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

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 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

R_{c} = - [\frac{1}{Le} + (1 + \frac{Le α_{k_{c}}^{2}}{24}) \frac{α_{1} α_{T} h^{2}}{Le α_{q} α_{k_{c}}^{2} κ_{T}}] \tilde{R} + α_{k_{c}}^{4} \frac{r_{0}^{2}}{k^{2}} (α_{k_{c}}^{2} + \frac{1}{r_{0}^{2}}) + \frac{α_{k_{c}}^{2}}{r_{0}^{2}} .

As in the expression of the thermal critical Rayleigh number R_c, the coefficient of the humidity Rayleigh number

\tilde{R}

is negative; this indicates that the presence of humidity lowers the critical 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 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.

2. Model for atmospheric motion with humidity

(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=(u_φ,u_θ,u_r), 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

\begin{aligned} ρ [\frac{\partial u}{\partial t} + \nabla_{u} u + 2 Ω \times u] + \nabla p + k ρ g = μ △ u, \end{aligned}

2.1

\begin{aligned} \frac{\partial ρ}{\partial t} + div (ρ u) = 0, \end{aligned}

2.2

\begin{aligned} ρ c_{v} [\frac{\partial T}{\partial t} + u \cdot \nabla T] + p div u = \tilde{Q} + {\tilde{κ}}_{T} △ T, \end{aligned}

2.3

\begin{aligned} ρ [\frac{\partial q}{\partial t} + u \cdot \nabla q] = \tilde{S} + {\tilde{κ}}_{q} △ q \end{aligned}

2.4

\begin{aligned} and & p = R ρ T . \end{aligned}

2.5

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,

{\tilde{κ}}_{T}, {\tilde{κ}}_{q}, c_{v}, R

are constants,

\tilde{Q}

and

\tilde{S}

are heat and humidity sources, and k=(0,0,1). The differential operators used are as follows:

	(1) The gradient and divergence operators are given by $\begin{aligned} \nabla & = (\frac{1}{r \cos θ} \frac{\partial}{\partial φ}, \frac{1}{r} \frac{\partial}{\partial θ}, \frac{\partial}{\partial r}) \\ and div u & = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} u_{r}) + \frac{1}{r \cos θ} \frac{\partial (u_{θ} \cos θ)}{\partial θ} + \frac{1}{r \cos θ} \frac{\partial u_{φ}}{\partial φ} \end{aligned}$
	(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 Δ for both of them $\begin{aligned} Δ u & = (Δ u_{φ} + \frac{2}{r^{2} \cos θ} \frac{\partial u_{r}}{\partial φ} + \frac{2 \sin θ}{r^{2} \cos^{2} θ} \frac{\partial u_{θ}}{\partial φ} - \frac{u_{φ}}{r^{2} \cos^{2} θ}, \\ Δ u_{θ} + \frac{2}{r^{2}} \frac{\partial u_{r}}{\partial θ} - \frac{u_{θ}}{r^{2} \cos^{2} θ} - \frac{2 \sin θ}{r^{2} \cos^{2} θ} \frac{\partial u_{φ}}{\partial φ}, \\ Δ u_{r} - \frac{2 u_{r}}{r^{2}} - \frac{2}{r^{2} \cos θ} \frac{\partial (u_{θ} \cos θ)}{\partial θ} - \frac{2}{r^{2} \cos θ} \frac{\partial u_{φ}}{\partial φ}) \\ and Δ f & = \frac{1}{r^{2} \cos θ} \frac{\partial f}{\partial θ} (\cos θ \frac{\partial}{\partial θ}) + \frac{1}{r^{2} \cos^{2} θ} \frac{\partial^{2} f}{\partial φ^{2}} + \frac{1}{r^{2}} \frac{\partial f}{\partial r} (r^{2} \frac{\partial}{\partial r}) . \end{aligned}$
	(3) The convection terms are given by $\begin{aligned} \nabla_{u} u & = (u \cdot \nabla u_{φ} + \frac{u_{φ} u_{r}}{r} - \frac{u_{φ} u_{θ}}{r} \tan θ, \\ u \cdot \nabla u_{θ} + \frac{u_{θ} u_{r}}{r} + \frac{u_{φ}^{2}}{r} \tan θ, u \cdot \nabla u_{r} - \frac{u_{φ}^{2} + u_{θ}^{2}}{r}) . \end{aligned}$
	(4) The Coriolis term 2Ω×u is given by $2 Ω \times u = 2 Ω (\cos θ u_{r} - \sin θ u_{θ}, \sin θ u_{φ}, - \cos θ u_{φ}) .$ 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):

ρ = ρ_{0} [1 - α_{T} (T - T_{0}) - α_{q} (q - q_{0})],

2.6

where ρ₀ is the density at T=T₀ and q=q₀, and α_T and α_q 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 $S_{a}^{2} \times (a, a + h)$ with the product metric

d s^{2} = a^{2} d θ^{2} + a^{2} \sin^{2} θ d ϕ^{2} + d z^{2} .

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 diffusion equation of the humidity. These equations are restricted on the lower latitude region where the meridional velocity component u_θ 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=(u_ϕ,u_z), the temperature function T, the humidity function q and the pressure p. Then, the equations governing the motion and states of the atmosphere read

\begin{aligned} \frac{\partial u_{ϕ}}{\partial t} + (u \cdot \nabla) u_{ϕ} + \frac{u_{ϕ} u_{z}}{a} = ν (Δ u_{ϕ} + \frac{2}{a^{2}} \frac{\partial u_{z}}{\partial ϕ} - \frac{u_{ϕ}}{a^{2}}) - 2 Ω u_{z} - \frac{1}{ρ_{0} a} \frac{\partial p}{\partial ϕ}, \\ \frac{\partial u_{z}}{\partial t} + (u \cdot \nabla) u_{z} - \frac{u_{ϕ}^{2}}{a} = ν (Δ u_{z} - \frac{2}{a^{2}} \frac{\partial u_{ϕ}}{\partial ϕ} - \frac{2 u_{z}}{a^{2}}) + 2 Ω u_{ϕ} - \frac{1}{ρ_{0}} \frac{\partial p}{\partial z} \\ - g (1 - α_{T} (T - T_{0}) - α_{q} (q - q_{0})), \\ \frac{\partial T}{\partial t} + (u \cdot \nabla) T = κ_{T} Δ T + Q, \\ \frac{\partial q}{\partial t} + (u \cdot \nabla) q = κ_{q} Δ q + S \\ and & \frac{1}{a} \frac{\partial u_{ϕ}}{\partial ϕ} + \frac{\partial u_{z}}{\partial z} = 0, \end{aligned}\}

2.7

where Ω is the Earth’s rotating angular velocity, g the gravitational constant, ρ₀ the density of air at T=T₀, ν=μ/ρ₀,

κ_{T} = {\tilde{κ}}_{T} / ρ_{0} c_{ν}

κ_{q} = {\tilde{κ}}_{q} / ρ_{0}

Q = \tilde{Q} / ρ_{0} c_{ν}

S = \tilde{S} / ρ_{0}

and α_T and α_q are the coefficients of thermal and humidity expansion. However, the differential operators used in (2.7) are as follows:

\begin{aligned} (u \cdot \nabla) & = \frac{u_{ϕ}}{a} \frac{\partial}{\partial ϕ} + u_{z} \frac{\partial}{\partial z}, \\ Δ f & = \frac{1}{a^{2}} \frac{\partial^{2}}{\partial ϕ^{2}} f + \frac{\partial^{2}}{\partial z^{2}} f \\ and Δ u & = Δ (u_{ϕ} e_{ϕ} + u_{z} e_{z}) \\ = (Δ u_{ϕ} + \frac{2}{a^{2}} \frac{\partial u_{z}}{\partial ϕ} - \frac{u_{ϕ}}{a^{2}}) e_{θ} + (Δ u_{z} - \frac{2}{a^{2}} \frac{\partial u_{ϕ}}{\partial ϕ} - \frac{2 u_{z}}{a^{2}}) e_{z}, \end{aligned}

where Δf is the Laplacian for scalar functions, and Δu 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 = α_{0} + α_{1} q,

2.8

and the humidity source is taken as zero

S = 0.

2.9

The boundary conditions are periodic in the ϕ-direction

(u, T, q) (ϕ + 2 π, z) = (u, T, q) (ϕ, z),

2.10

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

\begin{aligned} u_{z} & = 0, \frac{\partial u_{ϕ}}{\partial z} = 0, T = T_{0}, q = q_{0}, at z = a \\ and u_{z} & = 0, \frac{\partial u_{ϕ}}{\partial z} = 0, T = T_{1}, q = 0, at z = a + h, \end{aligned}\}

2.11

where T₀, T₁, q₀ and q₁ are constants satisfying

T_{0} > T_{1} and q_{0} > 0.

