 r20 αkj Pr 6 2 4 + Pr (1 + Le + Pr)αkj + 2 (1 + Le)αkj + 2 Le(1 + Le)αkj k Pr r0 (1 + Le) 2 k2 Le Pr Pr k2 2 + αkj + 4 + 4 (1 + Pr)αkj + 2 , (3.16) r20 r0 r0 r0

2 is as defined in (3.8). where αkj According to (3.16), we define the critical Rayleigh number for the complex PES as follows:

R∗c = min

1

2 + (Pr + 1)αkj α1 αT h2 Pr 1 Pr 2 2 ˜ × + (Pr + 1)α + (Pr + Le)α 1− − R kc∗ kc∗ αq κT 24 r20 r20 2 r20 αkj Pr 6 2 4 (1 + Le)αkj + 2 Le(1 + Le)αkj + Pr (1 + Le + Pr)αkj + 2 k Pr r0 (1 + Le) 2 k2 Le Pr Pr k2 2 + αkj + 4 + 4 (1 + Pr)αkj + 2 . r20 r0 r0 r0

k,j≥1

Pr/r20

Then we have the following complex PES.

(3.17)

...................................................

= (Pr + Le

4 Pr + Le)αkj


ρ02

11


Furthermore, for all complex eigenvalues βkj of (3.1), we have Reβkj (R∗c ) < 0,

for (k, j) = (kc∗ , j∗c ).

(c) Proof of lemmas 3.1 and 3.3 We note that all eigenvalues of (3.1) are determined by (3.9) and the eigenvalue equations

T + αq = βT Le q = βq,

and

(3.18)

with the boundary condition (2.19). It is clear that the eigenvalues of (3.18) are real and negative. Hence, it suffices to prove lemmas 3.1 and 3.3 for eigenvalues βkj of (3.9). We rewrite (3.9) as β 3 + b2 β 2 + b1 β + b0 = 0,

(3.19)

where 2 + b2 = (1 + Le + Pr)αkj

b1 = (Pr + Le

and

b0 = Le

r20

4 Pr + Le)αkj

4 αkj

Pr

2 Pr αkj

+

Pr r20

+

+

+

k2 Pr 2 r40 αkj

Pr r20

,

(1 + Le)

Pr k2 2 r20 αkj

2 αkj

+

k2 r20

Pr k2

−

2 r20 αkj


˜ (γ R + R)

2 R+ αkj

4 Le αkj

r20

.

We see that b0 , b1 , b2 > 0 at R = 0 for all k, j ≥ 1,

(3.20)

which implies that all real eigenvalues of (3.9) are negative at R = 0. By the definition of Rc , we find ⎫ b0 > 0, ∀(k, j) = (kc , 1), 0 ≤ R ≤ Rc ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎬ ⎪ > 0, R < R ⎪ c ⎨ (3.21) ⎪ and b0 = 0, R = Rc , for (k, j) = (kc , 1).⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ < 0, R > R , c As b0 = −βkj1 βkj2 βkj3 , lemma 3.1 follows from (3.20) and (3.21). Next, let βkc∗ j∗c , the solution of (3.9), take the form βkc∗ j∗c (R) = λ(R) + iσ (R) and

λ(R) → 0,

⎫ ⎬

σ (R) → σ0 , as R → R∗c .⎭

(3.22)

Inserting (3.22) into (3.19), we obtain ⎫ (−3σ 2 + b1 )λ + b0 − b2 σ 2 + o(λ) = 0 ⎬ and

− σ 3 + σ b1 + 2σ b2 λ + o(λ) = 0.⎭

(3.23)

...................................................

c

12


Lemma 3.3. Let kc∗ , j∗c be the integers satisfying (3.17), and βk1∗ j∗ and βk2∗ j∗ be the pair of complex c c c c eigenvalues of (3.9) with (k, j) = (kc∗ , j∗c ) near R = R∗c . Then, ⎧ ∗ ⎪ ⎪ ⎨< 0, R < Rc , Reβk1∗ j∗ = Reβk2∗ j∗ = 0, R = R∗c , c c c c ⎪ ⎪ ⎩> 0, R > R∗ .


As σ0 = 0, we infer from (3.23) that

Thus, we have λ(R) =

b0 − b1 b2 − 2b22 λ + o(λ). 2b1 + 6b2 λ

(3.24)

Under the condition (3.15), we see b1 > 0. It follows from (3.22) and (3.24) that ⎧ ⎪ ⎪ ⎨< 0, Reβkc∗ j∗c (R) = λ(R) = 0, ⎪ ⎪ ⎩> 0,

b0 < b1 b2 , b0 = b1 b2 ,

(3.25)

b0 > b1 b2 ,

for R near R∗c . By the definition of R∗c , it is easy to see that ⎧ ⎪ ⎪ ⎨< b1 b2 , b0

=b b , ⎪ 1 2 ⎪ ⎩> b b , 1 2

R < R∗c , R = R∗c ,

(3.26)

R > R∗c .

