Radiative–convective equilibrium (RCE) describes an idealized state of the atmosphere in which the vertical temperature profile is determined by a balance between radiative and convective fluxes. While RCE has been applied extensively over oceans, its application over the land surface has been limited. The present study explores the properties of RCE over land using an atmospheric single-column model (SCM) from the Laboratoire de Météorologie Dynamique–Zoom, version 5B (LMDZ5B) general circulation model coupled in temperature and moisture to a land surface model using a simplified bucket model with finite moisture capacity. Given the presence of a large-amplitude diurnal heat flux cycle, the resultant RCE exhibits multiple equilibria when conditions are neither strictly water nor energy limited. By varying top-of-atmosphere insolation (through changes in latitude), total system water content, and initial temperature conditions the sensitivity of the land RCE states is assessed, with particular emphasis on the role of clouds. Based on this analysis, it appears that a necessary condition for the model to exhibit multiple equilibria is the presence of low-level clouds coupled to the diurnal cycle of radiation. In addition the simulated surface precipitation rate varies nonmonotonically with latitude as a result of a tradeoff between in-cloud rain rate and subcloud rain reevaporation, thus underscoring the importance of subcloud layer processes and unsaturated downdrafts. It is shown that clouds, especially at low levels, are key elements of the internal variability of the coupled land–atmosphere system through their feedback on radiation.
The concept of radiative–convective equilibrium (RCE) was introduced by Manabe and Wetherald (1967), following the earlier work of Gold (1909) and Goody (1949), to describe an idealized, statistical state of the atmosphere in which a balance between radiative cooling and convective heating determines the vertical temperature profile. RCE postulates that, on average, convective scale motions compensate for the destabilization of the atmosphere by radiation. RCE represents a powerful tool for estimating convective sensitivity to ocean surface temperature and for diagnosing the possible mechanisms through which deep convection is maintained and interacts with the surface fluxes in the absence of large-scale flow. RCE has been applied to a wide range of problems, including estimation and scaling of convective mass fluxes (Tompkins and Craig 1998a), the organization of tropical deep convection (Tompkins and Craig 1998b; Tompkins 2001a,b), and climate sensitivity to greenhouse gas forcing (Muller et al. 2011; Romps 2011).
In most of the applications of RCE in either single-column models (SCMs) or cloud-resolving models (CRMs), the surface boundary has been an ocean, often with prescribed surface temperature. Complications over land stem from the greater complexity associated with interacting components such as vegetation, soil moisture, and soil temperature. However, some studies have simplified the land system to address these complications. For instance, RCE has been considered for a swamp surface (Manabe and Wetherald 1967; Renno 1997) or with a constrained hydrologic cycle (i.e., with prescribed evapotranspiration) and nudging of 10-cm soil temperature toward a prescribed value (Prigent et al. 2011). Tompkins and Craig (1998a) and Schlemmer et al. (2011) performed an experiment involving a CRM coupled to land surface in which the system achieves a “diurnal equilibrium” state: that is, a quasi-stationary regime in which the surface and the boundary layer temperatures exhibit diurnal oscillations. In these studies, the soil and atmospheric temperature profiles were relaxed toward climatological values at points far from the surface.
In our study, we consider an extension of the concept of RCE applied to a land surface with a closed (water conserving) hydrologic budget. As far as we are aware, ours is the first study to evaluate equilibrium land–atmosphere coupling in a single-column model fully coupled to a land surface model. Of particular interest in our exploration of RCE over land is the potential existence of multiple equilibria. For prescribed-SST conditions, the RCE state is uniquely determined by the SST and radiative forcing; that is, in the absence of energy exchange through the surface (and associated advective transport), the isolated, fixed SST RCE system possesses a unique equilibrium solution (Hilbert 1912; Tompkins and Craig 1998b; Renno 1997), with the ocean acting as both a thermostat and an infinite water source. On the other hand, Renno (1997) and Tompkins (2001a,b) showed that introduction of a surface hydrologic cycle through a swamp ocean, in which the surface temperature is interactively determined through the balance of surface fluxes assuming zero soil heat capacity, permits the existence of multiple equilibria. Even if multiple equilibria are not realized in the real climate system (e.g., because of the presence of internal variability or seasonal evolution), they may nonetheless provide insights regarding the evolution of the coupled land–atmosphere system.
Over land the existence of multiple equilibria has been explored in the context of land–atmosphere feedbacks. For instance, large-scale continental recycling forced by stochastic advection exhibits two distinct equilibria comprising dry and moist surface states (Rodriguez-Iturbe et al. 1991a,b; Entekhabi et al. 1992). Land and boundary layer interactions can also induce bimodality in the surface Bowen ratio (Entekhabi and Brubaker 1995; Brubaker and Entekhabi 1995). Similarly, Wang and Eltahir (2000) demonstrated the emergence of multiple equilibria in simulations of the West African climate including biosphere–atmosphere interactions. The Global Land–Atmosphere Coupling Experiment (GLACE) (Koster et al. 2004) suggests the existence of land–atmosphere coupling hotspots, which typically occur in the transition zones between arid and humid regions. The physical mechanisms that produce such hotspots are still not completely understood, although some progress has been made using some simplified analysis of the land surface coupled to the atmosphere (e.g., Guo et al. 2006; Koster et al. 2006; DelSole et al. 2009; Lintner et al. 2013). Recently, Aleina et al. (2013) demonstrated the emergence of multiple equilibria (desert or forest) for a toy model of a planet when interactive vegetation is included.
Apart from the general circulation model (GCM)-based analyses of GLACE, most studies of the feedbacks of soil moisture and precipitation over land have been performed over relatively short time scales from one to several days (Hohenegger et al. 2009; Seneviratne et al. 2010; Gentine et al. 2013) or by aggregating diurnal-scale processes over the summer season (D’Odorico and Porporato 2004; Findell et al. 2011). A key challenge for the analysis of observations and complex GCMs is that weather and climate variability may overwhelm or otherwise mask signatures of land–atmosphere coupling (Phillips and Klein 2014). Thus, we believe that analyses performed using idealized frameworks such as the model considered here can stimulate improved understanding of long-term (seasonal and annual) land–atmosphere interactions: what such analyses may lack in terms of realism is leveraged against the ease and transparency of diagnosis.