2.12

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

\begin{aligned} \tilde{u} & = 0, \\ \tilde{T} & = \frac{γ_{1}}{6 κ_{T}} z^{3} - \frac{γ_{0}}{2 κ_{T}} z^{2} + c_{1} z + c_{0}, \\ \tilde{q} & = - \frac{q_{0}}{h} z + \frac{q_{0}}{h} (h + a) \\ and \tilde{p} & = - \int_{0}^{z} ρ_{0} g (1 - α_{T} (\tilde{T} - T_{0}) - α_{q} (\tilde{q} - q_{0})) d z, \end{aligned}\}

2.13

where

\begin{aligned} γ_{0} & = α_{0} + \frac{α_{1} q_{0} (h + a)}{h}, γ_{1} = \frac{α_{1} q_{0}}{h}, \\ c_{0} & = \frac{a + h}{h} (- \frac{γ_{1}}{6 κ_{T}} a^{3} + \frac{γ_{0}}{2 κ_{T}} a^{2} + T_{0}) - \frac{a}{h} (- \frac{γ_{1}}{6 κ_{T}} {(a + h)}^{3} + \frac{γ_{0}}{2 κ_{T}} {(a + h)}^{2} + T_{1}) \\ and c_{1} & = - \frac{1}{h} (T_{0} - T_{1}) - \frac{1}{κ_{T}} (\frac{γ_{1}}{6} (h^{2} + 3 h a + 3 a^{2}) - \frac{γ_{0}}{2} (h + 2 a)) . \end{aligned}

To obtain the non-dimensional form, let

\begin{aligned} x & = h x^{'}, a = h r_{0}, \\ t & = \frac{h^{2} t^{'}}{κ_{T}}, u = \frac{κ_{T} u^{'}}{h}, \\ T & = (T_{0} - T_{1}) T^{'} + \tilde{T}, q = q_{0} q^{'} + \tilde{q} \\ and p & = \frac{ρ_{0} ν κ_{T} p^{'}}{h^{2}} + \tilde{p}, \end{aligned}

and consider

(x_{1}, x_{2}) = (r_{0} ϕ, z) and (u_{1}, u_{2}) = (u_{ϕ}, u_{z}) .

Omitting the primes, equations (2.7) become

\begin{aligned} \frac{\partial u_{1}}{\partial t} & = Pr (Δ u_{1} + \frac{2}{r_{0}} \frac{\partial u_{2}}{\partial x_{1}} - \frac{1}{r_{0}^{2}} u_{1} - \frac{\partial p}{\partial x_{1}}) \\ - ω u_{2} - (u \cdot \nabla) u_{1} - \frac{1}{r_{0}} u_{1} u_{2}, \\ \frac{\partial u_{2}}{\partial t} & = Pr (Δ u_{2} - \frac{2}{r_{0}} \frac{\partial u_{1}}{\partial x_{1}} - \frac{2}{r_{0}^{2}} u_{2} + R T + \tilde{R} q - \frac{\partial p}{\partial x_{2}}) \\ + ω u_{1} - (u \cdot \nabla) u_{2} + \frac{1}{r_{0}} u_{1}^{2}, \\ \frac{\partial T}{\partial t} & = Δ T - \frac{1}{T_{0} - T_{1}} \frac{d \tilde{T} (h x_{2})}{d x_{2}} u_{2} + α q - (u \cdot \nabla) T, \\ \frac{\partial q}{\partial t} & = Le Δ q + u_{2} - (u \cdot \nabla) q \\ and \frac{\partial u_{1}}{\partial x_{1}} + \frac{\partial u_{2}}{\partial x_{2}} & = 0, \end{aligned}\}

2.14

where the non-dimensional physical parameters are

\begin{aligned} Pr & = \frac{ν}{κ_{T}} the Prandtl number, \\ R & = \frac{α_{T} g (T_{0} - T_{1}) h^{3}}{κ_{T} ν} the thermal Rayleigh number, \\ \tilde{R} & = \frac{α_{q} g q_{0} h^{3}}{κ_{T} ν} the humidity Rayleigh number, \\ Le & = \frac{κ_{q}}{κ_{T}} the Lewis number, \\ ω & = \frac{2 Ω h^{2}}{κ_{T}} the Earth rotation \\ and α & = \frac{α_{1} q_{0} h^{2}}{κ_{T} (T_{0} - T_{1})}, \end{aligned}\}

2.15

and

\begin{aligned} - \frac{1}{T_{0} - T_{1}} \frac{d \tilde{T} (h x_{2})}{d x_{2}} & = 1 + \frac{γ_{0} h^{2}}{κ_{T} (T_{0} - T_{1})} (x_{2} - r_{0} - \frac{1}{2}) \\ - \frac{α_{1} q_{0} h^{2}}{2 κ_{T} (T_{0} - T_{1})} (x_{2}^{2} - r_{0}^{2} - r_{0} - \frac{1}{3}), \end{aligned}

2.16

where r₀<x₂<r₀+1, and r₀ is the non-dimensional radius of the Earth. For simplicity, we take the average approximation

x_{2} = r_{0} + \frac{1}{2}

. In this case, (2.16) is given by

- \frac{1}{T_{0} - T_{1}} {\frac{d \tilde{T}}{d x_{2}}|}_{x_{2} = r_{0} + 1 / 2} = 1 + \frac{α_{1} q_{0} h^{2}}{24 κ_{T} (T_{0} - T_{1})} .

Thus, equation (2.14) is approximated by

\begin{aligned} \frac{\partial u_{1}}{\partial t} & = Pr (Δ u_{1} + \frac{2}{r_{0}} \frac{\partial u_{2}}{\partial x_{1}} - \frac{1}{r_{0}^{2}} u_{1} - \frac{\partial p}{\partial x_{1}}) \\ - ω u_{2} - (u \cdot \nabla) u_{1} - \frac{1}{r_{0}} u_{1} u_{2}, \\ \frac{\partial u_{2}}{\partial t} & = Pr (Δ u_{2} - \frac{2}{r_{0}} \frac{\partial u_{1}}{\partial x_{1}} - \frac{2}{r_{0}^{2}} u_{2} + R T + \tilde{R} q - \frac{\partial p}{\partial x_{2}}) \\ + ω u_{1} - (u \cdot \nabla) u_{2} + \frac{1}{r_{0}} u_{1}^{2}, \\ \frac{\partial T}{\partial t} & = Δ T + γ u_{2} + α q - (u \cdot \nabla) T, \\ \frac{\partial q}{\partial t} & = Le Δ q + u_{2} - (u \cdot \nabla) q \\ and div u & = 0, \end{aligned}\}

2.17

where

γ = 1 + \frac{α}{24} .

2.18

The domain is M=[0,2πr₀]×(r₀,r₀+1), and the boundary conditions are given by

\begin{aligned} (u, T, q) (x_{1} + 2 π r_{0}, x_{2}) = (u, T, q) (x_{1}, x_{2}) \\ and & u_{2} = 0, \frac{\partial u_{1}}{\partial x_{2}} = 0, T = 0, q = 0, at x_{2} = r_{0}, r_{0} + 1. \end{aligned}\}

2.19

For the problem (2.17)–(2.19), we set the spaces

\begin{aligned} H & = {(u, T, q) \in L^{2} (M, R^{4}) ∣ div u = 0, u_{2} = 0 at r_{0}, r_{0} + 1} \\ and H_{1} & = {(u, T, q) \in H^{2} (M, R^{4}) \cap H ∣ (u, T, q) satisfies (2.12)} . \end{aligned}\}

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:

\begin{aligned} Pr (Δ u_{1} + \frac{2}{r_{0}} \frac{\partial u_{2}}{\partial x_{1}} - \frac{1}{r_{0}^{2}} u_{1} - \frac{\partial p}{\partial x_{1}}) - ω u_{2} = β u_{1}, \\ Pr (Δ u_{2} - \frac{2}{r_{0}} \frac{\partial u_{1}}{\partial x_{1}} - \frac{2}{r_{0}^{2}} u_{2} + R T + \tilde{R} q - \frac{\partial p}{\partial x_{2}}) + ω u_{1} = β u_{2}, \\ Δ T + γ u_{2} + α q = β T, \\ Le Δ q + u_{2} = β q \\ and & \frac{\partial u_{1}}{\partial x_{1}} + \frac{\partial u_{2}}{\partial x_{2}} = 0 \end{aligned}\}

3.1

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:

ψ^{1} = \{\begin{cases} u_{1}^{1} = - u_{k} (x_{2}) \sin \frac{k x_{1}}{r_{0}}, \\ u_{2}^{1} = v_{k} (x_{2}) \cos \frac{k x_{1}}{r_{0}}, \\ T^{1} = T_{k} (x_{2}) \cos \frac{k x_{1}}{r_{0}}, \\ q^{1} = q_{k} (x_{2}) \cos \frac{k x_{1}}{r_{0}}, \\ p^{1} = A_{k} (x_{2}) \cos \frac{k x_{1}}{r_{0}} - B_{k} (x_{2}) \sin \frac{k x_{1}}{r_{0}} \end{cases}

3.2

and

ψ^{2} = \{\begin{cases} u_{1}^{2} = u_{k} (x_{2}) \cos \frac{k x_{1}}{r_{0}}, \\ u_{2}^{2} = v_{k} (x_{2}) \sin \frac{k x_{1}}{r_{0}}, \\ T^{2} = T_{k} (x_{2}) \sin \frac{k x_{1}}{r_{0}}, \\ q^{2} = q_{k} (x_{2}) \sin \frac{k x_{1}}{r_{0}}, \\ p^{2} = A_{k} (x_{2}) \sin \frac{k x_{1}}{r_{0}} + B_{k} (x_{2}) \cos \frac{k x_{1}}{r_{0}}, \end{cases}

3.3

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

u_{k} = \frac{r_{0}}{k} \frac{d v_{k}}{d x_{2}} .

