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  • View in gallery

    Equilibrated vertical profiles of (a) potential temperature and (b) water vapor mixing ratio from the RCE simulation.

  • View in gallery

    Domain- and time-mean precipitation as a function of the NEI to the atmospheric column for (a) WTG and (b) DGW methods using nonzero initial moisture profile from the RCE experiment. The control run has NEI = 60 W m−2. On the red curve, radiative heating is being perturbed while surface fluxes are kept fixed at the control-run magnitude of 206 W m−2; on the blue curve, surface fluxes are perturbed while radiative heating is kept fixed at the control-run magnitude of −145 W m−2.

  • View in gallery

    Normalized gross moist stability (M) for the precipitating statistically steady states above RCE (i.e., NEI > 0), using (a) WTG and (b) DGW methods. Red symbols indicate experiments with radiative heating perturbations relative to the control run and blue symbols indicate those with surface flux perturbations.

  • View in gallery

    As in Fig. 2, but using zero moisture initial conditions.

  • View in gallery

    Time series of domain-mean precipitation for the experiments with NEI = (a) 70, (b) 80, and (c) 90 W m−2 using WTG method. In red are shown radiative heating perturbations and in blue are shown surface flux perturbations relative to the control run.

  • View in gallery

    As in Fig. 5, but for the DGW method for the case NEI = 40 W m−2.

  • View in gallery

    Vertical profiles of time-mean large-scale vertical velocity for precipitating (NEI ≠ 0 cases) NEI = −20, 20, 40, 60, 80, and 100 W m−2 for a set of experiments where surface fluxes are perturbed (radiative heating perturbations produce very similar vertical motion; not shown): (a) WTG and (b) DGW methods.

  • View in gallery

    Maxima of large-scale vertical velocity using (a) WTG and (b) DGW methods as a function of NEI for experiments initialized with nonzero moisture profile. In red are the radiative heating perturbations and in blue are the surface flux perturbations relative to the control run.

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Effect of Surface Fluxes versus Radiative Heating on Tropical Deep Convection

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  • 1 Lamont-Doherty Earth Observatory, Palisades, and Department of Earth and Environmental Sciences, Columbia University, New York, New York
  • 2 Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York
  • 3 Lamont-Doherty Earth Observatory, Palisades, and Department of Applied Physics and Applied Mathematics, and Department of Earth and Environmental Sciences, Columbia University, New York, New York
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Abstract

The effects of turbulent surface fluxes and radiative heating on tropical deep convection are compared in a series of idealized cloud-system-resolving simulations with parameterized large-scale dynamics. Two methods of parameterizing the large-scale dynamics are used: the weak temperature gradient (WTG) approximation and the damped gravity wave (DGW) method. Both surface fluxes and radiative heating are specified, with radiative heating taken as constant in the vertical in the troposphere. All simulations are run to statistical equilibrium.

In the precipitating equilibria, which result from sufficiently moist initial conditions, an increment in surface fluxes produces more precipitation than an equal increment of column-integrated radiative heating. This is straightforwardly understood in terms of the column-integrated moist static energy budget with constant normalized gross moist stability. Under both large-scale parameterizations, the gross moist stability does in fact remain close to constant over a wide range of forcings, and the small variations that occur are similar for equal increments of surface flux and radiative heating.

With completely dry initial conditions, the WTG simulations exhibit hysteresis, maintaining a dry state with no precipitation for a wide range of net energy inputs to the atmospheric column. The same boundary conditions and forcings admit a rainy state also (for moist initial conditions), and thus multiple equilibria exist under WTG. When the net forcing (surface fluxes minus radiative heating) is increased enough that simulations that begin dry eventually develop precipitation, the dry state persists longer after initialization when the surface fluxes are increased than when radiative heating is increased. The DGW method, however, shows no multiple equilibria in any of the simulations.

Corresponding author address: Usama Anber, Lamont-Doherty Earth Observatory, 61 Route 9W, Palisades, NY 10964. E-mail: uanber@ldeo.columbia.edu

Abstract

The effects of turbulent surface fluxes and radiative heating on tropical deep convection are compared in a series of idealized cloud-system-resolving simulations with parameterized large-scale dynamics. Two methods of parameterizing the large-scale dynamics are used: the weak temperature gradient (WTG) approximation and the damped gravity wave (DGW) method. Both surface fluxes and radiative heating are specified, with radiative heating taken as constant in the vertical in the troposphere. All simulations are run to statistical equilibrium.

