Asymptotic approaches to convective quasi-equilibrium

Q. J. R. Meteoml. SOC. (2000), 126,pp. 1861-1887

Asymptotic approaches to convective quasi-equilibrium By JUN-ICHI YANO’*, WOJCIECH W. GRABOWSKI’, GREGORY L. ROW’ and BRIAN E. MAPES3 Monash University,Australia 2Nationa1Centerfor AtmosphericResearcht, USA University of Colorado at Boulder; USA

’

(Received 7 December 1998; revised 10 September 1999)

SUMMARY The physical principle of convective quasi-equilibrium proposed by Arakawa and Schubert states that the atmosphere is effectively adjusted to equilibrium by an active role of convectiveheating against large-scale forcing (physical convective quasi-equilibrium, or PCQ).A simple consequence of this principle is that the rate of change of the thermodynamic field (typically measured by the convective available potential energy (CAPE)) is much smaller than the rate of change of the large-scale forcing (diagnosticconvective quasi-equilibrium, or DCQ).Such a diagnostic state is generally observed in the tropical atmosphere at the synoptic-scale, and this is often taken as a proof for the physical mechanisms behind Arakawa and Schubert’s convective quasi-equilibrium: however, theoretically,there are several alternative physical mechanisms that are also able to establish this diagnostic state. The paper examines the approach of the tropical atmospheric system to DCQ with increasing time-scale in order to distinguish various alternatives to PCQ.The latter predicts that the system approaches DCQ exponentially with a time-scale characteristic of convection. However, the alternatives considered in the paper predict algebraic asymptotesto DCQ with increasing time-scale. First it is demonstratedthat PCQ is not required to achieve DCQ by considering a linear primitive-equation system with arbitrary convective heating, in which the roles of convective heating and large-scale forcing are completely reversed; algebraic asymptotes are achieved. An even simpler analogue is to assume that the rate of generating CAPE is controlled by white-noise forcing. More generally, such an algebraic asymptote is obtained by any system with a power-law spectrum both for CAPE and large-scale forcing, although a restriction must be applied to ensure a decreasing asymptote with increasing time-scale. The approach to DCQ is examined for both the Maritime Continent Thunderstorm Experiment data and cloud-resolving model simulation data, and both indicate no tendency for exponential adjustments in the short time limit. KEYWORDS: Cumulus parametrization Scale-separation principle Tropical convection

1. INTRODUCTION

The purpose of cumulus parametrization is to provide a representation of the moist convective processes that are otherwise not directly available in ‘large-scale’ models due to their limited spatial resolution. The basic premise is that although it would be impossible to predict every detail of convective processes, such a parametrization could predict their bulk behaviour in a large-scale average sense. Arakawa and Schubert (1974) proposed the two guiding principles that enables such a procedure. The first of them is ‘scale separation’, which assumes that the characteristic scale for moist cumulus convection is sufficiently smaller than that for the large-scale motions. This principle enables us to consider the convective processes in a statistical sense, or only in terms of their ensemble average. The second principle, ‘convective quasi-equilibrium’, can be thought of as a consequence of the first principle. It assumes that the cumulus convective system is maintained almost in equilibrium, or in ‘quasi-equilibrium’, against large-scale forcing. Here, the cumulus convective system is defined as an ‘ensemble’ average of cumulus convection over a large horizontal scale and over a sufficiently long time-scale, being consistent with the scale-separation principle. The hypothesis of convective quasi-equilibrium insists that such an ensemble state is slaved to the large-scale environmental state, so that we can determine it in a closed form in terms of the large-scale variables. Stating it slightly differently, this latter principle maintains that the role of cumulus convection * Comsponding author, present address: Meteorologisches Institut, Universitaet Hamburg, Bundesstrasse 55, D-20146Hamburg, Germany. e-mail: [email protected] The National Center for Atmospheric Research is sponsored by the National Science Foundation (USA). 1861

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is to maintain the atmosphere in equilibrium against the large-scale forcing, F, that induces convective instabilities. Arakawa and Schubert argued, by evoking the scaleseparation principle, that the response of cumulus convection to forcing, i.e. convective damping D,is relatively short compared with the large-scale variation. This argument constitutes the physical basis of the convective quasi-equilibrium hypothesis. This principle of convective quasi-equilibrium (or statistical equilibrium) is often considered as central to the theoretical studies for the tropical large-scale circulations (cf. Emanuel et al. 1994). However, its physical basis has not had a great deal of critical examination (cf. Mapes 1997), and so little effort has been made in order to verify this principle from the observational point of view. A diagnostic relation (see Eq. (2.3)) follows from this principle, and this can be used as an observational test for the convective quasi-equilibrium principle. Arakawa and Schubert applied this test to the Marshall Island sounding dataset (see also Lord and Arakawa (1980), Lord (1982), and Wang and Randall (1994)). A similar test is applied to a mid-latitude case by Grell et al. (1991) and to cloud-resolving model simulations by Xu and Arakawa (1992). These analyses have shown that this diagnostic relation is indeed satisfied in various states of the convective atmosphere. Hence, these results are consistent with the convective quasi-equilibrium principle proposed by Arakawa and Schubert, and indeed are often taken as a proof for this physical principle. In the present paper, however, we would like to emphasize an important distinction between these two types of convective quasi-equilibrium: the one proposed by Arakawa and Schubert as a physical principle and the other defined in the diagnostic sense by Eq.(2.3). Logically speaking, the latter follows from the former, but the former does not necessarily follow from the latter. Note that physical convective quasi-equilibrium proposed by Arakawa and Schubert is based on a specific physical principle (causality), in which D actively works to balance with F. Hence, F is considered as a cause almost totally determining the effect of D.On the other hand, the diagnostic relationship can be satisfied, in principle, regardless of the specific causality assumed. In fact, it was suggested by Mapes (1998; hereinafter BEM) that this diagnostic relationship can be established even by a model assuming a completely opposite causality. More specifically, a model can be designed to maintain this diagnostic relationship by letting the large-scale ‘uplifting’ (forcing, F) freely responding to a random convective heating (damping, D:see section 2(b) for more details). Thus, the question arises as to how can we diagnose, from the observational data, the physical processes controlling the adjustment of convection to the large-scale forcing or, conversely, the response of the large-scale circulations to convective heating? The main claim of the present paper is that in order to answer this question, we have to examine carefully the approach of a given system to diagnostic convective quasiequilibrium as a larger and larger scale is considered. Thus, the present paper asks the question: how is a system expected to approach convective quasi-equilibrium in the diagnostic sense with increasing scale? It is normally understood in Arakawa and Schubert’s argument that convective quasi-equilibrium is established at the large-scale limit. For this reason, the observational (and some modelling) diagnoses quoted above typically utilize datasets with three-hour time intervals and spatial average-scales of the order of 100 km. However, it is suggested that it is not sufficient to diagnose the dataset only in this limit. Although some analyses along these lines have been performed (Grell et al. 1991; Xu and Arakawa 1992), this issue has not been systematically examined. By quantitatively analysing the approach of the system to convective quasiequilibrium with increasing scale, the robustness of the scale-separation principle can