This proves lemma 3.3. By lemmas 3.1 and 3.3, we immediately obtain the following theorem which provides a criterion to determine the equilibrium and the spatio-temporal oscillation transitions. Theorem 3.4. Let Rc and R∗c be the parameters defined by (3.13) and (3.17) respectively. Then, the following assertions hold true. (i) When Rc < R∗c , the first critical-crossing eigenvalue of the problem (3.1) is βk1c given by lemma 3.1, i.e. ⎧ ⎪ ⎪ ⎨< 0, if R < Rc , 1 (3.27) βkc (R) = 0, if R = Rc , ⎪ ⎪ ⎩> 0, if R > R c

and Reβ(Rc ) < 0,

(3.28)

for all other eigenvalues β of (3.1). (ii) When R∗c < Rc , the first critical-crossing eigenvalues are the pair of complex eigenvalues βk1∗ j∗ and c c

βk2∗ j∗ given by lemma 3.3, namely, c c

⎧ ⎪ ⎪< 0, ⎨

Reβk1∗ j∗ (R) = Reβk2∗ j∗ (R) = 0, c c c c ⎪ ⎪ ⎩> 0,

if R < R∗c , if R = R∗c , if

(3.29)

R > R∗c

and Reβ(R∗c ) < 0, for all other eigenvalues β(R) of (3.1).

(3.30)

...................................................

b0 − b2 σ 2 b0 − b1 b2 − 2b22 λ = + o(λ). 2b1 + 6b2 λ 3σ 2 − b1


λ(R) + o(λ) =

13


R∗c =

1 Pr/r20 + (Pr + 1)αk2∗ c α1 αT h2 Pr 1 Pr 2 2 ˜ × + (Pr + 1)αk∗ + (Pr + Le)αk∗ 1− − R c c αq κT 24 r20 r20 r20 αk2∗ Pr c 6 2 (1 + Le)αk4∗ + 2 Le(1 + Le)αk∗ + Pr (1 + Le + Pr)αk∗ + 2 c c c k Pr r0 (1 + Le) 2 kc∗ 2 Le Pr Pr kc∗ 2 + α + + (1 + Pr)αk2∗ + 2 , ∗ kc 2 4 4 c r0 r0 r0 r0

(3.32)

where αk2c = π 2 +

kc2 r20

and αk2∗ = π 2 + c

kc∗ 2 r20

.

(3.33)

4. Transition theorem Inferring from theorem 4.1, system (2.17)–(2.19) has a transition to equilibria at R = Rc provided Rc < R∗c and has a transition to spatio-temporal oscillation at R = R∗c provided R∗c < Rc , where the critical values Rc and R∗c are defined by (3.13) and (3.17), respectively. Theorem 4.1. For the problem (2.17)–(2.19), we have the following assertions. (1) When R < min{Rc , R∗c }, the equilibrium solution (u, T, q) = 0 is stable in H. (2) If Rc < R∗c , then this problem has a continuous transition at R = Rc , and bifurcates from ((u, T, q), R) = (0, Rc ) to an attractor ΣR = S1 on R > Rc which is a cycle of steady-state solutions. (3) As R∗c < Rc , the problem has a transition at R = R∗c , which is either of continuous type or of jump type, and it transits to a spatio-temporal oscillation solution. In particular, if the transition is continuous, then there is an attractor of three-dimensional homological sphere S3 is bifurcated from ((u, T, q), R) = (0, R∗c ) on R > R∗c , which contains no steady-state solutions. Remark 4.2. Although the introduction of the humidity effect leads to substantial difficulty from the mathematical point of view, we see from theorem 4.1 that the humidity does not affect the type of dynamic transition the system undergoes. In particular, as Ma & Wang [17,18], the random transition behaviour of the system in the general case explains general features of the observed abrupt changes between strong El Nino and strong La Nina states. Also, with the deterministic model considered in this article, the randomness is closely related to the uncertainty/fluctuations of the initial data between the narrow basins of attractions of the corresponding metastable events, and the deterministic feature is represented by a deterministic coupled atmospheric and oceanic model predicting the basins of attraction and the SST. From the predictability and prediction point of view, it is crucial to capture more detailed information on the delay feedback mechanism of the SST. For this purpose, the study of an explicit multi-scale coupling mechanism to the ocean is inevitable. In fact, the new mechanism strongly suggests the need and importance of the coupled ocean–atmosphere models for ENSO predication in [1,3–11]. Finally, by (3.31), the critical thermal Rayleigh number is slightly smaller than that in the case without moisture factor [17,18].

