Atmospheric Instability and Its Associated Oscillations in the Tropics

Abstract

The interaction between tropical clouds and radiation is studied in the context of the weak temperature gradient approximation, using very low order systems (e.g., a two-column two-layer model) as a zeroth-order approximation. Its criteria for the instability are derived in the systems. Owing to the connection between the instability (unstable fixed point) and the oscillation (limit cycle) in physics (phase) space, the systems suggest that the instability of tropical clouds and radiation leads to the atmospheric oscillations with distinct timescales observed. That is, the instability of the boundary layer quasi-equilibrium leads to the quasi-two-day oscillation, the instability of the radiative convective equilibrium leads to the Madden–Julian oscillation (MJO), and the instability of the radiative convective flux equilibrium leads to the El Niño–southern oscillation. In addition, a linear model as a first-order approximation is introduced to reveal the zonal asymmetry of the atmospheric response to a standing convective/radiative heating oscillation. Its asymmetric resonance conditions explain why a standing ~45-day oscillation in the systems brings about a planetary-scale eastward travelling vertical circulation like the MJO. The systems, despite of their simplicity, replicate the oscillations with the distinct timescales observed, providing a novel cloud parameterization for weather and climate models. Their instability criteria further suggests that the models can successfully predict the oscillations if they properly represent cirrus clouds and convective downdrafts in the tropics.

1. Introduction

The views of atmospheric predictability changed from Laplace’s determinism to the deterministic chaos of Lorenz (1963), and recently to the coexistence of determinism and chaos (Shen et al. 2021; see Shen et al. 2022 for review) [1,2,3]. Since the previous studies focused on the Bénard convection and the geostrophic baroclinic model in the middle latitudes (Lorenz 1963, 1969, 1990) [1,4,5], the present study focuses on tropical convection, the engine of atmospheric circulations (Riehl and Malkus 1958) [6].

Lorenz (1963) used a nonlinear system of three differential equations to study the predictability of dry convection with a timescale of minutes [1], referring to the predictability of weather and climate change. Since the Lorenz system included a convective instability, its prognostic variables changed periodically and/or non-periodically. Once the system had no instability, its prognostic variables eventually became steady, which exhibited the importance of the convective instability in the predictability.

In this paper, simple systems like the Lorenz system were constructed to address the predictability of weather and climate change in the tropics. The systems incorporated two key components of weather and climate change: moisture (or clouds), which involves the release of potential energy and radiation, which controls the generation of potential energy and is affected by the clouds. In addition, the systems use the instability of tropical clouds and large-scale circulation (Raymond and Zeng 2000) in place of the convective instability in the Lorenz system, where the instability of Raymond and Zeng (2000) has a much longer timescale than the convective instability (see Section 2) [7].

In the proposed systems, the instability was usually connected to oscillations, where one of them is illustrated as

$$\frac{d{X}_2}{d\tau }=-70X_2-35(X_2-6.3)(X_2+1)(X_2-Y_2)\mathrm{and}$$

(1a)

$$\frac{d{Y}_2}{d\tau }=140(X_2+1)(X_2-Y_2),$$

(1b)

where X₂ and Y₂ represent the dimensionless air surface entropy (or energy) in one region and the air saturation moist entropy (or temperature) at an altitude of 2 km in another neighboring region, respectively. The system of Equation (1) describes the charge and discharge of entropy in the planetary boundary layer (PBL) (see Appendix C for details). It has an unstable fixed point of X₂ = Y₂ = 0 in the phase space (X₂, Y₂), which corresponds to the instability of the boundary layer quasi-equilibrium. It also has a limit cycle that corresponds to a quasi-two-day oscillation (see Figure 1). Once its coefficients are changed so that the fixed point becomes stable, its limit cycle expires. This connection between the unstable fixed point and the limit cycle is generalized to all the continuous dynamical systems on the plane by the Poincaré—Bendixson theorem [8], suggesting that the quasi-two-day oscillation is connected to the instability of the boundary layer quasi-equilibrium.

The instability of the boundary layer quasi-equilibrium in the system functions similarly to the convective instability in the Lorenz system, and, therefore, is key to understand the quasi-two-day oscillation. In this paper, the connection between the instability and the oscillation is extended to other two cases: the Madden–Julian oscillation (MJO) and the El Niño–southern oscillation (ENSO).

The paper is organized as follows. Section 2 uses the moist entropy to distinguish two opposite views on convective clouds and large-scale circulations. Section 3 illustrates the instability of the radiative convective equilibrium in the context of the weak temperature gradient approximation. Section 4 introduces a two-column two-layer model of the interaction between tropical clouds and large-scale circulations. Section 5 extends the model to the other two cases with distinct timescales observed. Section 6 concludes the paper.

2. Are Convective Clouds Active or Passive?

There are two opposite views on atmospheric convection in the tropics. Deep convective clouds, as an active factor, penetrate through the level of the minimum moist entropy into the upper troposphere (UT) as hot towers and subsequently drive large-scale vertical circulations (e.g., the Hadley circulation, hurricanes) (Riehl and Malkus 1958) [6]. On the other hand, convective clouds occur as a passive factor to balance atmospheric radiation losses and consequently their spectrum is controlled by radiative cooling (Manabe and Strickler 1964) [9]. In this section, the two opposite views are distinguished via a series of ideal experiments on the radiative convective equilibrium (RCE), beginning with the expression of the moist entropy.

2.1. Expression of Moist Entropy

Consider an air parcel with water vapor, liquid drops and ice particles. Its moist entropy per unit mass of dry air equals the sum of the entropy of dry air and the entropies of vapor, liquid and solid water. The moist entropy thus is expressed as [10]

$$s=(C_p+c_l{q}_t)\ln \frac{T}{T_{ref}}-R_d\ln \frac{p-e}{p_{ref}}+\left[\frac{L_v(T)}{T}-R_v\ln f\right]q_v-\left[\frac{L_f(T)}{T}-R_v\ln \frac{E_{sw}(T)}{E_{si}(T)}\right]q_i$$

(2)

where T is the air temperature, p is the total pressure of the moist air, e is the partial pressure of the water vapor, q_t is the total mixing ratio of the airborne water (i.e., q_t = q_v + q_c + q_i), q_v/q_c/q_i is the mixing ratio of the water vapor/cloud water/cloud ice, L_v/L_f is the latent heat of the vaporization/freezing, C_p/c_l is the specific heat of the dry air/liquid water, R_d/R_v is the gas constant of the dry air/water vapor, E_sw/E_si is the saturation vapor pressure over water/ice and f = e/E_sw is the relative humidity of the air. In addition, T_ref = 273.15 K and p_ref = 10⁵ pa are the reference temperature and pressure, respectively.