3.4

Let

B_{k} = \frac{ω r_{0}}{k Pr} v_{k} (x_{2}) and A_{k} = p_{k} + \frac{2}{r_{0}} v_{k} (x_{2}) .

3.5

Owing to (2.19), plugging (3.2) or (3.3) into (3.1), we obtain the following system of ordinary differential equations:

\begin{aligned} Pr (D_{k}^{2} u_{k} - \frac{1}{r_{0}^{2}} u_{k} - \frac{k}{r_{0}} p_{k}) = β u_{k}, \\ Pr (D_{k}^{2} v_{k} - \frac{2}{r_{0}^{2}} v_{k} + R T_{k} + \tilde{R} q_{k} - D p_{k}) = β v_{k}, \\ D_{k}^{2} T_{k} + γ v_{k} + α q_{k} = β T_{k}, \\ Le D_{k}^{2} q_{k} + v_{k} = β q_{k} \\ and & D u_{k} = 0, v_{k} = 0, T_{k} = 0, q_{k} = 0, at x_{2} = r_{0}, r_{0} + 1, \end{aligned}\}

3.6

where

D = \frac{d}{d x_{2}} and D_{k}^{2} = \frac{d^{2}}{d x_{2}^{2}} - \frac{k^{2}}{r_{0}^{2}} .

Plugging

v_{k} = \sin j π (x_{2} - r_{0})

into (3.6), we see that the eigenvalue β satisfies the cubic equation

\begin{aligned} (α_{k j}^{2} + β) (Le α_{k j}^{2} + β) (Pr α_{k j}^{2} + \frac{Pr}{r_{0}^{2}} + β) α_{k j}^{2} + \frac{k^{2} Pr}{r_{0}^{4}} (α_{k j}^{2} + β) (Le α_{k j}^{2} + β) \\ - \frac{k^{2} Pr R}{r_{0}^{2}} (γ Le α_{k j}^{2} + γ β + α) - \frac{k^{2} Pr \tilde{R}}{r_{0}^{2}} (α_{k j}^{2} + β) = 0, \end{aligned}

3.7

where

α_{k j} = {(j^{2} π^{2} + \frac{k^{2}}{r_{0}^{2}})}^{1 / 2} (j \geq 1, k \geq 1) .

3.8

(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

\begin{aligned} β^{3} + [(1 + Le + Pr) α_{k j}^{2} + \frac{Pr}{r_{0}^{2}} + \frac{k^{2} Pr}{r_{0}^{4} α_{k j}^{2}}] β^{2} \\ + [(Pr + Le Pr + Le) α_{k j}^{4} + \frac{Pr}{r_{0}^{2}} (1 + Le) (α_{k j}^{2} + \frac{k^{2}}{r_{0}^{2}}) - \frac{Pr k^{2}}{r_{0}^{2} α_{k j}^{2}} (γ R + \tilde{R})] β \\ + Le α_{k j}^{4} (Pr α_{k j}^{2} + \frac{Pr}{r_{0}^{2}}) + \frac{Pr k^{2}}{r_{0}^{2} α_{k j}^{2}} (- α_{k j}^{2} \tilde{R} - α R - γ Le α_{k j}^{2} R + \frac{Le α_{k j}^{4}}{r_{0}^{2}}) \\ = 0. \end{aligned}

3.9

We see that β=0 is a solution of (3.9) if and only if

Le α_{k j}^{4} (Pr α_{k j}^{2} + \frac{Pr}{r_{0}^{2}}) + \frac{Pr k^{2}}{r_{0}^{2} α_{k j}^{2}} (- α_{k j}^{2} \tilde{R} - α R - γ Le α_{k j}^{2} R + \frac{Le α_{k j}^{4}}{r_{0}^{2}}) = 0.

3.10

Inferring from (2.15) and (2.18),

\begin{aligned} α R & = \frac{α_{1} α_{T} g q_{0} h^{5}}{κ_{T}^{2} ν} = \frac{α_{1} α_{T} h^{2}}{α_{q} κ_{T}} \tilde{R} \\ and γ R & = R + \frac{α_{1} α_{T} g q_{0} h^{5}}{24 κ_{T}^{2} ν} = R + \frac{α_{1} α_{T} h^{2}}{24 α_{q} κ_{T}} \tilde{R} . \end{aligned}\}

3.11

Hence, we rewrite (3.10) as

R = - (\frac{1}{Le} + (1 + \frac{Le α_{k j}^{2}}{24}) \frac{α_{1} α_{T} h^{2}}{Le α_{q} α_{k j}^{2} κ_{T}}) \tilde{R} + α_{k j}^{4} \frac{r_{0}^{2}}{k^{2}} (α_{k j}^{2} + \frac{1}{r_{0}^{2}}) + \frac{α_{k j}^{2}}{r_{0}^{2}} .

3.12

We define the critical Rayleigh number R_c as

\begin{aligned} R_{c} & = min_{k, j \geq 1} [- (\frac{1}{Le} + (1 + \frac{Le α_{k j}^{2}}{24}) \frac{α_{1} α_{T} h^{2}}{Le α_{q} α_{k j}^{2} κ_{T}}) \tilde{R} + α_{k j}^{4} \frac{r_{0}^{2}}{k^{2}} (α_{k j}^{2} + \frac{1}{r_{0}^{2}}) + \frac{α_{k j}^{2}}{r_{0}^{2}}] \\ = min_{k \geq 1} [- (\frac{1}{Le} + (1 + \frac{Le α_{k 1}^{2}}{24}) \frac{α_{1} α_{T} h^{2}}{Le α_{q} α_{k 1}^{2} κ_{T}}) \tilde{R} + α_{k 1}^{4} \frac{r_{0}^{2}}{k^{2}} (α_{k 1}^{2} + \frac{1}{r_{0}^{2}}) + \frac{α_{k 1}^{2}}{r_{0}^{2}}], \end{aligned}

3.13

where

α_{k 1}^{2} = π^{2} + \frac{k^{2}}{r_{0}^{2}} .

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 β_jkbe the eigenvalues of (3.1) that satisfy (3.9). Letk_c≥1 be integer minimizing (3.13), and $β_{k_{c}}^{i}$ (1≤i≤3) be solution of (3.9) with (k,j)=(k_c,1), and

R e β_{k_{c}}^{3} \leq R e β_{k_{c}}^{2} \leq R e β_{k_{c}}^{1} .

then

β_{k_{c}}^{1}

must be real nearR=R_c, and

\begin{aligned} β_{k_{c}}^{1} (R) \{\begin{cases} < 0, R < R_{c}, \\ = 0, R = R_{c}, \\ > 0, R > R_{c}, \end{cases} \\ R e β_{k_{c}}^{j} (R_{c}) < 0, for j = 2, 3. \end{aligned}

Moreover, we have

β_{j k} (R_{c}) < 0 for β_{j k} being real and β_{j k} \neq β_{k_{c}}^{1} .

Remark 3.2

The value R_c 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 ΔT_c=T₀−T₁ given by

\frac{α_{T} g h^{3} Δ T_{c}}{κ_{T} ν} = R_{c} .

Equivalently, we have

\begin{aligned} Δ T_{c} & = \frac{κ_{T} ν}{α_{T} g h^{3}} [- (\frac{1}{Le} + (1 + \frac{Le α_{k_{c} 1}^{2}}{24}) \frac{α_{1} α_{T} h^{2}}{Le α_{q} α_{k_{c} 1}^{2} κ_{T}}) \tilde{R} \\ + α_{k_{c} 1}^{4} \frac{r_{0}^{2}}{k_{c}^{2}} (α_{k_{c} 1}^{2} + \frac{1}{r_{0}^{2}}) + \frac{α_{k_{c} 1}^{2}}{r_{0}^{2}}], \end{aligned}

3.14

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

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

\begin{aligned} ρ_{0}^{2} & = (Pr + Le Pr + Le) α_{k j}^{4} + \frac{Pr}{r_{0}^{2}} (1 + Le) (α_{k j}^{2} + \frac{k^{2}}{r_{0}^{2}}) - \frac{Pr k^{2}}{r_{0}^{2} α_{k j}^{2}} (γ R + \tilde{R}) \\ and ρ_{0}^{2} & = [Le α_{k j}^{4} (Pr α_{k j}^{2} + \frac{Pr}{r_{0}^{2}}) + \frac{Pr k^{2}}{r_{0}^{2} α_{k j}^{2}} (- α_{k j}^{2} \tilde{R} - α R - γ Le α_{k j}^{2} R + \frac{Le α_{k j}^{4}}{r_{0}^{2}})] / \\ [(1 + Le + Pr) α_{k j}^{2} + \frac{Pr}{r_{0}^{2}}] . \end{aligned}

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

\begin{aligned} [(1 + Le + Pr) α_{k j}^{2} + \frac{Pr}{r_{0}^{2}}] [(Pr + Le + Le Pr) α_{k j}^{4} \\ + \frac{Pr}{r_{0}^{2}} (1 + Le) (α_{k j}^{2} + \frac{k^{2}}{r_{0}^{2}}) - \frac{Pr k^{2}}{r_{0}^{2} α_{k j}^{2}} (γ R + \tilde{R})] \\ = [Le α_{k j}^{4} (Pr α_{k j}^{2} + \frac{2 Pr}{r_{0}^{2}}) + \frac{Pr k^{2}}{r_{0}^{2} α_{k j}^{2}} (- α_{k j}^{2} \tilde{R} - α R - γ Le α_{k j}^{2} R + \frac{Le α_{k j}^{4}}{r_{0}^{2}})] \\ > 0. \end{aligned}