...................................................

and

14


Remark 3.5. In the atmospheric science, the integer j∗c in (3.29) is 1. Hence, the critical Rayleigh numbers Rc and R∗c are given by 2 Le αk2c αk2c 1 1 α1 αT h2 4 r0 2 ˜ + 1+ Rc = − + (3.31) R + α α + kc 2 kc Le 24 k Le αq αk2c κT r20 r20


ψ = (u, T, q) ∈ H1 . Thus, the problem (2.17)–(2.19) is expressed in form of dψ = LR ψ + G(ψ), dt

(4.2)

and the eigenvalue problem (3.1) with the condition (2.19) is rewritten as LR ψ = βψ.

(4.3)

Step 2. We shall calculate the centre manifold reduction for (4.2) in this step. Let ψk1c and ψk2c be the

eigenfunctions of (4.3) corresponding to βk1c (R), where βk1c is the eigenvalue of (4.3) in the case of ∗

∗

(3.27). Denote the conjugate eigenfunctions of ψk1c and ψk2c by ψk1c and ψk2c , i.e. ∗

∗

L∗R ψki c = βk1c ψki c , The corresponding equations of (4.4) read 2 ∂u∗2 1 − 2 u∗1 − Pr u∗1 + r0 ∂x1 r0 2 ∂u∗1 2 − 2 u∗2 − Pr u∗2 − r0 ∂x1 r0

i = 1, 2.

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ⎪ ∂p ∗ ∗ ∗ ∗⎪ ⎪ + γ T + q − ωu1 = βu2 ,⎪ ⎪ ⎬ ∂x2 ∂p∗ ∂x1

+ ωu∗2

= βu∗1 ,

T∗ + Pr Ru∗2 = βT∗ , ˜ ∗ + αT∗ = βq∗ Le q∗ + Pr Ru 2 and

(4.4)

∂u∗1 ∂u∗ + 2 = 0. ∂x1 ∂x2

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.5)

Express ψ ∈ H1 as ψ = xψk1c + yψk2c + Φ(x, y, R),

(4.6)

where (x, y) ∈ R2 and Φ(x, y, R) is the centre manifold function of (4.2) near R = Rc . For the reference of centre manifold functions, see, for example, Section 2.4 of [23]. Then, the reduction equations of (4.2) are ⎫ dx 1 1∗ ⎪ ⎪ = βk1c x +

G(ψ), ψ ⎪ ∗ kc ⎪ dt ⎬

ψk1c , ψk1c (4.7) ⎪ 1 dy 1 2∗ ⎪ ⎪ = βkc y + and ∗ G(ψ), ψkc .⎪ ⎭ dt

ψk2c , ψk2c

...................................................

where P : L2 (M, R4 ) → H is the Leray projection and

15


Proof of theorem 4.1. We shall prove this theorem with several steps. Step 1. Let H and H1 be the spaces defined by (2.20). We define the operators LR = A + BR : H1 → H and G : H1 → H by ⎫ 2 ∂u2 2 ∂u1 ⎪ ⎪ , Pr u2 − , T, Le q , Aψ = P Pr u1 + ⎪ ⎪ r0 ∂x1 r0 ∂x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ Pr 2 ˜ (4.1) BR ψ = P − 2 u1 − ωu2 , −Pr 2 u2 − RT − Rq + ωu1 , γ u2 + αq, u2 ⎪ r0 r0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎭ and G(ψ) = −P (u · ∇)u1 + u1 u2 , (u · ∇)u2 − u21 , (u · ∇)T, (u · ∇)q) , ⎪ r0 r0

Downloaded from http://rspa.royalsocietypublishing.org/ on December 22, 2014 ∗

Step 3. Computation of ψki c and ψki c (i = 1, 2). Based on (3.2)–(3.8), we can derive

16

⎧ kc x1 ⎪ 2 ⎪ ⎪u1 = ukc cos π (x2 − r0 ) cos r , ⎪ ⎪ 0 ⎪ ⎪ kc x1 ⎪ ⎪ ⎨u22 = vkc sin π (x2 − r0 ) sin , r0 ψk2c = kc x1 ⎪ ⎪ , ⎪T2 = Tkc sin π (x2 − r0 ) sin ⎪ ⎪ r0 ⎪ ⎪ ⎪ x k ⎪ ⎩q2 = qkc sin π (x2 − r0 ) sin c 1 , r0

(4.9)

and

where ukc = and

Tkc =

r0 π , kc

⎫ ⎪ ⎪ ⎪ ⎬

vkc = 1,

α + γ (Le αk2c + βk1c ) (αk2c + βk1c )(Le αk2c + βk1c )

,

qk =

1 Le αk2c + βk1c

⎪ ⎪ .⎪ ⎭

(4.10)

