JAMES, VOL. ???, XXXX, DOI:10.1002/,

Predicting convective rainfall over tropical oceans from

environmental conditions

David J. Raymond1 and Marcos Flores1

Key points:• Rainfall in modeled tropical convection is pre-

dictable.• Stronger surface moist entropy flux increases rain-

fall.• Greater moist convective instability decreases

rainfall.

Abstract. A cloud resolving model in spectral weak temperature gradient mode is usedto explore systematically the response of mean convective rainfall to variations tropicalenvironmental conditions. A very large fraction of the variance in modeled rainfall is ex-plained by three variables, the surface moist entropy flux, the instability index (a mea-sure of low to mid-level moist convective instability), and the saturation fraction (a kindof column-averaged relative humidity). The results of these calculations are comparedwith the inferred rainfall from 37 case studies of convection over the tropical west Pa-cific, the tropical Atlantic, and the Caribbean, as well as in the NCEP FNL analysis andthe ERA-Interim reanalysis. The model shows significant predictive skill in all of thesecases. However, it consistently overpredicts precipitation by about a factor of three, duepossibly to simplifications made in the model. These calculations also show that the sat-uration fraction is not a predictor of rainfall in the case of strong convection. Instead,saturation fraction co-varies with the precipitation as a result of a moisture quasi-equilibriumprocess.

1. Introduction

Prediction of convective rainfall is one of the most im-portant functions of cumulus parameterizations in large-scale numerical models. The problem with many cumu-lus parameterizations is that their performance is diffi-cult to untangle from other effects in a large-scale model.However, this situation is changing with the employmentof various cloud-resolving numerical modeling techniques.Such models are not the real world, but results from superparameterization schemes, in which a small cloud modelis run in each large-scale grid box, are often better thanconventional cumulus parameterizations in their predic-tions, indicating that they have something to teach us[Khairoutdinov et al., 2005, 2008].

Cumulus parameterizations are generally constrainedto use only current, resolved-scale conditions availablefrom the encompassing large-scale model. These poten-tially include surface fluxes of heat, moisture, and mo-mentum, sea surface temperature (over oceans), radiativeforcing, and the vertical profiles of temperature, humid-ity, and wind.

Treatments of cumulus convection have traditionallybeen deterministic, in that the behavior of the convec-

1Physics Department and Geophysical Research Center,New Mexico Tech, Socorro, New Mexico, USA.

Copyright 2016 by the American Geophysical Union.

tion is completely specified by the resolved-scale condi-tions. However, there is increasing interest in incorporat-ing random fluctuations in parameterizations so as to re-produce the observed variability of convection and precip-itation [Palmer, 2001; Lin and Neelin, 2000, 2003; Plantand Craig, 2008; Khouider et al., 2010; Groenemeijer andCraig, 2012; Peters et al., 2013; Dorrestijn et al., 2015].

Non-determinate fluctuations in convection and rain-fall are an important part of the cumulus parameteri-zation problem. However, our goal in this paper is tounderstand the determinate part of these phenomena, asthis forms the mean state about which fluctuations takeplace.

There is a temptation to use gradient quantities suchas mass and moisture convergence to drive cumulus pa-rameterizations in models. However, this is likely to beproblematic, as these convergence-related variables are atleast partly a result of the convection itself rather thana cause [see, e.g., Raymond and Emanuel, 1993], at leastin the tropics where quasi-geostrophic forcing is minimal.Even in middle latitudes the total convergence is the sumof the large-scale balanced contribution (mostly quasi-geostrophic or boundary layer in origin) and convergenceinduced by the convection itself. Models such as wave-CISK (convective instability of the second kind), e.g.,Lindzen [1974] that drive parameterized convection withgradient quantities like mass or moisture convergence, of-ten exhibit an “ultraviolet catastrophe” in which instabil-ity growth rates asymptote to infinity for large wavenum-bers. Ooyama [1982] characterizes these as aliased mod-els of convective instability of the first kind in disguise.

1

X - 2 RAYMOND AND FLORES: PREDICTING CONVECTIVE RAINFALL

We therefore hypothesize that the average state of con-vection, and the precipitation in particular, is a functiononly of non-gradient environmental variables.

Ooyama [1982] further asserted that convection canonly be parameterized in terms of the balanced part ofthe environmental flow. This part of the flow is derivedfrom the potential vorticity distribution and generally hasan evolution time scale longer than that of the convectionitself, indicating that the arrow of causality points fromthe balanced flow to the convection. The unbalanced partof the flow is related in an intimate and complex mannerto the convection itself and it is impossible to say thatone causes the other – they co-evolve. This co-evolutionand the fact that the convection is turbulent means thatthe unbalanced part of the flow (including the convection)has a chaotic component. Only the statistical behavior ofthe convection can be predicted and this prediction mustcome from the balanced part of the environmental flow.

The particular mechanism of convective control ofmost interest to Ooyama [1982] was the “cooperative in-tensification mechanism” that occurs in the core of trop-ical storms. In this mechanism boundary layer frictionalconvergence drives convection, which in turn increases theboundary layer circulation via the hydrostatic decreasein surface pressure due to convective warming aloft. Thisonly acts reliably when the vorticity is strong enough tomake the Rossby radius comparable to the spatial scale ofa convective cell. In less extreme situations, the balancedflow may still control the statistical characteristics of theconvection, but the mechanism is likely to be different.As Ooyama [1982] points out, frictional convergence maystill exist in this case, but it is typically less effective thanother mechanisms, e.g., surface heat and moisture fluxes,in destabilizing the atmosphere to moist convection.