In the precipitating equilibria, which result from sufficiently moist initial conditions, an increment in surface fluxes produces more precipitation than an equal increment of column-integrated radiative heating. This is straightforwardly understood in terms of the column-integrated moist static energy budget with constant normalized gross moist stability. Under both large-scale parameterizations, the gross moist stability does in fact remain close to constant over a wide range of forcings, and the small variations that occur are similar for equal increments of surface flux and radiative heating.

With completely dry initial conditions, the WTG simulations exhibit hysteresis, maintaining a dry state with no precipitation for a wide range of net energy inputs to the atmospheric column. The same boundary conditions and forcings admit a rainy state also (for moist initial conditions), and thus multiple equilibria exist under WTG. When the net forcing (surface fluxes minus radiative heating) is increased enough that simulations that begin dry eventually develop precipitation, the dry state persists longer after initialization when the surface fluxes are increased than when radiative heating is increased. The DGW method, however, shows no multiple equilibria in any of the simulations.

Corresponding author address: Usama Anber, Lamont-Doherty Earth Observatory, 61 Route 9W, Palisades, NY 10964. E-mail: uanber@ldeo.columbia.edu
Keywords: Deep convection

1. Introduction

Surface turbulent heat fluxes and electromagnetic radiation are the most important sources of moist static energy (or moist entropy) to the atmosphere. In the idealized state of radiative–convective equilibrium (RCE), the source due to surface fluxes must balance the sink due to radiative cooling (negative radiative heating). In this state, the surface evaporation and precipitation also balance, and there is no large-scale circulation. In a more realistic situation in which there is a large-scale circulation, the strength of that circulation’s horizontally divergent component can be viewed as proportional, in a column-integrated sense, to the net moist static energy source (surface fluxes plus column-integrated radiative heating), with the proportionality factor being known as the gross moist stability (e.g., Neelin and Held 1987; Neelin and Zeng 2000; Sobel 2007; Raymond et al. 2009).

The gross moist stability is the rate at which the circulation exports moist static energy from a column for a given rate of mass circulation through the column (or alternately, as used below, a given dry static energy export or moisture import to the column). In general it is a function of time and position, and in a closed dynamical theory we might expect it to be an interactive function of atmospheric state variables or other quantities predicted by the theory.

In fact, there is no accepted theory that satisfactorily predicts the gross moist stability. One option is to attempt to derive a parameterization of it from either observations or numerical simulations. Numerical simulations allow greater control than do analyses of observations, but in principle a large set of simulations is required to determine how the gross moist stability depends on all environmental factors that might potentially be relevant. A dramatic simplification would be possible, however, if we could assume that the gross moist stability were constant under some circumstances.

Among studies that consider tropical phenomena through the lens of the vertically integrated moist static energy (or, similarly, moist entropy) budget, the constancy or variability of the gross moist stability arises regularly as an issue. In the case of the Madden–Julian oscillation (MJO), for example, Kuang (2011) argues that variations in gross moist stability are important to MJO dynamics. Sobel and Maloney (2012, 2013), on the other hand, assume a constant gross moist stability, and Inoue and Back (2015) present evidence that this is a defensible assumption for the MJO in particular despite considerable variability in the gross moist stability [see also Wang et al. (2013)]. In the case of tropical cyclogenesis, arguments that dynamic variations in gross moist stability are important have been made by Raymond and Sessions (2007), Raymond et al. (2011), and Gjorgjievska and Raymond (2014).

If the gross moist stability could be considered constant for the purposes of studying some specific set of phenomena, then not only would the divergent circulation (i.e., the large-scale vertical motion) in those phenomena be predictable as a function of the surface fluxes and radiative heating, but surface fluxes and radiative heating would influence that circulation in the same way. All that would matter would be the sum of the two, which is the forcing due to the column-integrated net moist static energy. In other words, we could reduce the problem of knowing the functional dependences of the gross moist stability on state variables or other environmental factors to the much simpler problem of determining only a single scalar free parameter. This study investigates, in an idealized setting, whether this is the case.

In statistical equilibrium simulations—best thought of as being relevant to the time-mean tropical circulation, rather than to transient phenomena such as the MJO or tropical cyclones—we ask whether surface fluxes and radiative heating influence the circulation differently. We might expect that they would; given that surface fluxes act at the surface while radiation acts throughout the column. Such a difference would necessarily be expressed (at least in the time mean) as a difference in the gross moist stability between two situations in which the net moist static energy source is the same but its partitioning between surface fluxes and radiative heating is different.