3.15

It follows from (3.11) and (3.15) that

\begin{aligned} (\frac{Pr}{r_{0}^{2}} + (Pr + 1) α_{k j}^{2}) R & = [\frac{α_{1} α_{T} h^{2}}{α_{q} κ_{T}} (1 - \frac{1}{24} (\frac{Pr}{r_{0}^{2}} + (Pr + 1) α_{k_{c}^{*}}^{2})) - (\frac{Pr}{r_{0}^{2}} + (Pr + Le) α_{k_{c}^{*}}^{2})] \tilde{R} \\ + \frac{r_{0}^{2} α_{k j}^{2}}{k^{2} Pr} (Le (1 + Le) α_{k j}^{6} + Pr [(1 + Le + Pr) α_{k j}^{2} + \frac{Pr}{r_{0}^{2}}] [(1 + Le) α_{k j}^{4} \\ + \frac{(1 + Le)}{r_{0}^{2}} α_{k j}^{2} + \frac{k^{2} Le}{r_{0}^{4}}] + \frac{Pr k^{2}}{r_{0}^{4}} [(1 + Pr) α_{k j}^{2} + \frac{Pr}{r_{0}^{2}}]), \end{aligned}

3.16

where

α_{k j}^{2}

is as defined in (3.8).

According to (3.16), we define the critical Rayleigh number for the complex PES as follows:

\begin{aligned} R_{c}^{*} & = min_{k, j \geq 1} \frac{1}{Pr / r_{0}^{2} + (Pr + 1) α_{k j}^{2}} \\ \times \{[\frac{α_{1} α_{T} h^{2}}{α_{q} κ_{T}} (1 - \frac{1}{24} (\frac{Pr}{r_{0}^{2}} + (Pr + 1) α_{k_{c}^{*}}^{2})) - (\frac{Pr}{r_{0}^{2}} + (Pr + Le) α_{k_{c}^{*}}^{2})] \tilde{R} \\ + \frac{r_{0}^{2} α_{k j}^{2}}{k^{2} Pr} (Le (1 + Le) α_{k j}^{6} + Pr [(1 + Le + Pr) α_{k j}^{2} + \frac{Pr}{r_{0}^{2}}] [(1 + Le) α_{k j}^{4} \\ + \frac{(1 + Le)}{r_{0}^{2}} α_{k j}^{2} + \frac{k^{2} Le}{r_{0}^{4}}] + \frac{Pr k^{2}}{r_{0}^{4}} [(1 + Pr) α_{k j}^{2} + \frac{Pr}{r_{0}^{2}}])\} . \end{aligned}

3.17

Then we have the following complex PES.

Lemma 3.3

Let $k_{c}^{*},$ $j_{c}^{*}$ be the integers satisfying (3.17), and $β_{k_{c}^{*} j_{c}^{*}}^{1}$ and $β_{k_{c}^{*} j_{c}^{*}}^{2}$ be the pair of complex eigenvalues of (3.9) with $(k, j) = (k_{c}^{*}, j_{c}^{*})$ nearR=R*_c. then,

R e β_{k_{c}^{*} j_{c}^{*}}^{1} = R e β_{k_{c}^{*} j_{c}^{*}}^{2} \{\begin{cases} < 0, R < R_{c}^{*}, \\ = 0, R = R_{c}^{*}, \\ > 0, R > R_{c}^{*} . \end{cases}

Furthermore, for all complex eigenvalues β_kj of (3.1), we have

R e β_{k j} (R_{c}^{*}) < 0, for (k, j) \neq (k_{c}^{*}, j_{c}^{*}) .

(c) Proof of lemmas thm3.1 and thm3.3

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

\begin{aligned} Δ T + α q & = β T \\ and Le Δ q & = β q, \end{aligned}\}

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 β_kj of (3.9). We rewrite (3.9) as

β^{3} + b_{2} β^{2} + b_{1} β + b_{0} = 0,

3.19

where

\begin{aligned} b_{2} & = (1 + Le + Pr) α_{k j}^{2} + \frac{Pr}{r_{0}^{2}} + \frac{k^{2} Pr}{r_{0}^{4} α_{k j}^{2}}, \\ b_{1} & = (Pr + Le Pr + Le) α_{k j}^{4} + \frac{Pr}{r_{0}^{2}} (1 + Le) (α_{k j}^{2} + \frac{k^{2}}{r_{0}^{2}}) - \frac{Pr k^{2}}{r_{0}^{2} α_{k j}^{2}} (γ R + \tilde{R}) \\ and b_{0} & = Le α_{k j}^{4} (Pr α_{k j}^{2} + \frac{Pr}{r_{0}^{2}}) + \frac{Pr k^{2}}{r_{0}^{2} α_{k j}^{2}} (- α_{k j}^{2} \tilde{R} - α R - γ Le α_{k j}^{2} R + \frac{Le α_{k j}^{4}}{r_{0}^{2}}) . \end{aligned}

We see that

b_{0}, b_{1}, b_{2} > 0 at R = 0 for all k, j \geq 1,

3.20

which implies that all real eigenvalues of (3.9) are negative at R=0. By the definition of R_c, we find

\begin{aligned} b_{0} > 0, \forall (k, j) \neq (k_{c}, 1), 0 \leq R \leq R_{c} \\ and & b_{0} \{\begin{cases} > 0, R < R_{c} \\ = 0, R = R_{c}, \\ < 0, R > R_{c}, \end{cases} for (k, j) = (k_{c}, 1) . \end{aligned}\}

3.21

b_{0} = - β_{k j}^{1} β_{k j}^{2} β_{k j}^{3}

, lemma 3.1 follows from (3.20) and (3.21).

Next, let $β_{k_{c}^{*} j_{c}^{*}}$ , the solution of (3.9), take the form

\begin{aligned} β_{k_{c}^{*} j_{c}^{*}} (R) = λ (R) + i σ (R) \\ and & λ (R) \to 0, σ (R) \to σ_{0}, as R \to R_{c}^{*} . \end{aligned}\}

3.22

Inserting (3.22) into (3.19), we obtain

\begin{aligned} (- 3 σ^{2} + b_{1}) λ + b_{0} - b_{2} σ^{2} + o (λ) & = 0 \\ and - σ^{3} + σ b_{1} + 2 σ b_{2} λ + o (λ) & = 0. \end{aligned}\}

3.23

As σ₀≠0, we infer from (3.23) that

λ (R) + o (λ) = \frac{b_{0} - b_{2} σ^{2}}{3 σ^{2} - b_{1}} = \frac{b_{0} - b_{1} b_{2} - 2 b_{2}^{2} λ}{2 b_{1} + 6 b_{2} λ} + o (λ) .

Thus, we have

λ (R) = \frac{b_{0} - b_{1} b_{2} - 2 b_{2}^{2} λ}{2 b_{1} + 6 b_{2} λ} + o (λ) .

3.24

Under the condition (3.15), we see b₁>0. It follows from (3.22) and (3.24) that

R e β_{k_{c}^{*} j_{c}^{*}} (R) = λ (R) \{\begin{cases} < 0, b_{0} < b_{1} b_{2}, \\ = 0, b_{0} = b_{1} b_{2}, \\ > 0, b_{0} > b_{1} b_{2}, \end{cases}

3.25

for R near R*_c. By the definition of R*_c, it is easy to see that

b_{0} \{\begin{cases} < b_{1} b_{2}, R < R_{c}^{*}, \\ = b_{1} b_{2}, R = R_{c}^{*}, \\ > b_{1} b_{2}, R > R_{c}^{*} . \end{cases}

3.26

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 R_cand R*_cbe the parameters defined by (3.13) and (3.17) respectively. Then, the following assertions hold true.

(i) When $R_{c} < R_{c}^{*},$ the first critical-crossing eigenvalue of the problem (3.1) is $β_{k_{c}}^{1}$ given by lemma 3.1, i.e.

β_{k_{c}}^{1} (R) \{\begin{cases} < 0, & if R < R_{c}, \\ = 0, & if R = R_{c}, \\ > 0, & if R > R_{c} \end{cases}

3.27

and

R e β (R_{c}) < 0,

3.28

for all other eigenvalues β of (3.1).

(ii) When R*_c<R_c, the first critical-crossing eigenvalues are the pair of complex eigenvalues $β_{k_{c}^{*} j_{c}^{*}}^{1}$ and $β_{k_{c}^{*} j_{c}^{*}}^{2}$ given by lemma 3.3, namely,

R e β_{k_{c}^{*} j_{c}^{*}}^{1} (R) = R e β_{k_{c}^{*} j_{c}^{*}}^{2} (R) \{\begin{cases} < 0, & if R < R_{c}^{*}, \\ = 0, & if R = R_{c}^{*}, \\ > 0, & if R > R_{c}^{*} \end{cases}

3.29

and

R e β (R_{c}^{*}) < 0,

3.30

for all other eigenvalues β(R) of (3.1).

