weighted essentially non-oscillatory scheme for cloud edge problem

Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 139: 1374–1388, July 2013 A

Weighted essentially non-oscillatory scheme forcloud edge problem

Yuya Baba*and Keiko TakahashiEarth Simulator Center, Japan Agency for Marine-Earth Science and Technology, Yokohama, Japan

*Correspondence to: Y. Baba, Earth Simulator Center, JAMSTEC, 3173-25 Showa-machi, Kanazawa-ku, Yokohama236-001, Japan. E-mail:[email protected]

The weighted essentially non-oscillatory (WENO) scheme is applied to a cloud-resolving model and is used for the cloud edge problem. Validity is tested usingthree idealized experiments and the results are compared with those of recent flux-corrected transport (FCT) schemes. In all three experiments, the WENO schemesimulates the energy properties of cloud edges better than the FCT schemes do.Because the WENO scheme can capture cloud edge more sharply, the buoyancy ofthe cloud edge is simulated without numerical diffusion. The increased buoyancyalso causes enhancement of the convection cycle in deep convection through thecold pool, which is formed below the convective cell. This fact indicates thatunderestimations of convection strengths can be avoided using the WENO scheme.

Key Words: cloud edge problem; advection scheme; cloud-resolving model

Received 1 February 2012; Revised 24 July 2012; Accepted 31 July 2012; Published online in Wiley Online Library30 October 2012

Citation: Baba Y, Takahashi K. 2013. Weighted essentially non-oscillatory scheme for cloud edge problem.Q. J. R. Meteorol. Soc. 139: 1374–1388. DOI:10.1002/qj.2030

1. Introduction

The cloud-resolving model (CRM) is becoming a usefultool for simulating and understanding cloud behaviourson various scales. In the cloud-resolving simulation usingCRM, cloud microphysical schemes are considered mainlyresponsible for simulating cloud behaviours, and havebeen validated in many intercomparisons. However, theresponsibilities of other components such as an advectionscheme for the mixing ratio of a water species hasbeen pointed out and has recently been developed. Penget al. (2005) showed that model results relating to watervapour are improved by employing a high-accuracy semi-Lagrangian scheme. Schroeder et al. (2006) applied ENO(essentially non-oscillatory scheme; Shu and Osher, 1989)and WENO (weighted ENO; Jiang and Shu 1996) schemesto their dynamical core in order to introduce a sharpchange in resolution, and reported improvements byapplying the schemes. Miura (2007) proposed an upwind-biased advection scheme on hexagonal–pentagonal gridsand showed its superiority. These studies focused onconservation of water species and accuracy of the advectionschemes.

However, monotonicity and positivity are also requiredfor advection schemes, since the mixing ratio is a positivescalar variable, which sometimes suddenly and discontinu-ously increases or decreases. When the monotonicity andpositivity are not preserved, spurious water species increaseis known to occur (Skamarock and Weisman, 2009). Theserequirements used to be satisfied using the flux-correctedtransport (FCT) scheme (Zalesak, 1979) in CRMs. The lat-est FCT schemes are positive-definite (PD) and monotonic(MO) flux limiters proposed by Skamarock (2006). Thefeasibility of the PD flux limiter in a realistic cloud-resolvingsimulation has been validated; the results showed that thepositive bias of precipitation is improved by employing thescheme (Skamarock and Weisman, 2009). Wang et al.(2009)also demonstrated the validity of the MO flux limiter in high-resolution cloud-resolving simulations with a high-orderupwind scheme.

PD/MO flux limiters are considered state-of-the-artschemes for CRMs, but some questions remain unanswered.One question is their ability for capturing cloud edge incloud-resolving simulations, i.e. the scheme can capture thecloud edge sharply whether non-oscillatory or not. Thesefeatures are required for advection schemes to overcome or

c© 2012 Royal Meteorological Society

WENO Scheme for Cloud Edge Problem 1375

mitigate the cloud edge problem, the details of which areexplained as follows.

The ‘cloud edge problem’ has long been recognized,as Klaassen and Clark (1985) reported that nonlinearflow experienced instability when cloud passed acrossthe grid, and the instability leads to inaccurate solution.Grabowski (1989) identified the problem as due toerrors originating from artificial oscillations, i.e. oscillatoryapproximation to thermodynamic fields and oscillatingsupersaturation at cloud boundaries. Later, based on theresults of Grabowski (1989) and using one-dimensionaltests, Grabowski and Smolarkiewicz (1990) summarized thecauses of the cloud edge problem as follows: (1) spuriouscloud water evaporation at the outer part of cloud edge(outer problem); and (2) oscillation in temperature andmoisture (i.e. relative humidity) at the inner part of thecloud edge (inner problem). The first problem used tobe solved using the monotonic scheme for computingadvection, but the second problem was not completelysolved by the advection scheme, although the advectionscheme has a mitigation effect on the problem. They notedthat the second problem is attributed to the inconsistencyexisting between advected temperature and moisture fields.Stevens et al. (1996) concluded that different rates of subgridscale mixing in temperature and moisture fields cause theinconsistency. To solve this problem, several schemes andmethods have been proposed such as coupling advectionscheme, and mitigation methods (e.g. Ovtchinnikov andEaster, 2009; Grabowski and Morrison, 2008). On theother hand, in order to overcome the cloud edge problem,alternative methods, which use a different cloud boundarytracking method instead of using FCT schemes, have beenproposed. These methods include a Lagrangian approach(Andrejczuk et al., 2008), volume-of-fluid (VOF) method(Margolin et al., 1997; Kao et al., 2000) and the smooth cloudmodel (Reisner and Jeffery, 2009). Unfortunately, becausethe order of the monotonic advection scheme was generallyfixed to less than third-order, the effect of a high-order non-oscillatory scheme on the cloud edge problem has not yetbeen clarified. A non-oscillatory scheme with high accuracyhas possibilities for mitigating both outer and inner cloudedge problems due to its features, i.e. the scheme excludesnumerical diffusion and oscillation in computing advectionfor both outer and inner parts of the cloud edge.