The paper is structured as follows. Section 2 provides an overview of the Laboratoire de Météorologie Dynamique–Zoom, version 5B (LMDZ5B) general circulation model SCM used in this study and the experimental setup employed to obtain RCE solutions over land. In section 3, we document the existence of multiple equilibria in a set of experiments in which we vary latitude, total (soil plus atmosphere) moisture content, and initial soil temperature, whereas section 4 provides a more in-depth analysis of the RCE solutions and how these relate to land surface, cloud–radiative, and convective processes in the model. In section 5, we present the results of sensitivity experiments to assess how the diurnal cycle of radiation and cloud–radiative feedbacks impact the existence of multiple equilibria. The final section summarizes the key findings of this study and discusses some implications of land region RCE for interpreting land region climate.
2. Model description and setup
a. Model description
We use the SCM version of the LMDZ5B GCM developed by the Laboratoire de Météorologie Dynamique (Hourdin et al. 2013). LMDZ5B has been used to perform climate simulations for the Intergovernmental Panel on Climate Change (IPCC) Fifth Assessment Report. Here we give a brief description of the model; the reader is referred to Hourdin et al. (2013) for a more extensive discussion.
The model has 39 nonuniformly spaced levels in the vertical. The first grid point is at 35 m, with 8 grid points distributed in the first kilometer. Between 1 and 20 km, the mean vertical resolution is 800 m and the model top is located at 40 km. Separate parameterizations of shallow and deep convection are included. For shallow convection, the eddy diffusive scheme of Mellor and Yamada (1982) is combined with a mass-flux representation of boundary layer thermals (Hourdin et al. 2002; Rio and Hourdin 2008; Rio et al. 2010) to account for turbulence in the surface and inversion layers and nonlocal convective transport induced by boundary layer coherent structures, respectively. Thermals are represented using a bulk entraining–detraining plume approach (Simpson and Wiggert 1969; Betts 1973) to compute the properties of a mean characteristic thermal representing the dry and cloudy (if saturation level is reached) boundary layer thermals present in a model grid cell. The plume model diagnoses the heights of cumulus base and top, as well as the vertical profiles of the plume vertical velocity, thermodynamic properties, and fractional coverage, through the vertical evolution of mass flux.
The Emanuel deep convection scheme (Emanuel 1991) is added to this scheme for the treatment of precipitating deep convection. The scheme has been modified by Grandpeix and Phillips (2004) to improve the sensitivity of the simulated deep convection to tropospheric relative humidity (Derbyshire et al. 2004). The triggering criterion of deep convection is based on the concept of available lifting energy (ALE) provided by boundary layer thermals: deep convection is triggered whenever ALE overcomes the convective inhibition (CIN). A cold pool (or wake) parameterization has also been added to Emanuel’s scheme with cold pools fed by the unsaturated downdrafts resulting from rain reevaporation (Betts 1976; Tompkins 2001b) outside the cloud. These cold pools also provide updraft lifting energy that may retrigger deep convection by exceeding the CIN (Tompkins 2001b). The closure follows Grandpeix and Lafore (2010) and relates the cloud-base mass flux to the available lifting power (ALP) provided by subcloud processes, CIN, and the vertical velocity at the level of free convection (see details in Grandpeix et al. 2010; Grandpeix and Lafore 2010).
In the LMDZ GCM, precipitation is divided into 1) a convective part, generated by Emanuel’s convection scheme (i.e., cumulonimbus clouds), and 2) a stratiform part, generated by (i) large-scale condensation related to a grid-scale ascent, (ii) boundary layer thermal plumes related to cumulus clouds, and (iii) turbulent diffusion related to fog. However, since there is no large-scale ascent in the present RCE SCM framework, clouds and precipitation are completely determined by the parameterized subgrid-scale processes: that is, turbulence, shallow convection and deep convection. The radiation scheme (Morcrette 1991) fully interacts with clouds and other components of the atmosphere. In LMDZ5B, maximum cloud overlapping is applied to compute radiative forcing when adjacent layers are cloudy, as for cumulus clouds, whereas the random overlapping is applied when two cloudy layers are separated by at least one clear layer, as for stratiform clouds.
For both stratiform and convective clouds, cloud cover is computed following a statistical cloud scheme with a lognormal probability density function, representing the subgrid-scale variability of total water content (Bony and Emanuel 2001). In this scheme, the in-cloud water content qinc, condensed water qc, and cloud fraction are deduced from the distribution and average saturation of specific humidity. In the current standard version of LMDZ5B, ice thermodynamics is not taken into account in the deep convection scheme. Inclusion of ice increases the cold pool intensity and thereby strengthens deep convection via the ALP closure, although this has been found to have little effect on upper-level heating rates.
2) Soil model
The soil model uses a diffusion scheme for heat propagation, assuming a diffusivity of 1.06 × 10−6 m2 s−1. A zero ground heat flux condition is imposed at infinite depth. The dynamics of soil water content Qsoil is represented with a simple bucket model (Manabe 1969; Koster and Suarez 1994) and includes precipitation, evaporation, and runoff generation. A soil saturation threshold is prescribed at Qmax = 1.5 m, above which the excess of water is removed completely via runoff. Under these conditions, total water content of the land–atmosphere system, Qtot = Qsoil + W, where W is precipitable water, is no longer conserved. However, in our experiments, Qmax is set to a sufficiently large value (1.5 m) to avoid runoff and therefore nonconservation. Physically Qmax corresponds to an effective rooting depth (Rodriguez-Iturbe et al. 1999; Laio et al. 2001), although the model contains no explicit representation of vegetation. The surface albedo is taken as α = 0.19. Although this highly reduced soil model may affect the coupling between the soil and the atmosphere, we show below that the simplified system still permits nonlinearities and multiple equilibria.