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also be established. While scale separation can be established either on time or space scales, we will focus on the time-scale separation principle, and so the approach of a system to convectivequasi-equilibrium with increasing time-scale will be systematically investigated. The paper is constructed as follows. In the next section we introduce some basic concepts while defining our general strategy, and then propose several physical and mathematical models that lead to convective quasi-equilibrium in the diagnostic sense in the long time-scale limit. One of the models is that proposed by BEM, and another closely follows the physical principle of Arakawa and Schubert. Section 3 is devoted to the data analyses of numerically generated and observational datasets that test the theories presented in section 2. Section 4 contains a summary and a discussion on the physical and theoretical basis of the convectivequasi-equilibrium hypothesis in the light of our results. Conclusions are given in section 5. 2. THEORIES

Some basic concepts are introduced in section 2(a), in which the standard diagnostic test for convective quasi-equilibrium is first reviewed. Our method resides on the use of the so-called ‘standard deviation’, and will examine the dependence of this quantity on moving time-averages. It transpires that this ‘standard deviation’ is a good measure of a typical magnitude of a variable, which is, from the theoretical point of view, best estimated by scale analysis. First we demonstrate the usefulness of our approach by analysing BEM’s counter-example model in sections 2(b) and (c). The relation of our approach to the standard spectrum analysis is remarked on in section 2(d), where we also introduce a rather mathematical model to describe the approach of a system to convective quasi-equilibrium. The results for BEM’s case are contrasted with Arakawa and Schubert’s case, which is discussed in section 2(f). Another model is introduced in section 2(e) which provides a preparation for this analysis, and which also applies to the cloud-resolving model simulation data analysed in section 3(c). In order to make our theoretical points as clear as possible, the simplest possible analogues are chosen throughout this section, where the minimum physics (sections 2(b), (c), (e) and (f) or even no physics (section 2(d)) is retained. Thus, although we do not consider surface processes explicitly, this is solely for the sake of mathematical simplicity, as scale analysis which includes surface moist potential temperature could not be readily calculated in a closed form and, hence, was not further pursued. Radiative cooling is also neglected, as this term is not dominant on time-scales of less than 10 days, which is our prime focus. These initial restrictions become less critical as the analysis is generalized into more conceptual models in later subsections, where identical arguments can be applied by simply replacing potential temperature with convective available potential energy (CAPE).

(a) Basic concepts Conceptually, the magnitude of moist convective instability may be measured by CAPE. Arakawa and Schubert’s argument for convective quasi-equilibrium can be reconstructed by starting from the CAPE budget equation i.e. d -CAPE = F D, (2.1) dt where the dominant term in F is typically the adiabatic cooling by upward motion (uplifting), with surface fluxes and radiative cooling also contributing, and is opposed

+

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by D due to convective heating. Note that Arakawa and Schubert's original formulation used the cloud work function, which reduces to CAPE in the zero entrainment-rate limit. The quasi-equilibrium hypothesis states that convective damping always approximately balances large-scale forcing, i.e.

F+D-O,

(2.2)

because cumulus convection almost instantaneously responds to large-scale forcing to maintain the balance. As a result, the rate of change of CAPE is expected to remain almost stationary with time, d -CAPE 2 0, dt or more precisely remain much less than the magnitude of large-scale forcing, i.e.

Hence, from the diagnostic point of view, the degree to which the atmosphere is in a state of convective quasi-equilibrium can be tested by Eq.(2.3). A scatter plot of observed values plotted on the phase plane of F and d(CAPE)/dt (see Figs. 1 and 2) is usually used graphically to measure the degree to which a system satisfies convective quasi-equilibrium in the diagnostic sense. The degree of scatter of the points along the F-axis and the d(CAPE)/dt-axis are compared, and the extent to which the former is larger than the latter is interpreted as the degree to which convective quasi-equilibrium as defined by Eq. (2.3) is satisfied. Arakawa and Schubert (1974) and other references cited in the introduction used this method to test convective quasiequilibrium. In order to investigate the approach of a system to convective quasi-equilibrium with increasing time-scale, we could thus construct a series of scatter plots using the various time-scales. However, this cumbersome approach can be simplified if we note that the purpose of the scatter plot is to see the relative scatter of the two variables from the coordinate axes on the phase plane. This can be conveniently measured by the root-mean-square of the two variables, i.e. ( F 2 )'I2 and ( (d(CAPE)/dt)2)lj2, and if the relation ( (d(CAPE)/dt)2)V2 L/Cg. On substituting typical TOGA F A * sounding array parameters (cg 2: 50 m s-l, L 2: lo3 km, At 2: 1 day -lo5 s), we obtain (aO’/at)/F 2: 4 x

This ratio can be directly compared with the scatter plot in Fig. 15 of BEM. Thus, the diagnostic validity of convectivequasi-equilibrium is demonstrated by scale analysis for the BEM case. * Tropical Ocean and Global Atmosphere programme (World Climate Research Programme) Intensive Flux h Y .

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We may also re-write Eq. (2.1 1) as

( a e ’ / a t ) / F= ( T / A t ) 2 ,

(2.12)

where T = L / c g is the period for the gravity waves. The conditions for diagnostic convective quasi-equilibrium can then be stated in two complementary ways: (1) the measurement time interval At must be considerably larger than the characteristic gravitywave period T (Eq. (2.12)); and (2) the array size L must be considerably smaller than the distance cgAt that gravity waves can propagate during the measurement interval At (Eq. (2.1 1)). The second condition may appear to be counter-intuitive, because convective quasi-equilibrium is normally considered as a large-scale dynamical concept. However, this can be understood if we note that the vertical uplifting responding to convective heating is much less efficient, due to mass continuity, in the large-scale limit with a given wind profile. This last statement reiterates the point emphasized by BEM that spatially, convective quasi-equilibrium is a concept applicable to the small-scale rather than the largescale. However, it is also emphasized here that, temporally, it is a long time-scale concept rather than a short time-scale concept. Interestingly, Yano (1999) found that this statement is also true for Arakawa and Schubert’s formulation. It is also consistent with the data analysis by Grell et al. (1991) and the cloud-resolving model simulations by Xu and Arakawa (1992). Note also that the square in Eq. (2.1 1) (and also Eq. (2.12)) is not associated with the horizontal dimension of the system, even though it can be interpretated as the ratio of the horizontal spread (c,At)2 of the gravity wave to the analysis area L2 on a twodimensional horizontal domain. We will test this scaling in section 3(a), including the dependence of a system on its dimensionality.

(c) Heating-response model: rotating case When the rotation effect is included, the scale analysis can be classified into two where gravity-wave dynamics dominate and (ii) At > f where regimes: (i) At < f geostrophy dominates. Note that this should be the case regardless of the horizontal scale L in this idealized linear system, and as the first regime reduces to the non-rotating case of the previous subsection, we will only consider the second regime, here. The scale-analysis procedure in this case still has geopotential fluctuationsestimated by Eq. (2.8), but geostrophic balance leads to a new estimate for the horizontal velocity

-’

and substitution of Eq. (2.8) into this gives 6u = g H 68.