Remark 3.5

In the atmospheric science, the integer $j_{c}^{*}$ in (3.29) is 1. Hence, the critical Rayleigh numbers R_c and R*_c are given by

R_{c} = - (\frac{1}{Le} + (1 + \frac{Le α_{k_{c}}^{2}}{24}) \frac{α_{1} α_{T} h^{2}}{Le α_{q} α_{k_{c}}^{2} κ_{T}}) \tilde{R} + α_{k_{c}}^{4} \frac{r_{0}^{2}}{k^{2}} (α_{k_{c}}^{2} + \frac{1}{r_{0}^{2}}) + \frac{α_{k_{c}}^{2}}{r_{0}^{2}}

3.31

and

\begin{aligned} R_{c}^{*} & = \frac{1}{Pr / r_{0}^{2} + (Pr + 1) α_{k_{c}^{*}}^{2}} \\ \times \{[\frac{α_{1} α_{T} h^{2}}{α_{q} κ_{T}} (1 - \frac{1}{24} (\frac{Pr}{r_{0}^{2}} + (Pr + 1) α_{k_{c}^{*}}^{2})) - (\frac{Pr}{r_{0}^{2}} + (Pr + Le) α_{k_{c}^{*}}^{2})] \tilde{R} \\ + \frac{r_{0}^{2} α_{k_{c}^{*}}^{2}}{k^{2} Pr} (Le (1 + Le) α_{k_{c}^{*}}^{6} + Pr [(1 + Le + Pr) α_{k_{c}^{*}}^{2} + \frac{Pr}{r_{0}^{2}}] [(1 + Le) α_{k_{c}^{*}}^{4} \\ + \frac{(1 + Le)}{r_{0}^{2}} α_{k_{c}^{*}}^{2} + \frac{{k_{c}^{*}}^{2} Le}{r_{0}^{4}}] + \frac{Pr {k_{c}^{*}}^{2}}{r_{0}^{4}} [(1 + Pr) α_{k_{c}^{*}}^{2} + \frac{Pr}{r_{0}^{2}}])\}, \end{aligned}

3.32

where

α_{k_{c}}^{2} = π^{2} + \frac{k_{c}^{2}}{r_{0}^{2}} and α_{k_{c}^{*}}^{2} = π^{2} + \frac{{k_{c}^{*}}^{2}}{r_{0}^{2}} .

3.33

4. Transition theorem

Inferring from theorem 4.1, system (2.17)–(2.19) has a transition to equilibria at R=R_c provided $R_{c} < R_{c}^{*}$ and has a transition to spatio-temporal oscillation at R=R*_c provided R*_c<R_c, where the critical values R_c 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 $R < min {R_{c}, R_{c}^{}},$ the equilibrium solution* (u,T,q)=0 is stable in H.
	(2) If R_c<R_c, then this problem has a continuous transition at R=R_c, and bifurcates from ((u,T,q*),R)=(0,R_c) to an attractor Σ_R=S¹on R>R_cwhich is a cycle of steady-state solutions.
	(3) As R_c<R_c, the problem has a transition atR=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 S³is 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].

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 L_R=A+B_R:H₁→H and G:H₁→H by

\begin{aligned} A ψ & = P (Pr (Δ u_{1} + \frac{2}{r_{0}} \frac{\partial u_{2}}{\partial x_{1}}), Pr (Δ u_{2} - \frac{2}{r_{0}} \frac{\partial u_{1}}{\partial x_{1}}), Δ T, Le Δ q), \\ B_{R} ψ & = P (- \frac{Pr}{r_{0}^{2}} u_{1} - ω u_{2}, - Pr (\frac{2}{r_{0}^{2}} u_{2} - R T - \tilde{R} q) + ω u_{1}, γ u_{2} + α q, u_{2}) \\ and G (ψ) & = - P ((u \cdot \nabla) u_{1} + \frac{1}{r_{0}} u_{1} u_{2}, (u \cdot \nabla) u_{2} - \frac{1}{r_{0}} u_{1}^{2}, (u \cdot \nabla) T, (u \cdot \nabla) q)), \end{aligned}\}

4.1

where

P : L^{2} (M, R^{4}) \to H

is the Leray projection and

ψ = (u, T, q) \in H_{1} .

Thus, the problem (2.17)–(2.19) is expressed in form of

\frac{d ψ}{d t} = L_{R} ψ + G (ψ),

4.2

and the eigenvalue problem (3.1) with the condition (2.19) is rewritten as

L_{R} ψ = β ψ .

4.3

Step 2. We shall calculate the centre manifold reduction for (4.2) in this step. Let $ψ_{k_{c}}^{1}$ and $ψ_{k_{c}}^{2}$ be the eigenfunctions of (4.3) corresponding to $β_{k_{c}}^{1} (R)$ , where $β_{k_{c}}^{1}$ is the eigenvalue of (4.3) in the case of (3.27). Denote the conjugate eigenfunctions of $ψ_{k_{c}}^{1}$ and $ψ_{k_{c}}^{2}$ by ${ψ_{k_{c}}^{1}}^{*}$ and ${ψ_{k_{c}}^{2}}^{*}$ , i.e.

L_{R}^{*} {ψ_{k_{c}}^{i}}^{*} = β_{k_{c}}^{1} {ψ_{k_{c}}^{i}}^{*}, i = 1, 2.

4.4

The corresponding equations of (4.4) read

\begin{aligned} Pr (Δ u_{1}^{*} + \frac{2}{r_{0}} \frac{\partial u_{2}^{*}}{\partial x_{1}} - \frac{1}{r_{0}^{2}} u_{1}^{*} - \frac{\partial p^{*}}{\partial x_{1}}) + ω u_{2}^{*} = β u_{1}^{*}, \\ Pr (Δ u_{2}^{*} - \frac{2}{r_{0}} \frac{\partial u_{1}^{*}}{\partial x_{1}} - \frac{2}{r_{0}^{2}} u_{2}^{*} - \frac{\partial p^{*}}{\partial x_{2}}) + γ T^{*} + q^{*} - ω u_{1}^{*} = β u_{2}^{*}, \\ Δ T^{*} + Pr R u_{2}^{*} = β T^{*}, \\ Le Δ q^{*} + Pr \tilde{R} u_{2}^{*} + α T^{*} = β q^{*} \\ and & \frac{\partial u_{1}^{*}}{\partial x_{1}} + \frac{\partial u_{2}^{*}}{\partial x_{2}} = 0. \end{aligned}\}

4.5

Express ψ∈H₁ as

ψ = x ψ_{k_{c}}^{1} + y ψ_{k_{c}}^{2} + Φ (x, y, R),

4.6

where

(x, y) \in R^{2}

and Φ(x,y,R) is the centre manifold function of (4.2) near R=R_c. For the reference of centre manifold functions, see, for example, Section 2.4 of [23]. Then, the reduction equations of (4.2) are

\begin{aligned} \frac{d x}{d t} & = β_{k_{c}}^{1} x + \frac{1}{⟨ ψ_{k_{c}}^{1}, {ψ_{k_{c}}^{1}}^{*} ⟩} ⟨ G (ψ), {ψ_{k_{c}}^{1}}^{*} ⟩ \\ and \frac{d y}{d t} & = β_{k_{c}}^{1} y + \frac{1}{⟨ ψ_{k_{c}}^{2}, {ψ_{k_{c}}^{2}}^{*} ⟩} ⟨ G (ψ), {ψ_{k_{c}}^{2}}^{*} ⟩ . \end{aligned}\}

4.7

Step 3. Computation of $ψ_{k_{c}}^{i}$ and ${ψ_{k_{c}}^{i}}^{*}$ (i=1,2). Based on (3.2)–(3.8), we can derive

ψ_{k_{c}}^{1} = \{\begin{cases} u_{1}^{1} = - u_{k_{c}} \cos π (x_{2} - r_{0}) \sin \frac{k_{c} x_{1}}{r_{0}}, \\ u_{2}^{1} = v_{k_{c}} \sin π (x_{2} - r_{0}) \cos \frac{k_{c} x_{1}}{r_{0}}, \\ T_{1}^{1} = T_{k_{c}} \sin π (x_{2} - r_{0}) \cos \frac{k_{c} x_{1}}{r_{0}}, \\ q^{1} = q_{k_{c}} \sin π (x_{2} - r_{0}) \cos \frac{k_{c} x_{1}}{r_{0}} \end{cases}

4.8

and

ψ_{k_{c}}^{2} = \{\begin{cases} u_{1}^{2} = u_{k_{c}} \cos π (x_{2} - r_{0}) \cos \frac{k_{c} x_{1}}{r_{0}}, \\ u_{2}^{2} = v_{k_{c}} \sin π (x_{2} - r_{0}) \sin \frac{k_{c} x_{1}}{r_{0}}, \\ T^{2} = T_{k_{c}} \sin π (x_{2} - r_{0}) \sin \frac{k_{c} x_{1}}{r_{0}}, \\ q^{2} = q_{k_{c}} \sin π (x_{2} - r_{0}) \sin \frac{k_{c} x_{1}}{r_{0}}, \end{cases}

4.9

where

\begin{aligned} u_{k_{c}} & = \frac{r_{0} π}{k_{c}}, v_{k_{c}} = 1, \\ and T_{k_{c}} & = \frac{α + γ (Le α_{k_{c}}^{2} + β_{k_{c}}^{1})}{(α_{k_{c}}^{2} + β_{k_{c}}^{1}) (Le α_{k_{c}}^{2} + β_{k_{c}}^{1})}, q_{k} = \frac{1}{Le α_{k_{c}}^{2} + β_{k_{c}}^{1}} . \end{aligned}\}

