a. Specification of pressure levels

The thermodynamic variables of the model, including temperature T, water vapor mass mixing ratio r_υ, and moist static energy h_m, are defined on fixed full pressure levels. The boundaries between full pressure levels are referred to as half pressure levels. The surface pressure, or lowest half pressure level p_half(1), is assigned an initial value 1000 hPa. For p > 40 hPa, the remaining half pressure levels are given by

In the standard version of the model, Δp_trop = 50 hPa and f_p = 0.97. The difference between half pressure levels is 50 hPa near the surface and 25 hPa near 100 hPa. The full pressure levels are given by

The model has 37 full levels, with 26 in the troposphere (p > 100 hPa). The model top is at 1.56 hPa. The temperature and relative humidity profiles generated by the model are essentially unchanged when the model is run at higher vertical resolution.

b. Surface heat and moisture fluxes

The heat and moisture fluxes at the surface are calculated according to bulk aerodynamic formulae. The temperature tendency of the lowest full level of the model due to a flux of heat from the surface is given by

where C_D = 0.002 is a dimensionless exchange coefficient, |V| is the surface wind speed, T_sst is the sea surface temperature, and dz(1) is the thickness of the bottom layer. The water vapor mass mixing ratio tendency of the bottom model level is given by

where r_s,sst is the saturated water vapor mixing ratio at the ocean temperature and surface pressure.

c. Radiative cooling

The clear-sky radiative cooling rate is calculated interactively every time step using a radiative transfer model (Fu and Liou 1992), and the most recent temperature and water vapor mixing profiles of the model as inputs. In the default version of the model, the ozone profile is fixed at a tropical annual mean profile generated using 12 stations from the Southern Hemisphere Additional Ozonesondes (SHADOZ) archive (Thompson et al. 2003).

d. Mass flux closure

Updraft parcels originate from model levels near the surface with positive reversible density CAPE. CAPE is calculated in the standard way, except that the contributions from model levels closer to the surface are weighted more strongly. This makes the updraft mass flux less responsive to changes in the background temperature of the upper troposphere:

The sum over i extends from near surface layer n to the level of neutral buoyancy (LNB), T_υ(i) is the virtual temperature of the background atmosphere at layer i, and T_dp(i, n) is the density temperature of a parcel starting from layer n, lifted reversibly to layer i. The density temperature takes into account the effects of condensate loading and water vapor on density. The weighting parameter z_scale is set equal to 5000 m; p(i) is the pressure of full level i; and R_d is the gas constant for dry air.

Convection is initiated from a layer n near the surface when cape_d (n) > 0. The total number of updraft parcels released from a level per time step is n_launch. In these simulations n_launch = 16. The total dry mass of all n_launch parcels is given by

where (with default values in brackets) t_step is the time step of the model (20 min), t_scale is a prescribed time scale over which convection is assumed to remove positive CAPE air from the boundary layer (6 h), cape_scale is a parameter that sets the scale of CAPE in the boundary layer (200 J kg⁻¹), g is the gravitational acceleration, and dp_d (i) is the dry pressure increment of layer n:

In these models runs, cape_d (1) ∼ 80 J kg⁻¹, so that convection typically removes about 2.2% of the mass in the bottom layer every 20-min time step. The updraft mass flux and mean rainfall will typically increase in response to increases wind speed and sea surface temperature.

e. Updraft parcel mass and moist static energy spectrum

The moist static energy per unit dry mass of a parcel can be defined (Emanuel 1994)

where c_pd is the specific heat of dry air, c_l the specific heat of liquid water, T_p the air parcel temperature, l_υ(T_p) the latent heat of vaporization, l_f (T_p) the heat of melting, z the altitude, r_υp the water vapor mass mixing ratio, r_ip the ice mass mixing ratio, and r_tp the total water mass mixing ratio. The liquid water mass mixing ratio of the parcel can be defined r_lp = r_tp − r_υp − r_ip. The model is formulated in such a way that cloud ice and snow are permitted. However, the simulations discussed here do not consider this possibility (i.e., r_ip = 0 always).

Equation (6) specifies the total mass of all updraft parcels from a convecting level during a time step. This total mass is divided into a number n_launch parcels of varying mass and moist static energy. The introduction of an updraft mass and moist static energy spectrum is an attempt to represent subgrid-scale variability and to generate a range of updraft outflow altitudes. The moist static energy h_m of the updraft parcels at starting level n were assumed to be equally distributed between h_m(n) − 0.5Δh_m and h_m(n) + 0.5Δh_m, where Δh_m = 4 kJ kg⁻¹. Let n_l refer to a parcel index ranging from 1 to n_launch, and the parameter f_u have the definition

Here f_u ranges from 0 to 1 and specifies the position of a parcel within the updraft h_m spectrum. The equally spaced moist static energy spectrum of the updraft parcels starting at level n is then given by

Each updraft parcel is assumed to have the same initial relative humidity. To conserve total water, the dry mass-weighted mean r_υp of the parcel spectrum is constrained to equal the background r_υ of the initial layer.

The probability distribution function of moist static energy in the boundary layer is roughly triangular (Folkins and Braun 2003). We therefore adopt a triangular mass spectrum in which the mass of the two parcels with the lowest and highest moist static energy (i.e., n_l = 1 and n_l = n_launch) is m_ratio times smaller than the mass of the parcel whose h_m is at the center of the moist static energy spectrum (i.e., has f_u = 0.5). The total initial dry mass of all updraft parcels is constrained to equal m_d,up,all(n), as defined in Eq. (6).