When an air parcel ascends or descends adiabatically, its moist entropy changes with time t as

$$ds/dt=Q_s,$$

(3)

where Q_s represents the increase in entropy from the irreversible processes (e.g., the heat flow from high to low temperature, liquid drop evaporation and ice particle melting; see [10] for its expression). When T_ref = 273.15 K (i.e., at the triple point of pure water), Q_s is small. Hence Q_s ≈ 0 (or the conservation of moist entry) is usually assumed in a conceptual model.

The moist entropy is related to the equivalent potential temperature θ_se by s = C_pln(θ_se/T_ref) [11]. It functions similar to the moist static energy (or C_pT + gz + L_vq_v where z is the height and g is the acceleration due to gravity), although the moist static energy is not conserved in the convective regions with a strong vertical velocity. In addition, Equation (3) can be used to simulate an air parcel with the same accuracy as the energy equation represented in terms of the (potential) temperature [12].

The moist entropy yields conceptual models of atmospheric convection [6]. In contrast, other variables (e.g., temperature or its variation) often yield ill-posed models by artificially separating the water and energy cycles in the atmosphere. One ill-posed model, for example, is the convective instability of the second kind (CISK) [13], which is reasoned as follows. Consider a horizontally uniform atmosphere with convective available potential energy (CAPE). No matter how large the low-level horizontal convergence is and the strength of the convective clouds it induces, the mid-tropospheric temperature never surpasses the temperature of an air parcel that moves adiabatically from the surface to the middle troposphere. In other words, the mid-tropospheric temperature has an upper limit, which is contrary to the prediction of the unlimited mid-tropospheric temperature in the CISK [14,15]. Since the moist entropy does not need an artificial separation of the water and energy cycles, it cannot yield an ill-posed model of moist convection like the CISK.

2.2. Approach of the RCE

Consider a tropical atmosphere subject to radiative cooling over the oceanic equator. Since the radiative cooling brings about the CAPE, convection occurs and transports the energy upward, leading to a balance between the radiative cooling and the convective heating, which is referred to as the RCE [16,17,18,19].

The RCE has three forcing factors: the radiative cooling rate, air surface wind speed and sea surface temperature (SST). If the forcing factors are fixed, coupling does not occur between the convection, atmospheric radiation and surface wind in the atmosphere, providing a simple case to determine the timescales in the approach of the RCE.

Suppose that the radiative cooling rate is 1 °C/day below the height z = 9 km and decreases linearly with height to zero at z = 15 km. In addition, the surface wind speed $w_s$ = 4 m s⁻¹ and the SST = 302 K are set, which are used for determining the sensible heat flux from the sea to the air and the latent heat flux due to sea surface evaporation.

The RCE is simulated with a cloud resolving model (CRM), using the periodic lateral boundary conditions and the vertical velocity w = 0 at the sea surface [19]. Figure 2 displays the time series of the three modeled variables: the flux of entropy from the sea to the air due to sea surface evaporation and the sensible heat transfer from the sea to the air, the flux of entropy from the air to space due to the radiative cooling, and the internal generation of entropy due to cloud microphysics (e.g., rainwater evaporation) and the heat transfer from high to low temperature (or mixing). In the figure, the surface entropy flux from the sea to the air approaches the top entropy flux from the air to space, arriving at the RCE eventually, where the internal entropy generation is negligible.

Figure 3 displays the time series of the air temperature and the relative humidity at z = 0 and 4.5 km, showing two timescales of the temperature in the approach of the RCE. The short timescale is ~2 days and is associated with the timescale of the relative humidity. Hence, the short timescale measures the approach of the water quasi-balance between the sea surface evaporation and precipitation (figure omitted). In contrast, the long timescale measures the approach of the RCE while the atmosphere remains at the water quasi-balance.

The long timescale of the temperature ${\tau }_t$= 29.2 days, obtained by fitting the mid-tropospheric temperature after day four in Figure 3 with $\exp (-2t/{\tau }_t)$, is explained with the budget equation of the tropospheric entropy. Consider a small perturbation of the troposphere from the RCE whose deviation of the moist entropy of the air at z = 0 is denoted as $\Delta s_s$. Thus, $\mu \left(p_s-p_t\right)g^{-1}\Delta s_s$ represents the deviation of the total entropy of the troposphere, where $p_s$ and $p_t$ are the pressure at the sea surface and tropopause, respectively; and μ is the vertically averaged deviation of the moist entropy relative to that at z = 0. After overlooking the internal generation of entropy, the budget equation of the moist entropy of the troposphere becomes

$$\frac{\mu (p_s-p_t)d\Delta s_s}{gdt}={\rho }_s{c}_E{w}_s(s_{sea}-s_{s0}-\Delta s_s)-F_R,$$

(4)

where the first and second terms on the right-hand side represent the surface and top entropy fluxes, respectively; ${\rho }_s$ is the air density near the sea surface, $c_E$ = 1.1 × 10⁻³, $w_s$ is the surface wind speed, $s_{sea}$ is the entropy of the sea surface water as a function of the SST, $s_{s0}$ is the surface moist entropy at the RCE; and F_R is the entropy flux from the air to space due to radiative cooling.

Since the top and surface entropy fluxes are balanced at the RCE or ${\rho }_s{c}_E{w}_s(s_{sea}-s_{s0})=F_R$, Equation (4) is solved with $\Delta s_s\propto \exp (-2t/{\tau }_t)$, where the following timescale

$${\tau }_t=\frac{2\mu (p_s-p_t)}{{\rho }_s{c}_{E}g{w}_s}$$

(5)

shows that ${\tau }_t$ is inversely proportional to $w_s$.

The CRM simulations support the inverse relationship between ${\tau }_t$ and the surface wind speed [19]. Figure 4 displays the modeled ${\tau }_t$ against the dimensionless number

$$N_w={\rho }_s{c}_E{L}_v{w}_s/{\overline{Q}}_R,$$

(6)

where $\overline{Q_R}$ is the total loss of energy of the troposphere due to radiative cooling. The figure also displays the theoretical timescale estimated with Equation (5) and μ = 3.6 for comparison. Generally speaking, Equation (5) agrees with the CRM results except at some low values of N_w ~100 where the initial status is so close to the RCE that the long timescale in the approach of the RCE is not clear and, consequently, its estimation is “polluted” by the short timescale.

In addition, the modeled ${\tau }_t$ decreases with an increase in the radiative cooling rate, which is attributed to the sensitivity of μ to the radiative cooling rate. If the CAPE = 0 were maintained strictly, μ = 1. However, convection usually lags the generation of the CAPE, bringing about μ ≠ 1. When the radiative cooling rate varies with a very short forcing period, for example, μ >> 1 because the surface entropy does not have sufficient time to respond to the cooling of the troposphere. The CRM simulations show that ${\tau }_t$ at the radiative cooling rate of 2 °C/day is about one half of that at the radiative cooling rate 1 °C/day [19]. Hence, μ = 1.8 at the radiative cooling rate of 2 °C/day is obtained that is about one half of μ = 3.6 at the radiative cooling rate 1 °C/day.

