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\title{Quasi-equilibrium Thinking} 
\author{Kerry Emanuel \\
Program in Atmospheres, Oceans and Climate \\
Massachusetts Institute of Technology \\
Cambridge, MA 02139, USA}

\date{}

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\centerline{\bf Abstract}

It has been almost a quarter century since the idea of quasi-equilibrium was 
introduced by Arakawa and Schubert. During this time, the idea has gained general 
acceptance as a closure for cumulus convection parameterizations, but its 
implications for the physics of large-scale circulations in convecting atmospheres 
have taken longer to be fully appreciated. In this article these implications are 
explored and contrasted with ``non-equilibrium thinking", which is still arguably 
the dominant mode of thinking about the interaction of cumulus convection with 
larger scales of motion. 

\section{Introduction}

   Statistical equilibrium thinking is natural to us in most contexts. In  
fluid 
problems for which the scales of interest are much larger than the mean free 
path between molecular collisions, we are comfortable with  dealing with the 
statistics of molecules rather than their individual  dynamics, so that 
concepts 
such as pressure and temperature are natural  and well developed. The great 
conceptual simplification brought about by  statistical thinking arises from 
explicit assumptions that the space and time scales we are 
interested in are large compared to  those characterizing the elementary 
particles or processes we are  averaging over.    

    In large-scale geophysical fluid dynamics, we have 
become comfortable with a variety of scaling assumptions that  greatly 
simplify 
thinking about the dynamics and formulating them in  numerical models. Among 
the 
first approximations we become acquainted  with are the hydrostatic and 
anelastic approximations, which filter out  sound waves. It is important to 
remember here that these approximations  are by no means equivalent to 
pretending that sound waves do not exist or  that they are not important; 
rather, we simply assume that adjustments  brought about by them are so fast 
compared to weather systems that we may  think of them as occurring infinitely 
fast. If we had to consider flows  with speeds close to or exceeding the speed 
of sound, we would be forced  to abandon these approximations and the special 
kind of thinking that  goes with them. Similarly, for flows of small Rossby 
number, we can stop  thinking about gravity waves and take it for granted that 
they bring  about adjustments that are very fast compared to the time scale of  
weather systems of synoptic scale and larger. Once again, this mode of  
thinking 
should not be regarded as pretending that gravity waves do not  exist; rather, 
we approximate their adjustment time scale as zero. The  enormous 
simplification 
that this and a few other assumptions bring about  is beautifully entailed in 
``PV thinking", as described by Hoskins et al. (1985).    

     Statistical equilibrium ideas play a crucial role in thinking about  and 
accounting for turbulence at all scales. Almost all successful  theories and 
parameterizations of three-dimensional turbulence rely on the  idea that 
kinetic 
energy is cascaded so rapidly to small scales, where it  is dissipated, that 
generation and  dissipation are nearly in equilibrium. Even the so called 
``one-and-a-half order" closure, popular  in a variety of applications, allows 
for only small departures from this  kind of equilibrium. Turbulence kinetic 
energy can respond with a small time lag to changes in generation and 
dissipation, and can be advected away from regions where it is generated.  

   Statistical equilibrium thinking is also the natural mode of thinking about 
ordinary dry convection. We regard the intensity of boundary layer convection 
as 
a statistical quantity that is directly related to the intensity of the surface 
heat flux. To a first approximation, we think of such convection establishing 
on 
a very short time scale a dry adiabatic lapse rate through the depth of the 
convecting layer. If we were asked why a circulation develops between a dry, 
sandy field and an adjacent irrigated pasture, we reply that the air over the 
pasture is cooler, owing to evaporation. We take it for granted that 
small-scale 
convection over the dry field distributes heat through the boundary layer on a 
short time scale. Few would state that the field-scale circulation arises from 
a 
spontaneous organization of small-scale convective elements. 