4.10

Similarly, we can also derive from (4.5) the conjugate eigenfunctions as follows:

{ψ_{k_{c}}^{1}}^{*} = \{\begin{cases} {u_{1}^{1}}^{*} = - u_{k_{c}}^{*} \cos π (x_{2} - r_{0}) \sin \frac{k_{c} x_{1}}{r_{0}}, \\ {u_{2}^{1}}^{*} = v_{k_{c}}^{*} \sin π (x_{2} - r_{0}) \cos \frac{k_{c} x_{1}}{r_{0}}, \\ {T_{1}^{1}}^{*} = T_{k_{c}}^{*} \sin π (x_{2} - r_{0}) \cos \frac{k_{c} x_{1}}{r_{0}}, \\ {q^{1}}^{*} = q_{k_{c}}^{*} \sin π (x_{2} - r_{0}) \cos \frac{k_{c} x_{1}}{r_{0}} \end{cases}

4.11

and

{ψ_{k_{c}}^{2}}^{*} = \{\begin{cases} {u_{1}^{2}}^{*} = u_{k_{c}}^{*} \cos π (x_{2} - r_{0}) \cos \frac{k_{c} x_{1}}{r_{0}}, \\ {u_{2}^{2}}^{*} = v_{k_{c}}^{*} \sin π (x_{2} - r_{0}) \sin \frac{k_{c} x_{1}}{r_{0}}, \\ {T^{2}}^{*} = T_{k_{c}}^{*} \sin π (x_{2} - r_{0}) \sin \frac{k_{c} x_{1}}{r_{0}}, \\ {q^{2}}^{*} = q_{k_{c}}^{*} \sin π (x_{2} - r_{0}) \sin \frac{k_{c} x_{1}}{r_{0}}, \end{cases}

4.12

where

\begin{aligned} u_{k_{c}}^{*} & = \frac{r_{0} π}{k_{c}}, v_{k_{c}}^{*} = 1, \\ and T_{k_{c}}^{*} & = \frac{Pr R}{α_{k_{c}}^{2} + β_{k_{c}}^{1}}, q_{k_{c}}^{*} = \frac{α Pr R - Pr \tilde{R} (α_{k_{c}}^{2} + β_{k_{c}}^{1})}{(α_{k_{c}}^{2} + β_{k_{c}}^{1}) (Le α_{k_{c}}^{2} + β_{k_{c}}^{1})} . \end{aligned}\}

4.13

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

G (ψ, ϕ) = - P (\begin{matrix} u_{1} \frac{\partial v_{1}}{\partial x_{1}} + u_{2} \frac{\partial v_{1}}{\partial x_{2}} + \frac{1}{r_{0}} u_{1} v_{2} \\ u_{1} \frac{\partial v_{2}}{\partial x_{1}} + u_{2} \frac{\partial v_{2}}{\partial x_{2}} - \frac{1}{r_{0}} u_{1} v_{1} \\ u_{1} \frac{\partial \tilde{T}}{\partial x_{1}} + u_{2} \frac{\partial \tilde{T}}{\partial x_{2}} \\ u_{1} \frac{\partial \tilde{q}}{\partial x_{1}} + u_{2} \frac{\partial \tilde{q}}{\partial x_{2}} \end{matrix}),

4.14

where ψ=(u₁,u₂,T,q) and

ϕ = (v_{1}, v_{2}, \tilde{T}, \tilde{q})

. Direct calculation shows

\begin{aligned} G (ψ_{k}^{1}, ψ_{k}^{1}) & = - P [{(\frac{r_{0} π^{2}}{2 k} \sin \frac{2 k x_{1}}{r_{0}}, \frac{π}{2} \sin 2 π (x_{2} - r_{0}) - \frac{r_{0} π^{2}}{4 k^{2}} [1 + \cos 2 π (x_{2} - r_{0})], 0, 0)}^{t} \\ + {(- \frac{π}{4 k} \sin 2 π (x_{2} - r_{0}) \sin \frac{2 k x_{1}}{r_{0}}, \frac{r_{0} π^{2}}{4 k^{2}} \cos 2 π (x_{2} - r_{0}) \cos \frac{2 k x_{1}}{r_{0}}, 0, 0)}^{t} \\ + {(0, \frac{r_{0} π^{2}}{4 k^{2}} \cos \frac{2 k x_{1}}{r_{0}}, \frac{T_{k} π}{2} \sin 2 π (x_{2} - r_{0}), \frac{q_{k} π}{2} \sin 2 π (x_{2} - r_{0}))}^{t}] . \end{aligned}

4.15

As the first two terms on the right-hand side of (4.15) are gradient fields, we obtain

G (ψ_{k}^{1}, ψ_{k}^{1}) = {(0, - \frac{r_{0} π^{2}}{4 k^{2}} \cos \frac{2 k x_{1}}{r_{0}}, - \frac{T_{k} π}{2} \sin 2 π (x_{2} - r_{0}), - \frac{q_{k} π}{2} \sin 2 π (x_{2} - r_{0}))}^{t} .

4.16

Similarly, we can derive

\begin{aligned} G (ψ_{k}^{1}, ψ_{k}^{2}) & = {(- \frac{r_{0} π^{2}}{2 k} \cos 2 π (x_{2} - r_{0}) + \frac{π}{4 k} \sin 2 π (x_{2} - r_{0}), - \frac{r_{0} π^{2}}{4 k^{2}} \sin \frac{2 k x_{1}}{r_{0}}, 0, 0)}^{t}, \\ G (ψ_{k}^{2}, ψ_{k}^{1}) & = {(\frac{r_{0} π^{2}}{2 k} \cos 2 π (x_{2} - r_{0}) - \frac{π}{4 k} \sin 2 π (x_{2} - r_{0}), - \frac{r_{0} π^{2}}{4 k^{2}} \sin \frac{2 k x_{1}}{r_{0}}, 0, 0)}^{t} \\ and G (ψ_{k}^{2}, ψ_{k}^{2}) & = {(0, \frac{r_{0} π^{2}}{4 k^{2}} \cos \frac{2 k x_{1}}{r_{0}}, - \frac{T_{k} π}{2} \sin 2 π (x_{2} - r_{0}), - \frac{q_{k} π}{2} \sin 2 π (x_{2} - r_{0}))}^{t} \end{aligned}\}

4.17

and

\begin{aligned} G (ψ_{k}^{1}, {ψ_{k}^{1}}^{*}) & = {(0, - \frac{r_{0} π^{2}}{4 k^{2}} \cos \frac{2 k x_{1}}{r_{0}}, - \frac{{T_{k}}^{*} π}{2} \sin 2 π (x_{2} - r_{0}), - \frac{{q_{k}}^{*} π}{2} \sin 2 π (x_{2} - r_{0}))}^{t}, \\ G (ψ_{k}^{1}, {ψ_{k}^{2}}^{*}) & = {(- \frac{r_{0} π^{2}}{2 k} \cos 2 π (x_{2} - r_{0}) + \frac{π}{4 k} \sin 2 π (x_{2} - r_{0}), - \frac{r_{0} π^{2}}{4 k^{2}} \sin \frac{2 k x_{1}}{r_{0}}, 0, 0)}^{t}, \\ G (ψ_{k}^{2}, {ψ_{k}^{1}}^{*}) & = {(\frac{r_{0} π^{2}}{2 k} \cos 2 π (x_{2} - r_{0}) - \frac{π}{4 k} \sin 2 π (x_{2} - r_{0}), - \frac{r_{0} π^{2}}{4 k^{2}} \sin \frac{2 k x_{1}}{r_{0}}, 0, 0)}^{t} \\ and G (ψ_{k}^{2}, {ψ_{k}^{2}}^{*}) & = {(0, \frac{r_{0} π^{2}}{4 k^{2}} \cos \frac{2 k x_{1}}{r_{0}}, - \frac{{T_{k}}^{*} π}{2} \sin 2 π (x_{2} - r_{0}), - \frac{{q_{k}}^{*} π}{2} \sin 2 π (x_{2} - r_{0}))}^{t} . \end{aligned}\}

4.18

By (4.16) and (4.17), we have

⟨ G (ψ_{k}^{i_{1}}, ψ_{k}^{i_{2}}), {ψ_{k}^{i_{3}}}^{*} ⟩ = 0,

4.19

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

\begin{aligned} ⟨ G (ϕ_{1}, ϕ_{2}), ϕ_{3} ⟩ & = - ⟨ G (ϕ_{1}, ϕ_{3}), ϕ_{2} ⟩ \\ and ⟨ G (ϕ_{1}, ψ_{k}^{i}), {ψ_{k}^{i}}^{*} ⟩ & = 0, \end{aligned}\}

4.20

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

\begin{aligned} ⟨ G (ψ), {ψ_{k_{c}}^{1}}^{*} ⟩ & = - x ⟨ G (ψ_{k_{c}}^{1}, {ψ_{k_{c}}^{1}}^{*}), Φ ⟩ - y ⟨ G (ψ_{k_{c}}^{2}, {ψ_{k_{c}}^{1}}^{*}), Φ ⟩ \\ + y ⟨ G (Φ, ψ_{k_{c}}^{2}), {ψ_{k_{c}}^{1}}^{*} ⟩ + o (| x |^{3} + | y |^{3}) \\ and ⟨ G (ψ), {ψ_{k_{c}}^{2}}^{*} ⟩ & = - x ⟨ G (ψ_{k_{c}}^{1}, {ψ_{k_{c}}^{2}}^{*}), Φ ⟩ - y ⟨ G (ψ_{k_{c}}^{2}, {ψ_{k_{c}}^{2}}^{*}), Φ ⟩ \\ + x ⟨ G (Φ, ψ_{k_{c}}^{1}), {ψ_{k_{c}}^{2}}^{*} ⟩ + o (| x |^{3} + | y |^{3}) . \end{aligned}\}

4.21

Let the centre manifold function be denoted by

Φ = \sum_{β_{k j}^{i} \neq β_{k_{c}}^{1}} Φ_{β_{k j}^{i}} (x, y) ψ_{k j}^{i} .