2.3. Ideal Experiments of the RCE

To show whether the convective clouds are active or passive, two ideal experiments of the RCE were designed with prescribed forcing. The first experiment used a fixed radiative cooling rate of

$$w_s=w_{s0}\left[1+A\sin (\omega t)\right],$$

(7)

where $w_{s0}$ is the mean surface wind speed, A is the dimensionless amplitude and ω is the angular frequency of the forcing. Substituting Equation (7) for Equation (4) and then solving the resulting equation yields

$$\Delta s_s=\frac{A{\tau }_t}{4+{(\omega {\tau }_t)}^2}\frac{g{F}_R}{\mu (p_s-p_t)}\left[2\sin (\omega t)-\omega {\tau }_t\cos (\omega t)\right]$$

(8)

and the surface entropy flux with the aid of Equation (5).

Figure 5 displays the amplitudes of the air surface temperature and the surface energy flux from Equation (8) against the forcing period of the surface wind. For the forcing period 2π/ω << ${\tau }_t$, Equation (8) degenerates into the term of $\omega {\tau }_t\cos \left(\omega t\right)$, which is referred to here as the active mode. In this mode, $w_s$ leads to a sine variation of the surface entropy flux and a cosine variation of the surface moist entropy (or the CAPE). The convective clouds then distribute the cosine variation of the surface moist entropy into the mid-troposphere, bringing about a cosine variation of the mid-tropospheric moist entropy (or temperature). Hence, the convective clouds are active, leading to the variation of the mid-tropospheric moist entropy (or temperature).

For 2π/ω >> ${\tau }_t$, Equation (8) degenerates into the term of 2sin(ωt), which is referred to as the passive mode. In this mode, the atmosphere is always close to the RCE and, thus, the surface entropy flux is controlled by the fixed radiative cooling rate. Since the surface entropy flux or $w_s\left(s_{sea}-s_s\right)$ is constant, $s_{sea}-s_s$ is inversely proportional to $w_s$. As a result, the surface entropy $s_s$ changes with $w_s$ but the surface entropy flux does not, indicating that the convective clouds are passive and controlled by the radiative cooling rate.

As a general case, Equation (8) possesses both the active and passive modes. When the forcing period 2π/ω is close to ${\tau }_t$ or

$$2\pi /\omega =\pi {\tau }_t,$$

(9)

the two terms of $\omega {\tau }_t\cos \left(\omega t\right)$ and 2sin(ωt) in Equation (8) have the same amplitude, indicating the active and passive modes have the same importance. Their relative importance is measured by the phase delay of the variation in the mid-tropospheric entropy (or temperature) with respect to the variation in the forcing.

To test the analytic solution of Equation (8), three CRM simulations were carried out [19]. The simulations used the forcing period 2π/ω = 4, 32 and 128 days in Equation (7), respectively. In addition, the simulations set $w_{s0}$ = 6 m s⁻¹, A = 1/3, and the radiative cooling rate at 2 K/day. Their results are displayed in Figure 5, supporting the analytic solution of Equation (8).

The second ideal experiment, in contrast to the first one, was carried out to test the response of the convective clouds to the radiative cooling. The experiment used a fixed surface wind speed $w_s$, but the radiative cooling rate was

$$Q_r=Q_{r0}\left[1+A\sin (\omega t)\right],$$

(10)

where Q_r0(z) equals 1.5 °C/day below the height z = 9 km and decreases linearly with the height to zero at z = 15 km. Substituting Equation (10) for Equation (4) yields

$$\Delta s_s=\frac{A{\tau }_t^{}}{4+{(\omega {\tau }_t)}^2}\frac{g{F}_{R0}}{\mu (p_s-p_t)}\left[-2\sin (\omega t)+\omega {\tau }_t\cos (\omega t)\right]$$

(11)

and the surface entropy flux, where F_R0 denotes the entropy flux from the air to space with the radiative cooling rate Q_r0.

The amplitudes of the air surface temperature and the surface energy flux were computed using Equation (11). They were displayed against the forcing period of the radiative cooling in Figure 6. To support the analytic solution of Equation (11), three CRM simulations were carried out with the forcing period 2π/ω = 4, 32 and 128 days in (10), respectively [19]. In addition, the CRM simulations used $w_s$ = 4 m s⁻¹ and A = 1/3. Their results are displayed in Figure 6 for comparison, supporting the analytic solution of Equation (11).

The second experiment supported the first one regarding the separation between the active and passive modes (see Figure 5 and Figure 6). When the forcing period was long (or 2π/ω >> ${\tau }_t$), the top entropy flux determined the surface entropy flux whether the surface wind $w_s$ changed or not, indicating the radiative cooling rate controlled the convective clouds. In contrast, when the forcing period was short (or 2π/ω << ${\tau }_t$), the top entropy flux had almost no effect on the surface entropy flux.

In summary, the active and passive modes were separated at the forcing period of the surface wind and radiative cooling at 2π/ω = π${\tau }_t$ ≈ 45 days (if ${\tau }_t$ = 14 days; see Figure 4 for the variation of τ_t), which corresponds to the period of the MJO [20,21,22]. Roughly speaking, the convective clouds were active in the weather systems with a timescale shorter than ~ 45 days, they are passive in the climate systems with a timescale longer than ~45 days and they possessed dual roles in the systems with a timescale of ~45 days (e.g., the MJO).

3. Instability of the RCE

The RCE in a wide region is unstable once its clouds are organized via a large-scale vertical circulation into two sub-regions: one with deep convective clouds and the other with shallow (or no) clouds (Raymond and Zeng 2000; Sessions et al. 2010) [7,23], which explains the strong Hadley circulation observed (Raymond 2000) [24]. In this section, the instability of the RCE is discussed with its key processes, beginning with its foundation in the weak temperature gradient (WTG) approximation (Bretherton and Smolarkiewicz 1989; Sobel and Bretherton 2000; Bretherton and Sobel 2002) [25,26,27].

3.1. The WTG and Its Application

The WTG is the foundation for understanding the interaction between tropical convection and large-scale vertical circulations (Nilsson and Emanuel 1999; Raymond and Zeng 2000; Sobel and Bretherton 2000) [7,26,28]. In a three-dimensional stratified atmosphere with a large Rossby number (e.g., in the tropics), gravity waves damp the temperature perturbations excited by clouds so that the perturbations disappear eventually, which is referred to as buoyancy damping [19]. On the other hand, the imbalance of the convective heating and radiative cooling generates a forced vertical circulation. Since the horizontal scale of convective clouds is small compared to the Rossby radius of deformation, the amplitude of the forced circulation and the horizontal temperature gradient in the mid-troposphere are small, which is referred to as the WTG [26,27,29]. In brief, buoyancy damping and the WTG represent the process and result of gravity waves, respectively. They are analogous to the damping of the surface waves and the little slope of the pond surface, respectively, when pebbles are tossed into a pond (see Zeng et al. 2020 for the connection between the WTG and buoyancy damping [30]).