Raymond et al. [2015] proposed a model for the in-teraction between convection and balanced circulationsthat are not strong enough to activate Ooyama’s coopera-tive intensification mechanism. In this “thermodynamic”regime, which covers many if not most tropical distur-bances, convection is hypothesized to be forced primarilyby thermodynamic effects. A key diagnostic in this caseis the gross moist stability. A particular version of thisvariable called the normalized gross moist stability wasdefined by Raymond and Sessions [2007] and expandedupon by Raymond et al. [2009]. In the steady state, thisquantity becomes

Γ =TR(Fs −R)

L(P − E)(1)

where Fs is the surface moist entropy flux, E is the surfaceevaporation rate, R is the radiative sink of entropy inte-grated over the troposphere, and P is the precipitationrate. Normalizing constants are a reference temperatureTR and the latent heat of condensation L. Solving thisfor the net precipitation rate (precipitation minus evapo-ration) yields

P − E =TR(Fs −R)

LΓ. (2)

For (2) to be useful, a model is needed for the thenormalized gross moist stability. This comes from therealization that in a steady state Fs −R equals the verti-cally integrated lateral detainment of moist entropy. (The

irreversible generation of entropy is assumed to be negli-gible here, but can be added to the analysis if needed.)This detrainment rate is a function of the structure andareal density of convection; the smaller this detrainmentrate for a single convective cell, the larger is the numberof cells per unit area needed to balance the steady stateentropy budget.

Assuming positive Γ, the net precipitation rate is onlypositive when the surface entropy flux exceeds the radia-tive loss of entropy from the troposphere. Thus, strongsurface entropy flux is a prerequisite for intense mean pre-cipitation in this model. Furthermore, the effects of thesurface entropy flux are amplified by low values of thegross moist stability. Observations, theory, and model-ing have shown that the surface moist entropy flux exertsstrong control over the amount of deep convection andmean rainfall rates [Ooyama, 1969; Emanuel, 1986, 1987;Yano and Emanuel, 1991; Raymond, 1995; Raymond etal., 2003; Maloney and Sobel, 2004; Back and Bretherton,2005; Raymond and Zeng, 2005; Raymond et al., 2006;Raymond and Sessions, 2007; Sobel et al., 2009; Wangand Sobel, 2011, 2012]. This dependence is important forthe development of both tropical cyclones and intrasea-sonal oscillations.

Observations also show that small changes in the ver-tical profiles of temperature and humidity have dramaticeffects on the gross moist stability and hence the rain-fall rate over tropical oceans [Raymond et al., 2011; Ko-maromi, 2013; Gjorgjievska and Raymond, 2014; Sentic etal., 2015]. In particular, increased saturation fraction (akind of column-averaged relative humidity) and decreasedlow to mid-level moist convective instability (character-ized by the instability index) are associated with an in-crease in the mean convective precipitation rate in de-veloping tropical storms. In addition, decreased insta-bility is related to an increase in convective mass fluxesat low levels and a decrease aloft, i.e., it makes thesemass fluxes more bottom-heavy. Both of these effectsact by decreasing the gross moist stability, as illustratedby (2). Basically, this occurs by increasing the verticallyintegrated moisture convergence, which increases precip-itation in the steady state.

Weak temperature gradient (WTG) convective simu-lations [Sobel and Bretherton, 2000; Raymond and Zeng,2005; Raymond and Sessions, 2007; Wang et al., 2013;Herman and Raymond, 2014; Sessions et al., 2015; Senticet al., 2015] are in agreement with these observations,suggesting that this modeling technique is a useful toolin understanding deep convection over tropical oceans.

This paper reports on a systematic exploration of thesensitivity of time and space averaged precipitation tovariations in environmental conditions in convection sim-ulated using a WTG convective model. By taking anaverage, we exclude the chaotic variations in convectionresulting from small-scale turbulent dynamics and con-centrate on those properties of convection constrainedby balanced dynamics and thermodynamic conservationlaws. Thus, comparison of model precipitation with realworld values requires a large degree of smoothing of ob-servations.

An important aspect of any convective modeling ex-ercise is determining how to relate model results to thereal world. This is particularly tricky when questionsof causality arise. For instance, are the surface moistentropy fluxes, the instability index, and the saturationfraction imposed by the environment or are they artifacts

RAYMOND AND FLORES: PREDICTING CONVECTIVE RAINFALL X - 3

of the convection itself? Considerable effort is addressedto this issue here.

This investigation is preliminary in that certain simpli-fying assumptions are made, to be discussed later in thepaper. The results are nevertheless interesting in that themodel for precipitation rate resulting from these simula-tions exhibits significant skill in reproducing actual meanprecipitation rates via mechanisms observed in the realworld.

The experimental design of the study is reported insection 2 and the results of the modeling are presentedin section 3. A comparison with observations is given insection 4 and conclusions are presented and discussed insection 5.

2. Experiment design

The spectral weak temperature gradient (SWTG)model [Herman and Raymond, 2014] is used in two-dimensional mode with fixed radiation and sea surfacetemperature (SST). A radiative-convective equilibriumcalculation is first made to provide a base state aboutwhich perturbations in temperature and humidity pro-files, sea surface temperature, and imposed wind speedare made to produce a wide variety of reference profilesand surface conditions for the weak temperature gradi-ent calculations. The results are then categorized ac-cording to surface moist entropy flux, the instability in-dex, and the saturation fraction. These parameters havebeen shown to be important for precipitation production[Gjorgjievska and Raymond, 2014]. The instability indexII is defined

II = s∗lo − s∗hi, (3)

where s∗lo is the saturated moist entropy averaged overthe range [1, 3] km and s∗hi is this quantity averaged over[5, 7] km. It is a measure of low to mid-troposphere moistconvective instability. The saturation fraction SF is

SF =

∫rdp

/∫r∗dp (4)

where r is the mixing ratio and r∗ is the saturation mixingratio. It is a kind of column-averaged relative humidity.