    In 1974, Arakawa and Schubert formally introduced their application of 
statistical equilibrium to wet convection, using virtually the same general 
idea 
that had met with some success in ordinary three-dimensional turbulence: the 
idea that generation and dissipation of turbulence kinetic energy are nearly in 
equilibrium. This followed more than a decade of false starts by quite a few 
distinguished researchers, grappling with the construction of an appropriate 
closure for wet convection. These failed largely because they did not regard 
convection as an equilibrium process, or because they assumed that 
water 
vapor, rather than energy, is the quantity in equilibrium. Among the few physically 
consistent 
representations of convection that appeared before this time was moist 
convective adjustment (Manabe et al, 1965) which, while it is not based on 
statistical equilibrium, acts in such a way as to preserve it. 

It is somewhat surprising that, almost a quarter century after the introduction 
of the idea of quasi-equilibrium, very little of its conceptual content has 
influenced the thinking of most tropical meteorologists, even while the 
parameterization itself is enjoying increasing use. It is still very common to 
hear statements to the effect that latent heating drives tropical circulations, 
or that such circulations arise from a cooperative interaction between cumulus 
clouds. In the following sections, I attempt to show why such statements are 
inconsistent with the notion of quasi-equilibrium and to trace the history of 
thinking about the interaction of cumulus convection with large-scale 
circulations.

\section{Is ``latent heating" a useful concept?}

We are all taught that the condensation of water vapor releases a comparatively 
large quantity of heat to the air in which the condensate is suspended, and are 
use to thinking of this just like any other heat source, like radiation, for 
example. The problem with this way of thinking is that it fails to recognize 
and 
take advantage of the fact that virtually all condensation in the atmosphere is 
very nearly reversible, and so may be usefully incorporated into the definition 
of the entropy of a system consisting of dry air, water vapor and condensed 
water. (Of course, the fallout of condensate is irreversible, but that is 
another matter. Note also that in contrast to condensation, freezing is not usually 
reversible as it occurs at temperatures well below 0 C.) In such a system, there is 
no ``latent heating"; phase changes 
between vapor and liquid droplets hardly affect the entropy of the system. 

The distinction between external heating and internal rearrangements of the 
terms 
that comprise the specific entropy is far from academic. For example, external 
heating applied to rotating, stratified fluid will result in a local increase 
in 
the temperature of the fluid in the vicinity of the source. But the presence of 
deep, precipitating cumulus convection in a rotating, stratified fluid may very 
well be associated with local reduction of temperature. In the first case, the 
correlation between heating and temperature is virtually guaranteed to be 
positive, while in the second case it is quite possible for the ``latent 
heating" 
to be negatively correlated with temperature, resulting in a reduction of 
kinetic energy. Thus the ``organization of convection" need not lead to the 
amplification of a disturbance. Despite this, the idea that certain types of 
tropical disturbance arise from an organization of convection persists.

We illustrate the fallacy of regarding latent heating as the cause of phenomena 
associated with convection by a few examples. 

\subsection{Dry convective turbulence}

It is helpful to start out with a problem uncomplicated by the presence of 
moisture. One relatively simple paradigm, a version of which dates back to 
Prandtl (1925), consists of a shallow layer of dry soil continuously heated by 
a 
constant imposed solar radiation, underlying a fluid cooled through a finite 
depth by a constant imposed radiative cooling. In equilibrium, the incoming 
solar radiation at the top of the system matches the total outgoing radiation. 
But conduction of heat from the soil to the overlying fluid will destabilize 
the 
latter, resulting in convection. In statistical equilibrium, the convergence of 
the convective heat flux matches the radiative cooling of the fluid. This 
paradigm is illustrated in Figure 1. 