3) Surface fluxes
Sensible heat flux and evaporation are computed via bulk formulations: and , where is the potential evaporation computed as and β is the evapotranspiration coefficient reflecting linear soil moisture stress. Here, ρ = 1.17 kg m−3 is the surface air density, V0 is the first-level wind speed, Cd,υ is the neutral drag coefficient for a land surface, Cd,h is the stability correction based on the local Richardson number (see Hourdin et al. 2013), Cp=1004 J K−1 kg−1 is the dry air heat capacity, Ts is the surface skin temperature, T1 is the first atmospheric layer temperature, is the saturation specific humidity at the surface, and q1 is the first-layer specific humidity. The β varies linearly between 0 and 1 for soil moisture content between Qsoil and Qmax/2 and saturates at β = 1 for Qsoil > Qmax/2. This relationship actually mimics idealized vegetation, for which stomatal opening depends quasi linearly on soil moisture up to a maximum value (saturation) (Porporato et al. 2001). If Qsoil > Qmax, runoff is generated to maintain Qsoil = Qmax (see previous subsection).
The LMDZ5B SCM is integrated for 10 years with a time step Δt = 450 s, the LMDZ standard time step used for CMIP5 simulations. An initial atmospheric profile [extracted from the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment (FIRE) campaign; see Cox et al. 1987] is prescribed, as well as initial, vertically uniform ground temperature T0 and soil moisture Q0. The RCE framework is applied without large-scale velocity or tendencies, as in most prior studies of oceanic RCE. A diurnal cycle of incoming solar radiation is imposed at the top of the atmosphere (TOA), corresponding to the annual-mean value at the prescribed latitude. The model computes a residual surface wind, driven by convective-scale motions, that maintains consistent surface fluxes. Over oceanic surfaces, some surface wind must be included to ensure heat and moisture transfer; indeed, surface wind provides the mechanical forcing (forced convection) that critically contributes to surface fluxes over oceans. Over land, however, daytime heating of the surface provides surface buoyancy instability (free convection) that thermally drives surface energy exchanges. Thus, surface wind is much less critical over lands than over the ocean. In section 5, we perform a sensitivity analysis on the impact of relaxing the atmosphere toward a nonzero geostrophic wind and found that it does not modify the conclusions reached here.
The sensitivity experiments discussed in the following sections consider changes in (i) latitude λ, (ii) total water content Qtot, and (iii) initial ground temperature T0. Note that variation of total water content is performed by varying the initial soil water content Q0 while maintaining the same initial precipitable water W0 = 25 kg m−2.
c. Characteristics of the baseline LMDZ5B RCE state
We run the model for the following baseline conditions: λ = 35°, T0 = 300 K, and Qtot = 40 mm (i.e., Q0 = 15 mm). When the soil–atmosphere system reaches TOA radiative equilibrium, a diurnal-mean equilibrium surface temperature is reached. The system may require several years to achieve equilibrium because of the large soil depth and its corresponding thermal inertia. In this study, we assume that 10 yr is a sufficiently long period for the soil to approximately equilibrate with the atmosphere: that is, with an imbalance inferior to 4 W m−2.
The atmosphere is in RCE when the net atmospheric radiative cooling balances the net convective heating. In fact, an atmospheric equilibrium is reached within a few weeks (see Fig. 1a) and thereafter holds throughout most of the simulation. Indeed, the daily averaged time series of the radiative cooling (Qrad) and convective heating (Qcon) profile nearly balance each other. The time scale of convection (few hours) is much shorter than the radiative time scale (around 40 days), so any fluctuation in the surface energy or TOA radiation budgets is rapidly eliminated by convection (Cronin and Emanuel 2013). Consequently, on daily time scales the integrated atmospheric energy budget is zero in all simulations at equilibrium, as depicted by the solid line in Fig. 1b. In other words, RCE guarantees the same net energy flux at the surface and at TOA since there is no energy accumulation in the atmosphere. Consequently, the dashed line in Fig. 1b not only represents the net energy flux but also the net energy flux at TOA. Periods where the two curves are superimposed in Fig. 1b correspond to periods where the atmospheric energy loss (gain) corresponds to a similar energy gain (loss) for the soil. However, because of its large depth, the soil requires some time to balance atmospheric perturbations and the whole soil–atmosphere system may not be at equilibrium. For example, in simulation year 1984, the atmosphere experiences a strong perturbation that induces a departure from the preceding RCE state. This large atmospheric fluctuation drives a soil response at longer time scales through a surface temperature increase (see Fig. 1c). Again, before the final stable RCE is reached, the entire land–atmosphere system may be out of equilibrium as the soil column adjusts. In this unstable regime, the inherent system internal variability can generate jumps from one equilibrium state to another one. Hence, the system can pass through different equilibria states before reaching its final steady RCE solution. Hereafter, we only consider the RCE state reached after 10 years of simulation (assuming it is close enough to the final RCE).
Figure 2 depicts the diurnal course of energy fluxes for the control run (CTL). The bucket soil model (bucket) used in the LMDZ5B SCM may be compared with a slab ocean model with a small heat capacity, which allows diurnal variations in surface temperature, but with a larger Bowen ratio and limited water holding capacity. Because of the small thermal soil inertia the system oscillates around the equilibrium state on daily time scales (see Fig. 2a).
As already seen in Fig. 1, the surface and TOA energy budgets are close to zero in the baseline simulation. The RCE over a land surface reaches a steady periodic regime, in which the diurnal solar forcing drives a periodic response of the land–atmosphere system, in terms of both surface temperature and precipitation, as seen in Figs. 2a,b. At equilibrium, surface temperature and precipitation exhibit very little day-to-day variability (not shown). The diurnal cycle of Ts exhibits a 1–2-h lag with respect to solar forcing, characteristic of land surfaces for which the maximum is typically reached between 1300 and 1600 local time (LT) (Gentine et al. 2010). Regarding precipitation, prior studies have documented the presence of an afternoon peak in land region precipitation, especially in the tropics, in observations and CRMs (e.g., Bechtold et al. 2004; Guichard et al. 2004; Dai 2006; Rio et al. 2009, 2012). Thus, the land region RCE in the LMDZ5B SCM produces consistent diurnal cycle phasing relative to observations and high-resolution models.