We study this problem using a cloud-resolving model (CRM). CRMs have proven to be very powerful tools for studying deep moist convection. One set of useful studies involves simulations of RCE (e.g., Emanuel 2007; Robe and Emanuel 2001; Tompkins and Craig 1998; Bretherton et al. 2005; Muller and Held 2012; Popke et al. 2012; Wing and Emanuel 2014). While RCE has provided many useful insights, it entirely neglects the influences of the large-scale circulation. Another approach is to parameterize the large-scale circulation (e.g., Sobel and Bretherton 2000; Mapes 2004; Bergman and Sardeshmukh 2004; Raymond and Zeng 2005; Kuang 2011; Romps 2012a,b; Wang and Sobel 2011; Anber et al. 2014; Edman and Romps 2014) as a function of variables resolved within a small domain. This approach is computationally inexpensive (compared to using domains large enough to resolve the large scales present on the real Earth) and still provides a two-way interaction between cumulus convection and large-scale dynamics.

One method to parameterize the large-scale dynamics is called the weak temperature gradient (WTG) approximation (Sobel and Bretherton 2000). As the name suggests, this method relies explicitly on the smallness of the horizontal temperature gradient in the tropics, which is a consequence of the small Coriolis parameter there. Any temperature, or density, anomaly in the free troposphere generated by diabatic processes is rapidly wiped out by means of gravity wave adjustment to restore the temperature profile to that of the adjacent regions. Hence, the dominant balance is between diabatic heating and adiabatic cooling, and the tropospheric temperature is constrained to remain close to a target profile which is interpreted as that of surrounding regions.

While WTG captures the net result of the gravitational adjustment, it does not simulate the gravity waves themselves. Another method of representing the large-scale dynamics in CRMs represents those dynamics as resulting explicitly from such waves, with a single wavenumber, interacting with the simulated convection. This method was introduced by Kuang (2008) and Blossey et al. (2009) and is called the damped gravity wave (DGW) method.

Both methods have been shown to produce results qualitatively similar to observations in some settings; for example, Wang et al. (2013) compared the two methods with observations produced during the TOGA COARE field experiment.

Utilizing both of the above two methods, as we do here, allows us to explore a variety of mechanisms and parameters affecting the interaction between deep convection and large-scale dynamics, among which are the surface turbulent fluxes and radiative heating.

In numerical experiments using the WTG method, Sobel et al. (2007) and Sessions et al. (2010) found that the statistically steady solution is not unique for some forcings: the final solutions can be almost entirely dry, with zero precipitation, or rainy, depending on the initial moisture content. We have interpreted this behavior as relevant to the phenomenon of “self-aggregation” in large-domain RCE simulations (Bretherton et al. 2005; Muller and Held 2012; Wing and Emanuel 2014), with the two states corresponding to dry and rainy regions within the large domain. Tobin et al. (2012) find evidence of this behavior in observations. In the present study, we perform sets of simulations with different initial conditions to look for multiple equilibria and to determine whether their existence or persistence is influenced differently by surface fluxes and radiation.

This paper is organized as follows: in section 2 we describe the model and the experiment setup. In section 3 we show results. We highlight some implications of our results and conclude in section 4.

2. Model configuration and experimental setup

a. Model configuration

We use the Weather Research and Forecast (WRF) Model version 3.3, in three spatial dimensions, with doubly periodic lateral boundary conditions. The experiments are conducted with Coriolis parameter f = 0. The domain size is 192 × 192 km2, with a horizontal grid spacing of 2 km. There are 50 vertical levels in the domain, extending to 22 km high, with 10 levels in the lowest 1 km. Gravity waves propagating vertically are absorbed in the top 5 km to prevent unphysical wave reflection off the top boundary using the implicit damping vertical velocity scheme (Klemp et al. 2008). The two-dimensional Smagorinsky first-order closure scheme is used to parameterize the horizontal transports by subgrid-scale eddies. The Yonsei University (YSU) first-order closure scheme is used to parameterize boundary layer turbulence and vertical subgrid-scale eddy diffusion (Hong and Pan 1996; Noh et al. 2003; Hong et al. 2006). The microphysics scheme is the Purdue–Lin bulk scheme (Lin et al. 1983; Rutledge and Hobbs 1984; Chen and Sun 2002) that has six species: water vapor, cloud water, cloud ice, rain, snow, and graupel.