Another question concerns properties of FCT schemessuch as spatial accuracy and non-oscillatory characteristicscompared with high-order non-oscillatory schemes, whichcan avoid numerical diffusion of the FCT schemes. AlthoughSokol (1999) concluded that Bott’s scheme (Bott, 1989),which contains the basic concept of the FCT scheme, issuperior to the ENO scheme, the properties of the FCTscheme have not been compared with those of the WENOscheme, whose accuracy is higher than that of the ENOscheme. Moreover, the WENO scheme has been continuallyimproved until recently in terms of accuracy and gridconvergence rate (e.g. Ren et al., 2003; Henrick et al., 2005).Although Blossey and Durran (2008) partly introduced theconcept of the WENO scheme to the FCT scheme, andshowed the superiority of their scheme in preserving thepositiveness of the advected value, the feasibility of theWENO scheme for the cloud edge problem has not yetbeen verified and compared with those of the recent FCTschemes.

The purpose of this study is to show the validity of theWENO scheme, which has a non-oscillatory feature andhigh spatial accuracy, for capturing cloud edge in a cloud-resolving simulation. In particular, the effect of the WENOscheme on the thermodynamic field is focused on, becausethe non-oscillatory feature of the WENO scheme was orig-inally aimed at capturing a discontinuous profile of depen-dent (prognostic) variables except mixing ratio. For thispurpose, PD/MO flux limiters are also employed as recentFCT schemes, and the results simulated by each scheme arecompared with those simulated by the WENO scheme, usingseveral idealized experiments for the cloud edge problem.

The formulations of the WENO scheme and PD/MO fluxlimiters are described in the next section. The differences andcharacteristics of the schemes are described in detail. In sec-tion 3, three idealized experiments, i.e. the one-dimensionaladvection–condensation problem, shallow cumulus con-vection and squall line experiments, are introduced, andthe results simulated by different schemes are analysed. Asummary and conclusion are presented in section 4.

2. Advection schemes

2.1. Weighted essentially non-oscillatory (WENO) scheme

The WENO scheme was proposed by Jiang and Shu (1996)as an improved ENO scheme, which has non-oscillatory,non-dissipative characteristics, and higher accuracy thanthe TVD (total variation diminishing) scheme. The WENOscheme can capture sharp gradients of scalar variableswithout smoothing the profile, while the scheme maintainshigh-order accuracy for continuous profiles. Owing to theseadvantages, the scheme has been used to simulate multiphaseand shock-wave problems to capture sharp interfaces. Theformulation of the standard WENO scheme is

∂F

∂x

∣∣∣∣xi

= F̂i+1/2 − F̂i−1/2

�xi,

with F̂i±1/2 =r−1∑k=0

ωkFki±1/2, (1)

where �xi, ωk and Fki±1/2 are spatial intervals of the

ith direction, weights of fluxes and reconstructed fluxes,respectively. r indicates that the scheme is either a third-(r = 2) or fifth-order (r = 3) scheme. The weights of thefluxes are

ωk = αk∑r−1i=0 αi

, with αk = ω̄k

(ε + βk)p,

for k = 0, . . . r − 1, (2)

where ε = 1.0 × 10−6 and βk are smooth indicators. ω̄k

denotes the ideal weights and the values are summarizedin Table 1 for each scheme. The power p = 2 is knownto be accurate for both the third- and fifth-order schemes(Jiang and Shu, 1996) and is also set to 2 in this study.Reconstructed and interpolated fluxes Fk

i+1/2 to be used inEq. (1) are formulated as

Fki+1/2 =

r−1∑l=0

ark,lFi+k+l−(r−1), (3)

c© 2012 Royal Meteorological Society Q. J. R. Meteorol. Soc. 139: 1374–1388 (2013)

1376 Y. Baba and K. Takahashi

Table 1. Ideal weights ω̄k for third- (r = 2) and fifth-order (r = 3) WENOschemes.

r k = 0 k = 1 k = 2

21

3

2

3–-

31

10

6

10

3

10

Table 2. Reconstruction coefficients ark,l for third- (r = 2) and fifth-order

(r = 3) WENO schemes.

r k l = 0 l = 1 l = 2

2 0 − 1

2

3

2–-

11

2

1

2–-

3 01

3− 7

6

11

6

1 − 1

6

5

6

1

3

21

3

5

6− 1

6

where ark,l are the reconstruction coefficients given in Table 2.

Fi+k+l−(r−1) are the cell centre fluxes computed at the gridpoints. Using Eq. (3), cell face fluxes are obtained from cellcentre fluxes. The smooth indicators βk are defined for r = 2of the third-order scheme as

βk = (F [i + k − 1, 1])2 , k = 0, 1, (4)

and are defined for r = 3 of the fifth-order scheme as

βk = 1

2(F[i + k − 2, 1])2 + (F[i + k − 1, 1])2

+ (F[i + k − 2, 2])2 , k = 0, 1, 2, (5)

where

F[i, 0] = Fi,

F[i, l] = F[i + 1, l − 1] − F[i, l − 1]. (6)

The third- and fifth-order WENO schemes are con-structed using the above formulations. For each flux in bothschemes, Lax-Friedrichs flux splitting is adopted (see Jiangand Shu, 1996). In addition, a mapping procedure (Henricket al., 2005) that maintains a constantly high accuracy forflux computation is implemented in the formulations. Thisprocedure only modifies the weights of the WENO schemesdescribed in Eq. (2).

Since the WENO scheme was originally developed tocapture discontinuous profiles and to suppress numericaloscillation, the flux correction step is unnecessary afterthe advection fluxes are computed. In addition, theflux reconstruction step is simpler than those of otherschemes that were originally developed to capture cloudedge. However, because both reconstruction and weightcalculations are repeated r times before computing advectionflux, the WENO scheme’s computational cost tends to behigher than those of ordinary advection schemes.

Although several WENO-derived schemes exist, such asthe weighted compact nonlinear scheme (WCNS; Deng andZhang, 2000) and compact WENO scheme (Ren et al., 2003),these schemes are not treated here, for simplicity.

2.2. Flux-corrected transport (FCT) schemes

The PD flux limiter (Skamarock, 2006) is an optionalscheme that can be used with various advection schemes.Formulations of the recent PD flux limiter scheme are givenby Skamarock and Weisman (2009). Advection terms aregenerally evaluated in the following form:

(ρq

)t+�t = (ρq

)t − �t∑

i

δxi Fxi , (7)

where q is the mixing ratio, t the time, �t the time interval,xi the ith direction coordinate, δxi the derivative operatorin xi and Fxi the flux computed from high-order advectionschemes along with the ith coordinate. Using a flux limiter,Fxi is assumed to consist of both F1

xiand Fcor

xias

Fxi = F1xi

+ rxi Fcorxi

, (8)

where F1xi

is the fluxes computed from the 1st-order upwindscheme, rxi the renormalization factor (0 ≤ rxi ≤ 1), andFcor

xi= Fxi − F1

xi. rxi is determined as follows. Advection is

fractionally evaluated using F1xi

as

ρ̃q = (ρq)t − �t∑

i

δxi F1xi

, (9)

where ρ̃q is the predictor of (ρq)t computed using F1xi

.