(2.13) L f 00 The magnitude of the vertical velocity is estimated from mass continuity (2.5b) but, because the goestrophic wind is non-divergent to leading order, we need to multiply by the Rossby number ER in order to obtain a proper estimate of the vertical velocity, i.e. L

where ER

= 1/f

A t . Substitution of Eq. (2.13) gives

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which estimates the magnitude of the large-scale forcing as

in place of Eq. (2.10). Hence, the ratio of the rate of change of CAPE to F is

(ae’/af ) / F = (Lf/Cgl2.

(2.14)

A rather surprising aspect of this scale analysis is that the degree of quasi-equilibrium

no longer depends on the time-scale At. The estimate (2.14) can be re-written by introducing the radius of deformation LR

= Cg/f

, (ae’/at)/F = ( L / L R ) ~ ,

or equivalently,by introducing wg = c g / L ( = 1/ T ) , (ae’/at)/F = (f/wgl2. Consequently, in order to satisfy convective quasi-equilibrium under geostrophy, we require LR >> L, or wg >> f . This result means that L must be considerably smaller than LR to ensure quasi-equilibrium, again demonstrating the small-scale spatial nature of the convective quasi-equilibrium hypothesis,

(d) Mathematical remarks Some general statements are made in this subsection before we perform the scale analysis for different cases in the following subsections. The ‘standard deviation’, say, ((gAt)2)1/2 of the moving average introduced in section 2(a) is related to the power spectrum P ( 8 , w ) by

-

((gAf)2)

(2.15)

P(0, w ) dw, 0

-

where the cut-off frequency wg may be estimated by wg l/At. Note that this relationship is more accurate when we remove the mean value in the definition of the lefthand side. A well known relation between standard deviation and power spectrum is recovered in the limit wg l/At + +oo, as a special case of the Wiener-Khinchin theorem. Note that moving averaging is equivalent to removing the contribution of higher frequencies in the Fourier space, so that the above expression is ensured. A natural extension of the results from the previous two subsections(see Eqs. (2.1 l), (2.12) and (2.14)) is to assume that the ‘standard deviation’ of a variable asymptotically reduces with At by, say, ( (8At)2) ‘I2 (At)-” w i with LJ a constant. Then, the power spectrum asymptotically follows

-

-

P ( e , 0)

-

-

0-y

with the power exponent y=-2u+1

(2.16)

defined from Eq. (2.15) (cf. Voss 1985). If a spectrum decreases with frequency, as is often the case (i.e. y > 0), then we expect u < 1 /2 from the above formula. Furthermore, if the slope LJ for the standard deviation suddenly changes from LJ -= 1/2 to u > 1/2 over

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a certain time-scale, we expect to find a peak at this time-scale in the power spectrum. Note, more specifically for white noise where y = 0, we have u = 1/2. Conversely, if the ‘standard deviation’ does not change with the averaging, i.e. u = 0, we have y = 1. Such a time series is often called l/f-noise (cf. Schuster 1988, section 4.3). The extension of the present analysis to the nonlinear case is not straightforward. In particular, when the system is fully nonlinear, a simple scale analysis is not applicable due to the multiple scales involved. Nevertheless, the spectrum often follows a power law in such a situation. It follows from the argument of the foregoing paragraph that 68 (At)-” and F (At)-”’ with u and u’ constant. Hence, as long as u - u’ 1 > 0, convective quasi-equilibrium is asymptotically satisfied for large At, because (aO’/at)/F (At)-(”-”’+’). Although this argument has the obvious weakness of not being able to specify the parameters, it does demonstrate that convective quasiequilibrium can even be satisfied asymptotically for large At only by assuming a certain power law for the spectra without any specific physics.

-

-

-

+

(e) System under an imposed large-scaleforcing The analysis for the cloud-resolving model system considered in section 3(c) is presented in this subsection. The scale analysis applicable for this system is performed here under the linear dry approximation for simplicitly, although the simulation is for the fully nonlinear moist system. In the present case, the system is driven by an external large-scale forcing defined in terms of the large-scale domain-averaged vertical velocity W by

and a sounding observation is used for ZU. Hence, the thermodynamic equation is given bv

where w’is the internally predicted vertical velocity as distinct from ZU. Note that the quasi-equilibrium analysis will be performed with the model-domain averages in section 3(c). Because the model vertical velocity w’ vanishes by this averaging, we expect that the major balance becomes

Furthermore, we expect Qc = F when convective quasi-equilibrium is approximately satisfied. In this limit, slight deviations of the right-hand side from quasi-equilibrium drives the fluctuations of the potential temperature (and hence, CAPE). We may write such a system as

aef

(2.17)

-26R at

-

by setting 6 R = Qc - F. We expect that this perturbation forcing 6 R works like a random noise on the temperature evolution. We may set 6 R (At)-” with a constant u (and normally u > 0). The fluctuation of the temperature is, hence, given by 68 (At)’-”. In particular, when 6 R is white noise (u = 1/2: see section 2(c)), then the temperature behaves like Brownian motion, 68 (At)’/’. It is seen that as long as

-

-

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-

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F does not change its magnitude substantially with At (i.e. I; At-” and u >> iJ -” 0), convective quasi-equilibrium is satisfied asymptotically in this case, too, by (aO’/at)/F in the limit of large At.

-

(f) Arakawa and Schubert ’s physical convective quasi-equilibrium In this subsection, finally we ask: how does the system approach convective quasiequilibrium with ensemble cumulus convection playing an active role, as envisioned by Arakawa and Schubert (1974)? We wish to show that what is expected from the physical argument of Arakawa and Schubert can also be derived mathematically. As a standard closure assumption, we assume that convective damping D , or heating Qc(= - D ) , is completely defined in terms of large-scale variables such as 8 , VH,w , and 4, along with some cloud parameters, in the system (2.5). We express this assumption by Qc = Qc(8, W, t ) , where the cumulus mass-flux wc represents a possible cloud-parameter dependence, and the dependence of Qc on other variables is implicitly expressed by time t . A more specific expression for this term in Arakawa and Schubert’s formulation is found by referring to their Eq. (74). By referring to Q. (2.2), strict convective equilibrium is stated by Qc(8, wc, t ) = FThis relation diagnostically defines the evolution of convection, namely, wc. The evolution of the potential temperatureunder this strict equilibrium is computed by substituting this heating rate into Eq. (2.5a).* We designate this value by 8 = O, and its deviation from strict equilibrium by 68. The deviation of the potential temperature from strict equilibrium is predicted by

a

(2.18) -68 = s ~ ~ , at where 6 Qc is the convective heating associated with this fluctuation. Here, for demonstrative purposes, let us assume hypothetically that only the potential temperature deviates from the state of strict equilibrium. Hence, 6 Qc is slaved to 68, or more generally to the large-scale variables. We may also consider some dependence of 6Qc on 6wc.This still leads to qualitatively the same conclusion as the following as long as Sw, is also slaved to the large-scale variables. We approximate the associated convective heating by (2.19) assuming that the deviation 68 is sufficiently small. The substitution of Eq.(2.19) into Eq. (2.18) gives (2.20a) where (2.20b) * In fact, the potential temperature is stationary if BEMs formulation is strictly applied. Actually, it is also the case with Arakawa and Schubert’s formulation that the cloud work functions remain stationary with time, if the strict equilibrium (as stated by their Eq.(150)) is applied.