Many possible other modeling choices could be made than those above. Resolution, domain size, numerical schemes, and physical parameterizations could all be varied. Our results cannot, of course, be assumed to be invariant to all changes in these choices and should be interpreted as demonstrating one set of possible solutions resulting from one particular model configuration.

We first perform an RCE experiment at fixed sea surface temperature of 28°C until equilibrium is reached at about 60 days. Results from this experiment are averaged over the last 10 days after equilibrium to obtain statistically equilibrated temperature and moisture profiles. Figure 1 shows the resulting vertical profiles of (Fig. 1a) potential temperature and (Fig. 1b) moisture. These profiles are then used to initialize other runs with parameterized large-scale circulations, and the temperature profile is used as the target profile against which perturbations are computed in both the WTG and DGW methods. All WTG and DGW experiments are run for about 55 days, and mean quantities are averaged over the last 10 days.

Fig. 1.
Fig. 1.

Equilibrated vertical profiles of (a) potential temperature and (b) water vapor mixing ratio from the RCE simulation.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0253.1

We will call the RCE moisture profile the nonzero moisture profile, or wet conditions, to distinguish it from other moisture profiles (zero, in particular, or dry conditions) used in this paper.

b. Parameterized large-scale circulation

The large-scale vertical velocity is dynamically determined using either the WTG or the DGW method. In the relaxed form of WTG used in CRM simulations (Raymond and Zeng 2005; Wang and Sobel 2011; Wang et al. 2013; Anber et al. 2014) the vertical velocity W is obtained by
e1
where θ is the domain-mean potential temperature, is the reference temperature (from RCE run), h is the height of the boundary layer determined internally by the boundary layer scheme, and τ is the relaxation time scale, and can be thought of as the time scale over which gravity waves propagate out of the domain, taken here 3 h.
In DGW method (Kuang 2008; Blossey et al. 2009; Romps 2012a,b; Wang et al. 2013) the large-scale vertical velocity is obtained by solving the elliptic partial differential equation:
e2
where p the pressure; ω is the pressure vertical velocity; is the dry gas constant; is the domain-mean virtual temperature; is the target virtual temperature (from RCE); ε is the momentum damping, in general a function of pressure but here taken constant at 1 day−1; and k is the wavenumber taken .
The boundary conditions used for solving (3) are
eq1
Once the vertical velocity obtained from (2) or (3), it is used to vertically advect domain-mean temperature and moisture at each time step. Horizontal moisture advection is not represented.

The free parameters used here are chosen to give a reasonable comparison between the general characteristics of the two methods and to produce a close, but not exact, precipitation magnitude in the control runs.

c. Experiment design

All simulations are conducted with prescribed surface fluxes and radiative heating and no mean wind. The radiative heating rate is set to a constant value in the troposphere, while the stratospheric temperature is relaxed toward 200 K over 5 days as in Wang and Sobel (2011) and Anber et al. (2014).

The control runs have surface fluxes of 205 W m−2, latent heat flux (LH) of 186 W m−2 and sensible heat flux (SH) of 19 W m−2 (the ratio of the two corresponding to Bowen ratio of 0.1), and vertically integrated radiative heating of −145 W m−2, corresponding to a radiative heating rate of −1.5 K day−1 in the troposphere in both the WTG and DGW experiments.

We perform two sets of experiments with parameterized large-scale dynamics: one in which surface fluxes are varied by increments of 20 W m−2 from the control run while holding radiative heating fixed at −145 W m−2, and the other in which the prescribed radiative heating is varied in increments of 20 W m−2 while holding surface fluxes fixed. Perturbations in are performed by varying radiative heating rate while holding it in uniform in the vertical. Table 1 summarizes the control parameters of the numerical experiments.

Table 1.

Control parameters of net radiation, surface fluxes, and their components of sensible and latent heat flux (chosen to keep the Bowen ratio close to 0.1). The sum of surface fluxes and radiative heating is the NEI. All values are in W m−2. Bold marks the control run parameters (see text for more details on the experiment setup). NEIs of 70 and 90 W m−2 are for cases initialized with dry conditions using WTG method.

Table 1.

Another two sets of simulations (with two methods) are performed which are identical except that they are initialized with a zero moisture profile (or “dry conditions”).

All mean quantities are plotted as a function of the net energy input (NEI) to the atmospheric column excluding the contribution from circulation. Thus, NEI is the sum of surface fluxes (SF) and vertically integrated radiative heating NEI = SF + .