To make (ρq)t+�t positive, the following relation must besatisfied:

ρ̃q < �t∑

i

δxi

(Fcor

xi

)+, (10)

where (Fcorxi

)+ consists of only outgoing fluxes. The pairof outgoing fluxes is distinguished by a positive value ofdivergence. According to Eq. (10), renormalization for theoutgoing fluxes is performed as (following the formulationof Blossey and Durran, 2008)

rxi = min

[1.0,

ρ̃q

�t∑

i δxi (Fcorxi

)+

]. (11)

By computing the contributions of the renormalizedfluxes using Eq. (8), actual advection is evaluated.

In Skamarock (2006), an MO flux limiter is also proposed.Wang et al. (2009) demonstrated its impact on high-resolution cloud-resolving simulations and pointed out thatthe PD flux limiter tends to cause numerical oscillation; toimprove this deficiency, the MO flux limiter is recommendedinstead of the PD flux limiter.

After Eq. (9), the renormalization of the MO flux limiterrequires a small modification compared to that of the PD



flux limiter, and the renormalization is performed using thefollowing rxi :

rxi = min

[1.0,

q̃ − qmin

q̃ − q̃t+�tmin

,q̃ − qmax

q̃ − q̃t+�tmax

], (12)

where qmin and qmax are the minima and maxima of themixing ratio related to flux divergence (Skamarock, 2006).The maxima and minima with superscript t + �t are takenfrom the predictors of q̃, which are evaluated using thecorrected fluxes as

ρ̃qt+�tmin = ρ̃q − �t

∑i

δxi (Fcorxi

)+, (13)

ρ̃qt+�tmax = ρ̃q − �t

∑i

δxi (Fcorxi

)−, (14)

where (Fcorxi

)− consists of only incoming fluxes. The pairof incoming fluxes is distinguished by a negative value ofdivergence.

Because the FCT schemes are used in conjunction with theadvection scheme, the accuracy for a continuous profile isdetermined by the advection scheme employed. In addition,spatial accuracy of the FCT scheme for cloud edge is expectedto be first-order, since the advection flux is replaced by afirst-order upwind scheme at the cloud edge.

3. Experiments

3.1. One-dimensional advection–condensation problem

For validation of the WENO scheme in a cloud-resolving model, a highly idealized, one-dimensionaladvection–condensation problem (e.g. Grabowski andSmolarkiewicz, 1990; Margolin et al., 1997; Grabowski andMorrison, 2008) is first conducted. The governing equationsused for this problem are

∂ρoθ

∂t+ ∇ · (ρoθu) = ρo

Lθ̄

CpT̄Cd, (15)

∂ρoqv

∂t+ ∇ · (

ρoqvu) = −ρoCd, (16)

∂ρoqc

∂t+ ∇ · (

ρoqcu) = ρoCd, (17)

where ρo is the ambient atmospheric density, θ the potentialtemperature, u the velocity, qv the mixing ratio of watervapour, qc the mixing ratio of cloud water, L the latent heatof condensation, Cp the specific heat at constant pressure andCd the condensation rate. θ̄ and T̄ are potential temperatureand temperature of the environment, which have the relationT̄/θ̄ = const.

The second terms on the left-hand side in the aboveequations are computed from the advection schemes to beevaluated. Time integration is explicitly performed usingthe second-order Runge–Kutta method. The condensationrate Cd is computed implicitly so that the saturatedwater vapour is immediately converted into cloud water.In the computation, inverse conversion, i.e. cloud waterevaporation, is also considered. When negative mixing ratio

appears due to the artificial effect, only the positive partof the mixing ratio q′ is used in the cloud physics, i.e.q′ = max(q, 0) is adopted, whereas the negative part isconsidered in computing advection. The treatment of thenegative value means that the total mass of water species isconserved during the simulation.

The experimental set-up of the one-dimensional advec-tion–condensation problem is as follows. The domain is600 m deep and a 160 m initial perturbation is placed nearthe bottom. Relative humidity and potential temperature ofthe environment are 30% and 302 K, and perturbations foreach field are 100% and 0.1 K, respectively. The perturba-tion also includes 0.2 g kg−1 cloud water content. Verticalvelocity is set to 2 m s−1 and the time step is determined sothat the Courant number becomes 0.2. Time integration isperformed for 150 s.

We employed five different advection schemes, namely, afirst-order upwind scheme (UP), third/fifth-order upwindschemes of Wicker and Skamarock (2002) (WS3/WS5)and third/fifth-order WENO schemes (WE3/WE5). Wealso employed two flux limiters with the above advectionschemes, i.e. PD flux limiter using UP (denoted as PUP),and MO flux limiter using UP (denoted as MUP), as the low-order advection flux F1

xi. To test the resolution sensitivities

of the advection schemes, 4 m, 2 m and 1 m resolutionswere used.

Tables 3, 4 and 5 summarize errors based on normalized l1and l2 norms of the prognostic variables (e.g. Li et al., 2008)simulated in three different resolutions. The normalized l1and l2 norms using the analytical solution are defined asfollows:

l1(φ) =∑ 〈|φ(z) − φT(z)|〉∑ 〈|φT(z)|〉 , (18)

l2(φ) =[∑ 〈φ(z) − φT(z)〉2]1/2[∑ 〈φT(z)〉2]1/2

, (19)

where φ(z) and φT(z) are the calculated prognosticand analytical variables at coordinate z, respectively.

∑represents the values summed over the whole domain.

Table 3. Normalized l1 and l2 norms of 4 m resolution case.