Consider two adjacent regions: one with convective clouds and the other with a clear sky. Let T_e(z) denote the vertical profile of the temperature in the clear sky (or environment) region and let ΔT_e(z) denote the difference in the temperature between the convective region and the clear sky region. Obviously, ΔT_e(z) is caused by the imbalance between the convective heating and radiative cooling in the convective region. The analytic model of gravity waves showed that ΔT_e(z) was inversely (directly) proportional to the gravity wave phase speed (the heating intensity and the distance between the two regions) [19,31]. Since many gravity waves with different vertical wavelengths (or phase speeds) work at the same time, their ensemble effect on ΔT_e(z) can be obtained. Generally speaking, ΔT_e(z) is small in the mid-troposphere, working as the WTG. In addition, ΔT_e(z) is relatively large and thus the WTG is not suitable in the PBL, indicating the PBL can maintain its local characteristics. Additionally, ΔT_e(z) is large in the UT, too, because the tropopause with a strong static stability and the sea surface trap internal gravity waves in the troposphere as two “rigid” boundaries.

3.2. Origin of the Instability

Consider the individual clouds in the convective region. Let $\Delta T_c^{\left(i\right)}\left(z\right)$ denote the temperature of a cloud marked with superscript i. Since $\Delta T_e\left(z\right)$ represents the difference in the temperature between the two regions, the temperature of cloud i at the mid-tropospheric height $z_m$ is expressed as

$$T_c^{(i)}(z_m)=\Delta T_c^{(i)}(z_m)+\Delta T_e(z_m)+T_e(z_m).$$

(12)

Replacing $\Delta T_c^{\left(i\right)}\left(z\right)$ approximately with its ensemble average $\Delta \overline{T_c}\left(z\right)$ yields

$$T_c^{(i)}(z_m)=\Delta {\overline{T}}_c(z_m)+\Delta T_e(z_m)+T_e(z_m),$$

(13)

where the left-hand side has superscript i but the right-hand side does not.

Equation (13) has two different applications. (a) Given the surface entropy $s_s$, an air parcel near the sea surface ascends adiabatically to the mid-troposphere. Thus, $\Delta T_c^{\left(i\right)}\left(z_m\right)$ is determined using the entropy conservation and Equation (12). (b) Since the surface entropy $s_s$ below a cloud is changed by the local factors of the cloud itself as well as its neighboring clouds (e.g., convective downdrafts, surface wind gusts), the surface entropy $s_s$ cannot be known at first. Instead, given the right-hand side of Equation (13), $s_s$ is known using the entropy conservation or

$$s_s=s^{\ast }\left(\Delta {\overline{T}}_c(z_m)+\Delta T_e(z_m)+T_e(z_m),p_m\right),$$

(14)

where $p_m$ denotes the pressure at the mid-troposphere and s^*(T, p) is the saturation moist entropy of the air with temperature T and pressure p (or the entropy of the air that is saturated with respect to water at temperature T and pressure p).

Equation (14) resembles the model of Emanuel et al. (1994) in that the vertical temperature profile itself is controlled by convection and tied directly to the surface entropy [14]. The model of Emanuel et al. (1994) is based on the idea that convection is nearly in statistical equilibrium with its environment or $\Delta T_c^{\left(i\right)}\left(z_m\right)$ ≈ 0. After taking the WTG (i.e., $\Delta T_e\left(z_m\right)$ ≈ 0) and $\Delta T_c^{\left(i\right)}\left(z_m\right)$ ≈ 0, Equation (14) is approximated as a concept or

$$s_s\approx s^{\ast }\left(T_e(z_m),p_m\right).$$

(15)

That is, the surface entropy in the convective region is determined by the mid-tropospheric temperature in the clear sky region.

Other models of the convective atmosphere in a tropical β channel complement Equation (15) with higher-order accuracy. The first model with Equation (15) in Section 4, as the zeroth-order approximation, produces a small variation of the imbalance between convective heating and radiative cooling and, thus, a small variation of $T_e\left(z_m\right)$ in a region. The second model in Appendix D, as the first-order approximation, represents a global response (i.e., vertical circulations) to the small thermal forcing in the first model. The third model, as the second-order approximation, represents the feedback between the forced vertical circulations and convective clouds especially those beyond the region of the first model.

If a large-scale vertical circulation was not present between the two regions, the atmosphere in the convective region would stay at the RCE with the surface entropy $s_{s0}$. Since $s_{s0}$ is determined by the convective downdrafts, SST, surface wind and other local factors (see Section 2),

$$s_{s0}\ne s^{\ast }\left(\Delta {\overline{T}}_c(z_m)+\Delta T_e(z_m)+T_e(z_m),p_m\right)$$

(16)

because the right-hand side is determined by a non-local factor, the mid-tropospheric temperature in the clear sky region. The inequality (16) leads to the instability of the RCE. Specifically, the large-scale vertical circulation rises to modulate deep convective clouds that, in turn, change $s_s$ until $s_s$ satisfies Equation (16). In other words, the instability exhibits a movement of $s_s$ from $s_{s0}$ to $s^{\ast }\left(T_e\left(z_m\right),p_m\right)$ with a growing large-scale vertical circulation.

3.3. Convective Downdrafts

Convective downdrafts embody the movement of $s_s$ from $s_{s0}$ to $s^{\ast }\left(T_e\left(z_m\right),p_m\right)$ by redistributing entropy vertically. Figure 7 displays the vertical profile of the buoyancy of an air parcel that ascends adiabatically from the surface to the UT, where the buoyancy is directly proportional to the difference in entropy between the surface entropy (green/black line) and the saturation moist entropy of the environmental air (red line). During their ascent from the surface to the UT, the air parcels detrain into the UT and then mix with their environment. Since the entropy of the mixed parcels falls between the blue and black lines, the cloud detrainment increases the UT entropy. If some mixed parcels have an entropy between the blue and red lines, their buoyancy becomes negative, forming convective downdrafts. Once the lower tropospheric downdrafts penetrate into the PBL, they bring down the mid-tropospheric air with low entropy into the PBL and, consequently, decrease the surface entropy.

Observations show that convective downdrafts are common in the tropics [32,33,34,35]. Figure 8, for example, displays the vertical distribution of downdrafts in a mature mesoscale convective system (MCS) observed by airborne nadir pointing Doppler radar during the Tropical Composition, Cloud and Climate Coupling (TC4) Experiment [36]. The figure displays the vertical cross sections of the observed radar reflectivity and Doppler velocity using the ER-2 Doppler radar (EDOP) on the NASA high-altitude (~20 km) ER-2 aircraft [37,38]. After taking the fall speed of the hydrometeors (ranging from 2.5 to 10 m s⁻¹ with a reflectivity from 0 to 50 dBZ for the drops; 1.2 to 1.8 m s⁻¹ with a reflectivity increasing from –10 to 30 dBZ for snow and 1.8 to 5.4 m s⁻¹ with a reflectivity from 30 to 45 dBZ for graupel) off the Doppler velocity [37], the figure clearly shows that convective downdrafts, especially those in the lower troposphere, are quite common.