3. Results

a. Precipitation and normalized gross moist stability

1) Mean precipitation

(i) Nonzero initial moisture conditions

Figure 2 shows the domain and time-mean precipitation as a function of the NEI using the nonzero moisture profile as the initial condition with (Fig. 2a) WTG and (Fig. 2b) DGW. At zero NEI, in one set of (red) experiments the radiative heating rate is increased from that in the control to balance surface fluxes (205 W m−2), while in the other (blue) surface fluxes are reduced to balance radiative heating (145 W m−2). The former gives more precipitation in both WTG and DGW experiments.

Fig. 2.
Fig. 2.

Domain- and time-mean precipitation as a function of the NEI to the atmospheric column for (a) WTG and (b) DGW methods using nonzero initial moisture profile from the RCE experiment. The control run has NEI = 60 W m−2. On the red curve, radiative heating is being perturbed while surface fluxes are kept fixed at the control-run magnitude of 206 W m−2; on the blue curve, surface fluxes are perturbed while radiative heating is kept fixed at the control-run magnitude of −145 W m−2.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0253.1

In all these experiments, the precipitation rate varies linearly over a broad range of NEI values. The precipitation rate produced for a given increment of surface fluxes exceeds that produced for the same increment of vertically integrated radiative heating. For example, increasing surface fluxes by 40 W m−2 from the control run (i.e., at NEI = 100 W m−2 or surface fluxes exceeds radiative heating by 100 W m−2) there is more precipitation (blue curve) than if we increase radiative heating by 40 W m−2 (red curve).

It is straightforward to understand this difference in the slopes of the precipitation responses from the point of view of the column-integrated moist static energy budget. We use the steady state diagnostic equation for precipitation as in, for example, Sobel (2007), Wang and Sobel (2011), or Raymond et al. (2009):
e3
where is the mass weighted vertical integral from the bottom to the top of the domain. The quantities P, L, H, and QR are precipitation, latent heat flux, sensible heat flux, and radiative heating, respectively.

We define as the normalized gross moist stability, which represents the export of moist static energy by the large-scale circulation per unit of dry static energy export (e.g., Neelin and Held 1987; Sobel 2007; Raymond et al. 2009; Wang and Sobel 2011; Anber et al. 2014). Here h is the moist static energy (sum of the thermal, potential, and latent energy), s is the dry static energy (thermal and potential energy), and the overbar is the domain mean and time mean.

The second and third terms (combined) on the right-hand side of (3) represent the precipitation that would occur in radiative convective equilibrium. The first term accounts for the contribution by the large-scale circulation, which arises from the discrepancy between surface fluxes and vertically integrated radiative heating. Therefore, contributes to P in two ways with opposite signs: to the dynamic part [the first term on the right-hand side of (3)], similar to the contribution from surface fluxes, and to the RCE precipitation [the second term on the RHS of (3)] in an opposite sense. Surface fluxes, on the other hand, contribute only positively.

Figure 3 shows the normalized gross moist stability (M) as a function of NEI > 0 for cases initialized with nonzero initial moisture conditions from (Fig. 3a) WTG and (Fig. 3b) DGW experiments. The M is a positive number less than 1 and exhibits only modest variations (measured, for example, by their influence on precipitation compared to that of the changes in diabatic forcing) under each forcing method, though the values under DGW are consistently smaller than those under WTG. The smallness of the variations in M is a nontrivial result; we know no a priori reason why M could not vary more widely. Even the variations that do occur as a function of NEI are similar for equal increments of surface flux or radiative heating, over most of the range, particularly in DGW. The most marked differences occur at NEI = 20 W m−2 under WTG, the value closest to RCE.

Fig. 3.
Fig. 3.

Normalized gross moist stability (M) for the precipitating statistically steady states above RCE (i.e., NEI > 0), using (a) WTG and (b) DGW methods. Red symbols indicate experiments with radiative heating perturbations relative to the control run and blue symbols indicate those with surface flux perturbations.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0253.1

At NEI = 0 the large-scale vertical velocity vanishes and M is undefined; however, M in that case is not needed to compute P. Equation (3) is derived by eliminating the vertical advection term between the moist and dry static energy equations, but NEI = 0 corresponds to RCE, in which the vertical advection vanishes. In that case, the precipitation is simply

When there is a large-scale circulation such that (3) is valid, we can see that if M and the surface fluxes are held fixed, the change in precipitation per change in radiative heating is
e4
where we have neglected the small contribution from H. As discussed above, Fig. 3 shows that constancy of M is a good approximation for all the numerical experiments.
On the other hand, the change in precipitation due to an increment in surface fluxes (SF) (holding radiative heating and M fixed) scales as
e5
Equations (4) and (5) show that a change in precipitation due to an increment in surface fluxes will exceed that due to an increment in radiative heating. The difference of unity, nondimensionally, means that for finite and equal increments of either surface fluxes or radiative heating, the excess precipitation due to surface fluxes is equal to the increment in forcing itself.