Scheme l1(θ) l1(qv) l1(qc)

UP 6.52×10−4 1.05×10−1 5.51×10−1

WS3-PUP 3.97×10−4 6.38×10−2 3.36×10−1

WS3-MUP 3.94×10−4 6.39×10−2 3.33×10−1

WS5-PUP 2.16×10−4 2.26×10−2 1.07×10−1

WS5-MUP 1.35×10−4 1.89×10−2 1.14×10−1

WE5 1.27×10−4 2.14×10−2 1.87×10−1

WE5-PUP 1.12×10−4 2.18×10−2 1.24×10−1

WE5-MUP 1.41×10−4 2.06×10−2 1.24×10−1


UP 1.44×10−3 1.77×10−1 6.66×10−1

WS3-PUP 1.12×10−3 1.38×10−1 5.20×10−1

WS3-MUP 1.12×10−4 1.38×10−2 5.20×10−1

WS5-PUP 7.44×10−4 7.48×10−2 2.75×10−1

WS5-MUP 6.33×10−4 7.35×10−2 2.96×10−1

WE5 5.13×10−4 7.85×10−2 3.61×10−1

WE5-PUP 5.06×10−4 7.82×10−2 2.93×10−1

WE5-MUP 6.18×10−4 7.63×10−2 3.03×10−1





UP 4.62×10−4 7.57×10−2 3.99×10−1

WS3-PUP 2.75×10−4 4.46×10−2 2.38×10−1

WS3-MUP 2.75×10−4 4.48×10−2 2.38×10−1

WS5-PUP 1.28×10−4 1.29×10−2 6.58×10−2

WS5-MUP 8.51×10−5 1.13×10−2 7.26×10−2

WE5 7.67×10−5 1.25×10−2 1.20×10−1

WE5-PUP 6.71×10−5 1.27×10−2 7.12×10−2

WE5-MUP 9.04×10−5 1.23×10−2 7.81×10−2


UP 1.20×10−3 1.50×10−1 5.66×10−1

WS3-PUP 9.38×10−4 1.16×10−1 4.39×10−1

WS3-MUP 9.38×10−4 1.16×10−1 4.39×10−1

WS5-PUP 6.02×10−4 5.85×10−2 2.16×10−1

WS5-MUP 4.78×10−4 5.54×10−2 2.32×10−1

WE5 3.89×10−4 6.07×10−2 2.95×10−1

WE5-PUP 3.78×10−4 6.06×10−2 2.25×10−1

WE5-MUP 4.76×10−4 5.79×10−2 2.38×10−1



UP 3.27×10−4 5.35×10−2 2.82×10−1

WS3-PUP 1.93×10−4 3.12×10−2 1.67×10−1

WS3-MUP 1.94×10−4 3.14×10−2 1.67×10−1

WS5-PUP 7.91×10−5 7.62×10−3 3.99×10−2

WS5-MUP 5.30×10−5 6.83×10−3 4.50×10−2

WE5 4.69×10−5 7.32×10−3 7.58×10−2

WE5-PUP 4.07×10−5 7.54×10−3 4.15×10−2

WE5-MUP 5.47×10−5 7.24×10−3 4.68×10−2


UP 1.01×10−3 1.26×10−1 4.76×10−1

WS3-PUP 7.85×10−4 9.71×10−2 3.67×10−1

WS3-MUP 7.85×10−4 9.73×10−2 3.67×10−1

WS5-PUP 4.84×10−4 4.55×10−2 1.68×10−1

WE5 2.97×10−4 4.72×10−2 2.39×10−1

WE5-PUP 2.82×10−4 4.69×10−2 1.72×10−1

WE5-MUP 3.66×10−4 4.38×10−2 1.87×10−1

It should be noted first here that the accuracy of theresults roughly depends on the order of the schemes. Theaccuracies of all the less than third-order schemes are lowerthan are those of fifth-order schemes. This fact indicates thatthe numerical diffusion of the high-order advection schemeused in FCT schemes has much greater impact on thesolution than does the flux limiter, and the cloud edge canbe diffused by a low-order advection scheme. Comparingthe results of cases using fifth-order schemes, cases using theWENO scheme show better accuracy in terms of both l1(θ)and l2(θ), while the accuracy for the water species dependson the flux limiters. The reason for these points will beexplained later. From the results of normalized norms basederrors, WE5-PUP is found to have the best accuracy amongall the cases and in all three resolutions.

The reason why WE5-PUP shows the best results isexplained by Figure 1, which compares the simulatedpotential temperature profile with the analytical solution.Numerical oscillation and undershoot occur when WS5-PUP is used (Figure 1(a)). This is because the PDflux limiter is not responsible for suppressing numericaloscillation (Wang et al., 2009). In contrast, numerical

oscillation is well suppressed when WS5-MUP is used,whereas undershoot occurs (Figure 1(b)). This undershootoriginates from cloud water evaporation, which is caused bynumerical diffusion at the outer part of cloud edge. (Thisundershoot does not occur if condensation/evaporation isnot considered.) As described in Heus and Jonker (2008),such artificial undershoot (which they note as ‘wiggle’)resembles the subsiding shell of clouds and can enhancenegative buoyancy. Thus it should be suppressed as faras possible; otherwise, updraught of clouds is artificiallyweakened. The undershoot is improved when WE5 is usedbecause numerical diffusion of the WE5 is smaller thanthat of the MO flux limiter (Figure 1(c)). Eventually, bothpositiveness and less diffusivity are simultaneously obtainedusing WE5-PUP (Figure 1(d)).

The effects of the advection schemes on water vapourand cloud water are different from those on the potentialtemperature. The simulated water vapour profiles are similar(Figure 2), but differences are found in the cloud waterprofiles indicating that positiveness for the cloud water isalmost preserved except for WE5 (Figure 3). When WE5-PUP is used, the negative value of the cloud water mixingratio is improved, with slightly larger oscillation comparedto WS5-PUP at the cloud edge (Figure 3(d)). Note that theoscillatory profile of the cloud water is not essentially derivedfrom the result of oscillatory supersaturation, because asimilar profile appears when cloud physics is not considered,and the relative humidity profile does not show oscillationat the inner part of the cloud edge in the present cases, justafter advection has been computed (Figure 4). However, thelarge negative value of cloud water is suppressed, and a smallnegative value still remains in WE5-PUP (and also in WS5-PUP). The negative value of the cloud water mixing ratiooriginates from excessive cloud water evaporation occurringat the outer part of the cloud edge. Because neither advectionscheme nor cloud physics is particularly affected by thenegative mixing ratio (only the flux limiter becomes active),an artificial effect results in a small water vapour increase. Itshould be noted here that the negative values of cloud waterin both WS5-PUP and WE5-PUP are of the same order asthey are less than 0.2% of maximum condensed cloud waterat 150 s. The impact on potential temperature is also of asimilar order, as negative buoyancy (negative from the initialstate) is observed at the last state. These facts mean that theirsignificance is less than the difference due to sharpness ofthe cloud edge.