2.1. SWTG model

WTG convective simulations relax the areally averagedconvective domain profile of potential temperature in acloud resolving model to a reference profile that is deemedto represent the environment surrounding the convection.This relaxation is assumed to be the result of gravitywave action that distributes buoyancy anomalies over alarge area [Bretherton and Smolarkiewicz, 1989]. Thetime scale τ for the relaxation is taken to be that of thegravity waves. A flaw in traditional WTG calculationsis that gravity waves of all vertical scales are assumed topropagate at the same speed. In the real world, gravitywaves of different vertical scales move at different speeds,resulting in multiple time scales.

In the SWTG model, the buoyancy anomaly in theconvective domain (i.e., the difference between the con-vective domain and reference profile potential tempera-tures) is subjected to a Fourier decomposition in the ver-tical. Each Fourier mode with a given vertical wavenum-ber mj = jπ/h is assigned a relaxation time proportional

to this wavenumber

τj = Lmj/N = jπL/(hN) j = 1, 2, 3, ... (5)

where h is the depth of the troposphere, N is the tropo-spheric Brunt-Vaisala frequency (assumed constant withheight), and L is an assumed horizontal length scale.Thus, shallower modes relax more slowly to the referenceprofile than deeper ones in agreement with the dynamicsof hydrostatic gravity waves. This spectral decomposi-tion yields a more physically realistic relaxation processthan is obtained by assuming that all vertical scales relaxat the same rate, as is done in classical WTG. The lengthscale L is assumed to be the distance traveled by a j = 1gravity wave over the convective life cycle time of order1 hr. We take L = 150 km in the present work.

Precipitating convection differs from other forms ofconvection in that the subsiding branch of the circula-tion has a much larger scale than the ascending branch.The SWTG vertical velocity is assumed to represent theascending branch of the convective circulation. It is givenby

wswtg(z) =∑j

Θj

τjsin(mjz) (6)

where

Θj =2

h

∫ h

0

θ′(z) sin(mjz)

dθ/dzdz. (7)

The quantity θ′(z) is the potential temperature anomaly,defined as the horizontal mean of potential temperaturein the convective domain θ(z) minus the reference profileθR(z), and Θj is the projection of θ′/(dθ/dz) onto the jthspectral mode.

Moisture and moist entropy are given horizontally uni-form source and sink terms in the convective domain con-sistent with the lateral entrainment and detrainment ofmass implied by wswtg, as described in Herman and Ray-mond [2014]. Reference mixing ratio and moist entropyprofiles rR(z) and sR(z) serve to specify the humidity andmoist entropy of the air being entrained into the convec-tive domain from the environment.

Wang and Sobel [2012] treat the effect of environmen-tal humidity on their WTG model by relaxing the con-vective domain toward a dry reference profile in order tomimic the effect of dry advection on the convection. The

Table 1. Important cloud model parameters.

parameter value commenteddy 1.0 coefficient for eddy mixingfilter 0.003 strength of horizontal 2*dx filterlambda 0.001 strength of vertical hypersmoothingcrain 0.01 s−1 rain production constantcsnow 0.01 s−1 ice production constantcevap 1.0 s−1 precipitation evaporation constantwtermw 5.0 m s−1 rain terminal velocitywtermi 5.0 m s−1 ice terminal velocityradcool 1.5 K d−1 radiative cooling ratecdrag 0.001 surface drag coefficientueffmin 3.0 m s−1 surface gustiness parameterhscale (L) 1.5 × 105 m SWTG horizontal scale

X - 4 RAYMOND AND FLORES: PREDICTING CONVECTIVE RAINFALL

philosophy taken in the present work is that the timeconstant for advective relaxation is much greater thanthe time constant of a convective cell. In this case theeffect of moisture advection is represented indirectly bythe actual reference profile and including an additionaldrying (or moistening) via this type of relaxation wouldbe redundant.

A two-dimensional version of the cloud model was usedto make possible a large number (≈ 100) of model runswith different environmental conditions. A domain of192 km in the horizontal by 20 km in the vertical wasused with a grid box size of 1000 m × 250 m. A spongelayer was used in the uppermost 5 km to damp gravitywave reflections and periodic lateral boundary conditionswere imposed.

Zero ambient wind in the x direction was specified andthe horizontal mean of the convective domain wind wasrelaxed toward this initial profile to suppress the effects ofcounter-gradient momentum transfer that tends to occurin two-dimensional models of convection.

Table 1 lists important cloud resolving model con-stants. See Herman and Raymond [2014] for more in-formation. Fixed radiation was used, with a constantradiative cooling rate of 1.5 K d−1 up to 12 km and alinear decay to zero at an elevation of 15 km.

The cloud physics parameterization is primitive by de-sign, allowing simple sensitivity tests to be made. Liquidand solid precipitation are treated equally with 5 m s−1

terminal velocities for both, so the cloud physics of snowthat is needed to produce stratiform rain areas is notcurrently included. A constant drag coefficient that isthe same for both momentum and thermodynamic trans-fers is also used in conjunction with constant gustinessparameter.

2.2. Model runs

Model runs were made in two dimensions on a192 km × 20 km domain with grid cells of dimensions

­1 ­0.5 0 0.5 1 1.5

amplitude

0

4000

8000

12000

16000

20000

he

igh

t (m

)

n = 0

n = 2

n = 4

Figure 1. Orthogonal basis functions for potential tem-perature and relative humidity reference profile pertur-bations.