Similarly, we can also derive from (4.5) the conjugate eigenfunctions as follows: ⎧ kc x1 ⎪ 1∗ ∗ ⎪ ⎪u1 = −ukc cos π (x2 − r0 ) sin r , ⎪ ⎪ 0 ⎪ ⎪ ⎪ 1 ∗ = v ∗ sin π (x − r ) cos kc x1 , ⎪ ⎨ u 2 0 2 kc ∗ r0 ψk1c = kc x1 ∗ ⎪ 1 ∗ ⎪ , ⎪T1 = Tkc sin π (x2 − r0 ) cos ⎪ ⎪ r0 ⎪ ⎪ ∗ ⎪ x k ⎪ ⎩q1 = q∗ sin π (x2 − r0 ) cos c 1 kc r0

(4.11)

⎧ kc x1 ∗ ⎪ ⎪ u21 = u∗kc cos π (x2 − r0 ) cos , ⎪ ⎪ r0 ⎪ ⎪ ⎪ ∗ x k ⎪ c 1 ⎪ ⎨u22 = vk∗c sin π (x2 − r0 ) sin , ∗ r0 ψk2c = x k ⎪ ⎪T2 ∗ = T∗ sin π (x2 − r0 ) sin c 1 , ⎪ ⎪ kc ⎪ r0 ⎪ ⎪ ⎪ 2∗ kx ⎪ ∗ ⎩q = q sin π (x2 − r0 ) sin c 1 , kc r0

(4.12)

and

where u∗kc = and

Tk∗c =

r0 π , kc

vk∗c = 1,

Pr R αk2c + βk1c

,

q∗kc =

⎫ ⎪ ⎪ ⎪ ⎬ ˜ 2 + β1 ) ⎪ αPr R − Pr R(α kc kc ⎪ .⎪ ⎭ (αk2c + βk1c )(Le αk2c + βk1c )

(4.13)

...................................................

(4.8)


⎧ kc x1 ⎪ ⎪ u11 = −ukc cos π (x2 − r0 ) sin , ⎪ ⎪ r0 ⎪ ⎪ ⎪ x k ⎪ c 1 ⎪ ⎨u12 = vkc sin π (x2 − r0 ) cos , r0 1 ψkc = ⎪ ⎪T1 = Tk sin π (x2 − r0 ) cos kc x1 , ⎪ c 1 ⎪ ⎪ r0 ⎪ ⎪ ⎪ 1 kx ⎪ ⎩q = qkc sin π (x2 − r0 ) cos c 1 r0


t π 2kx1 r0 π 2 2kx1 + − sin 2π (x2 − r0 ) sin , cos 2π (x2 − r0 ) cos , 0, 0 4k r0 r0 4k2 t ⎤ r0 π 2 2kx1 Tk π qk π sin 2π (x2 − r0 ), sin 2π (x2 − r0 ) ⎦ . + 0, cos , r0 2 2 4k2 As the first two terms on the right-hand side of (4.15) are gradient fields, we obtain t r0 π 2 2kx1 Tk π qk π 1 1 sin 2π (x2 − r0 ), − sin 2π (x2 − r0 ) . , − G(ψk , ψk ) = 0, − 2 cos r0 2 2 4k

(4.15)

(4.16)

Similarly, we can derive t ⎫ ⎪ r0 π 2 π r0 π 2 2kx1 ⎪ 1 2 ⎪ cos 2π (x2 − r0 ) + sin 2π (x2 − r0 ), − 2 sin , 0, 0 ,⎪ G(ψk , ψk ) = − ⎪ ⎪ 2k 4k r0 4k ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎬ 2 2 r0 π π r0 π 2kx1 2 1 (4.17) cos 2π (x2 − r0 ) − sin 2π (x2 − r0 ), − 2 sin , 0, 0 G(ψk , ψk ) = ⎪ 2k 4k r0 4k ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ 2 ⎪ r0 π 2kx1 Tk π qk π ⎪ 2 2 ⎪ ⎪ sin 2π (x sin 2π (x and G(ψk , ψk ) = 0, cos , − − r ), − − r ) 2 0 2 0 ⎭ 2 r0 2 2 4k and