Now consider the entropy budget of the system. By dividing the first law of 
thermodynamics through by temperature, $T$, one obtains

\beq
C_p\left( {dln(T)\over{dt}}\right)-R\left({dln(p)\over{dt}}\right) 
={Q_{rad}\over{T}}+\dot{s_{irr}}, 
\lab{1}
\eeq

\noindent where $C_p$ is the heat capacity at constant pressure, $R$ is the gas 
constant for dry air, $p$ is pressure, $Q_{rad}$ is the radiative (and 
conductive) heating, and $\dot{s_{irr}}$ represents various irreversible 
entropy 
sources. We consider the system to be closed in mass, so that integrating 
\eq{1} 
over the entire system and over a long enough time to average out the 
statistical fluctuations, we get

\beq
\int \dot{s_{irr}}=-\int {Q_{rad}\over{T}},
\lab{2}
\eeq

\noindent where the integral is over the entire system and time. Since, in 
equilibrium, the surface heating balances the net atmospheric cooling, we can 
express \eq{2} as

\beq
\int \dot{s_{irr}}= F_s\left({1\over{\overline{T}}}-{1\over{T_s}}\right),
\lab{3}
\eeq

\noindent where $F_s$ is the net radiative flux at the surface, $T_s$ is the 
surface temperature and $\overline{T}$ is the average temperature at which 
radiative cooling occurs. Now if we assume that dissipation of kinetic energy 
is 
the dominant irreversible entropy source, then the left side of \eq{3} is just 
the system integral of the dissipative heating divided by temperature. Since, 
in 
equilibrium, dissipation of kinetic energy must equal the rate of conversion of 
potential energy to kinetic energy, we can write \eq{3} as

\beq
\left( {1\over{T_{diss}}}\right) \int \overline{w'B'} = 
F_s\left({1\over{\overline{T}}}-{1\over{T_s}}\right),
\lab{4}
\eeq

\noindent where $\overline{w'B'}$ is the buoyancy flux, which is also the rate 
of conversion of potential to kinetic energy, and $T_{diss}$ is the mean 
temperature at which kinetic energy is dissipated. The expression \eq{4} tells 
us what the integrated buoyancy flux is as a function of the energy input to 
the 
system and something like a thermodynamic efficiency. Given that the 
temperature 
lapse rate is not likely to be too far off the dry adiabatic lapse rate, a very 
good estimate can be made of the mean temperature $\overline{T}$. The mean 
temperature at which kinetic energy is dissipated, $T_{diss}$ is not so easy to 
estimate, but as it appears only as an absolute value, errors in its estimate 
will not have a serious effect on the evaluation of \eq{4}. Thus the 
energy-entropy method yields an appropriate scale for the buoyancy flux in the 
system. This scale is proportional to the radiation absorbed by the surface and 
the difference between the surface temperature and a mean temperature of the 
free atmosphere. We can think of the convection as a heat engine, converting 
the 
absorbed heating into mechanical work with an efficiency proportional to the 
difference between the input and output temperatures. The engine does no work 
on its environment; instead, the mechanical energy is dissipated and locally turned 
back into enthalpy. 

Having conquered one aspect of the dry convection problem, let's apply the same 
methods to moist convection.  

\subsection{Moist convective turbulence: the naive approach}

We will use the same paradigm for moist convection, by replacing the dry soil 
used above with a thin layer of water. To make life simple, we shall assume 
that 
all of the net incoming radiation at the surface is balanced by evaporation, 
neglecting the sensible component of the turbulent surface enthalpy flux.
We shall allow the resulting moist convective clouds to precipitate, so we 
expect to see tall cumulonimbi separated by regions of clear, subsiding air. 
In 
spite of the possibly impressive appearance of such clouds, we shall continue 
to 
treat the convection statistically. The general picture is illustrated in 
Figure 
2.  

Here we are deliberately going to engage in sloppy thermodynamics, following 
the 
habits of many large-scale dynamists when they try to do thermodynamics. In 
particular, we shall forget about the dependencies of heat capacities and gas 
constants on water content, and not bother to distinguish between total 
pressure 
and the partial pressure of dry air. Following the same procedure as in the 
previous subsection, we get, from the first law,

\beq
C_p\left( {dln(T)\over{dt}}\right)-R\left({dln(p)\over{dt}}\right) 
=-{L_v\over{T}}\left( {dq\over{dt}}\right) +{Q_{rad}\over{T}}+\dot{s_{irr}}, 
\lab{5}
\eeq