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By solving Eq. (2,20a), we obtain SO = SO(t = 0) exp

(- s,’i?dt).

It is most illustrating to consider the evolution after onset of convection at t = 0. Because the convective equilibrium hypothesis anticipates SO + 0, then i? > 0. Furthermore, if i? is sufficiently smooth in time, i? dt 2: t / t c with a weak time-dependence in tc.Consequently,the system is adjusted to convective equilibrium exponentially with time-scale tc by the convective heating. The above formulation can be slightly generalized by including a possible largescale background noise or a white noise SR to Eq.(2.19). Then Eq. (2.20a) reduces to a linear regression model:

&

a

-88 = -i?Se at

+ SR.

(2.21)

Both systems (Eqs. (2.20) and (2.21)), predict that the ‘standard deviation’ of CAPE (or S O ) will decrease exponentially over the scale i?-l with increasing time-scale. The important point to note is that regardless of the precise expression of 6Qc (here given by Eq. (2.19)), as long as convective heating (or damping) actively responds to the change of CAPE (or S O ) with a characteristic time-scale tc,this conclusion qualitatively follows. This is exactly the physical picture presented by Arakawa and Schubert: cumulus convection responds to large-scale forcing on much shorter time-scales than the latter. The consequence must be, mathematically speaking, an exponential approach to convective quasi-equilibrium. Otherwise, we do not see a clear separation between the timescales for cumulus convection and large-scale dynamics, as postulated by Arakawa and Schubert. Randall and Pan (1993) considered more explicitly a relaxation of the quasiequilibrium assumption in the Arakawa-Schubert formulation and arrived at a similar exponential adjustment. 3. ANALYSES

This section is devoted to the data analyses used to test the theories presented in the last section. BEM’s model experiments are tested in section 3(a) against the scale analysis in section 2(b). This result is contrasted with a case satisfying Arakawa and Schubert’s physical principle in section 3(b). A similar analysis is then applied to more realistic datasets: the cloud-resolving mesoscale convection simulation in section 3(c) and sounding data obtained during the Maritime Continent Thunderstorm Experiment (MCTEX) in section 3(d). In these data analyses, the full definition of CAPE is used, in which the effect of the varying surface equivalent potential temperature is also taken into account. They are found to be consistent with the theoretical models presented in sections 2(e) and 2(d), respectively. (a) Synthetic sounding data analysis: heating-responseexperiment In this subsection, we analyse the synthetic sounding data generated in a similar manner as BEM, but restricting it to the non-rotating case. The governing equations for the system are given by Eq. (2.5) with no Coriolis effect (i.e. f = 0). The system is assumed to have either axisymmetric cylindrical or horizontally one-dimensional (slab symmetric) geometry, with horizontal coordinates designated by r and x, respectively. We only consider one vertical normal mode owing to the argument of BEM that only

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the leading mode dominates in this type of dynamical system. This approximation is equivalently obtained by assuming a homogeneous stratification (we set N = 1 x s-') with a fixed vertical scale H = 5 km. The half vertical wavelength of the system is defined by nH = 15.7 km; the gravity-wave speed for this system is cg = N H = 50 m s-'; and we also set 60 = 250 K. We idealize the synthetic data by assuming that the convective heating is provided by a single vertical column at the origin of the system (r = 0 in the axisymmetric case, and x = 0 in the slab-symmetric case). Consequently, mathematically speaking, the heating is given in terms of the Dirac's delta function. In choosing the time series of heating, we note that the scale analysis in the previous section implicitly assumes a broad spectrum in the heating, so that the dominant time-scale of the variability is always just above the time-averaging interval. As the simplest time series satisfying this property, we choose a white noise for the present analysis. The horizontal wind velocity is computed by using Green's functions given by Bretherton and Smolarkiewicz (1989). The domain-averaged vertical velocity w at any distance is estimated from the 'observed' radial wind and continuity. The temperature averaged over the area (r < R or 1x1 R, where R is a constant) is predicted by using the vertical velocity given by the Green's function and the external heating averaged over the area. The model is run for 160 hours and the result is recorded every 6 minutes for the analysis. The degree of quasi-equilibrium is usually measured by a scatter plot taking the large-scale forcing F and the rate of change of CAPE, -M/at, as the horizontal and the vertical axes, respectively. Such plots are given for the axisymmetric and slab-symmetric cases in Figs. 1 and 2, respectively, each for R = 222 and 555 km. The scatter plots are for moving-averaged data with Af = 1 hour and 12 hours. The data are sampled every Af in both cases. A dramatic improvement to attaining convective quasi-equilibrium is seen by increasing the scale from At = 1 hour to 12 hours in all cases. The approach to convective quasi-equilibrium is slower for a larger analysis domain, as indicated by Eq. (2.11). On the other hand, the difference between the axisymmetric and the slabsymmetric cases are surprisingly small. In order to see the At-dependence of this behaviour in a more compact manner, we plot the 'standard deviation' of a6/atA' and FA',i.e. ((M/af A')2) ' I 2and ( (FA')2) U2 against the averaging interval Af in Figs. 3 and 4 for the axisymmetric and the slabsymmetric cases, respectively. Again, the cases for R = 222 km and 555 km are shown. The magnitude of F changes relatively little with the change of the time-scale for the analysis, whereas the magnitude of the change of CAPE gradually decreases with increasing Af. The separation of these two curves indicates approach to convective quasi-equilibrium with increasing time-scale. ' I 2 more In order to inspect the asymptotic tendency of ((a6/atAt)2)' I 2 / ( quantitatively, we plot this quantity, multiplied by (Af/Afo)2 with At0 = 1 hour, in Figs. 3 and 4. The rate of the change of CAPE (relative to F) asymptotically follows (ae/at)/F in the axisymmetric case with R = 222 km for At > R/cg 1 hour (Fig. 3(a)) as expected from the scale analysis in section 2(b). Below the scale R / c g , the rate of decrease of ((a6/atAf)2)1/2with At is much weaker, as it is expected to balance with the damping according to the scale analysis. A similar transition over the scale R/cg 2: 3 hours is seen with R = 555 km (Fig. 3(b)). However, the asymptote beyond the scale R/cg is slightly weaker than the one expected from the scale analysis It is speculated that the transition from one balance to another is more