t ⎫ ⎪ 2kx1 Tk ∗ π qk ∗ π r0 π 2 ⎪ sin 2π (x2 − r0 ), − sin 2π (x2 − r0 ) ,⎪ , − 0, − 2 cos ⎪ ⎪ ⎪ r0 2 2 4k ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ 2 2 ⎪ r0 π π r0 π 2kx1 ∗ ⎪ cos 2π (x2 − r0 ) + sin 2π (x2 − r0 ), − 2 sin , 0, 0 , ⎪ G(ψk1 , ψk2 ) = − ⎪ ⎪ ⎬ 2k 4k r0 4k t ⎪ ⎪ ⎪ r0 π 2 π r0 π 2 2kx1 ∗ ⎪ ⎪ cos 2π (x2 − r0 ) − sin 2π (x2 − r0 ), − 2 sin , 0, 0 G(ψk2 , ψk1 ) = ⎪ ⎪ 2k 4k r0 ⎪ 4k ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ ∗ 2 ∗ ⎪ ⎪ π π π r 2kx T q ∗ 0 1 k k ⎪ 2 2 ⎪ sin 2π (x sin 2π (x and G(ψk , ψk ) = 0, cos , − − r ), − − r ) . ⎭ 2 0 2 0 r0 2 2 4k2 (4.18) By (4.16) and (4.17), we have ∗ G(ψk1 , ψk1 ) =

∗

G(ψki1 , ψki2 ), ψki3 = 0,

(4.19)

...................................................

˜ q˜ ). Direct calculation shows where ψ = (u1 , u2 , T, q) and φ = (v1 , v2 , T, ⎡ t r0 π 2 2kx1 π r0 π 2 sin , sin 2π (x2 − r0 ) − [1 + cos 2π (x − r )], 0, 0 G(ψk1 , ψk1 ) = −P ⎣ 2 0 2k r0 2 4k2

17


Step 4. Computation of the second-order approximation of the centre manifold function. The nonlinear operator G defined in (4.1) is bilinear, which can be expressed as ⎛ ⎞ ∂v1 ∂v1 1 u + u + u v 1 2 1 2 ⎜ ∂x1 ⎟ ∂x2 r0 ⎜ ⎟ ⎜ ⎟ ⎜ ∂v2 ⎟ ∂v2 1 ⎜u1 + u − u v 2 1 1⎟ ⎜ ∂x1 ⎟ ∂x r 2 0 ⎜ ⎟ (4.14) G(ψ, φ) = −P ⎜ ⎟, ⎜ ⎟ ˜ ˜ ∂ T ∂ T ⎜ ⎟ u + u 1 2 ⎜ ⎟ ∂x1 ∂x2 ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∂ q˜ ∂ q˜ u1 + u2 ∂x1 ∂x2


for i1 , i2 , i3 = 1, 2. Moreover, direct calculation shows

18

∗

(4.20)

⎭

G(φ1 , ψki ), ψki = 0,

for φ1 , φ2 , φ3 ∈ H1 and i = 1, 2. As the centre manifold function Φ(x, y) = O(|x|2 + |y|2 ), by (4.16)–(4.20), we obtain ⎫ ∗ ∗ ∗

G(ψ), ψk1c = −x G(ψk1c , ψk1c ), Φ − y G(ψk2c , ψk1c ), Φ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 1∗ 3 3 ⎪ ⎬ + y G(Φ, ψkc ), ψkc + o(|x| + |y| ) (4.21) ∗ ∗ ∗ ⎪ and

G(ψ), ψk2c = −x G(ψk1c , ψk2c ), Φ − y G(ψk2c , ψk2c ), Φ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ⎭ + x G(Φ, ψ 1 ), ψ 2 + o(|x|3 + |y|3 ). kc

kc

Let the centre manifold function be denoted by ! Φ= Φβ i (x, y)ψkji . By (4.15)–(4.21), only

(4.22)

kj

βkji =βk1c

⎫ 1 = (0, 0, 0, sin 2π (x2 − r0 )),⎪ ψ02 ⎪ ⎪ ⎬ 2 ψ02 = (0, 0, sin 2π (x2 − r0 ), 0) ⎪ ⎪ ⎪ 3 ψ02 = (cos 2π (x2 − r0 ), 0, 0, 0) ⎭

and

(4.23)

contribute to the third-order terms in evaluation of (4.21). Direct calculation shows that ⎫ qk 1 1

G(ψk1c , ψk1c ), ψ02 = − c π 2 r0 , G(ψk1c , ψk2c ), ψ02 = 0, ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ q ⎪ kc 2 2 1 1 2 2 1 ⎪

G(ψkc , ψkc ), ψ02 = 0, G(ψkc , ψkc ), ψ02 = − π r0 , ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ Tkc 2 1 1 2 1 2 2 ⎪ π r0 , G(ψkc , ψkc ), ψ02 = 0,⎪

G(ψkc , ψkc ), ψ02 = − ⎪ ⎪ 2 ⎬ T k c 2 2 π 2 r0 ,⎪ = 0, G(ψk2c , ψk2c ), ψ02 =−

G(ψk2c , ψk1c ), ψ02 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 2 3 ⎪ r0 π ⎪ 1 1 3 1 2 3 ⎪ ⎪