Every updraft parcel with positive reversible CAPE is forced to rise some initial distance z_up (800 m) from its starting level. After this forced ascent, parcels may spontaneously rise and mix with the background atmosphere. Parcels fully detrain into the background atmosphere once their buoyancy becomes less than b_detrain. In these simulations, b_detrain = −0.01 m s⁻² so that updraft parcels are permitted to be weakly negatively buoyant (Jorgensen and LeMone 1989).

f. Rain formation

At each model level, any liquid water condensate in excess of a prescribed threshold is removed from the updraft parcel and considered to be rain. This threshold is given by

where r_l,const and r_l,ratio are temperature dependent constants and r_sp(i) is the saturated water vapor mixing ratio of a parcel at level i. In practice, at lower altitudes, the maximum liquid water content is effectively fixed at r_l,const. At upper altitudes, it is fixed at some fraction of the saturated water vapor mass mixing ratio of the updraft parcel. Figure 1 shows vertical profiles of the updraft condensate r_lp in the model generated by implementing a prescribed variation in r_l,max; r_l,max is approximately constant at 0.0025 (kg of cloud water kg⁻¹ of dry air) below 8 km and rapidly decreases above this altitude. At some altitudes, condensate detrainment makes a significant contribution to the water vapor budget. The variation in r_l,max is therefore tuned, in part, to generate a mean relative humidity profile in agreement with observations. In particular, in the absence of reasonably large values of r_l,max to ∼10 km, the simulated relative humidity of the model in the 6–10-km height interval would be too dry.

g. Updraft mixing

Updraft parcels are lifted vertically in such a way as to conserve moist static energy h_mp and total water r_tp. Parcels are not permitted to mix for some initial distance from their starting level (z_nomix = 800 m). Otherwise, once an air parcel has ascended to the next highest level and has had possible condensate removal associated with rain formation, it undergoes buoyancy gradient mixing (Bretherton and Smolarkiewicz 1989). Buoyancy gradient mixing is designed to damp changes in the vertical buoyancy gradient of an updraft and helps equilibrate convective updrafts and the background atmosphere toward a common density profile.

The updraft parcel buoyancy at level i is calculated using

At any model level i, one could define a buoyancy gradient with respect to the adjacent level at i ± 1. The model calculates the buoyancy gradient with respect to both levels and does backward and forward buoyancy gradient mixing. The backward mixing coefficient is given by

where b_p(i) is the parcel buoyancy at level i prior to condensate removal; b_p(i − 1) is the parcel buoyancy at the previous lower level, after mixing and condensate removal at that level; b_scale is a parameter that helps set the updraft buoyancy scale. Smaller values of b_scale increase updraft entrainment and detrainment and tend to increase the relative humidity of the background atmosphere. Here b_scale = 0.23 m s⁻².

The introduction of the n_launch/n_l ratio in Eq. (13) is intended to decrease the mixing of the most energetic updraft parcels (those with larger n_l). In the absence of this ratio, the parcels with largest initial h_mp will have the largest buoyancy, the largest buoyancy gradients, and the largest σ_back mixing coefficients. Buoyancy gradient mixing would then quickly narrow the spread in any initial h_mp updraft parcel spectrum so that all updraft parcels would evolve in a similar manner and detrain at a similar height. In the model, the introduction of this ratio is needed to give a spread in updraft outflow altitudes.

An updraft parcel entrains a mass Δm_d of dry air from the background atmosphere when σ_back > 0 (increasing buoyancy) and detrains a mass Δm_d of dry air into the background atmosphere when σ_back < 0 (decreasing buoyancy). The change in mass Δm_d is calculated from σ_back using

in which m_d,up refers to the dry mass of the updraft parcel before mixing. In the case of entrainment, the new mixed moist static energy and total water mass mixing ratio of the parcel are given by

The new T_p(i), r_υp(i), and r_lp(i) of the mixed updraft parcel can then be determined from h′_mp and r′_tp. In the case of detrainment, the dry mass of the updraft parcel is reduced but the parcel properties are otherwise unchanged.

The forward updraft buoyancy gradient mixing coefficient is given by

Here b_p(i) is the buoyancy of the updraft air parcel at level i after both condensate removal and backward buoyancy gradient mixing, and b_p(i + 1) is the buoyancy the parcel at level i would have after undergoing a “virtual” ascent to level i + 1.

The value of b_p(i + 1) in Eq. (17) is sensitive to the treatment of condensate in going from level i to level i + 1. In reversible ascent, all condensate is assumed to be retained. This assumption tends to reduce b_p(i + 1) and favor detrainment. Alternatively, one could assume that condensate is removed according to Eq. (11). This assumption would increase the buoyancy at i + 1 and favor entrainment. In practice, some compromise between the two assumptions appears to work best. For the purpose of calculating b_p(i + 1) in Eq. (17), the condensate loading at i + 1 is assumed to equal

Here r_l,max(i + 1) is the condensate loading at i + 1 calculated using Eq. (11), and r_l,rev(i + 1) is the reversible condensate loading at i + 1. The parameter f_for is used to interpolate between these assumptions and is set equal to 0.6.