Given a positive M, (5) states that increasing surface fluxes always increases precipitation, but precipitation responses to changes in can be either negative or positive in principle, depending whether M is greater or less than 1. For a small M (M ≪ 1), the difference is small. In our experiments, where M is sufficiently large (~0.4, as shown below) that the difference is not negligible, surface fluxes have a significantly greater influence on precipitation than does radiative heating.

The difference we see in Fig. 2 is what we expect for constant M and might have been considered a null hypothesis. It indicates no fundamental difference in how surface fluxes and radiative heating influence the large-scale circulation in these simulations. We might have expected that forcing in the interior of the troposphere by radiative heating might induce differences in the thermodynamic profiles and the vertical motion profiles relative to forcing at the surface by turbulent fluxes, resulting in different values of M. However, the simulations here are well explained by the simplest vertically integrated theory, in which M remains constant. The simulations are needed, in this sense, only to provide the (single) value of M.

Quantitatively speaking, the slope of the rainfall in Fig. 2a (WTG, and similarly for DGW) for increasing the surface fluxes, , is about 2.3, which corresponds to M = 0.44 using (5). Similar magnitudes are obtained for . For DGW, however, is about 3.1 and corresponds to M = 0.32. Both values of M are close to those obtained in the respective control runs with NEI = 60 W m−2.

As in Fig. 3, the right-hand side of (5) with constant M gives a very good estimate of the changes in precipitation resulting from both types of forcings.

(ii) Zero initial moisture conditions

Figure 4 is analogous to Fig. 2 but now we initialize the simulations with a zero moisture profile while keeping everything else the same as above (including the reference temperature profile). For WTG (Fig. 4a) the system now exhibits multiple equilibria, staying in the dry state over a wide range of NEI (Sobel et al. 2007; Sessions et al. 2010). Precipitation does not occur for these dry initial conditions for NEI below a threshold value. Above this value, we obtain mean precipitation values identical to those found with nonzero initial moisture profile for the same forcings. However, the transition to a precipitating state (i.e., to the apparent inability of the dry state to be sustained) occurs at 60 W m−2 when radiative heating is varied but 70 W m−2 when surface fluxes are varied. In other words, starting from dry conditions the system requires less energy from radiation than from surface fluxes in order for precipitation to occur.

Fig. 4.
Fig. 4.

As in Fig. 2, but using zero moisture initial conditions.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0253.1

Under the DGW method (Fig. 4b), however, the system produces exactly the same precipitation rate as was produced with the nonzero initial moisture profile as an initial condition. This suggests that the DGW method does not allow multiple equilibria in the parameter range we have explored.

2) Precipitation time series

In this section we explore whether there are any differences between the influences of surface fluxes and radiative heating in the simulations starting with dry conditions other than those evident in the statistical equilibria which are eventually reached. Specifically, in the simulations where precipitation eventually does occur—but near the threshold value of NEI below, which the dry state can be sustained—we ask whether the time interval between the initial time and the time at which precipitation first occurs may be different.

Figure 5 shows domain-mean precipitation time series under WTG for the dry initial conditions with NEI of 70 (Fig. 5a), 80 (Fig. 5b), and 90 W m−2 (Fig. 5c), all of which are near the transition from existence to nonexistence of the dry solution (Fig. 4a). Two points are worth noticing here. First, the transition from the dry to precipitating state happens in a more dramatic fashion, overshooting the statistical equilibrium value, when radiative heating is perturbed. In contrast, perturbing surface fluxes leads to a much smoother transition to precipitation onset. Second, the precipitation lag due to perturbing surface fluxes versus radiative heating is apparent, not only in the time-mean picture, but also in the time series, as the time interval between initialization and precipitation onset is longer for a surface flux increase than an equivalent radiative heating increase. This lag decreases as the NEI increases (case of 100 W m−2 shows almost no lag, hence not shown).

Fig. 5.
Fig. 5.

Time series of domain-mean precipitation for the experiments with NEI = (a) 70, (b) 80, and (c) 90 W m−2 using WTG method. In red are shown radiative heating perturbations and in blue are shown surface flux perturbations relative to the control run.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0253.1

In Fig. 5a, case of NEI = 70 W m−2 in which precipitation occurs only from a radiative heating increment, the onset of precipitation is delayed by about 35 days, after which it takes less than 3 days to reach equilibrium.