1 km × 250 m. A long (107 s ≈ 116 d) radiative-convective equilibrium (RCE) integration of the modelwith an SST of 300 K and wind normal to the modelplane of vy = 5 m s−1 was done and initial RCE pro-files were obtained by averaging this output over the last8 × 106 s ≈ 93 d. This integration was itself initializedfrom a previous RCE run made under the same condi-tions, so the integration started out very close to RCE.These measures were needed to produce a very accurateinitial RCE profile, as small errors can produce deleteri-ous effects on subsequent SWTG runs.

Reference profiles for the SWTG calculations were ob-tained by perturbing the initial RCE temperature, hu-midity, and vy wind profiles as well as imposing a rangeof SSTs. (The x component of the reference wind is setto zero.) Values of vy = 0, 3, 5, 7, 10, 15, 20 m s−1 (inde-pendent of height) and SST = 299, 300, 301 K were used.Note that since the model is two-dimensional in the x−zplane and the Coriolis force is set to zero, The vy windcomponent has no effect on the model results except inthe calculation of the reference surface entropy fluxes.

The potential temperature and relative humidity ref-erence profiles were varied by imposing perturbationscomposed from basis functions consisting of the positivehalves of the n = 0, 2, 4 wave functions of the quantum

­5 0 5 10 15 20 25

imposed wind speed (m/s)

­2

­1

0

1

2

pote

ntial te

mp c

oeffs (

K)

T0 T2 T4

­5 0 5 10 15 20 25

­0.3

­0.2

­0.1

0

0.1

rela

tive h

um

idity c

oeffs

R0 R2 R4

Figure 2. Potential temperature and relative humiditycoefficients used in calculations with SST = 300 K.

RAYMOND AND FLORES: PREDICTING CONVECTIVE RAINFALL X - 5

mechanical harmonic oscillator [Schiff, 1955]. These func-tions are mutually orthogonal and are shown in figure 1.

Potential temperature and relative humidity pertur-bations were treated slightly differently. The potentialtemperature reference profiles θR(z) were obtained from

θR(z) = θ0(z) + T0M0 + T2M2 + T4M4 (8)

whereas the relative humidity reference profiles HR(z)were created using

HR(z) = H0(z)[1 +R0M0 +R2M2 +R4M4]. (9)

The quantities θ0(z) and H0(z) are the initial RCE pro-files of potential temperature and relative humidity, theMi(z) are the basis functions shown in figure 1, and theTi and Ri are coefficients determining the contributionsof each mode to the respective reference profiles. The rel-ative humidity perturbations are multiplicative to reducethe possibility of generating relative humidities less than0 or greater than 1.

The three basis functions used can represent only thecoarse vertical structure of the potential temperature andrelative humidity perturbations. However, experimentswith fine-scale perturbations of these quantities suggestthat they tend to be less important than the gross featuresof the profiles, at least if the perturbation amplitudes arerelatively small. In the tropics, one expects relativelyweak perturbations to the potential temperature profileto exist due to the tendency of buoyancy anomalies to besmoothed out by gravity wave action. The same may betrue of moisture profiles in the vicinity of deep convectiondue to a process of moisture quasi-equilibrium [Raymondet al., 2014], though not in convection-free regions.

The strategy used to minimize the parameter phasespace needing coverage is to assume that the rainfall rateis a linear function of the reference surface wind speedvy, the SST, and the thermodynamic profile coefficientsTi and Ri. Note that the surface evaporation rate andmoist entropy flux are implied by these variables.

The linearity assumption is likely to be true as longas the deviations from RCE are in some sense “small”,with the exception of cases in which the rainfall rate goesto zero. Other than that special case, what constitutes“small enough” is not known a priori and must be testeda posteriori. The linearity assumption allows parametersto be perturbed one at a time. However, since the as-sumed surface wind speeds cover a large range, we variedall of the other parameters for each value of wind speed.Figure 2 shows the values assumed for Ti and Ri as afunction of wind speed for the SST equal to 300 K. For299 K and 301 K we ran only the cases with Ti, Ri = 0.

SWTG calculations were run for 5 × 106 s, averagingover the last 3×106 s for all members of the above-definedparameter space. This long (≈ 35 d) averaging periodresults in accurate estimates of the average rainfall rateproduced for each simulation.

3. Results

The main purpose of this paper is to determine howenvironmental factors control space and time averagedprecipitation. The word control implies determination ofcausality. The reference profiles of temperature, humid-ity, and wind, SST, and in the present case radiation, areexternally imposed, and therefore ought to control the

behavior of the model. However, there are limits to thiscontrol.

The small space and time scale details of convection,and hence rainfall, are turbulent and hence unpredictable.However, in most cases the long-term time and space av-erages of convective behavior are uniquely determined bythe reference conditions, which can therefore be thoughtof as controlling average convective behavior.

An exception occurs where multiple equilibria exist insimulated convection, generally with two different equi-librium states, a moist, rainy equilibrium and a state oflow humidity and no rain [Sobel et al., 2007; Raymondet al., 2009; Sessions et al., 2010]. Using the traditionalWTG model, Sessions et al. [2015] showed that fixed ra-diation produces multiple equilibria over a smaller rangeof input parameters than interactive radiation. In addi-tion, our experience is that SWTG is even less susceptibleto multiple equilibria than traditional WTG for fixed ra-diation, so multiple equilibria are likely to be of little orno importance in the present results.

Based on previous work on tropical cyclone formation,e.g., Gjorgjievska and Raymond [2014], the dimensional-ity of the causal parameter space is vastly reduced by as-suming that the saturation fraction and instability index(both defined in section 2) characterize the temperatureand humidity profiles.