There are clearly a number of ambiguities in the implementation of buoyancy gradient mixing. In the backward mixing calculation, one could calculate b_p(i) before or after condensate removal at that level. In the forward mixing calculation, one could calculate b_p(i) before or after backward mixing and make various assumptions on the condensate loading at level i + 1. These choices can have a significant impact on the model simulations and are resolved here mainly by a process of determining which choices generate the most realistic temperature and relative humidity profiles.

h. Geometry and mass loading of rain shafts

The model converts any cloud condensate in excess of a prescribed maximum, as given by Eq. (11), to rain. Because the production of rain occurs within model layers, rain is considered a half-level variable, that is, defined at the boundaries between model layers. It is initially assumed that all rain remains within the cloud. At each height, however, some fraction of the in-cloud rain is converted to out-of-cloud rain. The fractional rate of conversion per kilometer of in-cloud rain to out-of-cloud rain is referred to as f_rem and is specified as a function of temperature. The dashed curve in Fig. 2 shows the dependence of f_rem on altitude in the model. The rapid increase in f_rem near 0°C is intended to represent the exit of stratiform rain from the base of rainy stratiform anvils.

The rainwater mass (kg water m⁻²) that exits a cloud at a model level is equally divided into num_shaft rain shafts of different lengths. The mean rain shaft length is specified as a function of temperature and plotted against altitude in Fig. 2. It is assumed that there is a rapid increase in the vertical coherence of rain shafts near the melting level. At each level, the longest rain shaft is assumed to be dz_shaft times longer than the shortest. The lengths of the remaining rain shafts are geometrically distributed between the shortest and longest. In these model runs, num_shaft = 10 and dz_shaft = 6. The main reason for using a variety of rain shaft lengths at every exit level is to introduce additional variance into the downdraft parcel trajectories.

Rain shafts that exit a cloud are assumed to have a particular local rain rate (kg water m⁻² s⁻¹). The value of rn_rate is determined by the temperature of the model level at which the rain shaft exits the cloud. The dependence of rn_rate on the altitude of origin is shown in Fig. 2. The rain rate of a rain shaft is assumed to be constant as it falls toward the surface.

The terminal fall speed w_rain of a raindrop depends mainly on the raindrop size and the local atmospheric density (e.g., Fowler et al. 1996). The dependence of w_rain on height used in the model is shown in Fig. 1. This fall speed is intended to be a representative average for out-of-cloud rain. For constant raindrop size, one would expect w_rain to increase with altitude. Above the melting level, however, w_rain should decrease in response to a reduction in mean hydrometeor size. Realistic changes in the prescribed raindrop fall speed do not appear to have a significant effect on the model simulations.

The rainwater mass mixing ratio lp_rain(i) (kg rain kg⁻¹ dry air) of a rain shaft at model level i that exited a cloud at model level i_exit is given by

in which ρ_d (i) refers to the background dry air density at level i. Figure 1 shows the variation in rainwater mass mixing ratios of various rain shafts as a function of altitude. At a given height, the variation in lp_rain(i) is due to changes in the imposed rain rate at the different exit altitudes.

The starting water mass of a rain shaft can be expressed in terms of the total in-cloud rain at the exit level, the fractional removal rate per kilometer of the in-cloud rain at that exit level, the width of the exit grid cell, and the total number of rain shafts originating at that level:

This rain mass can be used to assign a dimensionless fractional area to every rain shaft:

The areal fraction of the grid cell occupied by a rain shaft is proportional to the mass of the rain shaft and inversely proportional to the rainwater mass mixing ratio and imposed length.

Rain shafts evaporate as they fall toward the ground. The mass of a rain shaft at some model level i for i < i_exit will therefore be less than mass_shaft(i_exit). For i < i_exit the rainwater mass mixing ratio lp_rain(i) in Eq. (21) is constrained by Eq. (19). Reductions in the mass of a rain shaft as it falls toward the surface must therefore be compensated by some reduction in the fractional area or length of the shaft. Here we assume that the fractional area of a rain shaft is fixed at its exit value. Evaporation of a rain shaft therefore gives rise to reductions in rain shaft length.

i. Downdraft parcel initiation and descent

At every exit level, the fractional area of each rain shaft defines a subcolumn of the background atmosphere. At levels at and below the exit level, this fractional area is used to define the dry mass entrained from the background atmosphere into a downdraft parcel:

The dry pressure increment dp_d (i) of layer i is defined in Eq. (7). A rain shaft is then subdivided into num_layers = 15 layers. Each of these rain shaft layers successively falls through the downdraft parcels in the subcolumn beneath it. The rate of change of water vapor mixing ratio r_υp(i) within each downdraft parcel due to evaporation from the rain shaft layer is determined using a parameterized expression (Emanuel 1991):

where p(i) refers to the local pressure (Pa), and r_sp(i) is the saturated water vapor mass mixing ratio of the downdraft parcel. The mass mixing ratio lp_rain(i) of a rain shaft layer is recalculated at every height as the layer falls to the surface using Eq. (19). The increase in water vapor mass mixing ratio for the downdraft parcels is given by

The time t_eυap over which the downdraft parcel is exposed to the rain shaft layer is given by

The layer thickness dz_layer(i) is calculated using an expression similar to Eq. (21) except that the rainwater mass of a layer, rather than an entire shaft, is used. The water mass of a layer is updated after every evaporation event, with dz_layer(i) reduced accordingly. A rain shaft layer may entirely evaporate as it falls toward the surface. Alternatively, some fraction of the initial mass of a rain shaft layer may reach the surface. The relative humidity of the downdraft parcels tends to increase as rain shaft layers sequentially pass through them. This gives rise to a reduction in the evaporation of subsequent layers so that the uppermost layers of a rain shaft tend to fall farther than those near the bottom of a rain shaft. For numerical reasons, the relative humidity of a downdraft is not allowed to reach 100% but is assigned a maximum value of 0.96.