After 10 years of simulation, RCE applies to daily averages. Figure 2c demonstrates that the subdaily (hourly) behavior deviates from RCE. The maximum soil heating occurs early in the morning, when turbulent fluxes are relatively weak and cannot dissipate much of the radiative input; rather, most of the heating at this time is dissipated as ground heat flux (Gentine et al. 2011, 2012). In terms of hydrologic cycle, the daily averaged precipitation and evaporation balance each other with surface precipitation rates across the suite of simulations ranging from 1 to 3 mm day−1 (discussed below).
Figure 2d depicts the diurnal evolution of radiative and convective heating integrated over the atmospheric column. Convective heating exhibits a strong diurnal cycle imposed by the large-amplitude diurnal variations in surface turbulent heat fluxes. The net radiative energy budget of the atmosphere is positive between 0900 and 1500 LT because of the large shortwave absorption. Diurnal variations in longwave tendencies are an order of magnitude smaller than the shortwave ones. Continental RCE is thus achieved for daily averages: the atmosphere experiences a net heating during daylight hours compensated by an equivalent net cooling at night. This is a key difference with a prescribed-SST framework over the ocean, in which the fixed SST ensures a permanent energy balance (i.e., verified at each instant), with continuous precipitation. The large diurnal cycle over land surfaces, induced by the low continental heat capacity compared to the ocean, provides increased surface variability compared to the oceanic case with prescribed SST. Even at equilibrium there can be substantial subdiurnal changes in surface temperature, boundary layer depth, cloud cover, and convection, which can respond nonlinearly to the diurnal course of radiation. This added variability appears to play a role in determining the equilibrium states (Rodriguez-Iturbe et al. 1991a), as will be seen in section 5. The addition of the surface hydrological cycle, which limits the amount of available water at the surface, also increases the system’s degrees of freedom compared to the prescribed-SST RCE. The energy in the system is thus controlled by the latitude (and prescribed planetary albedo), and the hydrologic cycle is constrained by the initial total water content Q0.
3. Existence of multiple equilibria
The sensitivity of RCE to changes in solar forcing, total moisture, and initial surface temperature is investigated by modifying three parameters: the latitude λ; the total water content Qtot, through specification of initial soil water content Q0; and initial ground temperature T0. Each pair of latitude and total water contents may be viewed as distinct climatological conditions. The λ ranges from 30° to 40° in increments of 2.5°. As we will see in the following, this range is sufficient to cover the entire spectrum of possible RCE states. The initial Q0 ranges from 5 to 45 mm in increments of 10 mm; equivalently, Qtot ranges from 30 to 70 mm. For each of the 25 climate states, the LMDZ5B SCM is initialized with one of five values of T0 ranging from 280 to 320 K with an increment of 10 K, giving a total of 125 10-yr simulations. Note that all other model parameters are set to the baseline values. In what follows, we describe each simulation’s final state by its mean equilibrium surface temperature (K) and soil water content (mm). For some latitude and total water content pairs, distinct final states are obtained with different initial ground temperatures. Thus, the coupled soil–atmosphere system exhibits multiple equilibria (see next section).
Figure 3 provides an overview of the RCE combinations for the different λ, T0, and Q0. Before discussing the sensitivity analysis, we point out that a strong negative correlation exists between equilibrium soil surface temperature Ts and soil moisture content Qsoil. In the simplified bucket land surface hydrology the evapotranspiration efficiency β increases linearly with soil moisture [see section 2a(2)]. Low Qsoil generates low evaporation, so most of the net radiative heating at the surface must be balanced by sensible heat flux (or ground heat flux). On the other hand, latent heat release is a more efficient heat transfer mechanism than other heat fluxes at medium to high temperature (Bateni and Entekhabi 2012). For low soil moisture, surface temperature quickly rises since the available cooling mechanisms are not efficient. Such behavior is well known for the daily variations of surface skin temperature and is used as an indicator of water stress (Bastiaanssen et al. 1998; Castelli et al. 1999; Boulet et al. 2007).
No multiple equilibria are seen at either high (λ = 40°) or low (λ = 30°) latitudes. For these latitudes, the RCE is either “warm and dry surface” (see Fig. 3, red symbols) or “cool and wet surface” (Fig. 3, blue symbols). The warm–dry RCE corresponds to a high surface temperature associated with a nearly dry soil (Qsoil < 5 mm), while the cool–wet RCE corresponds to a low surface temperature associated with a high soil water content and low precipitable water (W ~ 5 mm). The warm and cool RCE equilibria can be found at all latitudes and total water contents. In other words, for any latitude and total water content, there is at least one T0 that can lead to a warm–dry or a cool–wet solution. However, the warm–dry and cool–wet solutions can exhibit different surface temperatures depending on the solar forcing and water availability.
As seen in Fig. 4 as an example, for Q0 = 5 mm (i.e., Qtot = 30 mm) increasing latitude (i) favors cool solutions and (ii) leads to a monotonic decrease of final soil temperature for both warm and cool solutions. Moreover, the sensitivity of the equilibrium solution to a change in latitude is more pronounced for warm solutions. Indeed, the final Ts ranges from 274 to 278 K for the cool states, whereas it ranges from 296 to 306 K for the warm solutions. Sensitivity to Q0 is nontrivial: at the extreme latitudes, the warm RCE state (λ = 30°) is dependent on total available moisture while the cool state (λ = 40°) is not. From Fig. 3a, it is clear that the warm states become warmer with increasing Qtot at low latitudes. At 30° (Fig. 3a), Ts increases from 306 to 316 K with Qtot increasing from 30 to 70 mm. However, at high latitudes (see Fig. 3e) Ts decreases from 274 to 272 K with Qtot increasing from 30 to 70 mm. Thus, at 30° the final state is “water limited”: incoming radiation is large and generates large surface evaporation so that the soil is effectively desiccated. Increasing Qtot similarly increases precipitable water, which in turn leads to a strong water vapor greenhouse effect causing Ts to increase. At high latitudes (λ = 40°) nearly all available water in the system resides in the soil, leaving the atmosphere nearly devoid of moisture. This corresponds to an “energy limited” regime: TOA incoming insolation is insufficient to generate substantial surface evaporation (or latent heat release). However, since β is an increasing function of Qsoil, Qsoil increases evaporation, which ultimately increases precipitable water. Under cold conditions, the humidity profile is very close to saturation; hence, even a small increase of precipitable water induces more low-level clouds, which reduces incoming surface shortwave radiation and results in a (small) decrease of Ts.