Other likely causal parameters are the surface evapo-ration rate and moist entropy flux as well as the SST.Variations of ±1 K in the SST account for a very smallfraction of the variance in the rainfall rate independent oftheir contribution to variations in surface thermodynamic

­20 0 20 40 60 80 100

predicted rainfall rate (mm/d)

0

20

40

60

80

100

actu

al ra

infa

ll ra

te (

mm

/d) R^2 = 0.96

Predictors: reference eflux, II, SF

Figure 3. Predicted vs. actual model rainfall based on alinear regression for the predicted rain with the referencevalues of surface moist entropy flux, instability index, andsaturation fraction as the predictor variables. 96% of thevariance is accounted for.

X - 6 RAYMOND AND FLORES: PREDICTING CONVECTIVE RAINFALL

fluxes in the present set of simulations. Furthermore, thesurface moist entropy flux and the surface evaporationrate are correlated at the 99% level, and are thereforenot independent. These two factors allow us to furthercollapse the parameter space by replacing the SST andthe surface wind speed by a single parameter, which wechoose to be the surface moist entropy flux. Thus, wehypothesize that the reference values of surface moist en-tropy flux (eflux), instability index (II), and saturationfraction (SF) are sufficient to characterize the time andspace averaged rainfall rate in the model.

Figure 3 shows a scatter plot of the predicted vs. theactual rainfall rate in the ensemble of runs with rainfallexceeding 1 mm d−1. The predicted rainfall is based on alinear regression with the above reference variables. Non-precipitating cases are excluded from the regression butincluded in the plot. The prediction equation for rainfallbased on this regression is

RP = −251 + 45.8 eflux − 1.02 II + 362 SF (10)

where the predicted rainfall RP is in millimeters per dayand the other quantities are in standard SI units.

The surface entropy flux by itself explains 69% of thevariance, while adding in the instability index results in84% explained. The saturation fraction raises the totalto 96%. Thus, these three reference variables account fora rather large fraction of average rain production in themodel.

Averages of these three parameters over the convectivedomain are not necessarily equal to the corresponding ref-erence profile values. We now explore how reference andconvective domain values compare with each other.

Figure 4 shows a scatter plot of reference vs. convec-tive domain surface moist entropy flux over the ensem-ble of model runs. The convective domain values are

0 0.5 1 1.5 2 2.5

reference entropy flux (J/K/m^2/s)

0

0.5

1

1.5

2

2.5

conv d

om

ain

entr

opy flu

x (

J/K

/m^2

/s)

rain < 0.01 mm/d

rain > 0.01 mm/d

Figure 4. Relationship between reference and convec-tive domain values of the surface moist entropy flux. Pre-cipitating and non-precipitating cases are distinguishedby red dots and blue triangles respectively.

generally larger than the reference profile values by ap-proximately 15%, especially when precipitation is occur-ring. This is presumably due to the fact that gustinessproduced by the convection can increase the fluxes overthose present in the undisturbed convective environment.Jabouille et al. [1996] found that convective gustinessproduced the strongest anomalies in surface latent heatfluxes in light wind situations, with values up to 50 m s−1

(≈ 0.17 J K−1 m−2 s−1 in terms of the moist entropyflux). Given other possible factors in our analysis, thisis a relatively minor effect. The non-precipitating casestended to have small values of the surface entropy fluxand large values of the instability index.

Figure 5 shows the convective domain vs. the referenceprofile instability indices. There is some random scatterbetween the two, though the main effect is a compres-sion in the range of convective domain values relative toreference values.

The situation for saturation fraction is somewhat morecomplicated. Figure 6 shows a scatter plot of referencevs. convective domain saturation fraction. Though thereis weak correlation between the two quantities for pre-cipitating cases, in the absence of precipitation there isbasically no correlation at all. Furthermore, the spread inconvective domain values is large compared to the spreadin reference values. This suggests that convective andradiative processes have large effects on the convectivedomain saturation fraction irrespective of the referencevalues.

Figure 7 shows plots of rainfall rate vs. reference andconvective domain saturation fraction. The latter ex-hibits a “hockey stick” curve with rapidly increasing rain-fall above a saturation fraction of about 0.7, as found inobservations by Bretherton et al. [2004] and many others.

0 5 10 15 20 25 30

reference instability index (J/K/kg)

0

5

10

15

20

25

30

co

nv d

om

ain

in

sta

bili

ty in

de

x (

J/K

/kg

)

rain < 0.01 mm/d

rain > 0.01 mm/d

Figure 5. Relationship between reference profile andconvective domain values of instability index. Precipi-tating and non-precipitating cases are distinguished byred dots and blue triangles respectively.

RAYMOND AND FLORES: PREDICTING CONVECTIVE RAINFALL X - 7

In contrast, the reference profile saturation fraction val-ues are limited to the range 0.65-0.75 with only a slightcorrelation with rainfall rate.

Figure 8 shows that convective domain saturation frac-tion and instability index are strongly anti-correlated forlarge values of the rainfall rate. This is in agreement withobservations of convection in potential precursor distur-

0 0.2 0.4 0.6 0.8 1

reference saturation fraction

0

0.2

0.4

0.6

0.8

1

conv d

om

ain

satu

ration fra

ction

rain < 0.01 mm/d

rain > 0.01 mm/d

Figure 6. Relationship between reference profile andconvective domain values of saturation fraction. Precip-itating and non-precipitating cases are distinguished byred dots and blue triangles respectively.