4.22

By (4.15)–(4.21), only

\begin{aligned} ψ_{02}^{1} & = (0, 0, 0, \sin 2 π (x_{2} - r_{0})), \\ ψ_{02}^{2} & = (0, 0, \sin 2 π (x_{2} - r_{0}), 0) \\ and ψ_{02}^{3} & = (\cos 2 π (x_{2} - r_{0}), 0, 0, 0) \end{aligned}\}

4.23

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

\begin{aligned} ⟨ G (ψ_{k_{c}}^{1}, ψ_{k_{c}}^{1}), ψ_{02}^{1} ⟩ & = - \frac{q_{k_{c}}}{2} π^{2} r_{0}, ⟨ G (ψ_{k_{c}}^{1}, ψ_{k_{c}}^{2}), ψ_{02}^{1} ⟩ = 0, \\ ⟨ G (ψ_{k_{c}}^{2}, ψ_{k_{c}}^{1}), ψ_{02}^{1} ⟩ & = 0, ⟨ G (ψ_{k_{c}}^{2}, ψ_{k_{c}}^{2}), ψ_{02}^{1} ⟩ = - \frac{q_{k_{c}}}{2} π^{2} r_{0}, \\ ⟨ G (ψ_{k_{c}}^{1}, ψ_{k_{c}}^{1}), ψ_{02}^{2} ⟩ & = - \frac{T_{k_{c}}}{2} π^{2} r_{0}, ⟨ G (ψ_{k_{c}}^{1}, ψ_{k_{c}}^{2}), ψ_{02}^{2} ⟩ = 0, \\ ⟨ G (ψ_{k_{c}}^{2}, ψ_{k_{c}}^{1}), ψ_{02}^{2} ⟩ & = 0, ⟨ G (ψ_{k_{c}}^{2}, ψ_{k_{c}}^{2}), ψ_{02}^{2} ⟩ = - \frac{T_{k_{c}}}{2} π^{2} r_{0}, \\ ⟨ G (ψ_{k_{c}}^{1}, ψ_{k_{c}}^{1}), ψ_{02}^{3} ⟩ & = 0, ⟨ G (ψ_{k_{c}}^{1}, ψ_{k_{c}}^{2}), ψ_{02}^{3} ⟩ = - \frac{r_{0}^{2} π^{3}}{2 k_{c}} \\ and ⟨ G (ψ_{k_{c}}^{2}, ψ_{k_{c}}^{1}), ψ_{02}^{3} ⟩ & = \frac{r_{0}^{2} π^{3}}{2 k_{c}}, ⟨ G (ψ_{k_{c}}^{2}, ψ_{k_{c}}^{2}), ψ_{02}^{3} ⟩ = 0. \end{aligned}\}

4.24

We note that

\begin{aligned} β_{02}^{1} & = - Le α_{02}^{2} = - 4 π^{2} Le, \\ β_{02}^{2} & = - α_{02}^{2} = - 4 π^{2} \\ and β_{02}^{3} & = - Pr α_{02}^{2} - \frac{2 Pr}{r_{0}^{2}} = - 4 π^{2} Pr - \frac{1}{r_{0}^{2}} Pr . \end{aligned}\}

4.25

By the approximation formula of centre manifold functions, see Section 2.4 of [23], we get

\begin{aligned} Φ_{β_{02}^{1}} (x, y) & = - \frac{q_{k}}{8 π Le} (x^{2} + y^{2}) + o (x^{2} + y^{2}), \\ Φ_{β_{02}^{2}} (x, y) & = - \frac{T_{k}}{8 π} (x^{2} + y^{2}) + o (x^{2} + y^{2}) \\ and Φ_{β_{02}^{3}} (x, y) & = o (x^{2} + y^{2}) . \end{aligned}\}

4.26

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

\begin{aligned} ⟨ G (ψ), {ψ_{k_{c}}^{1}}^{*} ⟩ & = - x ⟨ G (ψ_{k_{c}}^{1}, {ψ_{k_{c}}^{1}}^{*}), Φ ⟩ + o (| x |^{3} + | y |^{3}) \\ = - \frac{π r_{0}}{16} (\frac{q_{k_{c}} q_{k_{c}}^{*}}{Le} + T_{k_{c}} T_{k_{c}}^{*}) x (x^{2} + y^{2}) + o (| x |^{3} + | y |^{3}) \\ and ⟨ G (ψ), {ψ_{k_{c}}^{2}}^{*} ⟩ & = - y ⟨ G (ψ_{k_{c}}^{2}, {ψ_{k_{c}}^{2}}^{*}), Φ ⟩ + o (| x |^{3} + | y |^{3}) \\ = - \frac{π r_{0}}{16} (\frac{q_{k_{c}} q_{k_{c}}^{*}}{Le} + T_{k_{c}} T_{k_{c}}^{*}) y (x^{2} + y^{2}) + o (| x |^{3} + | y |^{3}) . \end{aligned}\}

4.27

By (4.10), (4.13) and (4.27), we evaluate (4.7) at R=R_c and obtain the reduction equation

\begin{aligned} \frac{d x}{d t} & = - \frac{b}{8} x (x^{2} + y^{2}) + o (| x |^{3} + | y |^{3}) \\ and \frac{d y}{d t} & = - \frac{b}{8} y (x^{2} + y^{2}) + o (| x |^{3} + | y |^{3}), \end{aligned}\}

4.28

where b is as defined as

\begin{aligned} b = \frac{{Le}^{3} α_{k_{c}}^{2} R_{c} + (((1 + {Le}^{2}) α_{1} α_{T} h^{2}) / α_{q} κ_{T} + α_{1} α_{T} α_{k_{c}}^{2} {Le}^{3} h^{2} / 24 α_{q} κ_{T} + α_{k_{c}}^{2}) \tilde{R}}{({Le}^{3} α_{k_{c}}^{6} / Pr) (1 + r_{0}^{2} π^{2} / k_{c}^{2}) + {Le}^{3} α_{k_{c}}^{2} R + Le ((1 + Le) α_{1} α_{T} h^{2} / α_{q} κ_{T} + α_{1} α_{T} α_{k_{c}}^{2} {Le}^{2} h^{2} / 24 α_{q} κ_{T} + α_{k_{c}}^{2}) \tilde{R}}, \end{aligned}

4.29

and $α_{K}^{2}$ and R_c 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:

\begin{aligned} \frac{\partial u_{1}}{\partial t} & = Pr (Δ u_{1} + \frac{2}{r_{0}} \frac{\partial u_{2}}{\partial x_{1}} - \frac{1}{r_{0}^{2}} u_{1} - δ_{0} u_{1} - \frac{\partial p}{\partial x_{1}}) \\ - ω u_{2} - (u \cdot \nabla) u_{1} - \frac{1}{r_{0}} u_{1} u_{2}, \\ \frac{\partial u_{2}}{\partial t} & = Pr (Δ u_{2} - \frac{2}{r_{0}} \frac{\partial u_{1}}{\partial x_{1}} - \frac{2}{r_{0}^{2}} u_{2} + R T + \tilde{R} q - δ_{1} u_{2} - \frac{\partial p}{\partial x_{2}}) \\ + ω u_{1} - (u \cdot \nabla) u_{2} + \frac{1}{r_{0}} u_{1}^{2}, \\ \frac{\partial T}{\partial t} & = Δ T + γ u_{2} + α q - (u \cdot \nabla) T, \\ \frac{\partial q}{\partial t} & = Le Δ q + u_{2} - (u \cdot \nabla) q \\ and div u & = 0, \end{aligned}\}

5.1

where

\begin{aligned} γ & = 1 + \frac{α}{24}, \\ δ_{i} & = \frac{C_{i} h^{4}}{ν}, i = 0, 1, \\ C_{0} & = 3.78 \times 10^{- 1} m^{- 2} s^{- 1} \\ and C_{1} & = 6.7 \times 10^{4} m^{- 2} s^{- 1} . \end{aligned}\}

5.2

The domain is M=[0,2πr₀]×(r₀,r₀+1), and the boundary conditions are given by

\begin{aligned} (u, T, q) (x_{1} + 2 π r_{0}, x_{2}) = (u, T, q) (x_{1}, x_{2}) \\ and & u_{2} = 0, \frac{\partial u_{1}}{\partial x_{2}} = 0, T = 0, q = 0, at x_{2} = r_{0}, r_{0} + 1. \end{aligned}\}