To summarize, the sensitivity of RCE states to changes in total water content is (i) of opposing sign for high and low latitudes and (ii) of larger magnitude at low latitudes. Such differential sensitivity can be explained through distinct responses of the atmospheric radiative properties to changes in precipitable water. Low incoming radiation induces cold conditions and high relative humidity, which strengthens low-level cloud cover and cloud optical thickness. High incoming radiation induces warm conditions associated with increased precipitable water.
Apart from the extreme latitudes, intermediate solutions (Fig. 3, green symbols) and multiple equilibria are found to occur at all other latitudes considered (see Figs. 3b–d). These intermediate RCE states are associated with water being more equitably partitioned between the soil and the atmosphere (Qsoil > 5 mm and W > 5 mm). In these experiments, the multiple equilibria appear to be either bimodal or trimodal. For intermediate latitudes, there is an optimum range of Qtot that favors multimodal solutions and the intermediate RCE states. Figure 3 shows that, for Qtot = 30 mm (i.e., Q0 = 5 mm), it is not possible to reach an intermediate RCE state for any of the latitudes considered (i.e., green symbols are absent). Only warm and cool equilibria are present for intermediate latitudes. Low values of Qtot appear to favor bimodality (warm and cool states) rather than multimodality (warm, intermediate, and cool). At large total water content, Qtot = 70 mm, the multiple equilibria disappear (see Fig. 3d), as the model tends to a cool RCE state regardless of the initial condition. In conclusion, we note here an optimum range of λ (energy) and Qtot (water) values that allow intermediate states with neither a completely dry, desertlike surface nor a fully wet, swamplike surface. This range corresponds to a regime that is neither energy limited (in terms of solar radiation) nor water limited (in terms of total water content). Within this range, the coupled land–atmosphere SCM system exhibits multimodal RCE states.
4. Investigation of equilibria states (dry, wet, and intermediate): Sensitivity to latitude and the role of clouds
a. Diagnosis of RCE state dependence on latitude
To understand the characteristics of the RCE states, including the genesis of multiple equilibria, we consider here how the system evolves toward these states at different latitudes. To do so, we focus on the simulations with Qtot = 60 mm and T0 = 280 K. From Fig. 3, we note that this set of conditions gives the range of RCE states—warm, cool, and intermediate—over 30°–40°.
Figure 5 depicts 10-day moving averages of the time series of surface temperature, soil water content, precipitation, and precipitable water at each latitude. For all latitudes except 30° these variables reach a steady state after ~1 yr. On the other hand, for λ = 30° there is a transient regime comparable to the one observed in Fig. 1. The coupled land–atmosphere system exhibits large, quasiperiodic oscillations at the monthly time scale (rather than the single big jump observed in Fig. 1a) during the first half of the simulation and then smoothly converges toward its stable solution. The coupled land–atmosphere system is capable of producing internal variability that can cause transitions from one equilibrium state to another one. Figure 5 also shows that the nontrivial latitude dependence of these variables. The warm (λ = 30°) and cool solutions (λ = 37.5° and 40°) are clearly distinguishable, whereas the two intermediate RCE states (λ = 32.5° and 35°) are quite close to each other, with equilibrium surface temperatures of 297.5 and 295 K for λ = 32.5° and 35°, respectively.
The (surface) precipitation rate provides an indication of the intensity of the hydrologic cycle at equilibrium: it is equal to the surface evapotranspiration when RCE is reached. The latitude dependence of surface precipitation is not only nonlinear but also nonmonotonic, as it increases from a minimum value λ = 30° (1.2 mm day−1) to a peak value at λ = 32.5° (~2.85 mm day−1) and then decreases toward higher latitudes. In the next subsection, we present and discuss mean vertical profiles at equilibrium to elucidate the behavior of precipitation with latitude.
b. Mean vertical profiles
Mean vertical profiles of heating tendencies, relative humidity, and precipitation averaged over the last 3 months of the simulation are depicted in Fig. 6. The heating tendencies (Fig. 6a) elucidate some important characteristics of the RCE states, including the strength and depth of convection. RCE is found to hold at essentially every point in the vertical, as, on average, turbulent diffusion and shallow and deep convection heating compensates radiative cooling at each level. It is interesting that some RCE solutions manifest similar column-integrated tendencies but with very different vertical heating profiles. Consequently, a continental RCE state is better defined by its surface temperature and soil moisture (or precipitable water) state than by its column-integrated radiative cooling (or convective heating). According to Fig. 6a, the principal differences in the radiative heating vertical distribution among the different equilibria, and thus in the convective heating profiles are readily seen in the vertical extent of the radiative–convective instability, with both convection height and strength decreasing with latitude. For the cool–wet and warm–dry cases, the convection heights are ~300 and ~150 hPa, respectively. This result is consistent with the monotonic decrease of the average surface temperature with latitude, which exerts a strong control on convection depth via control of the moist adiabatic temperature profile. It has been previously argued that the altitude at which radiative cooling drops to zero determines the anvil top: that is, the cloud-top detrainment zone.
Apart from the overall profile vertical extent, the largest heating tendency differences across the latitude range occur around the 750-hPa level and near the surface. At 37.5° and 40°, a peak in net radiative cooling is present at 600 hPa and is associated with net radiative heating immediately below. This radiative heating dipole is not present at the other latitudes. For λ = 35° and 37.5°, intense radiative cooling occurs in the lowermost atmospheric layer. In section 4c, we demonstrate that these features in the radiative heating profiles are related to the vertical distribution of cloudiness.