0.5 0.6 0.7 0.8

saturation fraction

0

10

20

30

40

50

60

70

80

90

rain

fall

rate

(m

m/d

)

reference SF

conv domain SF

Figure 7. Relationship between reference (blue trian-gles) and convective domain values (red dots) of satura-tion fraction and rainfall.

bances for tropical cyclones [Gjorgjievska and Raymond,2014]. We ascribe these results to a “moisture quasi-equilibrium” process as outlined by Singh and O’Gorman[2013]. If the environment is dry, then convection re-quires larger values of parcel buoyancy to withstand en-vironmental entrainment and evaporation of condensatethan when the environment is more moist. Larger parcelbuoyancy corresponds to larger instability index. If theenvironment is too dry to sustain convection for a givenvalue of instability index, then clouds detrain their mois-ture and decay, thus moistening the environment. On theother hand, if the environment is too moist, then con-vection is more vigorous and the resulting precipitationdries out the atmospheric column. The net effect is therelaxation of the humidity profile to some optimum valuebetween these two extremes. The optimum humidity be-comes drier as the instability becomes greater.

Modeling of convection by Singh and O’Gorman(2013) shows that convection adjusts the model environ-ment to produce very close to zero buoyancy for sub-sequent convection. In a model this could result fromthe convection altering either the environmental humid-ity or the moist convective instability. However, in thereal world the environment is continually adjusting toproduce a balanced temperature profile and associatedvalue of the instability index. Thus, on time scales longerthan the dynamical adjustment time scale, the convec-tion controls only the humidity since larger-scale dynam-ics regulates the convective instability. Greater instabil-ity therefore yields a drier atmosphere whereas less in-stability results in higher humidities. This explains theinverse relationship between instability index and satura-tion fraction seen for large rainfall rates in figure 8 andin observational results.

10 15 20 25

conv domain instability index (J/K/g)

0.5

0.6

0.7

0.8

0.9

conv d

om

ain

satu

ration fra

ction

rain > 10 mm/d

rain < 10 mm/d

Figure 8. Scatter plot of convective domain satura-tion fraction and instability index for the simulationsdiscussed in this paper. The cases are separated intotwo categories, those with rainfall rates greater than10 mm d−1 and those with less than this value.

X - 8 RAYMOND AND FLORES: PREDICTING CONVECTIVE RAINFALL

Raymond (2000) developed a simple model of whatis now called moisture quasi-equilibrium. The moisturerelaxation time in the case of weak convection character-istic of the radiative-convective equilibrium state is a fewweeks. However, this relaxation time scales as the inversesquare of the convective strength, as measured by the av-erage precipitation rate. For very strong convection itdecreases to less than one day. This explains why mois-ture quasi-equilibrium only holds for large rainfall ratesin figure 8.

If the convection is weak, then the above analysissuggests that the saturation fraction is an independentpredictor variable for precipitation. As the convectionbecomes stronger, then the saturation fraction becomesmore closely tied with the instability index. In this case,the saturation fraction becomes redundant as a predictor.

We now test this hypothesis with respect to our en-semble of model runs. Figure 9 shows that 91% of thevariance in the rainfall is predicted by the surface moistentropy flux and the instability index. This almost ashigh as the regression results for reference values of sur-face entropy flux, instability index, and saturation frac-tion shown in figure 3. However, there is a tendency fora few non-precipitating cases to exhibit large predictedvalues of precipitation. (As with reference value test,non-precipitating cases are omitted from the regression,though they are still displayed.) The predicted rain interms of the surface moist entropy flux and instabilityindex is

RP = 46 + 38.8 eflux − 3.49 II. (11)

A regression with just surface moist entropy flux re-sults in 79% of the variance explained, which means that

­40 ­20 0 20 40 60 80 100

predicted rainfall rate (mm/d)

0

20

40

60

80

100

actu

al ra

infa

ll ra

te (

mm

/d) R^2 = 0.91

Predictors: convective domain eflux, II

Figure 9. As in figure 3 with convective domain surfacemoist entropy flux and instability index taken as predic-tor variables. 91% of the variance in rainfall is explainedby the regression.

instability index adds significant information. On theother hand, adding the model domain saturation frac-tion to the list of predictor variables results in little newinformation (92% of variance explained) in comparisonto just the surface flux and instability index (see figure10). The main effect of the saturation fraction is in tidy-ing up the results for low rainfall rates. The cases withactual zero rain now take on negative predicted rainfall.These results are in accord with the hypothesis that sat-uration fraction is only an independent predictor at lowrainfall rates where the moisture relaxation time constantis large. The rain predicted by the regression is given by

RP = −28 + 36.4 eflux − 2.67 II + 83.3 SF. (12)

4. Comparison with observations

Comparing SWTG results with observations is tricky,as one must decide whether reference values of entropyflux, instability index, and saturation fraction or the cor-responding convective domain values should be used inthe comparison. In principle, the former should be usedif the observations are of the clear regions surroundingthe convection, while the latter would be more appropri-ate if detailed observations in the immediate vicinity ofthe convection were available.

If one is concerned about understanding the causes ofconvection, then the focus should be on reference val-ues, since these are unambiguously causal in the SWTGmodel; the convective domain values are significantly af-fected by the convection itself, especially the saturationfraction. The control of convection over the instability

­40 ­20 0 20 40 60 80 100

predicted rainfall rate (mm/d)

0

20

40

60

80

100

actu

al ra

infa

ll ra

te (

mm

/d) R^2 = 0.92

Predictors: convective domain eflux, II, SF

Figure 10. As in figure 9 except with the convective do-main saturation fraction added as an independent vari-able. 92% of the variance is explained.

RAYMOND AND FLORES: PREDICTING CONVECTIVE RAINFALL X - 9

index and the surface entropy flux is weaker but still sig-nificant.

Two comparisions are made below, one with the anal-ysis of mesoscale dropsonde observations of tropical con-vective disturbances, the other with the results of globalanalyses.