G(ψkc , ψkc ), ψ02 = 0, G(ψkc , ψkc ), ψ02 = − ⎪ ⎪ 2kc ⎪ ⎪ ⎪ ⎪ 2 3 ⎪ ⎪ π r ⎪ 0 2 1 3 2 2 3 ⎭ and

G(ψkc , ψkc ), ψ02 = , G(ψkc , ψkc ), ψ02 = 0. 2kc We note that

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

1 2 β02 = −Le α02 = −4π 2 Le, 2 2 β02 = −α02 = −4π 2

and

3 2 β02 = −Pr α02 −

2 Pr r20

= −4π 2 Pr −

(4.24)

1 r20

⎪ ⎪ ⎪ ⎪ Pr.⎪ ⎭

(4.25)

By the approximation formula of centre manifold functions, see Section 2.4 of [23], we get ⎫ qk Φβ 1 (x, y) = − (x2 + y2 ) + o(x2 + y2 ),⎪ ⎪ ⎪ 02 8π Le ⎪ ⎪ ⎬ Tk 2 2 2 2 (4.26) Φβ 2 (x, y) = − (x + y ) + o(x + y ) ⎪ 02 ⎪ 8π ⎪ ⎪ ⎪ ⎭ and Φβ 3 (x, y) = o(x2 + y2 ). 02

...................................................

and


⎫

G(φ1 , φ2 ), φ3 = − G(φ1 , φ3 ), φ2 ⎬


Plugging (4.22) into (4.21), by (4.18) and (4.26), we obtain

qkc q∗kc Le

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ 2 2 3 3 ⎪ ⎪ + Tkc Tkc y(x + y ) + o(|x| + |y| ).⎪ ⎭

∗ ∗

G(ψ), ψk2c = −y G(ψk2c , ψk2c ), Φ + o(|x|3

and

π r0 =− 16

qkc q∗kc Le

+ |y|3 )

By (4.10), (4.13) and (4.27), we evaluate (4.7) at R = Rc and obtain the reduction equation ⎫ b dx ⎪ = − x(x2 + y2 ) + o(|x|3 + |y|3 ) ⎪ ⎬ dt 8 ⎪ b dy ⎪ = − y(x2 + y2 ) + o(|x|3 + |y|3 ),⎭ and dt 8

(4.27)

(4.28)

where b is as defined as b=

˜ Le3 αk2c Rc + (((1 + Le2 )α1 αT h2 )/αq κT + α1 αT αk2c Le3 h2 /24αq κT + αk2c )R

, ˜ (Le3 αk6c /Pr)(1 + r20 π 2 /kc2 ) + Le3 αk2c R + Le((1 + Le)α1 αT h2 /αq κT + α1 αT αk2c Le2 h2 /24αq κT + αk2c )R (4.29)

and αK2 and Rc are given by remark 3.5. It is obvious that b > 0. As b > 0, due to the reduction equation (4.28), assertion (2) follows from Theorem 2.2.5 of [24]. Assertions (1) and (3) follow from Theorem 3.1 and Theorem 2.4.17 of [24]. This completes the proof of theorem 4.1.

5. Convection scales Under the same setting as (2.17)–(2.19), including the fluid frictions, we consider the following non-dimensional equation: ⎫ 2 ∂u2 1 ∂p ∂u1 ⎪ ⎪ ⎪ = Pr u1 + − 2 u1 − δ0 u1 − ⎪ ⎪ ∂t r0 ∂x1 ∂x1 ⎪ r0 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ − ωu2 − (u · ∇)u1 − u1 u2 , ⎪ ⎪ r0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ∂u1 2 ∂p ∂u2 ⎪ ˜ ⎪ = Pr u2 − − 2 u2 + RT + Rq − δ1 u2 − ⎪ ∂t r0 ∂x1 ∂x2 ⎪ ⎬ r0 (5.1) ⎪ 1 ⎪ ⎪ + ωu1 − (u · ∇)u2 + u21 , ⎪ ⎪ r0 ⎪ ⎪ ⎪ ⎪ ⎪ ∂T ⎪ ⎪ ⎪ = T + γ u2 + αq − (u · ∇)T, ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ∂q ⎪ ⎪ = Le q + u2 − (u · ∇)q ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎭ and div u = 0, where γ =1+ δi =

and

α , 24

Ci h4 , ν

i = 0, 1,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ C0 = 3.78 × 10−1 m−2 s−1 ⎪ ⎪ ⎪ ⎪ ⎪ 4 −2 −1 ⎭ C1 = 6.7 × 10 m s .

(5.2)

...................................................