The movement of $s_s$ from $s_{s0}$ to $s^{\ast }\left(T_e\left(z_m\right),p_m\right)$ is affected indirectly by a large-scale vertical circulation via deep convective clouds and their downdrafts. To be specific, the large-scale ascent (descent) raises (lowers) the top of deep convective clouds, which is discussed with the aid of Figure 7. A UT large-scale descent brings about an increase in entropy by vertical advection and an increase in the temperature due to the air compression in the UT. As a result, the descent leads to a shift of the blue and red lines in the UT to the right in Figure 7. Since the descent has no impact on the surface entropy (i.e., no impact on the green line in Figure 7), the descent lowers the top of the deep convective clouds (or the height with zero buoyancy). If the descent continues, the deep convective clouds decrease in population and, eventually, shallow ones take their place, which are simulated by the CRM simulations of Sessions et al. (2010) [23]. Similarly, the UT large-scale ascent decreases the UT entropy by vertical advection and, thus, leads to a shift of the blue and red lines in the UT to the left in Figure 7, raising the top of the convective clouds. In short, the large-scale ascent/descent affects the deep convective clouds that in turn vertically redistribute entropy, impacting $s_s$ indirectly.

3.4. Cirrus Clouds and Their Radiation

Cloud radiation, as a complement of the aforementioned dynamical factors, maintains the instability by fueling deep convective clouds. Cirrus clouds, including the cloud anvil from the MCSs, emit infrared radiation into space, bringing about a decrease in the UT temperature and entropy [39]. Meanwhile, the cirrus cloud amount is affected by large-scale vertical velocity. In general, the amount is increased (decreased) by the large-scale ascent (descent) [7]. In addition, the amount is changed indirectly by the large-scale ascent via UT convective downdrafts (see Figure 8 and Figure 9) [40,41,42,43].

The indirect effect of the large-scale ascent on cirrus clouds via UT downdrafts is discussed as follows. Consider an air parcel with super-cooled drops that undergoes a vertical circulation. When the parcel ascends, its temperature decreases first to activate many ice nuclei (IN), and then ice crystals form on the activated IN [44,45]. With the parcel descending, its super-cooled droplets evaporate first and then a part of the ice crystals disappear via sublimation. The sublimation of the ice crystals leaves new solid particles as residues that can be activated at warmer temperatures than the original IN [46,47]. As a result, the parcel has significantly more activated IN than the same parcel with no vertical circulation. In short, the recycling of IN brings about plentiful ice crystals (or activated IN). Roughly speaking, the number of the activated IN depends on the descent distance of convective downdrafts; it increases by 10 times when the descent distance increases by ~0.5 km [41,42,43].

The downdraft-induced activated IN lead to an increase in the ice crystal number, then a decrease in the precipitation efficiency and eventually an increase in cloud anvils (or cirrus clouds) in the UT [41,43]. This process is shown by two CRM simulations in Figure 9: one with a high number concentration of activated IN and the other with a low number concentration. The broad anvil of the MCSs in the CRM simulation with the plentiful activated IN agrees with the field observations, suggesting that the UT convective downdrafts play an important role in cloud anvils.

In summary, UT convective downdrafts are common in a convective region with a large-scale ascent but rare in a clear sky region with a large-scale descent, indicating that large-scale vertical circulation can significantly change the UT cirrus cloud amount via UT downdrafts. Since the cirrus cloud amount changes the radiative cooling of the UT, it modulates the population of deep convective clouds as feedback.

4. A Two-Column Two-Layer Model

A highly simplified model was constructed to assemble the aforementioned processes. In this section, the model is introduced with its structure first, then the criteria for stable RCE and, finally, a connection between the RCE’s instability and a 40-day oscillation of deep convective clouds.

4.1. Model Structure

Consider a two-column two-layer model on the interaction between convective clouds and large-scale vertical circulations in the tropics (see its schematic in Figure 10). The model, similar to Raymond and Zeng (2000) [7], consisted of two adjacent columns: one for testing and the other for reference. For simplicity, the two columns had the same area with width L. They used the same SST and surface wind speed so that the model focused on the interaction between cloud radiation and large-scale vertical circulation. Each column was characterized by two layers: one for the PBL and the other for the UT with cirrus clouds or cloud anvils that detrained from deep convective clouds (i.e., the layer from 7 to 12 km in Figure 7, Figure 8 and Figure 9). The two layers were bridged by deep convective clouds.

The two columns were connected by a large-scale vertical circulation with a mid-tropospheric vertical velocity w_m (or −w_m) in the test (or reference) column. Suppose the two columns stay at the RCE initially with entropy s_s0 in the PBL, s_u0 in the UT layer and w_m = 0. Once a vertical circulation occurs between them, it induces a perturbation of the surface entropy, denoted by Δs_s (or −Δs_s) in the test (or reference) column, and a perturbation of the UT entropy, denoted by Δs_u (or −Δs_u) in the test (or reference) column. Since the perturbation of the total entropy of the two columns was zero, only the entropy budget of the test column is discussed except for specification.

In the test column, w_m is associated with the deep convective clouds. To be specific, w_m is affected by two factors: Δs_s + s_s0 − s^*(T_m, p_m) and Δs_s − Δs_u, where s^*(T_m, p_m) denotes the saturation moist entropy with a mid-tropospheric temperature T_m and a pressure p_m. The first factor, Δs_s + s_s0 − s^*, represents the average mid-tropospheric buoyancy of the convective clouds (Figure 7). The second factor, Δs_s − Δs_u, represents the deviation of the average UT buoyancy of the convective clouds. Owing to a low water vapor content in the UT, the UT moist entropy is close to the saturation moist entropy (Figure 7) and, thus, can approximately represent its saturation moist entropy. Hence, Δs_s − Δs_u approximately represents the deviation of the average UT buoyancy of the convective clouds.).

When Δs_s > Δs_u, for example, more convective clouds reach the UT than at the RCE and, subsequently, w_m > 0. Hence, the mid-tropospheric vertical velocity is expressed as

$$w_m=\frac{H}{C_p^2{\tau }_c}(\Delta s_s-\Delta s_u)\left[\Delta s_s+s_{s0}-s^{\ast }(T_m,p_m)\right],$$

(17)

where H = 12 km is the depth of the troposphere and τ_c is a timescale to measure the upward transportation of the air mass into the UT induced by the deep convective clouds. The factor of Δs_s + s_s0 − s^*(T_m, p_m) is key for the model to embody the WTG, because Δs_s + s_s0 − s^*(T_m, p_m) ≥ 0 represents an essential condition that a cumulus cloud becomes a deep convective cloud.