4.1. TCS08 and PREDICT

In this section we compare the predictions derivedfrom the model to observations of convection in 37 casesof tropical disturbances in the western Pacific [Tropi-cal Cyclone Structure experiment (TCS08); Elsberry andHarr, 2008] and the western Atlantic and Caribbean [Pre-Depression Investigation of Cloud-Systems in the Tropics(PREDICT); Montgomery et al., 2012]. These observa-tions are taken from dropsonde grids over domains a fewdegrees in diameter with resolution of order one degreeand are described by Gjorgjievska and Raymond [2014].In these cases rainfall was not actually measured. Wetherefore use observed moisture convergence plus surfaceevaporation rate, averaged over a box typically a few de-grees in diameter, as a proxy for rainfall, which for con-venience we simply call “rainfall”. This proxy omits theeffect of the time tendency of precipitable water in themoisture budget, thus introducing noise. Additional noisecomes from the fact that the instantaneous rainfall rateitself is noisy relative to the long-term average rainfallextracted from the model. For this reason, the very highvalues of explained variance in the model results cannotbe expected in the application of the model predictionsof rainfall to the observational results.

Examination of fields of instability index and satura-tion fraction in the observational results shows extremevariability in these quantities over scales of 10-100 km.In many cases the averages of these quantities over the

­20 0 20 40 60 80 100 120

predicted rain (mm/d)

­20

0

20

40

60

80

actu

al ra

in (

mm

/d)

SL = 0.304

IC = 11.7

R^2 = 0.23

F = 10.2

Reference: eflux, II, SF

Figure 11. Actual rainfall from TCS08 and PREDICTplotted against the observed rainfall using the modelbased on reference values of parameters. SL and IC rep-resent the slope and y intercept of the blue regressionline while R2 and F represent the fraction of varianceaccounted for by the regression and the F statistic.

observed domain are dominated by the environment sur-

rounding the convection of interest. This supports the

use of the precipitation model based on reference rather

than convective domain values, though it is admittedly

difficult to make a clean separation between near and far

convective environment in the observations.

­20 ­10 0 10 20 30 40 50 60

Predicted Rain (mm/d)

­20

­10

0

10

20

30

40

50

60

Actu

al R

ain

(m

m/d

)

0

5

10

15

20

East Pacific FNL

Figure 12. Number density distribution of FNL valuesof the “actual” rainfall (moisture convergence plus sur-face evaporation) vs. the predicted rainfall from (10) forJune-September 2015 in the east Pacific ([110-90 W], [8-13 N]). The blue line indicates the weighted average ofthe actual rainfall as a function of the predicted rainfall.

­20 ­10 0 10 20 30 40 50 60

Predicted Rain (mm/d)

­20

­10

0

10

20

30

40

50

60

Actu

al R

ain

(m

m/d

)

0

30

60

90

120

150

West Pacific FNL

Figure 13. As in figure 12 except for the northwestPacific ([130-180 E], [0-23 N]).

X - 10 RAYMOND AND FLORES: PREDICTING CONVECTIVE RAINFALL

In order to evaluate the results of our precipitationpredictions against these observations, we compute a fur-ther regression between observed rainfall and rainfall pre-dicted from the observationally derived parameters usingthe reference profile model given by (10). Figure 11 showsactual rainfall as a function of model-predicted rainfall inthis case. 23% of the variance in observed rainfall is ex-plained by this model. The value of the F statistic is 10.2,which means that the correlation is significant at greaterthan the 99% level.

The rainfall prediction model therefore shows skill inpredicting rainfall for the TCS08-PREDICT cases. How-ever, the prediction has offset and scaling errors. Theoffset error can be easily ascribed to differences in thethermodynamics of the model compared to the real world.However the scaling error results in an over-prediction ofprecipitation by a factor of 3 once the offset error is ac-counted for. The error may be a result of simplificationsin the model such as two-dimensionality and fixed radia-tion. This bears further investigation.

The rainfall prediction using convective domain valuesof entropy flux, instability index, and saturation fractionwas also tested against this data set (not shown). The re-sults are similar to the reference profile case, though theskill is less, with 17% of the rainfall variance accountedfor. The offset and scaling errors are greater, with a scal-ing overprediction by a factor of 4. These results supportthe use of the rainfall prediction model based on referenceparameters, though extra skill exhibited in that case issmall.

4.2. Global analyses

We now test our precipitation prediction algorithmagainst the National Centers for Environmental Predic-tion’s Final Operational Global Analysis data set (FNL;http://rda.ucar.edu/datasets/ds083.2/). Since theFNL does not provide precipitation, we use the sameprecipitation proxy as in the TCS08-PREDICT dataset (“actual rainfall”), namely the vertically integratedmoisture convergence plus the surface evaporation rate.The reference parameter precipitation prediction equa-tion (10) is also used here.

Figure 12 shows the number density distribution ofvalues of the actual vs. predicted rainfall derived fromthe FNL for the summertime tropical east Pacific at andnorth of the intertropical convergence zone. The blueline shows the distribution-weighted average of the ac-tual rainfall as a function of predicted rainfall. A similarplot for the northern summer northwest Pacific is shownin figure 13.

The two cases are very similar, though the smaller datasample in the east Pacific case makes the number distri-bution noisier. In both cases the average actual rain islarger than the peak in the distribution because the dis-tributions are skewed toward larger values of actual rain-fall. The average lines in both cases are nearly linear, butwith offset and scaling errors. The offsets are small andthe scaling errors imply predicted values of order 3 timesactual values in both cases as in the TCS08-PREDICT re-sults. Similar results are obtained from the ERA-Interimreanalysis [Dee et al., 2011] (not shown).