WE5-PUP is found to be the best scheme for simulatingthe one-dimensional advection–condensation problem withcapturing the cloud edge. The results of the experimentalso show that the original WENO scheme for mixingratio is inferior to FCT schemes unless the flux limiter isapplied. To investigate the effect of the advection scheme onmultidimensional cloud development, a two-dimensionalshallow cumulus convection experiment is performed next.

3.2. Shallow cumulus convection experiment

Several cloud edge problems have been observed in variouspast cloud-resolving simulations. As described in theIntroduction, Klaassen and Clark (1985) and Grabowski(1989) reported that numerical instability originated fromboth advection errors and oscillation of supersaturationoccurring in shallow cumulus convection. Under differentsituations, similar artificial effects have been reported.



0 200 400 600

302

302.5

303

(a)

0 200 400 600

302

302.5

303

(b)

0 200 400 600

302

302.5

303

(c)

0 200 400 600

302

302.5

303

(d)

initial

Figure 1. Comparison with analytical solution for potential temperature: (a) WS5-PUP; (b) WS5-MUP; (c) WE5; and (d) WE5-PUP. Solid line,analytical solution; dashed line, simulation result. A 4 m resolution is used.

0 200 400 600

2

4

6

8

10

0 200 400 600

2

4

6

8

10

0 200 400 600

2

4

6

8

10

0 200 400 600

2

4

6

8

10

initial

(a) (b)

(c) (d)

Figure 2. Comparison with analytical solution for water vapour mixing ratio: (a) WS5-PUP; (b) WS5-MUP; (c) WE5; and (d) WE5-PUP. Solid line,analytical solution; dashed line, simulation result. A 4 m resolution is used.

Kao et al. (2000) showed that numerical instability at thecloud boundaries (cloud top) dissipates the entire cloudlayer within a half day, and Stevens et al. (2005) reportedthat oscillatory supersaturation makes the cloud deck ofstratocumulus continue to rise during the simulation. For

simplicity, to analyse the effect of the advection schemeon cloud development, the idealized shallow convectionexperiment of Klaassen and Clark (1985) that involvessome typical cloud edge problems is chosen in the presentstudy.



0 200 400 600

0

0.2

0.4

0.6

0.8

1

0 200 400 600

0

0.2

0.4

0.6

0.8

1

0 200 400 600

0

0.2

0.4

0.6

0.8

1

0 200 400 600

0

0.2

0.4

0.6

0.8

1

initial

(a) (b)

(c) (d)

Figure 3. Comparison with analytical solution for cloud water mixing ratio: (a) WS5-PUP; (b) WS5-MUP; (c) WE5; and (d) WE5-PUP. Solid line,analytical solution; dashed line, simulation result. A 4 m resolution is used.

0 200 400 600

40

60

80

100

120

rela

tive

hum

idity

%

z m

WS5- PUP WS5- MUPWE5- PUP WE5- MUP

initial

Figure 4. Comparison of relative humidity profiles. Only four cases whichshow high accuracy in comparison of normalized l1 and l2 norms arecompared. A 4 m resolution is used.

Our experimental set-up partly follows that of Klaassenand Clark (1985). The domain is a two-dimensional box witha size of 4.8 km × 4.8 km. The periodic boundary conditionis set to the horizontal direction and the slip wall conditionis set to both the top and bottom boundaries. The initialcondition is determined so that it consists of the stationaryatmosphere with a boundary layer, which has stability with apotential temperature gradient of 0.36 K km−1. The potentialtemperature gradients above the boundary layer are set to4.0 K km−1 from 1.4 to 1.8 km, and 4.5 K km−1 above1.8 km, respectively. The humidity profile is set as follows.The mixing ratio of water vapour decreases from 8.5 g kg−1

(bottom) to 8.0 g kg−1 (1.4 km). The humidity for theremaining parts is specified by the relative humidity as itdecreases from 91% to 39% between 1.4 and 1.8 km, and

as it gradually decreases to 23% at 3.0 km. Above 3.0 km,the relative humidity slowly decreases to 0% at the top ofthe domain. To force the initial updraught of the cumulus,the following heat flux is adopted for the domain duringcomputation:

F(x, z) = f0 exp

[− (x − x0)2

σ

]exp

(− z

α

), (20)

where x0 =2.4 km is the centre of the domain, andf0 = 150 W m−2, σ = 700 m and α = 300 m.

Simple cloud physics, which only considers conversionbetween water vapour and cloud water and is the same asthat employed in the previous experiment, is implementedin a non-hydrostatic dynamical core (Baba et al., 2010) toconduct the experiment. A standard Smagorinsky model isadopted in order to model the turbulence generated alongwith the shallow cumulus development.

Unlike the original experiment, only one domain is usedinstead of using two-stage nesting. In addition, two differentresolutions are used in the present study, i.e. 10 m and 5 mresolutions, in order to estimate both normalized l1 and l2norms-based errors of the simulated fields. Solutions with5 m resolution are first computed to obtain the referencesolution, and then solutions with 10 m are compared toestimate the errors. Resolutions coarser than 10 m are notused here, because the subgrid scale effect becomes moresignificant than does the effect of numerical diffusion, whichis originated from the advection schemes.

Table 1 summarizes the simulation cases conductedin the shallow cumulus convection experiments. In thisexperiment, the WENO scheme is applied not only to themixing ratio of the water species (tracer), but also to themass and energy variables (scalar) in the case of WE5-PUP2.



1200

1600

2000

2400

2800

3200

heig

ht m

1600 2400 3200

0 4 8 12 16 20

1200

1600

2000

2400

2800

3200

heig

ht m

1600 2400 3200

0 4 16 208 12

1200

1600

2000

2400

2800

3200

heig

ht m

1600 2400 3200

0 4 8 12 16 20

1200

1600

2000

2400

2800

3200

heig

ht m

1600 2400 3200

0 4 8 12 16 20

(a) (b)

(c) (d)

Figure 5. Comparison of instantaneous cloud water distributions at 35 min: (a) WE5: (b) WE5-PUP; (c) WS5-PUP: and (d) WS5-MUP. A 5 m resolutionis used.

In the other cases, WS5 is applied to all prognostic variablesexcept for water vapour and cloud water. First- and third-order upwind schemes are excluded here, since the accuracyof the fifth-order upwind scheme was found to be superiorto those schemes in the previous experiment.