Deep convective clouds decrease the entropy in the PBL via their downdrafts. Some of the downdrafts penetrate into the PBL (Figure 8) and, thus, bring the mid-tropospheric air with the minimum entropy s_min into the PBL (Figure 7). As a result, the vertical entropy flux from the lower troposphere to the PBL is proportional to $\Delta s_s+s_{s0}-s_{min}$. Since the deviation of the deep convective clouds from the RCE is proportional to w_m, the deviation of the vertical entropy flux from the lower troposphere to the PBL is also proportional to w_m. Hence, the entropy budget of the PBL is expressed as

$$\frac{d\Delta s_s}{dt}=-\frac{\Delta s_s}{2{\tau }_t}-U_s\frac{2\Delta s_s}{L}-\frac{k_d{w}_m}{H}(\Delta s_s+s_{s0}-s_{min}),$$

(18)

where the three terms on the right-hand side, in turn, correspond to the entropy flux from sea, the horizontal entropy flux from the reference column and the vertical entropy flux from the lower troposphere to PBL. U_s represents the horizontal wind speed of environmental air in PBL with respect to the test column. The dimensionless coefficient k_d represents the effect of the downdrafts on the surface entropy.

Deep convective clouds transport the surface air with high entropy into UT and consequently increase the entropy. Hence, the entropy budget of the UT layer is expressed as

$$\frac{d\Delta s_u}{dt}=-U_u\frac{2\Delta s_u}{L}+\frac{k_u{w}_m}{H}(\Delta s_s+s_{s0}-\Delta s_u-s_{u0})-\frac{g\sigma T_u^3\Delta {\sigma }_c}{p_m-p_t},$$

(19)

where the three terms on the right-hand side, in turn, correspond to the horizontal entropy flux from the reference column, the upward entropy flux from the PBL and the entropy flux to space via infrared radiation. U_u represents the UT horizontal wind speed of the environmental air with respect to the test column, and the dimensionless coefficient k_u represents the effect of deep convective cores on the UT entropy. σ is the Stefan–Boltzmann constant, T_u is the UT temperature, p_m/p_t are the pressure of the mid-troposphere/tropopause and Δσ_c is the deviation of the UT cloud fraction from σ_c0, the UT cloud fraction at the RCE.

The UT cloud fraction, Δσ_c +σ_c0, depends on the detrainment of the deep convective clouds in the UT and the precipitation formation [48]. Suppose σ_c0 = 0.5. Since 0 < Δσ_c + σ_c0 < 1, −0.5 < Δσ_c < 0.5, hence, the UT cloud fraction is described by

$$\frac{d\Delta {\sigma }_c}{dt}=k_c\frac{w_m}{H}[1-{(2\Delta {\sigma }_c)}^2]-\frac{\Delta {\sigma }_c}{{\tau }_p},$$

(20)

where the two terms on the right-hand side are attributed to the detrainment of deep convective clouds and the sink of cirrus clouds due to precipitation formation, respectively. The dimensionless coefficient k_c represents the effect of the detrainment of deep convective clouds on cirrus cloud fraction, and τ_p is the timescale of the sink of UT cirrus clouds.

4.2. Stability Criteria of the RCE

The two-column two-layer model is simplified into a very low order system of Equations (18)–(20). The system has three prognostic variables: Δs_s, Δs_u and Δσ_c. It is studied in the phase space (Δs_s, Δs_u, Δσ_c) similar to Lorenz (1963) [1]. Obviously, the system has a fixed point of the origin in the phase space that corresponds to the RCE, as shown as

Δs_s = Δs_u = Δσ_c = 0.

(21)

The criteria for the stability of the fixed point (or the RCE) are derived in Appendix A. They contained two factors of cloud radiation: the detrainment of deep convective clouds (or k_c) and the timescale of cirrus cloud sink (or τ_p). When the detrainment of deep convective clouds is strong and the cirrus cloud sink is slow, the RCE is unstable. In addition, the criteria also contain the width L of convective region. When L is small (e.g., ~100 km), the RCE is stable; when L is large (e.g., ~3,000 km), then the RCE is unstable. In other words, the criteria predict that the RCE is unstable over a wide convective region where the horizontal exchange in entropy between the columns is negligible.

4.3. Oscillation and an Unstable RCE

A specific case with an unstable RCE is discussed in Appendix A, where L = 3,000 km and τ_p = 10 days. Once its initial status was slightly away from the RCE, its status deviated from the RCE permanently and eventually approached a limit cycle (see Figure A1). In addition, its prognostic variables (e.g., surface entropy) oscillated with a period of 40 days (Figure 11). Since the peak of the UT cloud fraction corresponded to the onset of deep convective clouds, Figure 11 shows that the deep convective clouds also oscillated in number with a period of 40 days, providing a prototype of the MJO.

If L is decreased to 100 km, the RCE becomes stable and the oscillation disappears, indicating that the instability of the RCE is an essential condition of the oscillation. If τ_p is decreased from 10 to 5 days while maintaining L = 3,000 km, the RCE is still unstable, but the oscillation disappears. The system, instead, approaches a steady state where one column shows a large-scale ascent and deep convection and the other shows a large-scale descent and shallow clouds, which resemble the model of Raymond and Zeng (2000) [7]. This sensitivity of the system target to τ_p (or bifurcation in the phase space) indicates that the slow sink of UT cirrus clouds is another essential condition of the oscillation.

Recently, Zeng et al. (2022) showed the possibility of a slow sink of cirrus clouds in the tropics with a new precipitation process—the radiative effect on microphysics [48]. Consider relatively large ice crystals near cloud top. Owing to their radiative cooling, they grow to precipitating particles at the expense of small ones and then fall off their parent clouds. As a result, thick cirrus clouds (detrained from deep convective clouds) quickly become thin with a timescale of hours. Once the thin cirrus clouds form, their ice crystals undergo radiative warming (see Figure 3 of Zeng et al. 2022 [48]) and, consequently, survive with a slow gravitational deposition of ice crystals and/or a slow crystal sublimation in a large-scale descent.

The surface wind, of course, can be incorporated into the two-column model to represent its effects on the instability and the oscillation (Fuchs and Raymond 2005, 2017; Emanuel 2020; Wang and Sobel 2022) [49,50,51,52]. It is related to a large-scale vertical circulation via the surface wind gusts driven by cloud systems (see Section 5.1) and large-scale surface wind [50,53,54,55].

5. Other Forms of Instability

Instability behaves differently when its timescale is much shorter or much longer than ~45 days. In this section, the two-column two-layer model was extended to two cases with a timescale ~2 days and ~2 years, respectively. To minimize its complexity, the model was tuned for each case by adding or deleting processes.