After the evaporation associated with the passage of a rain shaft layer, the density temperature T_dp of a downdraft parcel is recalculated:

The downdraft parcel buoyancy b_p(i) is then determined using Eq. (12). Downdraft parcels are moved downward one level if their negative buoyancy is large enough to overcome the stability of the atmosphere. This is determined by first virtually advecting the parcel down one level and calculating buoyancy b_p(i − 1) at level i − 1. The average of the buoyancies at i and i − 1 is called the dynamical buoyancy:

Downdraft parcels are moved from level i to i − 1 if b_dyn < 0. The maximum number of evaporation events a downdraft may experience in a time step is num_layers. The maximum number of downward movements a downdraft parcel may experience within a time step is also equal to num_layers. If a downdraft parcel reaches the surface, it does not descend any further. It may, however, continue to be evaporatively cooled, if additional rain shaft layers pass through it, prior to detrainment into the bottom model level. Owing to differing vertical motions of downdraft parcels under a rain shaft, a rain shaft layer may evaporate into multiple downdraft parcels at a given model level, or none.

j. Brewer–Dobson circulation

Above 15 km, the Brewer–Dobson circulation plays an increasingly important role in regulating tropical temperature and chemical species profiles (Fueglistaler et al. 2009). The model imposes this circulation via a specified input of dry air between 400 and 200 hPa with the same mass of dry air removed from the model column between 40 and 5 hPa. Air being added or removed from the model is assumed to have the same water vapor mixing ratio as the ambient background profile. Because air is dehydrated as it rises up through the tropical tropopause, the imposition of the Brewer–Dobson circulation is associated with an external source of moisture to the model. This external moisture source is quite small, however, as the default Brewer–Dobson mass flux is specified as 27.3 Pa day⁻¹ (Rosenlof and Holton 1993), or approximately 100 times smaller than the overturning convective circulation. The Brewer–Dobson circulation is turned on in all simulations, except for those discussed in section 3m.

k. Calculation of the convective temperature and moisture tendencies

Updraft and downdraft parcels transport h_m and r_t vertically and mediate the exchange of h_m and r_t between the rain shafts and the background atmosphere. Briefly, the rates of change in the background temperature and water vapor mixing ratio profiles due to this vertical transport and mixing are calculated as follows.

It is assumed that there is no vertical flux of mass, h_m, or r_t through the top of the model. Entrainment and detrainment from updrafts and downdrafts occurring at a model level will add or remove moist static energy h_m, total water r_t, and dry mass m_d. The model calculates the residual (total) sum of each quantity. The net change in total mass (dry air plus water vapor) of the layer due to updraft and downdraft addition and removal can then be calculated. The pressure increment dp(i) and total mass of each layer are kept fixed. This constraint can be used to calculate the required flux of mass through the lower boundary of the layer. In the case of a need for an induced downward transport of air through the lower boundary of layer i, the r_υ and h_m of the subsiding air are assumed equal to the initial background values at layer i. Similarly, when there is an induced upward flux of air across the lower boundary of layer i from layer i − 1, the h_m and r_υ of the ascending air assume the values of their layer of origin. The model then calculates a new mixed total h_m, r_t, and m_d in layer i resulting from all horizontal updraft and downdraft inflows and outflows and vertical fluxes at the top and bottom boundaries. The new temperature and water vapor mixing ratio of the layer can then be calculated. The model successively works its way to the surface, using the solution at the lower boundary of layer i as an upper boundary condition for the solution of layer i − 1.

Note that, with the above procedure, all pressure levels of the model are fixed except the surface pressure p_half(1). While the total dry mass of the model is fixed, there will be a net reduction in column water when convection generates rainfall that reaches the surface. Within a time step, the associated reduction in surface pressure will be at least partially offset by a surface flux of water vapor and a small net amount of water vapor added to the model associated with the imposed Brewer–Dobson circulation. The surface pressure will converge to a constant value as radiative convective equilibrium is approached and the 24-h averaged column water becomes time independent. Within a 24-h period, the diurnal cycle in rainfall gives rise to a diurnal cycle in column water, surface evaporation, and surface pressure.

l. Moist static energy of rain

The total column moist static energy of all in-cloud and out-of-cloud rain is called H_m,rain (J m⁻²); H_m,rain is increased when the condensate r_l of an updraft parcel exceeds the local precipitation threshold r_l,max, and it is decreased during evaporation within downdrafts:

In this expression, T_p,up, z_p,up, r_l,up, and m_d,up refer to the updraft parcel temperature, height, liquid water mass mixing ratio, and dry mass (similarly for the downdraft parcels); Δr_d,dn refers to an increase in downdraft water vapor mixing ratio due to rainfall evaporation. The model does not attempt to explicitly convert the gravitational potential energy of rain to rain kinetic energy as it falls or dissipate the kinetic energy of the rain as heat. It also makes the assumption that, when evaporation occurs, the moist static energy of the rain is reduced as if the rain were at the same temperature and height as the downdraft air parcel. The model therefore makes no attempt to account for possible increases in downdraft cooling rates associated with increasing the temperature of the rain in a rain shaft to the temperature of the downdraft parcel through which it is falling. These simplifications in the moist static energy budget of the rain are not likely to have a significant effect on the simulations.

m. Enthalpy and water conservation

The moist static energy of a parcel is a sum of the moist enthalpy k_m and gravitational potential energy (Emanuel 1994):