3


π r0 =− 16

19

⎫ ⎪ + |y| ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ 2 2 3 3 ⎪ + Tkc Tkc x(x + y ) + o(|x| + |y| ) ⎪ ⎪ ⎪ ⎬

∗ ∗

G(ψ), ψk1c = −x G(ψk1c , ψk1c ), Φ + o(|x|3


(5.3)

(5.4)

Using the same analysis as in §3a, we obtain the following formula of the critical thermal Rayleigh number: 2 Le αkj 1 α1 αT h2 ˜ Rc = min − + 1+ R 2κ Le 24 k,j≥1 Le αq αkj T 2 2 2 δ0 π 2 r20 αkj αkj 1 4 r0 2 2 +αkj 2 αkj + 2 + 2 + δ1 αkj + k k2 r0 r0 2 Le αk1 1 α1 αT h2 + 1+ R˜ = min − 2 κ Le 24 k≥1 Le αq αk1 T 2 2 2 δ0 π 2 r20 αk1 αk1 1 4 r0 2 2 , (5.5) +αk1 2 αk1 + 2 + 2 + δ1 αk1 + k k2 r0 r0 where 2 = π2 + αk1

k2 r20

.

Let, y = k2 /r20 and

1 Le(y + π 2 ) α1 αT h2 1 1 2 2 2 ˜ + (y + π ) y + π + + 1+ g(y) = − R Le 24 y Le αq κT (y + π 2 ) r20 +

1 r20

(y + π 2 ) + δ1 (y + π 2 ) +

δ0 π 2 (y + π 2 ) . y

Taking the derivative of g(y), we get 2 π4 1 α1 αT h2 R˜ 2 6 4 . g (y) = 2y + 3π + 2 + δ1 − 2 π + 2 + δ0 π + y Le αq κT (y + π 2 )2 r0 r0 By (5.2), we have δ1 = 2.76 × 1020

and δ0 = 1.55 × 1015 .

(5.6)

Hence, δ1 δ0 1. Under this condition, the critical value of g(y) is approximated by δ0 π 2. yc δ1

(5.7)

(5.8)

...................................................

The corresponding eigenvalue problem reads ⎫ ⎪ 2 ∂u2 1 ∂p ⎪ ⎪ Pr u1 + − 2 u1 − δ0 u1 − − ωu2 = βu1 , ⎪ ⎪ r0 ∂x1 ∂x1 ⎪ r0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ∂u1 2 ∂p ˜ ⎪ − 2 u2 + RT + Rq − δ1 u2 − Pr u2 − + ωu1 = βu2 ,⎪ ⎪ ⎬ r0 ∂x1 ∂x r0 2 ⎪ ⎪ ⎪

T + γ u2 + αq = βT, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Le q + u2 = βq ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂u2 ∂u1 ⎪ ⎭ + = 0. and ∂x1 ∂x2

20


The domain is M = [0, 2π r0 ] × (r0 , r0 + 1), and the boundary conditions are given by ⎫ ⎪ (u, T, q)(x1 + 2π r0 , x2 ) = (u, T, q)(x1 , x2 ) ⎬ ∂u1 ⎭ and u2 = 0, = 0, T = 0, q = 0, at x2 = r0 , r0 + 1.⎪ ∂x2


Therefore, the critical thermal Rayleigh number is approximated by 2

21

21

(5.9)

⎫ κT = 2.25 × 10−5 m2 s−1 ,⎬

ν = 1.6 × 10−5 m2 s−1 , and

αT = 3.3 × 10−3 per ◦ C,

Pr = 0.71.

⎭

(5.10)

We note also that the height of the troposphere is h = 8 × 103 m. Hence, we get T0 − T1 = Tc =

κT ν Rc = 60◦ C. αT gh3

(5.11)

Here, we note that the above approximations agree with that of the model of tropical atmospheric circulations without humidity. However, as we can see from (5.5), the coefficient of R˜ is negative. This implies that the humidity factor lowers the critical thermal Rayleigh number a little bit. Next, as the non-dimensional radius of the Earth is r0 = 6 400 000/h = 800, we derive from k2 r20

=

δ0 δ1

1/2

π2

(5.12)

that the wavenumber kc and the convection length-scale Lc as kc 6 and

L=

(6400 × π ) = 3350 km. 6

This is consistent with the large-scale atmospheric circulation over the tropics. Acknowledgements. The authors are grateful for two referees for there insightful comments and suggestions. Funding statement. The work of S.W. was supported in part by the US Office of Naval Research and by the US National Science Foundation.