5.3

The corresponding eigenvalue problem reads

\begin{aligned} Pr (Δ u_{1} + \frac{2}{r_{0}} \frac{\partial u_{2}}{\partial x_{1}} - \frac{1}{r_{0}^{2}} u_{1} - δ_{0} u_{1} - \frac{\partial p}{\partial x_{1}}) - ω u_{2} = β u_{1}, \\ Pr (Δ u_{2} - \frac{2}{r_{0}} \frac{\partial u_{1}}{\partial x_{1}} - \frac{2}{r_{0}^{2}} u_{2} + R T + \tilde{R} q - δ_{1} u_{2} - \frac{\partial p}{\partial x_{2}}) + ω u_{1} = β u_{2}, \\ Δ T + γ u_{2} + α q = β T, \\ Le Δ q + u_{2} = β q \\ and & \frac{\partial u_{1}}{\partial x_{1}} + \frac{\partial u_{2}}{\partial x_{2}} = 0. \end{aligned}\}

5.4

Using the same analysis as in §3a, we obtain the following formula of the critical thermal Rayleigh number:

\begin{aligned} R_{c} & = min_{k, j \geq 1} [- (\frac{1}{Le} + (1 + \frac{Le α_{k j}^{2}}{24}) \frac{α_{1} α_{T} h^{2}}{Le α_{q} α_{k j}^{2} κ_{T}}) \tilde{R} \\ + α_{k j}^{4} \frac{r_{0}^{2}}{k^{2}} (α_{k j}^{2} + \frac{1}{r_{0}^{2}}) + \frac{α_{k j}^{2}}{r_{0}^{2}} + δ_{1} α_{k j}^{2} + \frac{δ_{0} π^{2} r_{0}^{2} α_{k j}^{2}}{k^{2}}] \\ = min_{k \geq 1} [- (\frac{1}{Le} + (1 + \frac{Le α_{k 1}^{2}}{24}) \frac{α_{1} α_{T} h^{2}}{Le α_{q} α_{k 1}^{2} κ_{T}}) \tilde{R} \\ + α_{k 1}^{4} \frac{r_{0}^{2}}{k^{2}} (α_{k 1}^{2} + \frac{1}{r_{0}^{2}}) + \frac{α_{k 1}^{2}}{r_{0}^{2}} + δ_{1} α_{k 1}^{2} + \frac{δ_{0} π^{2} r_{0}^{2} α_{k 1}^{2}}{k^{2}}], \end{aligned}

5.5

where

α_{k 1}^{2} = π^{2} + \frac{k^{2}}{r_{0}^{2}} .

Let,

y = k^{2} / r_{0}^{2}

and

\begin{aligned} g (y) & = - (\frac{1}{Le} + (1 + \frac{Le (y + π^{2})}{24}) \frac{α_{1} α_{T} h^{2}}{Le α_{q} κ_{T} (y + π^{2})}) \tilde{R} + \frac{1}{y} {(y + π^{2})}^{2} (y + π^{2} + \frac{1}{r_{0}^{2}}) \\ + \frac{1}{r_{0}^{2}} (y + π^{2}) + δ_{1} (y + π^{2}) + \frac{δ_{0} π^{2} (y + π^{2})}{y} . \end{aligned}

Taking the derivative of g(y), we get

g^{'} (y) = 2 y + (3 π^{2} + \frac{2}{r_{0}^{2}} + δ_{1}) - \frac{1}{y^{2}} (π^{6} + \frac{π^{4}}{r_{0}^{2}} + δ_{0} π^{4}) + \frac{α_{1} α_{T} h^{2} \tilde{R}}{Le α_{q} κ_{T} {(y + π^{2})}^{2}} .

By (5.2), we have

δ_{1} = 2.76 \times 10^{20} and δ_{0} = 1.55 \times 10^{15} .

5.6

Hence,

δ_{1} ≫ δ_{0} ≫ 1.

5.7

Under this condition, the critical value of g(y) is approximated by

y_{c} ≃ (\frac{δ_{0}}{δ_{1}}) π^{2} .

5.8

Therefore, the critical thermal Rayleigh number is approximated by

R_{c} ≃ δ_{1} π^{2} = 2.76 \times 10^{21} .

5.9

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

\begin{aligned} ν & = 1.6 \times 10^{- 5} m^{2} s^{- 1}, κ_{T} = 2.25 \times 10^{- 5} m^{2} s^{- 1}, \\ and α_{T} & = 3.3 \times 10^{- 3} {per}^{\circ} C, Pr = 0.71. \end{aligned}\}

5.10

We note also that the height of the troposphere is

h = 8 \times 10^{3} m .

Hence, we get

T_{0} - T_{1} = Δ T_{c} = \frac{κ_{T} ν}{α_{T} g h^{3}} R_{c} = 60^{\circ} C .

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 $\tilde{R}$ 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

\frac{k^{2}}{r_{0}^{2}} = {(\frac{δ_{0}}{δ_{1}})}^{1 / 2} π^{2}

5.12

that the wavenumber k_c and the convection length-scale L_c as

k_{c} ≃ 6 and L = \frac{(6400 \times π)}{6} = 3350 km .

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:

\frac{d u}{d t} = L_{λ} u + G (u, λ), u (0) = u_{0},

where

u : [0, \infty) \to X

is the unknown function,

λ \in R^{1}

is the system control parameter and X is a Banach space. We shall always consider u the deviation of the unknown function from some equilibrium state

\bar{u}

. Hence, L_λ:X₁→X is a linear operator, and

G : X_{1} \times R^{1} \to X

is a nonlinear 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 (A1). The underlying physical concept is the PES, leading to precise information on linear unstable modes. Let, ${β_{j} (λ) \in C ∣ j \in N}$ be the eigenvalues (counting multiplicity) of L_λ and assume that

Re β_{i} (λ) \{\begin{cases} < 0 & if λ < λ_{0}, \\ = 0 & if λ = λ_{0}, \\ > 0 & if λ > λ_{0} \end{cases} \forall 1 \leq i \leq m,

and

Re β_{j} (λ_{0}) < 0 \forall j \geq m + 1.

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 (A1) at (0,λ₀) is completely dictated by its reduction equation near λ=λ₀

\frac{d x}{d t} = J_{λ} x + g (x, λ) for x \in R^{m},

where g(x,λ)=(g₁(x,λ),…,g_m(x,λ)) and

g_{j} (x, λ) = ⟨G (\sum_{i = 1}^{m} x_{i} e_{i} + Φ (x, λ), λ), e_{j}^{*}⟩ \forall 1 \leq j \leq m .

Here, e_j and e*_j (1≤j≤m) are the eigenvectors of L_λ and L*_λ, respectively, corresponding to the eigenvalues β_j(λ), J_λ is the m×m order Jordan matrix corresponding to the eigenvalues, and Φ(x,λ) is the centre manifold function of (A1) 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 (A1). Suppose the nonlinear operator G to be of the form

G (u, λ) = G_{k} (u, λ) + o (∥ u ∥^{k}), as u \to 0 in X_{μ} .

for some integer k≥2, where G_k is a k-multilinear operator

G_{k} (u, λ) = G_{k} (u, \dots, u, λ) : X_{1} \times \dots \times X_{1} ⟶ X .

By Theorem A.1.1 in [24], the centre manifold function Φ(x,λ) can be expressed as

Φ (x, λ) = \int_{- \infty}^{0} e^{- τ L_{λ}} ρ_{ε} P_{2} G_{k} (e^{τ J_{λ}} x, λ) d τ + o (∥ x ∥^{k}),

where J_λ and

L_{λ}

are restrictions of L_λ to the linear invariant spaces generated by the first modes and the rest modes, respectively, G_k(u,λ) is the lowest order k-multiple linear operator, and

x = \sum_{i = 1}^{m} x_{i} e_{i}

. In particular, we have the following assertions:

(1) If J_λ is diagonal near λ=λ₀, then (A7) can be written as

- L_{λ} Φ (x, λ) = P_{2} G_{k} (x, λ) + o (∥ x ∥^{k}) + O (| β | ∥ x ∥^{k}) .

(2) Let m=2 and $β_{1} (λ) = \bar{β_{2} (λ)} = α (λ) + i ρ (λ)$ with ρ(λ₀)≠0. If G_k(u,λ)=G₂(u,λ) is bilinear, then Φ(x,λ) can be expressed as

\begin{aligned} [{(- L_{λ})}^{2} + 4 ρ^{2} (λ)] (- L_{λ}) Φ (x, λ) & = [{(- L_{λ})}^{2} + 4 ρ^{2} (λ)] P_{2} G_{2} (x, λ) \\ - 2 ρ^{2} (λ) P_{2} G_{2} (x, λ) + 2 ρ^{2} P_{2} G_{2} (x_{1} e_{2} - x_{2} e_{1}) \\ + ρ (- L_{λ}) P_{2} [G_{2} (x_{1} e_{1} + x_{2} e_{2}, x_{2} e_{1} - x_{1} e_{2}) \\ + G_{2} (x_{2} e_{1} - x_{1} e_{2}, x_{1} e_{1} + x_{2} e_{2})] + o (k), \end{aligned}

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

Footnotes

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

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This Issue

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences cover image

January 2015

Volume 471Issue 2173

Article Information

DOI:https://doi.org/10.1098/rspa.2014.0353
PubMed:25568615
Published by:Royal Society
Print ISSN:1364-5021
Online ISSN:1471-2946

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Manuscript received01/05/2014
Manuscript accepted31/10/2014
Published online08/01/2015
Published in print08/01/2015

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