The mean vertical profiles of relative humidity (Fig. 6b) and precipitation (Fig. 6c) are also useful for distinguishing RCE states, as they clearly show daily-mean boundary layer depth, cloud top, and levels of rainfall generation and reevaporation. Starting from the surface, the first peak in relative humidity indicates the daily mean boundary layer depth, or the daily mean cloud base, while the second peak indicates the daily mean cloud top. At λ = 30°, the boundary layer is significantly drier and higher than in other cases because of both the large surface sensible heat flux and boundary layer top dry air entrainment. Rain reevaporation in the subcloud layer is, in turn, dramatically increased (Fig. 6c, red curve). Consideration of the vertical profiles of precipitation points to the source of the maximum surface rain simulated λ = 32.5°. In particular, the vertical structure suggests a trade-off between convection maximum strength and rain reevaporation beneath cloud base. With increasing insolation, the vertical profile of convective heating shifts upward and strengthens. Thus, the maximum rain rate rises monotonically from ~1.7 mm day−1 at 750 hPa for λ = 40° to ~4.7 mm day−1 at 600 hPa for λ = 30° (see Fig. 6c). On the other hand, rain reevaporation increases as the cloud base rises. Thus, for the warmest case, the rain rate decreases from 4.7 mm day−1 at 600 hPa to 1.2 mm day−1 at the surface. In summary, convection strengthens as latitude decreases but boundary layer deepening and drying enhances evaporative cooling in the lower atmosphere. This strong evaporative cooling fuels very intense unsaturated downdrafts that spread at the surface as density currents (not shown).
In the cool cases (λ = 37.5° and 40°) relative humidity is larger, although there is much less precipitable water than in other cases (see Fig. 5). From the surface to the cumulonimbus top, relative humidity always exceeds 50% because the atmosphere is so cold that the partial pressure of saturation is low. Intermediate solutions (λ = 32.5° and 35°) exhibit very similar vertical profiles of relative humidity and precipitation, with little difference in the convection depth.
The three families of continental RCE states thus exhibit very distinct vertical profiles. Each one of the preferred states seems to correspond to a particular vertical structure of the atmosphere. The next subsection investigates the role of clouds in the establishment of these three families of solutions.
c. Cloud cover
The mean diurnal cycle of the vertical distribution of cloud fraction at equilibrium are displayed in Fig. 7. Total cloud cover, especially low-level cloud cover, increases with latitude. Oppositely, cloud top decreases with latitude, in agreement with the decreasing vertical extent of moist convection. During the daytime, cloud amount is also maximized because of moist convection. More importantly, cloud fractions clearly reveal the three different RCE types: warm, intermediate, and cool.
In the warm case (λ = 30°, Fig. 7a), cloud amount is low. Cumulus cloud base rises from 800 to 700 hPa from 1000 to 1400 LT, while cumulonimbus cloud top is located around 150 hPa and exhibits the strongest cloud fraction between 1300 and 1800 LT. In the LMDZ5B SCM, cumulus clouds are located at the detrainment zone of moist thermal plumes originating from the surface. When triggered, deep convection generates deep cumulonimbus that transports heat and moisture from the lower free troposphere to its top. In the model, the sudden disappearance of shallow cumuli synchronized with deep convection initiation is due to precipitation. Indeed, once deep convection is triggered, heavy rainfall and associated cold pools fed by unsaturated downdrafts tend to cool and stabilize the boundary layer (Tompkins 2001a,b; Grandpeix et al. 2010; Grandpeix and Lafore 2010). This stabilization likely accounts for the sudden disappearance of low-level clouds in the midafternoon, despite that convective precipitation is still present. Then, the time lag between low- and high-level clouds illustrates the succession between shallow and deep convective regimes in all cases listed in Fig. 7. However, we point to the shorter duration of convective events in the warm cases. In summary, two successive cloud layers characterize the warm RCE states: a cumulus layer followed by a cirrus layer collocated with the cumulonimbus cloud detrainment zone. They exert a net positive radiative forcing in the underlying atmosphere and at the surface (Bony et al. 2004; Bony and Dufresne 2005; Bony et al. 2006). We shall notice the absence of midlevel clouds. This lack of midlevel clouds is a long-standing issue in LMDZ as in many GCMs, which is related to the misrepresentation of congestus clouds.
Intermediate states (λ = 32.5° and 35°, Figs. 7b,c) are characterized by the presence of three distinct cloud layers: (i) a high one at the anvil top; (ii) a low one, which corresponds to the development of a shallow cumulus layer prior to deep convective onset; and (iii) a third one just above the surface that corresponds to morning fog occurring between 0300 and 0900 LT. This foggy layer represents a distinct behavior from an oceanic boundary since it is generated by rapid nocturnal cooling of the surface, which leads to the low condensation point observed in Fig. 6a (yellow and green curves) near the surface. This condensed layer is thus mostly induced by the diurnal evolution of the land surface energy budget and especially by its nighttime cooling—unlike stratocumuli over the ocean, which are induced by a combination of surface latent heat flux, a shallow boundary layer, and large-scale subsidence. This condensed layer plays a very important role in cooling the lower atmosphere by reflecting incoming sunlight. The layer delays sunlight surface heating and the morning deepening of the boundary layer, thereby modulating the entire diurnal cycle of convection. We shall notice that the absence of surface wind in our experiments favors the development and persistence of morning fog. In section 5 we evaluate the effect of wind on the RCE. At about 0900 LT, the fog disappears and shallow convection starts. Associated cumulus clouds develop until the deep convective onset, at around 1500 LT. Deep convection activity lasts for about 6 h.