<

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1874

.10 .08

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0

0

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- .02

-.04 -.06 .08

-

F (K/hour) Figure 1. The quasi-equilibriumscatter plot for the axisymmetrk case with (a) R = 222 km and, (b) R = 555 km (see text) for moving-averaged data with At = 1 hour ( 0 ) and 12 hours (+).The horizontal axis measures the large-scale forcing, F , and the vertical axis the rate of change of convective available potential energy, dO/dt. Unit slopes are added by dashed lines to assist in assessing the degree to which the system satisfies diagnostic quasi-equilibrium.

gradual with a larger domain size. The slab-symmetric case in Fig. 4 leads to similar conclusions but with a slower transition to the asymptote than expected from the scale analysis for both scales R. These results give some practical measure as to how much agreement can be expected with the scale analyses from actual time series. ( b ) A model for Arakawa and Schubert 's convective quasi-equilibrium In this subsection we seek to apply the 'standard deviation' analysis to a time series that satisfies Arakawa and Schubert's physical principle of convective quasiequilibrium. Producing such an example is not straightforward, as switching conditions for convection and large-scale evolution of CAPE need to be specified, and so, for simplicitly, we analyse a system with white noise given by Eq.(2.21). Note that this system behaves like Brownian motion in the short time-scale limit and white noise in the long time-scale limit. Nevertheless, by crossing the time-scale Z-',

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.02 .04 .06 .08 .10 .12 .14 .16 .18

F (K/hour) Figure 2. As Fig. 1 but for the slab-symmetric case.

we should observe an exponential decay of the ‘standard deviation’, ((i30/i3tAr)2)l/2. Results from the ‘standard deviation’ analysis for the case with G = 1, shown in Fig. 5(a), show this decay: existence of a characteristic time-scale is seen here, as with BEM model results in the last subsection; however, the decrease of the ‘standard deviation’ over this characteristic time-scale is qualitatively different between the two models. While the ‘standard deviation’ decreases with a constant rate (so that it becomes linear on the log-log plot) with the BEM model, it decreases with a faster rate with increasing time-scale, with the Arakawa-Schubert model, reflecting its tendency for exponential decay. This point is more easily seen by replotting the ‘standard deviation’ of (aO/i3t)Ar from Fig. 3(a) and Fig. 5(a) on a linear-logarithmic scale over the characteristic time-scale, as shown in Fig. 5(b), after some rescaling of the latter’s ‘standard deviation’. Although the point is rather subtle because the model time series for Arakawa and Schubert’s quasi-equilibrium turns into white noise at longer timescales (At =- lo), the ‘standard deviation’ tends to approach a straight asymptote with increasing At , which is consistent with exponential decay. On the other hand, the slope

J.-I. YANO et al.

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.

'

.

1oo

. . ' " I

10'

10'

average(hour) Figure 3. The change of the 'standard deviations' with the time-scale At for the axisymmetric case: ( (ae/atA')') (solid curve), (long-dash curve), and 'I2/ ( (FA')*) '/'(At /Afo)2 (short-dash curve) with At0 = 1 hour. Computed from the axisymmetric synthetic data with (a) R = 222 km and (b) R = 555 km. See text for further explanation.

of the 'standard deviation' continuously decreases with increasing time-scale, even after crossing the characteristic time-scale (At 21 1.2 hours) with the BEM model.

(c) Cloud-resolvingmodel data analysis The next test for convective quasi-equilibrium is performed by applying the scale analysis developed in section 2(e) to data generated by a mesoscale convective model. These simulations use a cloud-resolving model (CRM)which is run for 7 days using the evolving large-scale observations obtained during Phase I11 of the Global Atmosphere Research Programme (GARP) Atlantic Tropical Experiment (GATE), and performed in both two-dimensional and three-dimensional configurations. The former has a 400 km horizontal periodic domain, while the latter a doubly-periodic square domain with 400 km sides; details of these simulations are given by Grabowski et al. (1998). In this analysis, a more accurate CAPE, defined by a vertical integral of max(0, B) from the surface to 20 km, is used. Here, the buoyancy B = g { ( T " - T ) / T cq

+

CONVECTIVE QUASI-EQUILIBRIUM

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4

v I 1

&O;

. .

'

1oo

. * . ' . I

10'

10'

average(hour)

.,

C . . .

O0

t

.. .

_cc_----

average (hour) Figure 4. The same as Fig. 3 but for the slab-symmetric case.

(4: - qv)} is defined by the pseudo-adiabatic (irreversible) lifting of a parcel from the surface with cq = 0.608, and T and qv the temperature and the water vapour mixing ratio, respectively, for a given column, while the double-primes (") designate the corresponding variables for the lifted parcel. F is defined as the rate of generation of CAPE due to cooling and moistening associated with the combined effect of horizontal and vertical advection as deduced from the large-scale 'apparent' sources (see Grabowski et al. (1996) for a detailed discussion). Neither radiative tendencies nor surface fluxes are included in F, because they are not as important contributors as the other terms in this type of mesoscale convective simulation. Both CAPE and F are computed at every column, and then averaged over the model domain for the analysis. Statistics for the dependence of convectivequasi-equilibrium on At are presented in Fig. 6 in a similar manner as for the synthetic data shown in Figs. 3 and 4: the 'standard 1/2 are plotted for the two-dimensional deviations', ( (a(CAPE)/atA')2)' I 2and ( (FA')2) and the three-dimensional cases in (a) and (b), respectively. The magnitude of F seen in Fig. 6 is even less sensitive to At than the case with the BEM model. This reflects the fact that it is based on 3-hourly observations.

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0' O0

0" ld

a

d

ld

1

o4

4

m

1

time-scale

((mA')2)'/z

for a time series satisfying Arakawa and Schubert's Figure 5. (a) The 'standard deviation' physical principle (based on Eq.(2.21) with c = 1) plotted against the time-scale Af. The curve should follow the short-dashed line if the system is algebraically asymptotic to diagnostic quasi-equilibrium. (b) The same as (a) but in the linear-log scale. The additional chain-dash curve is the same quantity but from the solid curve in Fig. 3(a). The two straight lines are also added in order to compare the different asymptotic tendencies of the two time series. Arakawa-Schubert model (solid curve) is asymptotic to a straight line (long-dash) for At 2: 3 - 10, but the slope continuously deceases with the BEM model (chain-dash curve): compare it with the short-dash curve. Note that the scale of the vertical axis for the solid curve is arbitrary adjusted, and the nondimensional time-unit of this curve is taken to be 1 hour to match with BEM model. See text for further explanation.

A faster decrease of the rate of change of CAPE than the large-scale forcing indicates the approach of the system to convective quasi-equilibrium. The theory in section 2(e) predicts (see Eq. (2.17)) that ((a(CAPE)/arAf)2)1/2is asymptotic to -At-'12. In order to see this tendency, we also plot

in Fig. 6. This curve is almost flat for the At range from 2 to 20 hours, which is in agreement with the theory. Furthermore, according to this theory, the distribution of

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n

5 l 0

a, average(hour)

n

k

I G

\

x lo4:

W

W

Cn 4 u

lo9:

$ $

yo.