5. Discussion and conclusions

The spectral version of our cloud resolving model witha weak temperature gradient parameterization of thelarge-scale flow is used to evaluate the environmental con-ditions that control time and space-averaged convective

precipitation rates over tropical oceans. Over 100 two-dimensional simulations with fixed radiative cooling areused to span the range of variations in environmentalconditions that is likely to be found over warm tropi-cal oceans. The first step is to calculate closed domainradiative-convective equilibrium profiles of temperatureand humidity at fixed values of SST and imposed surfacewind speed. The second step is to use these thermody-namic profiles and this SST as reference conditions inweak temperature gradient calculations over the range ofimposed imposed winds of [0-20] m s−1. For each of thesewind values, further calculations are then made in whichthe thermodynamic profiles and SST are varied system-atically.

Even with this large number of simulations, filling theparameter space is only possible if certain assumptionsare made. The first is that temperature and humidityperturbations about the previously calculated radiative-convective equilibrium state are assumed to have a coarsevertical structure, i.e., with only 3 degrees of freedomeach. A posteriori tests suggest that perturbations withfine vertical structure are not important as long as theiramplitude is not too large. An additional critical assump-tion is that the imposed deviations are small enough thatresulting changes in the rainfall are linear functions of thedeviations. This allows us to explore the effect of chang-ing parameters one at a time, which results in a furtherlarge reduction in the size of the parameter space. Testsindicate that this assumption is largely justified, giventhe size of perturbations employed.

Observational results suggest that an additional reduc-tion in the parameter space is possible. The saturationfraction, which is essentially a column-averaged relativehumidity, has a close relationship to precipitation overwarm tropical oceans. The instability index, which mea-sures lower tropospheric moist convective instability, isclosely related to first baroclinic mode temperature per-turbations. Variations in SST and imposed surface windspeed collapse into variations in the surface evaporationrate and surface moist entropy flux. Furthermore, overtropical oceans, these two surface fluxes co-vary to a highdegree of accuracy. This leaves us with three importantparameters that potentially control precipitation, one ofthe two surface fluxes, which we choose to be the moistentropy flux, the instability index, and the saturationfraction.

We next address causality. Within the confines ofthe model, the values of entropy flux, instability index,and saturation fraction derived from reference thermody-namic and wind profiles are unambiguously causal; vari-ations in these externally imposed parameters explain96% of the variance in the precipitation over the rangeof model simulations used in the analysis. Convectivedomain values of the above parameters are affected bythe convection itself and to varying degrees co-vary withthe precipitation and are therefore not causal.

Comparison of our results with observations from twofield programs and with the FNL analysis and ERA-Interim reanalysis suggests that a linear precipitationmodel based on the reference values of surface moist en-tropy flux, instability index, and saturation fraction ex-hibits significant skill in predicting actual mean rainfall.However, the prediction has offset and scaling errors, the

RAYMOND AND FLORES: PREDICTING CONVECTIVE RAINFALL X - 11

most significant of which being the over-prediction of av-erage rain by roughly a factor of 3. This factor is consis-tent across the different observational cases. Simplifyingassumptions made in the model simulations, e.g., two-dimensionality and fixed radiation, could be responsiblefor this discrepancy. Further work is needed here.

Differences between reference and convective domainvalues of our parameters are enlightening. In our mod-eling results, the convective domain values of the surfacemoist entropy flux tend to be modestly larger than refer-ence values.

Somewhat more variability exists between referenceand convective domain values of the instability index;convective domain values tend to be larger than refer-ence values, especially for small reference values. Thetendency for gravity wave action to homogenize temper-ature profiles in the tropics also results in a tendency tohomogenize the instability index, since it depends solelyon the temperature profile. Thus, reference and convec-tive domain values of this parameter should not differ toomuch from each other in most cases.

The situation with saturation fraction is altogether dif-ferent. No large-scale restorative processes exist for watervapor and there is very little correlation between the ref-erence and convective domain values of the saturationfraction. There is, however, a strong negative correlationbetween saturation fraction and instability index that re-flects a convective process referred to as moisture quasi-equilibrium; moister atmospheres in convective regionsare more stable and vice versa. Therefore, convective do-main saturation fraction is not a predictive variable forprecipitation in regions of strong convection; it simply co-varies with the precipitation itself and in some ways canbe considered a surrogate for precipitation.

Moisture quasi-equilibrium breaks down when convec-tion is weak or non-existent. In this case saturation frac-tion becomes an important predictive parameter; if theenvironment is too dry, precipitation cannot occur nomatter how favorable the other predictive parameters are.

Since saturation fraction co-varies with instability in-dex in strong convection and becomes an independentpredictive parameter in weak convection, there is no harmin including it as a predictive parameter for precipita-tion. As a comparison of figures 9 and 10 shows, its in-clusion does basically nothing in heavy rain and plays theconstructive role of eliminating predictions of heavy rainwhen the actual rain is light or non-existent.

We conclude that the average rainfall in a weak tem-perature gradient convective model can be predicted bya combination of externally imposed values of only threeparameters, the surface moist entropy flux, the instabil-ity index, and the saturation fraction. Furthermore, theequation thus derived exhibits skill in predicting averageconvective rainfall over tropical oceans, though scalingand offset errors must be accounted for to make thesepredictions quantitative. Further work is needed to trackdown the source of these errors and to extend compar-isons to a broader range of observational cases.

Acknowledgments. Version entropy-029 of the cloudmodel used in these calculations is available at http://kestrel.nmt.edu/~raymond/tools.html. Reduced model out-put and observational data used in this paper are available onrequest from the author. Thanks to Zeljka Fuchs, Sharon Ses-sions, Stipo Sentic, and Michael Herman for useful commentson the paper. The comments of Ji Nie and two anonymous

reviewers greatly improved the paper. This work supportedby National Science Foundation Grant 1342001.

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Corresponding author: D. J. Raymond, Physics Depart-ment and Geophysical Research Center, New Mexico Tech,Socorro, NM 87801, USA. (raymond@kestrel.nmt.edu)