The delay in precipitation onset is reduced when NEI is increased. For NEI = 80 W m−2 (Fig. 5b), the delay is about 20 days for the radiative heating perturbation and about 27 days for the surface flux perturbation. For the case of NEI = 90 W m−2 (Fig. 5c), delays are about 16 and 18 days for radiative heating and surface flux perturbations, respectively.

The precipitation time series for the case NEI = 40 W m−2 in the DGW experiment with a dry initial moisture profile is shown in Fig. 6. Unlike the WTG experiment, the time the system takes to begin precipitating is indistinguishable for radiative heating and surface flux perturbations (the same is true for other NEI cases; not shown). This might be due to the vertical structure of the large-scale vertical velocity in WTG as we will discuss in the next section. When starting from dry conditions, precipitation is delayed for only 5 days compared to nonzero initial moisture conditions, and this lag is not dependent on the NEI.

Fig. 6.
Fig. 6.

As in Fig. 5, but for the DGW method for the case NEI = 40 W m−2.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0253.1

b. Large-scale vertical velocity

We now focus on the vertical profiles of large-scale vertical velocity W. We examine these for the experiments with moist initial conditions. Time-mean vertical profiles of W in the precipitating equilibrium under WTG and DGW and their corresponding maximum values are shown in Figs. 7 and 8 , respectively, for experiments in which surface fluxes are perturbed. Time-mean W from the experiments NEI = 0 is not shown because it is zero by design. As expected, the large-scale vertical velocity is more top heavy under WTG than DGW (Romps 2012a,b; Wang et al. 2013). The profiles’ peak values are almost identical for the same increment in radiative heating despite different precipitation magnitudes.

Fig. 7.
Fig. 7.

Vertical profiles of time-mean large-scale vertical velocity for precipitating (NEI ≠ 0 cases) NEI = −20, 20, 40, 60, 80, and 100 W m−2 for a set of experiments where surface fluxes are perturbed (radiative heating perturbations produce very similar vertical motion; not shown): (a) WTG and (b) DGW methods.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0253.1

Fig. 8.
Fig. 8.

Maxima of large-scale vertical velocity using (a) WTG and (b) DGW methods as a function of NEI for experiments initialized with nonzero moisture profile. In red are the radiative heating perturbations and in blue are the surface flux perturbations relative to the control run.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0253.1

All these experiments produce moist states with nonzero precipitation (Fig. 2). (Even moderate rates of subsidence need not, generally, be associated with completely dry states; if the condensation heating is nonzero but smaller than the radiative heating, descent will still occur.) When the NEI is negative, vertical motion is downward and precipitation falls below the RCE magnitude. The opposite happens when the NEI is positive.

c. Sensitivity experiments

Several additional sets of experiments were performed to explore the sensitivity to 1) initial moisture profiles, 2) the role of interactive surface fluxes, and 3) the role of randomness.

We have done simulations in which the initial moisture profile is neither equal to the RCE profile nor zero but different fractions of the RCE initial moisture profile. For moisture profiles greater than approximately 50% of the RCE profile, only the rainy equilibrium state is reached. For drier profiles, the dry equilibrium state can be reached, though over a narrower range of NEI (NEI < 60 W m−2 in Fig. 4) than when the completely dry initial conditions are used.

We also have done simulations with interactive surface fluxes with specified sea surface temperature. We obtain similar results to those with fixed surface fluxes if we compare simulations in which the actual values of the fluxes are similar—including the existence or nonexistence of multiple equilibria—as long as the NEI remains in the interval (−20, 60) W m−2.

Finally, we performed ensemble simulations similar to those described here for the zero moisture initial conditions but with random perturbations (positive and negative) in temperature and wind initial conditions. The time-mean picture is identical to Fig. 4. Variations in precipitation onset near the transition from dry to wet state in the time series are less than a day for the experiments with perturbed radiative heating. In other words, the transition of the red curve in Fig. 5 to the precipitating state, advances or retreats by a few hours. Experiments where surface fluxes are perturbed remain unchanged. This indicates that the differences in response are due to differences in forcing and not due to random variability.

4. Conclusions

Cloud-resolving model simulations have been conducted with parameterized large-scale circulation to contrast the effects of surface fluxes and radiative heating on deep tropical convection. Two different parameterizations of large-scale circulation—the weak temperature gradient (WTG) and damped gravity wave (DGW) methods—are used.