Figure 5 compares cloud water distributions at 35 min,simulated using 5 m resolution. When 5 m resolutionis used, almost identical solutions are obtained regardlessof the advection scheme, except for small differences inthe wake structures beside the cumulus. Indeed, with thisresolution, no clear differences such as buoyancy and totalcloud water content are found in the statistics. Therefore,henceforth, an averaged solution using the four simulationresults is used as the reference solution.

Comparing 10 m solutions with the reference solution,the differences in the simulated results depending onthe advection schemes are observed more clearly, sincenumerical errors originating from the employed advectionscheme begin to appear in coarser resolution. Figure 6compares the errors based on normalized l1 and l2 norms,i.e. l1(T) and l2(T), computed from the temperature fields.l1(T) and l2(T) are defined as

l1(T) =∑ 〈|T10m(x, z) − TT(x, z)|〉∑ 〈|TT(x, z)|〉 , (21)

l2(T) =[∑ 〈T10m(x, z) − TT(x, z)〉2]1/2[∑ 〈TT(x, z)〉2]1/2

, (22)

where T10m(x, z) and TT(x, z) are temperatures obtainedfrom 10 m and 5 m resolutions at coordinate points (x, z),respectively.

∑indicates that the errors are accumulated

over the entire computational domain. In the evolutionof l1(T) and l2(T), the errors similarly increase as thecumulus grows toward the upper atmosphere. Even thoughmost cases produce equivalent errors, WE5-PUP2 showssmaller errors compared to the other cases. This is becausethe WENO scheme can capture sharp mass and energygradients without numerical diffusion, whereas the FCTschemes cannot.

Along with the difference in temperature error, timevariations of total cloud water content are different



30 40

10-4

10-3

10-2

WE5WE5- PUP1WE5- PUP2WS5- PUPWS5- MUP

time min

(a)

30 40

10-3


time min

(b)

Figure 6. Time variations of (a) l1(T) and (b) l2(T). Simulation results with 5 m resolution are used for reference.

30 40

0

2

4

6

time min


Figure 7. Time variations of total cloud water content.

(Figure 7). Of the cases studied, WE5-PUP2 shows the largestcloud water content. However the difference in cloud waterby the advection scheme was unclear in the one-dimensionalproblem, the difference becomes clearer and appears to bemore significant in this two-dimensional problem. It isnatural that the cloud water content increases in WE5-PUP2, because when the cloud edge is captured sharply,condensational heat is generated and transported withoutartificial diffusion and that fact results in strengthening theconvection. Matheou et al. (2011) presented similar resultsin shallow convection and reported they had increasedcloud cover and precipitation using a conservative centraldifference scheme as used in Baba and Kurose (2008). Thepresent results differ from theirs, because their results mightbe affected by conserved kinetic energy, whereas the presentresults are obtained from non-diffusive thermodynamicadvection. In comparing the errors of the cloud water whichare estimated in the same manner, the errors of WE5-PUP2are found to be smaller than are those of other cases (notshown here).

Comparison of cross-sectional profiles reveals that cloudedge properties cause the error differences in the temperatureand cloud water fields (Figure 8). At 2.4 km height and at35 min, three cloud boundaries appear, namely at bothsides and the bottom of the shallow clouds. In all the

1600 2000 2400 2800 3200

1600 2000 2400 2800 3200

0

10

20

clou

d w

ater

x10

-4 k

g m

-3

WE5- PUP2 WS5- PUPWS5- MUP reference

-2

-1

0

1te

mpe

ratu

re d

iffer

ence

KWE5- PUP2 WS5- PUPWS5- MUP reference

(a)

(b)

Figure 8. Comparison of cross-sectional temperature (difference from theinitial state) and cloud water profiles at 2.4 km height and at 35 min: (a)temperature and (b) cloud water. WE5 and WE5-PUP1 are omitted in thisfigure.

three boundaries, WE5-PUP2 simulates a sharp temperatureprofile with small diffusivity and oscillation (Figure 8(a)).Although Grabowski and Morrison (2008) mentioned thattemperature fluctuation (such as that provided by negativebuoyancy due to excessive evaporative cooling) is diffused bysubgrid scale mixing, the fluctuation remains in the presentstudy, because of the small magnitude of the mixing effect



on the cloud edge. In the profiles of cloud water, all thecases show deviations, and the deviation at the centre of thedomain is the largest, i.e. much more cloud water evaporatedin both WS5-PUP and WS5-MUP (Figure 8(b)). The trendsare closely related to the negative temperature differenceat the centre, meaning that when the temperature of thecloud edge is not captured well at the region where resolvedscale transport is significant, large excessive cloud waterevaporation occurs, and the evaporative cooling also causestemperature decrease. This fact agrees with the feature of thecloud edge problem observed in the Eulerian approach thatthe impact becomes greater as the cloud edge passes acrossthe fixed grid (e.g. Stevens et al., 1996; Kao et al., 2000).

It is concluded that the WENO scheme is superior tothe FCT schemes in simulating cloud edge in the two-dimensional shallow cumulus convection, especially interms of capturing mass and energy jumps at the cloudedge. For further investigation, the effect of the WENOscheme in simulating deep convection behaviour is analysednext using a squall line experiment.

3.3. Squall line experiment

A two-dimensional squall line experiment (Redelspergeret al., 2000), which is considered one of the simplestexperiments for simulating convective system, is nextperformed. The experimental set-up of the present studyclosely follows that of original experiment. The domainhas horizontal 1000 km size with 1.25 km resolution and20 km height with 80 uniform vertical layers. A periodicboundary condition is applied to the horizontal directionand a slip wall condition is applied to both the top andbottom boundaries. The initial fields for the experiment areobtained from observation, and to initialize convection ofthe squall line, a circular cold pool of 2.5 km height is givenat x =400 km (where x is the horizontal coordinate) forthe first 20 min of computational time. A three-class cloudmicrophysics proposed by Grabowski (1998) is employedto consider the water cycle within the clouds instead of theformer simple cloud physics. Results of this simulation using100 km domain data are preliminarily compared with thosepresented in Redelsperger et al. (2000), and the results arefound to be quantitatively valid as the simulated propertiesare within the dispersion of the model intercomparisonresults. This fact does not mean that the results of our modelstrictly agree with observation, but means that our modelis able to simulate the squall line reasonably and similarlyto other models, i.e. there are still uncertainties betweenmodel results, and for further agreements, model parametertuning is required. In this experiment, simulation cases arealmost identical to those used in the previous experiment(see Table 6). However, WE5 and WE5-PUP1 are excludedbecause WE5-PUP2 was found to be superior to these twocases in the previous experiment.