1

1oo

10'

10'

average (hour) Figure 6. The change of the 'standard deviations' with the time-scale Ar: ((a(CAPE)/arAf)2)1/2(solid curve), ((FAr)2)'/2 (long-dash curve), and ((a(C~E)/arAf)2)/((FAr)2)1/z(Ar/Ar~)1/2 (short-dash curve) with Aro = 1 hour. (a) The two-dimensional, and (b) the three-dimensional cases of the cloud-resolvingsimulation. See text for further explanation.

change of CAPE should remain Gaussian with time, and its variance increase by ( (SCAPE)2)

-

Af.

(3.1)

We find that the probable density distribution for the change of CAPE over Af obtained by the simulation is reasonably close to Gaussian, and to remain fairly stationary with time after normalization by multiplying by (Afo/Af)1/2 (Fig. 7). This CRM system, overall, approaches convective quasi-equilibrium algebraically but with a different exponent than the BEM model. A major exception is for Af above 30 hours, where the 'standard deviation' of d(CAPE)/dt tends to decrease exponentially, indicating a realization of physical convectivequasi-equilibrium. However, this At-scale is much longer than typically believed to be that corresponding to convection.

J.-I. YANO eral.

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CAPE Variation

(b)'0'

y

W

ul 4

u n

c)

0

CAPE Variation Figure 7. The probable density distribution of change of convective available potential energy (CAPE), S CAPE, over the time Ar obtained by the cloud-resolving model simulation. The fluctuation is normalized by multiplying by the factor (Ato/At)'/* with At0 = 1 hour, so that the effect of the spread of the variance with time (Eq.(3.1)) is compensated. The results for Ar = 20 minutes, 3 hours, 6 hours, and 12 hours are shown by the solid, long-dash, short-dash, and the dot-dash curves, respectively, for (a) the two-dimensional and (b) the three-dimensional case.

( d ) MCTEX data analysis How do observed systems asymptotically approach convective quasi-equilibrium with increasing time-scale? We attempt to answer this question by an observational analysis using the MCTEX sounding dataset. The MCTEX soundings have an advantage over similar tropical datasets in that they were performed as frequently as every two hours during the intensive observation periods. The MCTEX occurred on the Tiwi Islands in the Northern Temtory of Australia from 6 November to 13 December 1995 (Keenan et al. 1994). During this period, 104 soundings were launched from Maxwell's Creek on the islands. From these soundings, we selected all the soundings which reached higher than 20 km altitude for the present analysis, 68 in total. The first successful flight was made at 8:07 UTC 20 November and the last at 6 5 6 UTC 4 December giving a 16 day analysis period.

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time(day)

W0'p

Figure 8. The Maritime Continent Thunderstorm Experiment data analysis. (a) The rate of change of irreversible convective available potential energy (CAPE) (solid), and of reversible CAPE (long-dash) computed from the soundings, and the large-scale forcing F due to uplifting (short-dash) and surface fluxes (chain-dash). (b) The change of the 'standard deviations' with the time-scale At: ((a(CAPE)/atA')2)1/2(solid and long-dash curves for the irreversible and the reversible cases, respectively), and ( (FAr)*)'I2 due to uplifting (short-dash) and surface fluxes (chain-dash). See text for further explanation.

Both irreversible (pseudoadiabatic) and reversible CAPE are computed from each of these soundings. In the latter case, the definition of the parcel buoyancy is replaced by B = g { (T" - T ) / T eq (4: - qv) - qg}, where q$ is the mixing ratio of the total water condensate of the lifted parcel with no precipitation, and CAPE is defined as the mean for the parcels lifted over the lowest 50 mb of a sounding. For each lifted parcel, max(0, B) is integrated vertically from the lifting level, where the parcel is lifted from, to the level of neutral buoyancy. The rate of change of CAPE is then computed by simply taking the finite differentiationof two adjacent soundings. The obtained rate of change of CAPE is plotted in Fig. 8(a) for the irreversible and reversible cases. The uneven distribution of the soundings in time is easily seen in the plot. Both large-scale uplifting and the surface flux effect are considered to gain an estimate of large-scale forcing. Forcing due to large-scale uplifting is estimated from

+

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the vertical velocity measured by the 920 MHz boundary-layer wind-profiler located at the same site, and operated during the experiment. No estimate of the vertical velocity can be made from the soundings as only a single site was available. Various limitations of the wind-profiler data (e.g. only boundary-layer measurements are available, this single-point measurement may not represent true ‘large-scale’ vertical motion, etc.) are granted, but this forcing estimate is only used as a crude reference level for comparison with the rate of change of CAPE. Wind-profiler observations were performed every few minutes ( 1-3 minutes usually) for 21 days from 19 November to 9 December, with a noticeable interruption for almost 1 day on the 6th and the 1lth days of the observation period. In order to avoid the unnecessary measurements of hydrometeor motion during rainfall periods, any singletime measurements containing data with a signalhoise ratio greater than -5 dB above 1 km are excluded*. Furthermore, in order to avoid erroneous values, any measurements of a single level with a signahoise ratio less than -20 dB as well as a spectrum width larger than 2.0 ps are excluded. The single-time measurements are vertically averaged from 0.2 km to 2.0 km, provided that they do not have more than 25%of the vertical levels missing. We define the large-scale forcing from this vertically averaged 20 by F = N2eow/g as in section 3(a), with the same parameters. Surface flux forcing is estimated from surface flux measurements made at Maxwell’s Creek. The measured total surface flux (the sum of the sensible, latent, and subtrate fluxes) is converted into forcing unit (K hour-’) by dividing it by phC,, where p = 1 kg m-3 is the air density, h = 1 km the boundary-layer depth, and C , = lo3 J kg-’K-’. This surface flux data, given every 20 minutes, is still to be fully calibrated and so it was suggested that total flux measurements greater than lo00 W m-2 were discarded as erroneous (Nigel Tapper, personal communication). These estimates of the two types of large-scale forcing (F),with a one-day moving average applied, are plotted in Fig. 8(a). We now perform the ‘standard deviation’ analysis on these time series as in the previous subsections, but note that the actual procedures are complicated by the uneven distribution of the data. The moving time-averagingis applied to every time span starting from a measurement (sounding) time, say, t = tj for the jth measurement and ending at t = tj A t . All the available measurements for this period tj 6 t ti At are averaged and the mean is defined as the moving average at the mean measurement time (i.e. the value obtained by averaging over all measurement times for all the available measurements of the period), say 7,. This moving-average process is terminated when t = tj At is found to be beyond the data period. The rate of change of CAPE is then defined as follows. Given a jth moving average measured at t = F,, we look for another moving average measured at time t (hours), where T j At - 0.5 6 t < 7 . At 0.5. ‘ I If such an average exists, we definethe rate of change by using the actual tlme difference in place of At in Eq. (2.4). The averaging time At is increased from 2 hours by 1 hour for the sounding data until the time differentiation is no longer definable. is The ‘standard deviation’ of the rate of change of CAPE, ((a(CAPE)/atAr)2)1/2, plotted against At for the irreversible and reversible cases in Fig. 8(b). This log-log scale plot shows that these ‘standard deviations’ follow - ( A t ) - ’ for At ‘v 2-80 hours. At At 2: 80 hours there is a distinct change of slope and, by crossing this scale, the asymptote changes to - ( A t ) - 4 . Following the argument of section 2(d), this change in slope

+

< +

+

+

+ +

* Nevertheless, this sampling procedure does not completely remove the gusty vertical winds of order of magnitude larger than 1 m s-’, presumably associated with the convective motions.