5.1. Boundary Layer Quasi-Equilibrium

The boundary layer quasi-equilibrium was proposed to describe the quasi-balance in the entropy flux between the surface and the PBL top (Emanuel 1995; Raymond 1995) [56,57]. It arrives with a timescale of ~1 day (or the short timescale of the temperature and relative humidity in Figure 2 and Figure 3). This subsection deals with the instability of the quasi-equilibrium in the context of the WTG, beginning with CRM simulations.

Consider a two-column model that represents clouds in each column with a three-dimensional CRM [19]. The two columns take the same forcing factors (e.g, the SST, radiative cooling rate) similar to the experiment in Figure 2 and Figure 3, except for maintaining the WTG via a large-scale vertical circulation between them. Surprisingly, the model replicates a quasi-two-day oscillation, such as those observed in the western Pacific [58,59,60,61].

To show how the oscillation works, Figure 12 displays the modeled time series of the surface entropy and the entropy flux from the sea to the air. In the test column with the large-scale ascent, the deep convective clouds spend and, thus, rapidly decrease the surface entropy while the sea slowly supplies entropy to the PBL.

The time series in Figure 12 are explained by the evolution of the horizontally averaged surface wind speed and the entropy flux at the PBL top in Figure 13. In the test column, the deep convective clouds generated strong surface gusts and subsequently increased the entropy flux from the sea to the air, providing energy to maintain the deep convective clouds. On the other side, in the reference column with the large-scale descent, few to no deep convective clouds brought about the small upward entropy flux at the PBL top and subsequently a high surface entropy. Once the surface entropy in the reference column becomes very high, deep convective clouds occur and, subsequently, invert the large-scale circulation between the two columns. Hence, the modeled two-day oscillation can be treated as an alternation of the charge and discharge of entropy in the PBL.

In the modeled two-day oscillation, the surface wind gusts induced by convective downdrafts changed the entropy flux from the sea to the air, bringing about the deviation of the boundary layer quasi-equilibrium, which resembled the wind-induced surface heat exchange (WISHE) model (Emanuel 1986; Rotunno and Emanuel 1987; Yano and Emanuel 1991) [53,54,55], to some extent. On the other hand, the population of convective downdrafts depended on the population of deep convective cores that, in turn, depended on the buoyancy at the cloud base. Generally speaking, the higher the buoyancy at the cloud base, the more deep convective cores were initiated, given the same external perturbations. Figure 14 displays the vertical profiles of the moist entropy and the saturation moist entropy near the MCS shown in Figure 8. An ascending surface air parcel, as shown by Figure 14, had a strong negative buoyancy at ~2 km, which explained the lack of scattered convective cores around the MCS.

To explain the modeled oscillation, the two-column two-layer model was modified based on the CRM simulation of Figure 12 and Figure 13. The modified model represents the surface wind gusts by introducing Δw_s (or −Δw_s), the deviation of the surface wind speed from w_s0 the surface wind speed at the RCE in the test (or reference) column. Since Δw_s changes the entropy flux from the sea to the PBL, the PBL entropy budget in the test column (18) becomes

$$\frac{d\Delta s_s}{dt}=\frac{c_E}{H_{PBL}}(s_{sea}-s_{s0})\Delta w_s-\frac{\Delta s_s}{2{\tau }_t}-U_s\frac{2\Delta s_s}{L}-\frac{k_d{w}_m}{H}(\Delta s_s+s_{s0}-s_{min}),$$

(22)

where H_PBL is the depth of the PBL.

To complement the thermodynamic Equation (22), the model introduced a dynamic equation on ΔT_2km, the deviation of the temperature or $\Delta s_{2km}^{\ast }$, the deviation of the saturation moist entropy from the RCE at z = 2 km in the reference column, where $\Delta s_{2km}^{\ast }$ corresponds to the peak of red line at z = 2 km in Figure 14. Since $\Delta s_{2km}^{\ast }$ is a function of ΔT_2km, differentiating Equation (2) with respect to time yields

$$\frac{d\Delta s_{2km}^{\ast }}{dt}=(C_p+\frac{L_v^2{q}_{vs}}{T_{2km}^2})\frac{d\Delta T_{2km}}{T_{2km}dt}$$

(23)

with the aid of the Clausius—Clapeyron equation, where q_vs is the saturation mixing ratio of water vapor at temperature T_2km and pressure p_2km. In addition, ΔT_2km is proportional to (Γ_d− Γ_s)/2 times the large-scale vertical velocity at z = 2 km, where Γ_d and Γ_s denote the dry and saturated adiabatic lapse rates in the reference and test columns, respectively. Thus, Equation (23) is simplified to

$$\frac{d\Delta s_{2km}^{\ast }}{dt}=b_1(C_p+\frac{L_v^2{q}_{vs}}{T_{2km}^2})\frac{{\Gamma }_d-{\Gamma }_s}{2T_{2km}}{w}_m,$$

(24)

where b₁ is the ratio between the large-scale vertical velocities at z = 2 km and the mid-troposphere.

Clearly, $\Delta s_s-\Delta s_{2km}^{\ast }$ represents the average buoyancy of the convective clouds at z = 2 km in the test column and, therefore, approximately describes whether the surface parcels can pass the lifted condensation level to form clouds. Hence, the large-scale vertical velocity at the mid-troposphere is related to $\Delta s_s-\Delta s_{2km}^{\ast }$ by

$$w_m=\frac{H}{C_p^2{\tau }_c}\left[\Delta s_s+s_{s0}-s^{\ast }(T_m,p_m)\right](\Delta s_s-\Delta s_{2km}^{\ast }).$$

(25)

Since the deep convective clouds generate downdrafts that, in turn, generate surface wind gusts, Δw_s is directly proportional to w_m (see Figure 13), which is expressed as

$$\Delta w_s=b_2{w}_m,$$

(26)

where b₂ is a dimensionless coefficient. In addition, Δs_u = 0 is set for simplicity because the radiative cooling slightly impacts the boundary layer quasi-equilibrium (see Section 2.3).

The system of Equation (22) and Equation (24) has a fixed point of $\Delta s_s=\Delta s_{2km}^{\ast }$ = 0 in the phase space that corresponds to the boundary layer quasi-equilibrium. The criteria for the stability of the fixed point of the origin are derived in Appendix C. The criteria showed that the boundary layer quasi-equilibrium is unstable when b₂ is large (or the downdraft-induced surface gusts are strong), which is supported by the CRM result in Figure 13. The criteria also show that the boundary layer quasi-equilibrium is unstable when L is large (or over a large convective domain), which is supported by the modeling of the MCS onset in horizontally uniform environments with no wind shear [43,62,63].

A specific case with an unstable boundary layer quasi-equilibrium is discussed in Appendix C, where L = 350 km. Once its initial status was away from the origin, its status deviated from the origin and eventually approached a limit cycle (see Figure 1). In addition, its prognostic variables (e.g., surface entropy) oscillated with a period of 2 days (Figure 15). This standing two-day oscillation can lead to a westward travelling two-day oscillation via the zonal asymmetry of the atmospheric response to the zonally symmetric convective heating (see Appendix D). Hence, the system of Equation (22) and Equation (24) provided a prototype of the observed quasi-two-day oscillation.