Appendix A. Recapitulation of dynamic transition theory The main results of this article are based on the dynamic transition theory developed recently by two of the authors [23,24]. Hereafter, we briefly recapitulate the basic ideas of the theory and refer the interested readers to these references for more details. First, dissipative systems are governed by differential equations—both ordinary and partial— which can be written in the following unified abstract form: du = Lλ u + G(u, λ), dt

u(0) = u0 ,

(A 1)

where u : [0, ∞) → X is the unknown function, λ ∈ R1 is the system control parameter and X is a Banach space. We shall always consider u the deviation of the unknown function from some ¯ Hence, Lλ : X1 → X is a linear operator, and G : X1 × R1 → X is a nonlinear equilibrium state u. operator. For example, for the classical incompressible Navier–Stokes equations, u represents the velocity function, and for the time-dependent Ginzburg–Landau equations of superconductivity, we have u = (ψ, A), where ψ is a complex-valued wave function, and A is the magnetic potential. Second, the dynamic transition of a given dissipative system is clearly associated with the linear eigenvalue problem for system (A 1). The underlying physical concept is the PES, leading to

...................................................

Next, as in [18,24], we adopt


Rc δ1 π = 2.76 × 10 .


and Re βj (λ0 ) < 0 ∀j ≥ m + 1.

(A 3)

Third, with PES, the dynamic transition is fully dictated then by the nonlinear interactions of the system. One important component of the dynamic transition theory is to establish a general principle, which classifies all dynamic transitions of a dissipative system into three categories, continuous, catastrophic and random, which are also called Type I, Type II and Type III. A continuous transition says that as the control parameter crosses the critical threshold, the transition states stay in a close neighbourhood to the basic state. A catastrophic transition corresponds to the case in which the system undergoes a more drastic change as the control parameter crosses the critical threshold. A random transition corresponds to the case in which a neighbourhood (fluctuations) of the basic state can be divided into two regions such that fluctuations in one of them lead to continuous transitions and those in the other lead to catastrophic transitions. Fourth, in the dynamic transition theory, the complete set of transition states is represented by local attractors. The identification and classification of these local attractors are important part of the theory. One crucial technical component is the approximation of the centre manifold of the underlying system, corresponding to the m unstable modes as described in the PES. In fact, using the centre manifold reduction, we know that the type of transitions for (A 1) at (0, λ0 ) is completely dictated by its reduction equation near λ = λ0 dx = Jλ x + g(x, λ) dt

for x ∈ Rm ,

where g(x, λ) = (g1 (x, λ), . . . , gm (x, λ)) and " m # ! xi ei + Φ(x, λ), λ , e∗j gj (x, λ) = G

(A 4)

∀1 ≤ j ≤ m.

(A 5)

i=1

Here, ej and e∗j (1 ≤ j ≤ m) are the eigenvectors of Lλ and L∗λ , respectively, corresponding to the eigenvalues βj (λ), Jλ is the m × m order Jordan matrix corresponding to the eigenvalues, and Φ(x, λ) is the centre manifold function of (A 1) near λ0 . The centre manifold function Φ is implicitly defined and is oftentimes hard to compute. A systematic approach is developed in [23,24] to derive approximations of Φ, which provide complete information on the dynamic transition of (A 1). Suppose the nonlinear operator G to be of the form G(u, λ) = Gk (u, λ) + o(uk ),

as u → 0 in Xμ .

(A 6)

for some integer k ≥ 2, where Gk is a k-multilinear operator Gk (u, λ) = Gk (u, . . . , u, λ) : X1 × · · · × X1 −→ X. By Theorem A.1.1 in [24], the centre manifold function Φ(x, λ) can be expressed as 0 e−τ Lλ ρε P2 Gk (eτ Jλ x, λ) dτ + o(xk ), Φ(x, λ) = −∞

(A 7)

where Jλ and Lλ are restrictions of Lλ to the linear invariant spaces generated by the first modes and the rest modes, respectively, Gk (u, λ) is the lowest order k-multiple linear operator, and x = $m i=1 xi ei . In particular, we have the following assertions:

...................................................

0

22


precise information on linear unstable modes. Let, {βj (λ) ∈ C | j ∈ N} be the eigenvalues (counting multiplicity) of Lλ and assume that ⎧ ⎪ ⎪ ⎨< 0 if λ < λ0 , (A 2) Re βi (λ) = 0 if λ = λ0 , ∀1 ≤ i ≤ m, ⎪ ⎪ ⎩> 0 if λ > λ


(1) If Jλ is diagonal near λ = λ0 , then (A 7) can be written as − Lλ Φ(x, λ) = P2 Gk (x, λ) + o(xk ) + O(|β|xk ).

23 (A 8)

− 2ρ 2 (λ)P2 G2 (x, λ) + 2ρ 2 P2 G2 (x1 e2 − x2 e1 ) + ρ(−Lλ )P2 [G2 (x1 e1 + x2 e2 , x2 e1 − x1 e2 ) + G2 (x2 e1 − x1 e2 , x1 e1 + x2 e2 )] + o(k),

(A 9)

where we have used o(k) = o(xk ) + O(|Re β(λ)|xk ).

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...................................................

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