Moist enthalpy is added to the model atmosphere via surface heat and moisture fluxes. It is removed from the model mainly via radiative cooling and precipitation. In the absence of mass transport across the boundaries, moist convection converts latent enthalpy into gravitational and internal energy but leaves the vertically integrated column moist enthalpy unchanged. The column moist enthalpy per unit area can be calculated by multiplying the moist enthalpy of every layer by the dry mass per unit area and summing over all model layers:

If the Brewer–Dobson circulation is turned off, the initial column enthalpy K_m,i prior to the convective portion of a time step should equal the sum of the column enthalpy K_m,f after convection plus the enthalpy lost from the model via transport of rain through the bottom level. The loss of enthalpy via rain exiting the column is equal to the rain moist static energy as calculated as in Eq. (28), with the understanding that the gravitational potential energy of the rain is converted to rain internal energy as it descends (i.e., increasing the effective temperature of the rain):

Within each model time step, there are several hundred events in which parcels are advected vertically, mixed, or exposed to evaporative cooling. The numerical routines used by the model enforce conservation of moist static energy and total water during each event and iteratively solve for the final temperature and water vapor mixing ratio. They are effectively forced to allow for some small inconsistency between the r_υ and r_s of saturated air parcels, the size of the inconsistency depending on the number of iterations. This inconsistency is preferable to the introduction of errors into conserved quantities.

Figure 3 shows the diurnal variation of some of the terms in the column and moist enthalpy budgets of the model during a simulation in which the Brewer–Dobson circulation is turned off. The residual column enthalpy error K_err of the model is ∼0.1 W m⁻². This is roughly equivalent to an error in column temperature of 0.001 K day⁻¹ (about 1000 times smaller than the convective tendencies). The loss of moist static energy from the model due to rain exiting the column during convection is ∼50 W m⁻². The net production of ∼25 W m⁻² of moist static energy during convection is due to an increase in column gravitational potential energy and is a reflection of the hydrostatic redefinition of z(i) during convection.

n. Buoyancy work

The model does not explicitly account for the dissipation of parcel kinetic energy. This source of enthalpy is, however, implicitly accounted for by the imposition of moist static energy conservation during parcel vertical displacements.

During an adiabatic process, there is a change dK in enthalpy associated with a pressure change dp:

Dividing both sides by the dry mass m_d gives

where k_m is the specific enthalpy per unit dry mass, and ρ_d = m_d/V is the dry air density. The total density can be expressed in terms of the dry air density or, alternatively, as a sum of a hydrostatic (i.e., background) density and a small deviation:

Using Eq. (34) for ρ_d in Eq. (33) gives

When δρ/ρ_hyd ≪ 1, and using the hydrostatic relation dp = −ρ_hydg dz,

Using Eq. (29), this can be written

where B_p = −δρ/ρ_hyd is the parcel buoyancy. Rather than using Eq. (37), the model imposes dh_m = 0 during the nonhydrostatic vertical displacement of updraft and downdraft air parcels. This is equivalent to adding a source of moist static energy to vertically displaced air parcels equal to the buoyancy work done on the air parcel, or effectively making the assumption that all parcel kinetic energy is immediately dissipated. For the purposes of the column enthalpy budget, it is therefore not necessary to explicitly calculate the kinetic energy of updraft and downdraft air parcels.

a. Updraft and downdraft mass fluxes

The model is typically run for 42 days to reach radiative convective equilibrium. Results are diurnally averaged over the last 10 days of a simulation. Figure 4 shows the updraft ω_up and downdraft ω_dn mass fluxes associated with the vertical movement of updraft and downdraft parcels between model levels. Figure 4 also shows the induced mass flux ω_ind. This is the mass transport between model levels required to keep the total mass between any two pressure half levels constant. In the real atmosphere, the induced descent (in the case of updrafts), or induced ascent (in the case of downdrafts), would occur within some horizontal distance of a convective event.

Figure 4 shows that there is a rapid increase in the downdraft mass flux below 6 km. This increase would contribute to the convergent inflow toward mesoscale convective systems near the melting level, as observed by aircraft radar (Mapes and Houze 1995). In the model, this increase is mainly due to imposed changes in the out-of-cloud rain rate and the imposed rain shaft geometry. In the upper troposphere, the out-of-cloud rain shafts have small vertical thickness and small rain rates. Rain shafts are distributed over a large horizontal fractional area, as are the downdraft parcels they entrain from the background atmosphere. The increase in downdraft water vapor mass mixing ratio r_υ due to the passage of a rain shaft layer through a downdraft parcel is proportional to the exposure time t_evap. The exposure times of updraft parcels exposed to vertically thin rain shaft layers (small dz_layer) are small. This reduces the net moistening and cooling of downdraft parcels exposed to rain shaft layers originating in the upper troposphere. As a result, the buoyancies of upper-tropospheric downdrafts are weak and rarely sufficient to overcome the stability of the atmosphere. Instead, downdraft parcels detrain into the background atmosphere at the same level as they were entrained. Evaporation of rain locally cools and moistens the atmosphere but does not give rise to a downdraft mass flux or any induced upward motion in the background atmosphere. This type of downdraft is referred to as “static.”

The model imposes rapid increases in the out-of-cloud rain rate and rain shaft length below 6 km. The evaporative cooling of rain shafts exiting clouds at these levels is focused into smaller fractional areas and, as shown in Fig. 5, generates downdraft parcel dynamical buoyancies that are negative. As a result, downdraft parcels move successively from one level to the next, advecting water vapor and other chemical tracers. This type of downdraft generates convective-scale downward motion and is referred to as “dynamic.”