In the precipitating equilibrium state, a given change in surface fluxes induces a greater change in precipitation in our simulations than does an equal change in radiative heating. This difference is a straightforward consequence of the column-integrated moist static energy budget with a constant normalized gross moist stability. The surface flux and radiative heating increments result in equal changes in the divergent circulation. The precipitation change, however, is a consequence of both that divergent circulation change and the change that would occur in its absence—that is, in radiative–convective equilibrium (RCE). In RCE, surface fluxes and radiative heating have opposite effects on precipitation; the overall precipitation change in our simulations is thus the sum of these opposite contributions from the RCE component and equal contributions from the induced circulations.

Aside from the differences in precipitation that result straightforwardly from their different roles in RCE, however, equal increments of surface fluxes and radiative heating influence our simulations identically for all practical purposes. The large-scale vertical motion and moist static energy export changes induced by equal increments of the two forcings are essentially identical. We might have expected, on the contrary, that the two forcings would induce different responses in such a way that the normalized gross moist stabilities would be different, allowing differences in large-scale vertical motion.

This does not occur; the gross moist stability, under both WTG and DGW, remains approximately constant under each method (though it is modestly smaller under DGW than WTG owing to the less top-heavy vertical motion profiles). Even the small variations that do occur are similar (over most of the range studied) for equal surface flux and radiation perturbations. If the normalized gross moist stability were precisely constant, the precipitation rate in all the experiments (under a given large-scale parameterization) could be predicted accurately from the moist static energy budget (3) after doing a single simulation to determine the gross moist stability, since the surface fluxes and radiative heating are both specified in these simulations. While this is a somewhat unrealistically constrained situation compared to the real one in which surface fluxes and radiation are interactive, it is nonetheless interesting that the one degree of freedom our simulations do have in the column-integrated moist static energy budget—the normalized gross moist stability—is exercised almost not at all.

A set of simulations was also conducted in which the model was initialized with zero moisture to determine whether multiple equilibria exist and whether surface fluxes and radiation affect their influence differently. Under WTG, a dry nonprecipitating equilibrium can be maintained over a wide range of NEI. To make a transition to a wet state, the system needs a smaller increase in radiative heating than surface fluxes. We interpret this as resulting from the distribution of radiative heating through the whole atmospheric column, such that reducing it reduces subsidence and increases humidity above the planetary boundary layer.

In the DGW method, only a precipitating equilibrium state is found. This may be understood (in the sense of proximate causes) in terms of the warm temperature anomalies in the free troposphere produced by this method that cause ascending large-scale vertical motion.

Studies of self-aggregation of convection in large-domain CRMs (Bretherton et al. 2005; Muller and Held 2012; Jeevanjee and Romps 2013; Wing and Emanuel 2014; Emanuel et al. 2014) show that interactive radiation is essential to the occurrence of self-aggregation. The multiple equilibria occurring in single-column or small-domain CRM simulations under WTG (e.g., Sobel et al. 2007; Sessions et al. 2010) have been interpreted as a manifestation of the same phenomenon, yet interactive radiation is not required for its occurrence in our WTG simulations. Herman and Raymond (2014) show that the occurrence of multiple equilibria in WTG simulations without interactive radiation is sensitive to the choice of the level used for the boundary layer top, a free parameter in the method, and that multiple equilibria do not occur in a new spectral WTG method, which—similarly to the DGW method—is nonlocal in the vertical and does not require a special treatment of the boundary layer. Their results and ours appear broadly consistent and suggest that the occurrence of multiple equilibria without interactive radiation under standard WTG method may be an artifact of the method’s locality in the vertical or (relatedly) its somewhat ad hoc treatment of the boundary layer. Which (if any) of these methods produces multiple equilibria in the presence of interactive radiation in CRMs, and whether the dynamics of those equilibria are faithful to the dynamics of self-aggregation seen in large-domain RCE simulations, remains as a question for future work.

Acknowledgments

This work was supported by NSF Grant AGS-10088 47. We would like to acknowledge high-performance computing support from Yellowstone (ark:/85065/d7wd3xhc) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation. We like to thank Sharon Sessions and two anonymous reviewers for their constructive comments.

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  • Wing, A. A., , and K. A. Emanuel, 2014: Physical mechanisms controlling self-aggregation of convection in idealized numerical modeling simulations. J. Adv. Model. Earth Syst., 6, 59–74, doi:10.1002/2013MS000269.

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