Table 6. Simulation cases of shallow cumulus convection experiment.

Case Tracer Scalar

WE5 WE5 WS5WE5-PUP1 WE5-PUP WS5WE5-PUP2 WE5-PUP WE5WS5-PUP WS5-PUP WS5WS5-MUP WS5-MUP WS5

5

10

15

5

10

15

300 400 500

0 2 4 6 8 10 12

300 400 500

0 2 4 6 8 10 12

5

10

15

300 400 500

0 2 4 6 8 10 12

(a)

(b)

(c)

Figure 9. Instantaneous cloud water profiles at 350 min: (a) WE5-PUP2;(b) WS5-PUP; and (c) WS5-MUP.

Figure 9 compares instantaneous cloud water profilessimulated using different advection schemes. The cloudwater profiles roughly represent vertical squall linestructures. It seems that WE5-PUP2 produces the largestsquall line structure and simulates larger cloud watercontent. In fact, when the time variation of total cloud watercontent is analysed, the total cloud water content of WE5-PUP2 is found to be the largest among the cases during thesimulation time (Figure 10(a)). When the WENO scheme isnot applied to advections of mass and energy, such differenceis not observed (the comparison is not shown here). Alongwith the cloud water increase in WE5-PUP2, contents ofother water species increase and become larger than thoseof other cases after the convective cell has been formed inWE5-PUP2 (Figure 10(b, c)), even though the time variationof total rain shows quantitatively good results with theintercomparison results (cf. Figure 4 of Redelsperger et al.,2000). Since the present study employs a lateral periodicboundary condition, existing water vapour in WE5-PUP2 isconsumed faster than in other cases. Therefore a difference in



0 100 200 300 4000

0.002

0.004

0.006

0.008

0.01

WE5- PUP2WS5- PUPWS5- MUP

time min

colu

mn-

inte

grat

ed c

loud

wat

er k

g m

-2

0 100 200 300 4000

0.002

0.004

0.006

time min

colu

mn-

inte

grat

ed c

loud

ice

kg m

-2


0 100 200 300 4000

5

10

time min

tota

l rai

n cm

day

-1


(a)

(b)

(c)

Figure 10. Comparison of time variations of column-integrated (a) cloudwater, (b) cloud ice (diagnosed) and (c) total rain. The total rain variationis estimated using 100 km domain data.

the statistics depending on the employed advection schememainly appears until the 300 min simulation time.

Structural difference of the squall line is considered tobe derived from the difference in buoyancy induced bycondensation. To clarify the difference, time variationsof column-integrated positive temperature differences(positive values obtained from temporal temperature minusinitial temperature) are compared (Figure 11). Results ofcases without subgrid scale mixing (i.e. the turbulence modelis switched off, as done in Tomita, 2008) are included here,and compared in order to investigate the effect of subgridscale mixing on the cloud edge at the present resolution. It isfirst found that WE5-PUP2 simulated the largest buoyancy

0 100 200 300 4000

0.2

0.4

0.6

0.8


tem

pera

ture

diff

eren

ce K

time min

Figure 11. Comparison of time variations of positive temperaturedifferences. The temperature differences are estimated by columnintegrating and averaging whole positive temperatures. Black line, withsubgrid scale mixing; grey line; without subgrid scale mixing.

0 0.005 0.010

5

10

15

20


cloud mass flux kg m-2 s-1

heig

ht k

m

stratiform updraught

convective updraught

Figure 12. Comparison of updraught cloud mass fluxes. Hour-averagedflow fields are used for the comparison. Black line, with subgrid scalemixing; grey line, without subgrid scale mixing.

during computation. This trend agrees well with the timevariation of total cloud water content in WE5-PUP2, wheremuch more condensation is considered to occur. It is alsofound, however subgrid scale mixing has effects on the totalbuoyancy; the relative significance of each advection schemedoes not change, and the effect of the advection scheme islarger than that of subgrid scale mixing.

The total buoyancy change is closely related to cloud massflux change. Figure 12 compares updraught convective andstratiform cloud mass fluxes which are estimated based onthe cloud mass flux partitioning method (Xu, 1995). Asdone in Figure 11, results without subgrid scale mixing areincluded in the comparison. The subgrid scale mixing effectis found to decrease updraught cloud mass flux, especiallyaround the peak updraught points (at 5 km height forthe convective ∼30% effect, and at 9 km height for thestratiform ∼18% effect, respectively), but the cloud massfluxes are found to be much more affected by the advectionschemes rather than the mixing effect. We can also find whichcloud mass fluxes are related to the buoyancy difference inFigure 12. Cloud mass flux increases are observed in the



5

10

300 400 500

−4 −3 −2 −1 0 1

5

10

300 400 500

−4 −3 −2 −1 0 1

5

10

300 400 500

−4 −3 −2 −1 0 1

heig

ht k

mhe

ight

km

heig

ht k

m

km

km

km

K

K

K

(a)

(b)

(c)

5

10

300 400 500

−0.3 −0.2 −0.1 0.0

5

10

300 400 500

−0.3 −0.2 −0.1 0.0

5

10

300 400 500

−0.3 −0.2 −0.1 0.0

heig

ht k

mhe

ight

km

heig

ht k

m

km

km

km

m s−1

m s−1

m s−1

Figure 13. Comparison of cold pool and downdraught profiles: (a) WE5-PUP2; (b) WS5-PUP; and (c) WS5-MUP. Left: cold pool; right: downdraught.Hour-averaged flow fields are used for the comparison. Cold pool profile is displayed, taking the difference between the temperature and the initialtemperature fields.

upper part of convective cloud and in the peak updraughtpoint of stratiform cloud, indicating that buoyancy (i.e.heated atmosphere) is transported to the upper atmospherewithout decrease. The trends are in agreement with thefeature of the WENO scheme that computes non-dissipativetransport of thermodynamic fields.