Figure 2 shows that, below 6 km, there is also a rapid increase in the rate of conversion of in-cloud rain to out-of-cloud rain. Changes in this conversion rate affect the magnitude of the downdraft evaporative cooling but have less effect on whether a downdraft is static or dynamic.

Static downdraft parcels do not themselves move between model levels but can have an indirect influence on the movement of updraft parcels. In radiative convective equilibrium, the temperatures tendencies at any model level must sum to zero. Increased evaporative cooling at any level must be balanced by some combination of decreased radiative cooling or increased updraft heating. Increased updraft heating is ordinarily associated with increased updraft mass flux.

b. Heating rates and diabatic mass fluxes

Figure 6 shows the heating rates associated with clear-sky radiation, updrafts, and downdrafts. In radiative convective equilibrium, Q_r + Q_up + Q_dn = 0. Updraft heating refers to the sum of temperature changes arising from induced subsidence, the evaporation of detrained updraft condensate, and the detrainment of updraft parcels whose temperature is different from the background temperature at that level. Similar considerations apply to the net temperature tendency from downdrafts. The downdraft cooling increases rapidly below 6 km. This is mainly a response to the prescribed increase in the in-cloud rain removal rate f_rem.

The diabatic, or cross-isentropic, mass flux of a heat source is equal to the heating rate divided by the local static stability σ. Figure 7 shows the updraft, downdraft, and radiative diabatic mass fluxes of the model.

Dynamic downdrafts cool the lower troposphere mainly by induced ascent. In this case, the convective downdraft mass flux ω_dn is approximately equal to the diabatic downdraft mass flux Q_dn/σ. Figures 4 and 7 indicate that these mass fluxes are, indeed, similar below the melting level.

The radiative mass flux divergence is given by

Figure 8 shows the diabatic mass flux divergence profiles associated with each of the three heat sources in the model. In the model, the mass flux divergence of the deep outflow mode between 10 and 15 km is balanced by the radiative mass flux convergence. The divergence of the congestus mode is balanced mainly by the melting-level downdraft convergence. The radiative mass flux divergence is weak between 2 and 8 km. In the model and observations (Folkins et al. 2008), the weakness of the vertical variation in the radiative mass flux in this interval is partly due to a covariation between the static stability and clear-sky radiative heating profiles.

c. Temperature profile

Figure 9 shows the deviation ΔT of the model temperature profile from a tropical climatology constructed from 14 SHADOZ radiosonde locations in the 20°S–20°N latitude band (Folkins and Martin 2005). The difference ΔT from the climatology is ∼1 K below 13 km. The largest model errors occur near the tropopause. There is a ∼6 K cold bias at 15 km. As discussed in a later section, this discrepancy is probably due to the convective heating profile of the model not extending to a sufficiently high altitude.

The left panel of Fig. 9 shows two lapse rate climatologies: one from SHADOZ and the other from Koror, an island within the west Pacific warm pool. It was obtained from the Stratospheric Processes and their Role in Climate (SPARC) radiosonde archive. The model is too unstable in the upper troposphere. Below 10 km, the lapse rate profile of the model is in good agreement with observations.

The gray curve in Fig. 9 is the lapse rate of a moist pseudoadiabat starting from the surface with θ_e = 352 K. Both model and observations closely follow the pseudoadiabat from 6 to 10 km. The model also successfully reproduces the deviation of the observed lapse rate from the pseudoadiabat between 3 and 5 km (Mapes 2001; Folkins and Martin 2005). This deviation is referred to here as the melting-level stability anomaly (MLSA).

d. Relative humidity profile

Figure 10 is a comparison of the relative humidity profile generated by the model with several satellite and radiosonde climatologies. The model successfully reproduces the overall C-shaped profile of tropical relative humidity. Some of this agreement arises from tuning. The model most accurately reproduces the slope and magnitude of observed relative humidities in the upper troposphere (10–15 km) when the maximum cloud condensate, as prescribed by Eq. (11), is set equal to a small fraction of the ambient saturated water vapor mixing ratio. In this region, the relative humidity profile of the model reflects a first-order balance between detrainment moistening and subsidence drying, as suggested by a previous diagnostic model (Folkins and Martin 2005).

Between 6 and 10 km, updraft detrainment is much weaker. In the absence of the larger values of updraft condensate, shown in Fig. 1, the relative humidity of the model in this height interval would be far too low. The retention of updraft condensate within tropical updrafts, to altitudes as high as 9 km (∼−20°C), may reflect the inefficiency of rainfall production via collision and coalescence of water droplets below 9 km or the lack of ice condensation nuclei active at temperatures warmer than −20°C.

Below the melting level, the main source of moisture to the background atmosphere is updraft vapor detrainment. Downdrafts also increase the relative humidity of the lower troposphere through induced uplift and through the detrainment of parcels with higher relative humidity than the background value.

e. Comparison of cloud and rain properties with observations

Figure 1 shows that the updraft condensate in the model is roughly constant at 2.5 g kg⁻¹ up to 8 km and then declines rapidly with height. This imposed variation is in rough agreement with observations. Liquid water content measurements within convective updrafts from aircraft suggest values of 0.5–1 g kg⁻¹ up to 6 km followed by a rapid decrease with altitude (Jorgensen and LeMone 1989). Cloud water content measurements from CloudSat indicate larger values of 3–13 mg m⁻³ above the boundary layer, a peak in cloud water near 8 km, and a transition from liquid water to ice between 5 and 8 km (Su et al. 2008).