\noindent where $L_v$ is the latent heat of vaporization and $q$ is the 
specific 
humidity. The first term on the right side is the latent heating term. Once 
again, we integrate this over the system, conserving mass, to get

\beq
\int \dot{s_{irr}}= -\int \left({Q_{rad}\over{T}}\right)+\int 
{L_v\over{T}}\left( {dq\over{dt}}\right).
\lab{6}
\eeq

Now we notice that, owing to the assumption that all of the absorbed solar 
radiation is compensated by evaporation, the terms on the right side of \eq{6} 
cancel when integrated through the thin layer of water. What we are left with 
is

\beq
\int \dot{s_{irr}}= -\int \left({Q_{cool}\over{T}}\right)+\int 
{L_v\over{T}}\left( {dq\over{dt}}\right)_{cloud},\
\lab{7}
\eeq

\noindent where the remaining terms on the right are the radiative cooling of 
the atmosphere and the latent heating inside clouds. Inside the clouds, the 
latent heat release shows up as an increase of potential temperature, so that

\beqd
-{L_v\over{T}}\left({dq\over{dt}}\right)\simeq 
{C_p\over{\theta}}\left({d\theta\over{dt}}\right),
\eeqd

\noindent where $\theta$ is the potential temperature. Outside the clouds, the 
radiative cooling causes a decrease in potential temperature:

\beqd
Q_{cool}=-C_p{T\over{\theta}}\left({d\theta\over{dt}}\right).
\eeqd

One can see that the two terms on the right side of \eq{7} cancel, leaving us 
with no irreversible entropy production. We have got nowhere, except to show 
that radiative cooling is balanced by radiative heating. Note also that, unlike 
the dry problem, the surface temperature vanished and plays no role. What 
happened?

\subsection{Moist convective turbulence: dotting the i's}

Let's start over again, this time being careful with the thermodynamics. We 
account for the effect of water substance on heat capacities and gas constants, 
and are careful to separate the total pressure into the partial pressure of dry 
air, $p_d$, and the partial pressure of water vapor (or ``vapor pressure"), 
$e$.
Instead of \eq{5}, we get (see Emanuel, 1994, for a derivation):

\begin{eqnarray}
&(C_{pd}(1-q_t)+C_{\em l}q_t)\left( 
{dln(T)\over{dt}}\right)-R_d(1-q_t)\left({dln(p_d)\over{dt}}\right) = \nonumber 
\\
&\ \ \ \ \ -{1\over{T}}\left( {dL_vq\over{dt}}\right) 
+qR_v\left({dln(e)\over{dt}}\right)+{Q_{rad}\over{T}}+\dot{s_{irr}}, 
\lab{8}
\end{eqnarray}    

\noindent where $C_{pd}$ is the heat capacity at constant pressure of dry air, 
$C_{\em l}$ is the heat capacity of liquid water, $q_t$ is the total (condensed 
plus vapor phase) specific water content, $R_d$ is the gas constant for dry air 
and $R_v$ is the gas constant for water vapor. Notice that, in addition to the 
modifications of the effective heat capacities and gas constants, there is an 
extra term on the right of \eq{8} that we neglected in \eq{5}: the part of the 
work done by expansion against the vapor pressure. This term does not integrate 
to zero through a closed system, owing to the variability of $q$. We can also 
re-express the latent heating term:

\beq
{1\over{T}}\left( 
{dL_vq\over{dt}}\right)={d\over{dt}}\left({L_vq\over{T}}\right)+{L_vq\over{T^2}
}
\left({dT\over{dt}}\right).
\lab{9}
\eeq

But, by the Clausius-Clapeyron equation (e.g. see Emanuel, 1994),

\beq
{L_vq\over{T^2}}\left({dT\over{dt}}\right)=R_vq\left({dln(e^*)\over{dt}}\right)
,
\lab{10}
\eeq