For cool RCE states (i.e., λ =37.5° and 40°, Figs. 7d,e), the succession of shallow and deep regimes is less obvious. A permanent stratocumulus layer is present at 750 hPa, which cools the lower atmosphere and strongly limits cumulus and cumulonimbus development. In the LMDZ5B SCM, this permanent cloud layer is collocated with turbulent motions associated with strong destabilization generated by cloud-top radiative cooling. Indeed, this cloud layer is nearly opaque to both incoming shortwave radiation and outgoing longwave radiation (Wood 2012), explaining the strong radiative cooling peak near cloud top (750 hPa) observed in Fig. 6a (light blue and blue curves). This dense cloud layer also traps longwave radiation in the lower atmosphere, leading to net radiative heating around 850 hPa (see Fig. 5a). The second peak is located much higher, between 500 and 250 hPa, and corresponds to the anvil cloud top. However, the cirrus radiative feedback is not sufficient to balance the strong surface cooling induced by the permanent stratocumulus layer.
In summary, in the cool RCE state, the atmosphere is so cold that the low saturation partial pressure is associated with large relative humidity values. The high relative humidity in turn favors the presence of a permanent low-level cloud layer, which is nearly opaque to daytime shortwave radiation heating of the surface and ultimately maintains the system in a cold state. The system is then locked into a very stable regime. Analogous stable, cold, and cloudy states have been found in other studies using SCMs such as the Cloud Feedback Model Intercomparison Project (CFMIP)/Global Atmospheric System Studies (GASS) Intercomparison of Large-Eddy and Single-Column Models (CGILS) (Zhang et al. 2012), which sought to enhance understanding of the transition from stratocumulus to cumulus regimes. Brient and Bony (2012) found that adding stochastic noise to the vertical motion field greatly improves a single-column model’s capacity to mimic the observed cloud cover vertical distribution. This suggests that maintaining a constant vertical wind velocity, as is the case here may favor “locked” cloudy regimes in SCMs.
To summarize the results, we have shown that the coupled land–atmosphere system supports multiple RCE states and can exhibit multiple equilibria based on the initial temperature of the system. The final equilibrium states fall into three main categories: (i) cool and wet surface, (ii) warm and dry surface, and (iii) intermediate temperature and moisture regimes. Each state corresponds to a stable cloud regime. Clouds and their interaction with radiation are suspected to play a key role in the establishment and maintenance of these equilibrium states. Each type of RCE corresponds to a certain vertical and temporal distribution of cloudiness: a two-layer (daytime cumulus and evening cirrus) configuration for warm RCEs, a three-layer configuration (morning fog, daytime cumulus, and evening cirrus) for intermediate RCEs, and a three-layer configuration (permanent stratocumulus, daytime cumulus, and evening cirrus) for cool RCEs. This result holds true for the 125 experiments conducted in section 3.
5. Role of cloud–radiative feedback on multiple equilibria
The relative importance of clouds for the emergence of multiple equilibria is now evaluated. A sensitivity experiment for the range of T0 and Qtot values at λ = 35° was performed without cloud–radiative forcing: that is, clouds were rendered transparent to both shortwave and longwave radiation. The RCE states for these sensitivity experiments are plotted in Fig. 8. Removing the cloud-radiative forcing eliminates multiple equilibria, at least for the conditions considered here. Moreover, none of the final states exhibits a warm–dry or a cool–wet RCE; rather, all final states are grouped around Ts = 300 K and water is present in both the soil and atmosphere, as with the intermediate RCE states described above. Hence, Fig. 8 strongly supports the hypothesis that cloud–radiative forcing is a necessary condition for the presence of multiple equilibria. The continental RCE framework allows a diurnal cycle of surface temperature that naturally introduces some variability into the system. For instance, the presence of morning fog is determined by the minimum nighttime surface temperature. Hence, if an internal perturbation results in a drop in the minimum nocturnal temperature, the morning fog may appear and delay the surface and atmospheric heating, increasing the probability of obtaining a colder surface on the following day. Ultimately, the system may fall into a colder equilibrium. Such a feedback loop involving clouds and radiation is rendered possible by the presence of the large diurnal variations of surface temperature. In the following, we attempt to identify the main cloud types contributing to the simulated nonlinearities and multiple equilibria.
We reproduce the sensitivity to latitude experiment conducted in section 4 under four different configurations, in order to quantify the relative role of low (i.e., below 600 hPa) and high (i.e., over 600 hPa) clouds. The original (fully coupled clouds) experiment is compared with three other experiments in which (i) all clouds are transparent to radiation (see Fig. 9b: “no clouds” experiment); (ii) only low clouds interact with radiation (see Fig. 9c: “low clouds only” experiment); and (iii) only high clouds interact with radiation (see Fig. 9d: “high clouds only” experiment). Since the no-cloud (Fig. 9b) and high-cloud-only (Fig. 9d) experiments exhibit similar behaviors, we conclude that high clouds are not a leading-order source of nonlinearity. Indeed, the equilibrium surface temperature decreases linearly with latitude when only the high-cloud radiative forcing is retained (see Fig. 9d). Compared to the no-cloud experiment (Fig. 9b), high clouds significantly heat the atmosphere and the surface through their longwave greenhouse effect. Without the cloud–radiative effect, equilibrium surface temperatures range from 289 to 308 K, whereas inclusion of high-cloud radiative forcing increases this range from 292 to 313 K.
When only low-cloud radiative forcing is included, the system may be attracted toward multiple preferred states, similar to what was observed in Figs. 5 and 9a. Therefore, we conclude that, within the LMDZ5B SCM, low-level clouds are the main source of nonlinearity leading to multiple equilibria. Low-level clouds are already known to be a principal source of GCM spread (Bony et al. 2004; Bony and Dufresne 2005; Bony et al. 2006); here their importance is further emphasized over land regions through the prism of the RCE framework. Of course, since LMDZ5B SCM tends to underestimate midlevel clouds, as do many GCMs, this conclusion should be viewed with caution. Indeed, undersimulation of midlevel clouds may exaggerate the role of low-level clouds.