5.2. Radiative Convective Flux Equilibrium

Consider a two-column model with the SST as a prognostic variable (Nilsson and Emanuel 1999) [28]. If a large-scale vertical circulation is not present between the two columns, the atmosphere approaches a balance between the radiative cooling, convective heating and surface energy flux in each column, which is referred to as the radiative convective flux equilibrium (RCFE). In contrast to the RCE, the RCFE focuses on the balance between the atmospheric radiative cooling and the surface energy flux, where the surface energy flux is related to the SST that is, in turn, related to the UT cirrus clouds via solar and infrared radiation.

The surface energy flux possesses three timescales: two timescales in the approach of the RCE (see Figure 3) and an additional timescale in the approach of the RCFE. Consider a mixed ocean layer with depth H_mix = 100 m that is situated below the atmosphere. Its thermal capacity is about 25 times as much as that of the atmosphere. Since the atmosphere has a long timescale τ_t of ~14 days, the surface energy flux has a very long timescale of 25 × 14 = 350 days (or ~1 year) in the approach of the RCFE. This third timescale of ~1 year measures the approach of the RCFE while the atmosphere remains at the RCE.

The two-column two-layer model in Section 4 was extended to incorporate the mixed ocean layer as the third layer (see Figure 10), addressing the instability of the RCFE and its connection to the ENSO [64]. Suppose that the test column sat over the central and eastern Pacific and the reference column over the maritime continent. The two columns would be connected by a large-scale vertical circulation (or Walker circulation).

Let $\Delta s_{sea}$ (or $-\Delta s_{sea}$) denote the deviation of the sea entropy from $s_{sea}$, the sea entropy at the RCFE in the test (or reference) column. Since $s_{sea}$ is related to the sea surface temperature $T_{sst}$ by $s_{sea}=c_l\ln \left(T_{sst}/T_{ref}\right)$ [10], $\Delta s_{sea}$ is related to ΔT_sst, the derivation of the SST, in the test column by

$$\Delta s_{sea}=c_l\Delta T_{sst}/T_{sst}.$$

(27)

In the test column, $\Delta s_{sea}$ is affected by three factors: infrared and solar radiation, the horizontal exchange in entropy between the columns induced by ocean circulations and the vertical exchange between the sea and air. Hence, its tendency is governed by

$$\frac{d\Delta s_{sea}}{dt}=\frac{p_s-p_t}{{\rho }_{sea}g{H}_{mix}}\frac{\Delta s_s-\Delta s_{sea}}{2{\tau }_t}-U_{sea}\frac{2\Delta s_{sea}}{L}-\frac{\sigma }{{\rho }_{sea}{H}_{mix}{T}_{sst}}\left[{(T_{sst}+\Delta T_{sst})}^4-T_{sst}^4-\Delta {\sigma }_c({\sigma }^{-1}{F}_{sun}-T_u^4)\right],$$

(28)

where ${\rho }_{sea}$ is the density of the sea water, U_sea is the flow speed of the sea surface with respect to the test column and F_sun is the average flux of the solar radiation absorbed by the sea with a clear sky.

The model introduces $\Delta s_{sea}$ into Equation (18), yielding a new PBL entropy budget equation

$$\frac{d\Delta s_s}{dt}=-\frac{\Delta s_s-\Delta s_{sea}}{2{\tau }_t}-U_s\frac{2\Delta s_s}{L}-\frac{k_d{w}_m}{H}(\Delta s_s+s_{s0}-s_{min}).$$

(29)

On the other hand, the model still uses Equation (19) and Equation (20) to describe the UT entropy and cloud fraction, respectively. The model overlooks the feedback of water vapor on radiation for simplicity [28].

In summary, the system of the RCFE represents a two-column three-layer model. It has four prognostic Equations (28), (29), (19) and (20). It has a fixed point of the origin in the phase space (Δs_s, Δs_u, Δσ_c, Δs_sea) that corresponds to the RCFE. Its criteria for the stability of the fixed point of the origin are presented in Appendix B. The criterion (A8) shows that the fixed point for the RCFE is unstable when $\left({\sigma }^{-1}{F}_{sun}-T_u^4\right)/T_{sst}^4$ is large.

A specific case with an unstable RCFE is discussed in Appendix B, where L = 10,000 km. Once its initial status was slightly away from the RCFE, the system deviated from the RCFE permanently and eventually approached a limit cycle (Figure A2). In addition, its prognostic variables (e.g., SST, UT cloud fraction) oscillated with a period of ~20 months (Figure 16). Since ${\tau }_t$ varies with the surface wind speed and radiative cooling rate (Figure 4), the oscillation in Figure 16 possesses a period close to that of the ENSO, providing a prototype of the ENSO.

6. Conclusions and Discussion

Convective clouds rapidly consume the CAPE and, thus, are self-destructive. Meanwhile, they modulate the external energy fluxes from the sea and to space, increasing the CAPE slowly for future convective clouds. If the convective clouds are treated as one group, they change themselves via the sea surface flux and radiative cooling. This nonlinear feedback between convective clouds and their external energy fluxes is discussed in this paper with three very low order systems.

The three systems are so simple that their criteria for a stable fixed point are derived. Furthermore, their connection between an unstable fixed point and a limit cycle provides a new clue for understanding the observed atmospheric oscillations. To be specific, the fixed point of the origin in the three systems corresponds to the boundary layer quasi-equilibrium, the RCE and the RCFE, respectively, and the limit cycle for three standing oscillations with a period of ~2 days, ~40 days and ~2 years, respectively. The three standing oscillations, due to the zonal asymmetry of the atmospheric response to the zonally symmetric convective/radiative heating (see Appendix D), bring about a westward travelling quasi-two-day oscillation, eastward travelling MJO and standing ENSO, respectively.

The systems show that the atmosphere is predictable via limit cycles if their parameters are known. On the other side, their parameters, especially the local variable of the surface wind speed, vary greatly. Since the parameters vary, the systems may frequently swing across the criteria for the stable equilibrium, bringing about bifurcations in the phase space. If so, their predictability is limited by their accuracy of process representation and the introduction of other processes, such as the effect of vertical wind shear on cloud anvil [65,66].

The current weather and climate models do not represent clouds well because of their common biases of “excessive water vapor” and “too dense clouds” [67,68,69,70]. Thus, they cannot represent the cloud-radiation interaction accurately and, consequently, cannot simulate the oscillations well [64,71]. The present theoretical study of the instability criteria suggests it is imperative to properly represent the feedback of clouds on radiation as well as convective downdrafts in the tropics. Recently, a new precipitation mechanism in cirrus clouds—the radiative effect on microphysics—has been proposed to remove the biases [48,72], raising a prospect that the models can predict the oscillations well.