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CONVECTIVE QUASI-EQUILIBRIUM

indicates a spectrum peak around this scale. No characteristic response time is detected in the shorter time-scales. The moving-averaged ‘standard deviation’ ( (FA‘)2)for the two types of large-scale forcing are also plotted in Fig. 8(b). It is seen to decrease following F for the uplifting forcing. This decrease is even slower for the surfaceflux forcing, and the ‘standard deviation’ for both of these forcing terms decreases much slower than that for d(CAPE)/dt . Consequently, convective quasi-equilibrium is also asymptotically satisfied by the MCTEX data for the large time-scale. This asymptote agrees with neither of the previous two cases, but rather lies somewhere in between. There is no immediately clear physical interpretation for this asymptote beside the general framework given in section 2(d) for fully nonlinear systems. With the help of Eq.(2.16), the power exponent is estimated to be y 2 1 and y 0.4 for CAPE and forcing, respectively. Note in particular that CAPE varies as a l/f-noise (cf. section 2(d)).

-

-

4. SUMMARY AND DISCUSSIONS

Theoretical and mathematical analogues were proposed, which produce convective quasi-equilibrium asymptotically in the long time-scale limit without evoking the physical principle proposed by Arakawa and Schubert (1974). In the first case, a linear dry primitive-equation system was considered, in which an arbitrary heating is applied externally. Since such heating mostly balances with the adiabatic cooling provided by the gravity-wave response, the system remains approximately in convective quasiequilibrium in the long time-scale limit. In this analogue, the roles of the two processes are completely reversed: quasi-equilibrium is established by the large-scale forcing responding to an arbitrary heating, instead of convective heating responding to large-scale forcing as proposed by Arakawa and Schubert. A simpler analogue was proposed to produce the same effect, in which CAPE is simply generated by white-noise forcing. The standard deviation of such a rate of change of CAPE (being identical to the white noise) decreases for the longer time-scale. Hence, convective equilibrium is again established asymptotically, provided the change of magnitude of the large-scale forcing over the time-scales is much weaker. Remarkably, in both cases, the asymptote to quasi-equilibrium is algebraic: i.e.

-

d -(CAPE)/F (At)-’, (4.1) dt where p is a positive constant. Scale analysis has shown that p = 2 and 0.5 for the first and the second analogues, respectively. It is emphasized that such an asymptote is also obtained for any system with a power-law spectrum both for CAPE and largescale forcing, provided p > 0. Such a power-law spectrum is known to be universal among strongly nonlinear systems such as fully developed turbulence. Note that, in these models, the physical connotations attached to the ‘forcing’ and ‘damping’ for F and D are completely lost. Generally speaking, a mere fact of obtaining an expected quulitutive relationship by an observation or a numerical model does not automatically guarantee the actual working of a presupposed physical principle. A mathematical model can relatively easily produce the identical result, and it is quite conceivable that various combinations of physical processes could be found which would reproduce such a mathematical model. For example, we showed that a CRM simulation of a mesoscale convective system with imposed large-scale forcing produces the same asymptote p = 0.5 as the second mathematical analogue. This is presumably because the imbalance between the largescale forcing and the convective damping behaves like white noise in this case.

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An observational test was performed using the MCTEX data, and an algebraic asymptote with j~ 0.7 was obtained. The standard deviation of CAPE was found to be close to that of l/f-noise (i.e. v = 0 and y = 1). This implies strong intermittency in the MCTEX CAPE time series due to either stochastic or deterministic but very complex, strongly nonlinear processes (cf. Schuster 1988). These algebraic asymptotes are, however, clearly distinguished from the asymptote obtained by applying the physical principle for convective quasi-equilibrium proposed by Arakawa and Schubert (1974). In the latter case, the atmosphere is adjusted to convective equilibrium exponentially with a characteristic time-scale, rc, by an active role of the convective heating. Hence, the ratio of the ‘standard deviations’ d(CAPE)/dt/F decays exponentially over the time-scale rc. However, such an exponential tendency is seen neither in the MCTEX data nor the CRM result. Note that the CRM data covers the convective time-scale rC, which is typically believed to be 20 minutes-3 hours (cf. Arakawa and Schubert 1974), by using a sampling rate of 20 minutes. The MCTEX dataset, which had a sampling rate of 2 hours (implying a minimum resolvable period of 4 hours), also marginally covers the convective time-scale. The result also contains an implication for another cornerstone of cumulus parametrization: the scale-separation principle. The exponential adjustment to convective quasi-equilibrium had automatically guaranteed this principle. The time-scale rc defined as a characteristic scale for the exponential adjustment to convective equilibrium can separate the cumulus convective processes from the processes with any longer timescales without ambiguity. However, the asymptotic tendency to quasi-equilibrium denies the existence of such a characteristic time-scale: the magnitude of convective variability (say, measured by CAPE) always decreases at a fixed rate whenever the time-scale increases at a fixed rate, say, by twice, and hence no objective threshold exists to define a characteristic time-scale. Consequently, the scale-separation principle also loses its empirical basis by the asymptotic approach to convective quasi-equilibrium. The convective quasi-equilibrium hypothesis has been, furthermore, the basis for the closure assumption that convective heating can be expressed solely in terms of the largescale variables, as so assumed in section 2(f). An analogy with statistical mechanics may be useful to explain this point. Because of the clear scale-separation between the macroscopic and the microscopic states, the statistical states of the microscopic (atomic) scales are in equilibrium with the macroscopic thermodynamic states and, hence, almost completely determined by the latter. The lack of such a clear scale-separation in cumulus parametrization implies that such an analogy is no longer valid: the ensemble state of cumulus convection is not completely determined by the large-scale variables, but rather depends to a substantial degree on the internal parameters of the ensemble. For example, initiation of convection by overcoming the convective inhibition in the boundary layer is likely to be more strongly determined by such internal parameters. Mapes (1997) called such an autonomous aspect of statistical cumulus dynamics as the ‘activation control’ principle in contrast to convective quasi-equilibrium. As a result, a new type of cumulus parametrization scheme becomes desirable, in which an internal state of the cumulus ensemble is explicitly considered. Such a type of parametrization has already been proposed by Yano et al. (1998) as the ‘convective life-cycle’ (CLC) scheme, in which a life-cycle of a mesoscale convective system is attempted to be described as a grid column process*. Furthermore, since this type of new parametrization is scale dependent, a new principle to replace with the scale-separation

=

* The concept of a ‘prognostic’ scheme introduced by Arakawa (1993), on the other hand, only means that the convective heating is predicted by a prognostic equation instead of a traditional diagnostic one. The relaxation of quasi-equilibrium by Randall and Pan (1993), for example, falls into this category, but is not considered as CLC.