Overall, these results underscore clouds as a critical component of the land–atmosphere system’s nonlinearity, with the diurnal cycle enhancing the internal variability that allows the presence of multiple equilibria. In the presence of interactive clouds, the RCE states correspond to distinct cloud vertical distributions as depicted in Fig. 7. Feedback pathways associated with convection, clouds, and radiative cooling may favor some convective regimes over others. That is, clouds strongly modulate the radiative cooling profiles with which the convective heating profiles must adjust to obtain RCE. In turn, vertical mixing of heat and moisture induced by moist convection ultimately leads to cloud formation, which affects radiation.
We performed similar experiments with a single-column model based on the version of the LMDZ GCM used for the IPCC Fourth Assessment Report (AR4) (Hourdin et al. 2006) to check the model dependency of the results (not shown). In this version, multiple equilibria were less common in the 30° to 40° range of latitudes, and fewer low clouds were observed in these simulations as well. Indeed, Hourdin et al. (2006) pointed out that an important bias of the AR4 version of the LMDZ GCM was its inability to represent low-level cumulus and stratocumulus. The absence of parameterization for coherent boundary layer structures (thermals) in this model version likely accounts for the lack of simulated cumulus clouds. At first glance, these results imply that low-level clouds are necessary for obtaining multiple equilibria, although additional analyses are required to substantiate this.
It is reasonable to consider how much the results presented here may depend on the details of the model configuration or the assumptions. Clearly, a comparison of multiple models would be useful in addressing some concerns about model dependence. Although a complete assessment regarding how model configuration or assumptions impact the results is beyond the scope of the current study, we reproduced the 125 experiments listed in section 3 with a wind forcing nudged toward a geostrophic value of 10 m s−1, in order to study the effect of large-scale forcing on the final equilibrium of the system. The wind profile computation results from the interaction between horizontal wind and surface roughness, turbulence, and thermals. Hence, we introduce an additional feedback loop into the system: the wind fully interacts with the SCM boundary layer parameterizations (but not deep convection). The resulting wind profile is constant in the free troposphere (V = 10 m s−1) and mimics the Ekman spiral in the boundary layer, with a parabolic decrease of the wind magnitude resulting in a surface wind V0 ~ 0.7 m s−1.
Figure 10 is the equivalent of Fig. 8 with a geostrophic wind forcing. The main conclusion is that multiple equilibria still exist as long as cloud–radiative feedback is retained. However, the number of final equilibria is slightly reduced and the final states are distinct from the equilibria obtained without wind. In particular, the morning fog disappears because of the added mechanical forcing that increases surface exchange and boundary layer entrainment. The disappearance of fog, in turn, reduces the vertical degrees of freedom in the cloud distribution and therefore the number of final RCE states. When clouds are made transparent to radiation, the multiple equilibria again disappear, similar to the model behavior in the absence of geostrophic wind. Thus, the interaction between the geostrophic wind and boundary layer processes does not diminish the importance of clouds, especially their radiative feedback, as the main source of internal variability. Finally, comparing Fig. 8b with Fig. 10b in the absence of cloud–radiative feedback, we observe that the RCE states are very close whether geostrophic wind is present. Indeed, the unique final equilibrium temperature is nearly 300 K in all cases, while soil water content is increased by ~5 mm with geostrophic wind forcing. This strongly suggests that wind influences the final states via cloud formation. In other words, when clouds are transparent to radiation, the wind stress does not considerably change the results, and more generally the wind–boundary layer feedback becomes of secondary importance. Overall, this experiment further supports the key role of cloud–radiation interactions in generating internal variability in the coupled soil–atmosphere system.
6. Summary and conclusions
In this study we have examined the applicability of radiative–convective equilibrium (RCE) over a land surface using a single-column atmosphere version of the LMDZ GCM coupled to an idealized land surface model. Relative to its oceanic counterpart, the land system has a finite moisture capacity corresponding to the total water content in the soil and in the atmosphere since atmospheric transport and runoff are assumed to be zero: that is, the hydrologic cycle is locally closed. Over the ranges of latitude and total water content explored, multiple equilibria can be obtained by varying initial soil temperature. Three classes of final RCE states are possible: namely, (i) a hot state with a hot and dry surface and most of the system water content residing in the atmosphere; (ii) an intermediate state with water partitioned between the soil and in the atmosphere; and (iii) a cold state with a wet surface and nearly no moisture present in the atmosphere.
By considering sensitivity experiments in which boundary layer diurnal cycle and cloud–radiative forcing are disabled, we show how these are necessary for the occurrence of multiple equilibria in the LMDZ5B SCM and how they determine the characteristics of the final RCE states. In particular, low-level clouds and fog appear to play a key role in the presence of multiple equilibria. For low total water content conditions, the system is bimodal, while increasing total water content allows the emergence of the RCE states with water in both the soil and atmosphere. These intermediate states correspond to either two- or three-layer cloud fraction distributions. In two-layer state, the succession of shallow and deep convection during daytime leads to cumulus and anvil clouds. In the three-layer state, morning fog develops before shallow convection onset. Above a threshold value for total water content multiple equilibria are no longer supported, and the system falls into a cold state. High relative humidity then favors the presence of a permanent, thick layer of low-level clouds. Outside of the latitude range emphasized here (30°–40°), all water evaporates from the surface (low-latitude hot states), corresponding to a water-limited regime, or precipitates (high-latitude cold states), corresponding to an energy-limited regime.
For a model configuration in which low-level clouds are rendered transparent to radiation, the multiple equilibria disappear, which emphasize the key role of clouds and of their radiative feedbacks in the land–atmosphere system. Overall, our results of the radiative–convective equilibrium over land indicate that the cloud–radiative feedback, interacting with the diurnal cycle of radiation, induces bifurcations in the land–atmosphere system and therefore determines the equilibrium conditions in the land–atmosphere system.
BRL and PG acknowledge funding support from NSF-AGS 1035843, PG acknowledges support from DOE-ASR DE-SC0008720, and AHS acknowledges support from NSF AGS-1008847. The authors thank Gilles Bellon, Christoph Schär, and Bjorn Stevens for stimulating discussions of this work and three anonymous reviewers for their careful reading and comments.