The out-of-cloud rain rate of the model is fixed at 100 mm day⁻¹ below the melting level. It then decreases rapidly with height above the melting level. Measurements of the stratiform rain rate from the Tropical Rainfall Measuring Mission (TRMM) show a similar variation with height, except that the average stratiform rain rate below the melting level is ∼50 mm day⁻¹ (Berg et al. 2002). Rain rates in the lower troposphere from convective precipitation are roughly 150 mm day⁻¹.

It is difficult to directly compare the imposed changes in rain shaft length with observations. However, this parameter has an effect on the amount of out-of-cloud rain. The evaporative moistening of longer rain shafts is focused over smaller areas, generates downdraft parcels of higher relative humidity, increases the likelihood of a rain shaft reaching the surface, and increases the out-of-cloud rain fraction. Figure 11 shows the vertical variation in the in-cloud and out-of-cloud rain. The ratio of out-of-cloud to in-cloud rain is roughly 1:3. Measurements from TRMM show that the proportion of stratiform to convective rain in the tropics between 2 and 4 km is roughly one to one (Takayabu 2002). This comparison suggests that the model underestimates out-of-cloud rain.

f. Diurnal cycle

The model simulations used a repetitive 24-h 1 June daily cycle at the equator. Due to solar heating during the day, atmospheric temperatures increase during the day and decrease at night. Figure 12 shows that the model has a ∼0.4 K diurnal variation in midtropospheric temperature, and a 1.6% variation in midtropospheric relative humidity (both averaged between 3 and 7 km). The diurnal variation in water vapor mixing ratio is small.

Over the oceans, tropical rainfall peaks at 0600 local time with an amplitude of 30% (Nesbitt and Zipser 2003). The black line in Fig. 12 shows the diurnal rainfall variation in the model. The timing of the early morning rainfall peak in the model is consistent with observations. However, the modeled diurnal amplitude of 18% is smaller than the observed amplitude.

a. Origin of the melting-level stability anomaly

Static downdrafts are those in which downdraft parcels are entrained from the background atmosphere, are moistened and evaporatively chilled by a rain shaft, and then detrain into the background atmosphere at the same level where they were entrained. For these types of downdrafts, it can be shown that the local cooling-to-moistening ratio of the background atmosphere associated with the downdraft is given by

Figure 13 shows the vertical variation of the downdraft cooling-to-moistening ratio in the model. Above 6 km, where downdrafts are static, the vertical variation in this ratio is due to the temperature dependence of the latent heat of vaporization. Below 6 km, dynamic downdrafts introduce a low-level vertically overturning circulation in which the downdraft dT/dr_υ ratio is no longer constrained by Eq. (39). The overall effect of downdrafts on the model becomes a nonlocal function of the downdraft parcel trajectories and the mean background r_υ and h_m profiles of the model. For example, dynamic downdrafts cool not only by the detrainment of evaporatively chilled air but also by induced ascent of the background atmosphere. Comparison of Figs. 4 and 13 indicates that the downdraft dT/dr_υ ratio tends to be reduced just below the melting level where downdrafts are convergent and increased below 4 km where downdrafts are divergent. The enhanced cooling just below the melting level destabilizes the tropical lower troposphere and brings the lapse rate profile of the model into better agreement with observations.

The melting level stability anomaly (MLSA) shown in Fig. 9 could also be reproduced using static downdrafts simply by tuning the vertical variation of the in-cloud rain removal rate f_rem to the observed stability profile. In this case, however, the relative humidity of the model near the melting level would be significantly larger than the observed climatology. In the model, simultaneous agreement with both observed relative humidity and stability profiles can only be obtained by a transition from static to dynamic downdrafts at the melting level in which there is a change in the downdraft dT/dr_υ ratio.

The amount of evaporation in the model is mainly controlled by the in-cloud rain shaft removal rate f_rem. The buoyancy of downdraft parcels, and in particular whether they are static or dynamic, is mainly determined by the out-of-cloud rain rate rn_rate and rain shaft length dz_rain. Above the melting level, changes in the vertical variation of rn_rate and dz_rain that do not generate dynamic downdrafts have no effect on the downdraft dT/dr_υ ratio. Such changes therefore also have no effect on the mean temperature or relative humidity profiles of the model. Similarly, below the melting level, where all downdrafts are to some degree dynamic, the effects of modest changes in rn_rate and dz_rain on the background temperature and relative humidity are small. The strategy adopted here was to impose a vertical variation in rn_rate that represented a rough compromise between observed stratiform and convective rain rates. The imposed vertical variation in dz_rain should be viewed as a particular mechanism in the model to force a static to dynamic downdraft transition at the melting level, rather than as an attempt to realistically represent an actual vertical variation in rain shaft geometry.

b. Origin of the congestus and deep outflow modes

Figure 14 shows a vertical profile of the updraft mixing coefficients σ_back and σ_for from one time step of a model run. At any given level, the signs of σ_for and σ_back may be opposite, so an updraft parcel may both entrain and detrain at the same altitude.

At the starting level, updraft parcels are initialized with an h_mp spectrum given by Eq. (10). As they rise in the atmosphere, the most energetic updraft parcels have the largest positive buoyancies, the largest buoyancy gradients, and therefore would tend to have the largest mixing coefficients. However, the updraft mixing coefficients given in Eq. (13) are also inversely proportional to the updraft index n_l so that, in practice, the most energetic updrafts have the smallest mixing coefficients.