\noindent where $e^*$ is the saturation vapor pressure. We now combine \eq{9} 
with \eq{10}, substitute the result into \eq{8} and integrate over the system 
as 
before. In doing so, we note that, because of fall out of precipitation, $q_t$ 
is not conserved following the motion of the air and this results in some 
additional, irreversible contributions to entropy production. Using some 
integrations by parts, we get

\beq
\int \dot{s_{irr}}=-\int {Q_{rad}\over{T}}+R_vln({\cal 
H})\left({dq\over{dt}}\right),
\lab{11}
\eeq

\noindent where ${\cal H}$ is the relative humidity, $\equiv e/e^*$.  The last 
term in \eq{11} is negative definite because the vapor content can only 
increase 
by evaporation into subsaturated air; condensation always occurs with ${\cal 
H}\ =\ 1$. Therefore, it belongs on the left side of the equation, as part of 
the irreversible entropy production term. 

What happened to the latent heating term? It cancelled with a 
term we left out when doing things the sloppy way..the work against the vapor 
pressure. There is no contribution of latent heating to mechanical energy 
production when the thermodynamics is done properly. What we are left with is 
an 
equation identical in form to \eq{3}, except that there are more contributions 
to the irreversible entropy production. (A relation like \eq{3} was first 
derived for the case of moist convection by Renn\'{o} and Ingersoll, 1996.) These 
include mixing of moist and dry air, evaporation of rain and surface water into 
subsaturated air, and frictional dissipation owing to falling rain. A complete 
scale analysis of these terms was performed by Emanuel and Bister (1996), who 
showed that mechanical dissipation still dominates, so that \eq{4} remains 
approximately true. The role of moisture is to some extent hidden; its primary 
function is possibly to modify the mean temperature, $\overline{T}$, at which 
radiative cooling occurs. {\em In no event is it sensible to regard moist 
convection, in equilibrium, as being driven by ``latent heat release".} Thus 
convective scheme closures that rely on the moisture budget are doomed to 
fail, because they violate causality. Convection is not caused by moisture, or 
``moisture convergence" any more than dry convection that happens to contain 
mosquitoes is caused by ``mosquito convergence". In neither case do we deny that 
there may be a very strong {\em association} between the two, but it is not 
causal in nature. 

Now one might argue that, when convection is far from being in equilibrium with 
large-scale processes, the concept of latent heating might be more useful. 
After 
all, the first paradigm of moist convection most of us hear about is the case 
of 
explosive, deep moist convection over middle latitude continents in spring and 
summer, when potential energy, stored in a conditionally unstable atmosphere with 
a ``lid", is suddenly released by some trigger. This may be true, but in that 
case, the interaction with the environment is largely one-way and it is not 
meaningful to think about parameterizing the convection as a function of 
large-scale variables. As put very succinctly by Arakawa and Schubert, 
``Unless a cumulus ensemble is in quasi-equilibrium with the large-scale 
processes, we cannot uniquely relate the statistical properties of the ensemble 
to the large-scale variables." 

\subsection{What does equilibrium convection look like?}

It is fine to imagine what moist convection in equilibrium with large-scale 
forcing looks like (Figure 2), but what does it {\em really} look like? In the 
last decade, it has become possible to numerically simulate whole ensembles of 
convection. Figure 3 shows the distribution of low-level upward motion in a doubly 
periodic box of $180\ km^2$, using a numerical cloud model developed by the 
Center for the Analysis and Prediction of Storms (CAPS). The model includes 
representations of cloud physical and turbulent processes and is here run with 
a horizontal resolution of 2 km. A radiative cooling of the troposphere is 
imposed, and the lower surface is an ocean with fixed surface temperature. The 
model is run long enough for the domain-average precipitation to come into 
statistical equilibrium. 

The convection is more or less randomly distributed, but a careful analysis 
(Islam et al., 1993) reveals that the spacing between clouds is more nearly regular 
than random. This means that clouds are {\em less} likely to clump together 
than would be true if their spatial distribution were random. {\em There is no 
tendency toward spontaneous organization of clouds}, at least at these scales. 
(One cannot rule out the possibility of spontaneous organization at scales 
larger than the domain size.) 