CONVECTIVE QUASI-EQUILIBRIUM

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principle is required. More specifically, the new guiding principle should be able to determine how the parameters in a parametrization depend on the minimum resolved scale. Yano et al. (1996) proposed renormalization as such a new principle. This principle may, furthermore, provide a clearer interpretation to the observed asymptotic tendency to convective quasi-equilibrium expressed by Eq. (4. l), because of its mathematical analogy (compare with Eq.(3) of Yano et al. (1996)). Nevertheless, the observational basis for the ‘asymptotic approach’ to convective quasi-equilibrium is still premature. Even the present MCTEX analysis was based only on 2-hourly sounding data. Hence the possibility of adjustment to convective quasiequilibrium on shorter time-scales cannot be excluded. The quality and universality of the MCTEX data can also be questioned. We have repeated the analysis for TOGACOARE* soundings to obtain a similar asymptote indicating 1/f-noise. This universality of l/f-noise is worth noting. Nevertheless, the short time-scale behaviour was not accessible by the 6-hourly (or 12-hourly) TOGA-COARE data. The present analysis was rather focused on the temporal asymptote to convective quasi-equilibrium, and no analysis is performed for the space-scale dependence of convective quasi-equilibrium. It can be well argued that the single-station data used in the present study is too heavily contaminated by small-scale noise to enable a characteristic time-scale for cumulus convection to be detected. In this respect, a spaceaveraging of the data may help better to depict the characteristictime-scale. On the other hand, both Mapes’s (1998) counterexample model and Arakawa and Schubert’s original formulation (see Yano ( 1999)) predict that convective quasi-equilibrium will deteriorate with increasing space-scale. This possibility is still to be tested carefully and we plan to analyse further the GATE and TOGA-COARE data from this point of view. The present study also points to the crucial importance of more frequent, say 1-hourly, soundings during an intensive observational period to establish this type of statistic in the tropical atmosphere. If such frequent soundings can be performed simultaneously over a multiple nested observational array, the issue of the spacedependence of convective quasi-equilibrium can be more systematically addressed. The development of a geostationary satellite with a wide range of multiple channels and high horizontal resolution, which measures the infrared radiation from the atmosphere with a high frequency (every 10 minutes, say), may be able to provide the same information as a heavily nested sounding system. Alternatively, a CRM could be run without largescale forcing, but using a sufficiently large horizontal domain.

5. CONCLUDING REMARKS

The relatively small size of the rate of change of CAPE compared with largescale forcing has been normally considered as observational proof for convective quasiequilibrium. Though it indeed establishes convective quasi-equilibrium in a diagnostic sense, it says nothing about a physical principle for convective quasi-equilibrium. We have shown by theoretical analogues that such diagnostic equilibrium can be established asymptotically by various mechanisms other than the adjustment by convective heating as proposed by Arakawa and Schubert (1974). Most notable to produce this effect is the linear primative-equation system with arbitrary convective heating. Both modelling and observational examples have shown that convective quasi-equilibrium is satisfied only asymptotically, as expected from these analogues. Such an asymptotic tendency * The Coupled Ocean-Atmosphere Response Experiment of the Tropical Ocean and Global Atmosphere programme (World Climate Research Programme).

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is clearly distinguished from the exponential adjustment to convective equilibrium anticipated by Arakawa and Schubert (1974)’s physical argument. Our finding is not inconsistent with that of Brown and Bretherton (1997), who examined the tendency of the tropical atmosphere to strict convective equilibrium in the long time and large space scale limits, and found that the tropical atmosphere is substantially distant from such a strict equilibrium state. Nevertheless, the observational basis for the asymptotic approach to convective quasi-equilibrium is still to be established by future intensive field experiments with, possibly, 1-hourly soundings. The physical significance of this asymptotic tendency is still to be further investigated. As Arakawa and Schubert (1974) pointed out, a statistical theory for the cumulus cloud ensemble is still to be constructed to answer the fundamental questions of cumulus parametrization. ACKNOWLEDGEMENTS

R. Schafer kindly provided the wind-profiler data from MCTEX with very helpful suggestions for its data quality control. Jason Beringer and Nigel Tapper provided the MCTEX surface flux data. J. I. Y.and G. L. R. are supported by the Australian Government cooperative Research Centres Program. W. W. G. is supported by the National Center for Atmospheric Research’s Clouds in Climate Program. B. E. M. was supported by a National Oceanic and Atmospheric Administration TOGA-COARE grant, jointly funded by the Office of Global Programs, National Oceanic and Atmospheric Administration and the Climate Dynamics Program, Division of Atmospheric Sciences, National Science Foundation. Comments by Kerry Emanuel, Akira Kasahara, and John L. McBride were helpful in preparing the manuscript. REFERENCES Arakawa, A.

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Emanuel, K. A., Neelin, J. D. and Bretherton, C. S. Grabowski, W. W., Wu, X.and Moncrieff, M. W.

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Grabowski, W. W., Wu, X., Moncrieff, M. W. and Hall, W. D. G M , G. A., KUO,Y.-H. and Pasch, R. J. Keenan, T., Holland, G., Rutledge, S., Simpson, J., McBride, J., Wilson, J., Moncrieff, M., Carbone, R., Frank, W., Sanderson, B., Tapper, N. and Hallett, J. Lord, S. J.

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Closure assumptions in the cumulus parameterization problem. Pp. 1-15 in The representation of cumulus convection in numerical models. Mefeoml. Mono., No. 46. American Meteorological Society Interaction of a cumulus cloud ensemble with the large-scale environment. Part I. J. Atmos. Sci., 31,674-701 Gravity waves, compensating subsidence and detrainment around cumulus clouds. J. Atmos. Sci., 46,740-759 A test of the strict quasi-equilibrium theory on long time and space scales. J. Atmos. Sci., 54,624-638 On large-scale circulations in convecting atmospheres. Q. J. R. Meteoml. Soc., 1u),1111-1143 Cloud-resolving modeling of tropical cloud systems during phase 111 of GATE. Part I: lko-dimensional experiments. J. Atmos. Sci., 53,3684-3709 Cloud-resolving modeling of tropical cloud systems during phase 111 of GATE. part II: Effects of resolution and the third spatial dimension. J. Atmos. Sci., 55,3264-3282 Semiprognostic tests of cumulus parameterization schemes in the middle latitudes. Mon. WeatherRev.,119,5-30 ‘Maritime Continent Thunderstorm Experiment (MCTEX)’. BMRC Research Report, No. 44 (available from BMRC, GPO Box 1289K. Melbourne, Victoria 3001, Australia)

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