Cloud water and temperature differences due to advectionschemes are also observed in the previous shallow cumulusconvection experiment, but the trends in this experimentseem to be enhanced (see Figures 7 and 10). Therefore, acertain mechanism involved in the convection cycle that doesnot occur in a short time integration is considered to causethe enhancement. It is known that not only condensationbut also wind shear and cold pool play an importantrule in maintaining the convective cell of a squall line

by updraughting moisture (Rotunno et al., 1988). Figure 13compares the structures of cold pool and downdraughtformed in the squall line. WE5-PUP2 seems to form a largerand colder cold pool compared to the other cases. Undersuch conditions, much more warm moisture is updraughtedover the cold pool, and the condensation is also enhanceddue to the updraught. Consequently, this situation meansthat the convection cycle is enhanced due to the cold pool,and the effect is derived from the simulated sharp cloudedge, because the cold pool is formed by the evaporation ofrain (droplets of which are formed from cloud water). Themechanism can explain why the buoyancy and cloud watercontent increased in WE5-PUP2.

The downdraught profiles corresponding to the coldpool formation indicate that rain formations of both



(a)

-0.0003 -0.0002 -0.00015

10

15

20

heig

ht k

m



(b)

-0.002 -0.001 05

10

15

20

heig

ht k

m



0

Figure 14. Comparisons of downdraught cloud mass fluxes: (a) convective; (b) stratiform.

WS5-PUP and WS5-MUP occur within only a smalldownstream region, whereas the downdraught of WE5-PUP2 located at low altitude continuously appears towardthe downstream (Figure 13(a) right). This fact means thatthe convective cell in WE5-PUP2 successively convertswater vapour into cloud water, and large cloud watercontent remains in the atmosphere. Figure 14 shows whichdowndraught cloud mass flux has influence on the coldpool formation. In the comparison, an apparent differenceis observed in the convective downdraught where WE5-PUP2 shows the largest downdraught, indicating thatconvective precipitation is responsible for enhancementof the convection cycle.

Although a larger and colder cold pool formed byprecipitation is considered to enhance water vapourconversion in the squall line, convection of the convectivecell is the most dominant process among the involvedprocesses, since convection precedes all other processesin squall line formation. To analyse the properties of theconvective cell, cross-sectional profiles of temperature andcloud water are compared at different heights (Figure 15).WE5-PUP2 simulates a sharper and colder cold poolboundary near the bottom of the domain (at 1 km height,Figure 15(a)). At this height, cloud water begins to form, andthe content is found to be largest in WE5-PUP2. Towardthe upper atmosphere, the buoyancy increases and thepeak temperature increase shows a sharper cloud edge inWE5-PUP2 (at 5 km height, Figure 15(b)). Simultaneously,the cloud edge represented by the cloud water shows asharper boundary. Similar trends are observed even at the10 km height, where the cloud edge is no longer obvious(Figure 15(c)). Although temperature increase from theinitial state and its gradient present widely spread profilesin WE5-PUP2, sharpness of the convective cell compared tothe other cases is preserved. From the results, the convectionis confirmed to be enhanced through the convection cycle,because of the difference in advection scheme.

4. Summary and conclusion

The validity of the WENO scheme for the cloud edgeproblem is tested using three idealized experiments and the

results are compared with those of recent FCT schemes, i.e.PD and MO flux limiters.

A one-dimensional advection–condensation problem isfirst performed to investigate the advection scheme’s abilityto capture the cloud edge. The original WENO scheme isfound to be able to capture the cloud edge well; however,it has a disadvantage in that it produces a negative mixingratio. The WENO scheme with PD flux limiter avoidsthis disadvantage, and the scheme shows the smallesterrors for all prognostic variables, especially for potentialtemperature. It should be noted that an ordinary PD fluxlimiter cannot capture the potential temperature jump, sinceit was originally developed to avoid a negative mixing ratio.On the other hand, the accuracy of the MO flux limiteris lower than that of the WENO scheme because it hasnumerical diffusion at the cloud edge.

A two-dimensional shallow cumulus convection experi-ment is performed next. When the WENO scheme is appliedto the advection of mass and energy, accuracies for tem-perature and cloud water increase, and a larger cloud watercontent is produced. Cross-sectional analysis on the shallowcumulus indicates that the cloud edge properties correspondto the overall temperature and cloud water trends, and theWENO scheme simulates less evaporative cooling and cloudwater evaporation. These facts indicate that the WENOscheme also works well to capture the cloud edge comparedto the FCT schemes in two-dimensional cloud-resolvingsimulations involving nonlinear cloud development.

Finally, a squall line experiment is performed to verify theeffect of advection schemes in simulating deep convection.Since coarse resolution is required for simulating deepconvection, analysis on the relative significance betweensubgrid scale mixing and advection scheme is performed.Although subgrid scale mixing has maximum 30% effecton the cloud-based transport, the relative significancerelating to the advection scheme does not change. In theresults, when the WENO scheme is applied to advectionsof mass and energy, enhancements of both buoyancy andcondensation, resulting in production of more water species,occur. It is found that the enhancements are caused by alarger and colder cold pool formed below the squall line,which is formed by the enhanced convective downdraughtcloud mass flux due to the employed advection scheme.



300 400 500

0

20

40

60

80

x km

clou

d w

ater

x10

-5 k

g m

-3

300 400 500

0

10

20

30

x km

clou

d w

ater

x10

-5 k

g m

-3

300 400 500

0

0.5

1

1.5

2(×10-5)

x km

clou

d w

ater

x10

-5 k

g m

-3

300 400 500

–2

–1

0

x km

tem

pera

ture

diff

eren

ce K

WE5-PUP2WS5-PUPWS5-MUP






300 400 500

0

0.5

1

1.5

x km

tem

pera

ture

diff

eren

ce K

300 400 500

0

0.5

1

x km

tem

pera

ture

diff

eren

ce K

(a)

(b)

(c)

Figure 15. Comparison of cross-sectional profiles of temperature and cloud water at different heights: (a) 1 km; (b) 5 km; and (c) 10 km. Hour-averagedflow fields are used for comparison. The average time is chosen from 2 to 3 h when the convective cell is almost stationary. The temperature difference iscomputed using the initial temperature.



Comparisons of cross-sectional profiles of the convectivecloud indicate that the WENO scheme simulates the sharpconvective cell boundaries in terms of both temperature andcloud water in the lower atmosphere, and the sharpness ispreserved even in the upper atmosphere where the cloudedge is no longer obvious.

In conclusion, the WENO scheme is applicable to cloud-resolving simulation. The WENO scheme can simulate thecloud edge more precisely and sharply than the FCT scheme,especially in terms of the temperature (mass and energy)of the cloud edge. It is also found that when the cloudedge is captured more sharply, the convection cycle in deepconvection is enhanced. In other words, underestimationof convection strength can be avoided with the WENOscheme.

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weighted essentially non-oscillatory scheme for cloud edge problem

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