The entrainment and detrainment of updraft parcels is governed by the local buoyancy gradient. Figure 9 shows that, between 2 and 4 km, the lapse rate of the model becomes progressively unstable with respect to the pseudoadiabat, which approximates the undilute density profile of the updrafts. In this height interval, the buoyancy of rising updraft parcels increases, so, as shown in Fig. 14, their mixing coefficients progressively favor entrainment. This tendency is reversed at 4 km where the modeled lapse rate profile starts to again approach the pseudoadiabat. As a result, the buoyancy of updraft parcels is reduced, their mixing coefficients become progressively more negative, and they preferentially detrain. The gray curve in Fig. 14 shows the updraft mass flux divergence −∂ω_up/∂p of the model. The congestus peak occurs at 6 km, near the top of the MLSA where the lapse rate of the model has rejoined the pseudoadiabat and detrainment is strongest.

Similar considerations apply to the deep outflow mode. The lapse rate of the model becomes increasingly stable with respect to the pseudoadiabat above 10 km. Updraft parcels detrain as they encounter this stability layer. This detrainment contributes to the large values of updraft divergence between 10 and 16 km.

Figure 15 shows the trajectories of updraft parcels in the moist static energy–mass (h_m − m_d,up) plane. The maximum number of parcels launched from the boundary layer during one time step (n_launch) is equal to 16. However, the four parcels with the smallest h_m do not have positive cape_d and are not subjected to an initial ascent. The eight parcel trajectories shown in black have larger initial h_m than the four trajectories shown in gray. All trajectories start at the lowest model level. Parcel h_m decreases in response to entrainment and loss of condensate (formation of precipitation). Parcel mass increases (decreases) in response to entrainment (detrainment). The mass of the most energetic parcels is virtually constant until several kilometers below their final detrainment height, indicating that their loss of moist static energy with altitude is almost entirely due to loss of condensate rather than mixing. The most energetic, but least massive, parcels detrain at highest final h_m and highest altitude. Reductions in parcel mass for h_m < 345 kJ are associated with the deep outflow mode. Reductions in parcel mass for h_m < 355 kJ are associated with the congestus mode.

Figure 14 shows that the congestus mode has a maximum divergence of ∼0.2 day⁻¹ and extends from 4 to 6.5 km. It is difficult to diagnostically infer the strength of the congestus mode from sounding observations. This is partly because, within a sounding array consisting of a mixture of cloud types, the divergence of the congestus mode can be offset by the melting-level convergence of rainy stratiform anvils. However, a sounding budget analysis near the Marshall Islands suggests an average peak divergence of 2 × 10⁻⁶ s⁻¹ (0.17 day⁻¹) near 500 hPa, with net divergence between 650 (3.5 km) and 400 hPa (7.5 km) (Schumacher et al. 2008).

c. Updraft mixing and the upper-tropospheric cold bias

In the tropics, convection is enhanced over areas of the ocean with high sea surface temperatures and over land. The spatial separation between areas of preferred convection and areas of preferred subsidence is associated with the Hadley–Walker circulation in which convection tends to occur under conditions of large-scale low-level convergence and upper-level divergence. On smaller spatial scales, within actively convecting regions, deep convection preferentially develops under conditions of enhanced low-level humidity (Sherwood 1999). The spatial and temporal variability of tropical convection can be expected to be associated with nonlinearities, which make it difficult to reproduce some aspects of the tropical mean state within the context of a one-dimensional steady-state model with no external forcing.

The model has little rainfall variance other than associated with the diurnal cycle. Deep convection therefore occurs in a background atmosphere that approximates the tropical mean. As a result, the entrainment of background air by updraft parcels reduces updraft moist static energy more rapidly than would occur within real deep-convective updrafts. The model attempts to compensate for the lack of lower-tropospheric moisture variance by reducing the mixing of the updraft parcels with the highest energy. If this were not done, deep convective outflow would still occur, but the variance in the updraft moist static energy and buoyancy spectrum would be much smaller, and the vertical width of the deep outflow profile shown in Fig. 8 would be reduced. It is likely, however, that despite the reduction in the mixing of the highest energy updraft parcels, the amount of very high convective outflow in the model is too weak. Diagnostic studies suggest that the updraft diabatic divergence profile extends to 17 km (Folkins and Martin 2005). However, Fig. 8 shows that the updraft diabatic divergence profile of the model is negative above 15 km. The lack of very high altitude convective outflow in the model contributes to the upper-tropospheric cold bias shown in Fig. 9.

d. Updraft buoyancies and the origin of the diurnal cycle

Figure 16 shows a mass-weighted probability distribution of the difference between the density temperature of the updraft parcels and the background atmosphere between 3 and 7 km. The density temperature incorporates the effects of changes in temperature, water vapor mixing ratio, and condensate loading on updraft buoyancy. In the model, the temperature excess ranges from −0.3 to 1.5 K. Observed values of virtual temperature excess within tropical updrafts range from −1.5 to 3.0 K (Jorgensen and LeMone 1989).

Figure 12 shows that the diurnal variation in midtropospheric temperatures of the model is roughly 0.4 K. As shown in Fig. 16, the temperature excess probability near zero buoyancy is roughly 6.0% per 0.1 K interval. An increase in midtropospheric temperatures of 0.4 K would therefore be expected to give rise to an approximately 24% reduction in midtropospheric mass flux in the absence of any other changes in updraft properties. This is similar to the modeled diurnal rainfall variation in the model of 18%. This comparison suggests that the ability of a model to replicate the diurnal rainfall cycle over the ocean may also be a test of its ability to replicate the observed updraft buoyancy distribution. Rainfall generated by weakly buoyant updrafts should be more sensitive to perturbations in background atmospheric temperatures associated with the diurnal cycle.