Figure 4 shows what happens, on the other hand, if we now impose a background 
vertical shear of the horizontal wind in the domain. (This is done by relaxing 
the domain horizontally averaged wind toward a prescribed value at each level.) 
Now we have very clear mesoscale organization of convection, with squall lines 
(or, more accurately, arcs) lined up across the direction of the background 
shear. The mechanism by which this happens was delineated by Thorpe et al. 
(1982) and Rotunno et al. (1988); it has to do with the interaction between the 
background vertical shear with the density currents created by cold, downdraft 
air spreading out at the surface. The spacing between the squall arcs is nearly 
that of the domain size, so that the domain may not be large enough to detect 
the true spacing such lines would have in an unbounded domain. (For different 
magnitudes of the shear, however, there can be several arcs within the present 
domain.)

One may reasonably ask whether a parameterization of moist convection should be 
able to explicitly simulate the actual shape of the convection; that is, to 
distinguish 
between the forms of convection in Figures 3 and 4. The answer is no. After all, the 
large-scale forcing imposed in both cases in 
identical. (The background wind shear is not a forcing in this sense; it does 
not contribute to destabilizing the atmosphere to convection.) Fortunately, 
there is hardly any detectable difference in the equilibrium, domain-averaged 
vertical profiles of temperature and relative humidity between Figures 3 and 4, 
so that if one is after the vertical heat and moisture fluxes, it may be 
permissible to neglect the background shear. The convective momentum fluxes are 
another matter, of course, and their parameterization remains an outstanding 
problem. 
(If the relaxation toward the background shear profile is suddenly stopped in 
the simulations above, the domain average shear relaxes toward zero on a 
surprisingly short time scale, indicating mostly down-gradient momentum 
transport by the convection.)

It might be possible, on the other hand, to formulate a representation of 
convection which regards only the active clouds as the subgrid-scale elements, 
and which takes the mesoscale cold pools to be explicitly simulated by the 
model. This form of mesoscale convective parameterization would be valid if 
quasi-equilibrium holds for the interaction between cumulus clouds and 
mesoscale circulations. That is, if the clouds forming at the leading edge of 
the cold pool behave in such a way that the rate of destabilization of the 
column owing to uplift at the leading edge of cold pools is nearly balanced by 
convective stabilization by the small-scale cumulus elements, than this kind of 
mesoscale parameterization is viable. But we emphaisze that in this case, the cold 
pools must be explicitly simulated. 

\subsection{Quasi-equilibrium and convective inhibition}

One peculiarity of moist convection, with no analog in dry convection, is the 
possibility of metastable states that are stable to small perturbations but 
unstable to sufficiently large ones. Text books almost always contain examples 
of metastable soundings from places like Texas, where the degree of convective 
inhibition can be exceptionally large, even when there is a large reservoir of 
convective available potential energy (CAPE). To what extent is the presence of 
convective inhibition (hereafter CIn) consistent with statistical equilibrium?

In numerical experiments such as those described in the previous subsection, 
the experimental design virtually assures statistical equilibrium when averaged 
over sufficient space-time subdomains. How small can one make the subdomain 
before statistical equilibrium fails? Figure 5 shows the ratio of the standard 
deviation of convective rainfall to the subdomain mean, as a function of the 
size of the space-time subdomain, for a pure convective-radiative equilibrium 
experiment (Islam et al., 1993) Clearly, the statistics in this case are stable 
down to remarkably small scales. But were the same thing done for the 
experiment with shear (Figure 4), surely the statistics would be less stable 
and bigger subdomains would be necessary for quasi-equilibrium to be valid. A 
careful examination of point soundings in these experiments reveals, that, 
indeed, there is some CIn in between active clouds in all the experiments. But 
it is noticeable larger in the experiments with shear. In this case, strong 
lifting at the leading edge of the cold pools forces convection there, but the 
total amount of convection over the domain is constrained by the radiative 
cooling. Thus the convection must be suppressed in between the squall lines. The 
magnitude of the CIn is part of the quasi-equilibrium state; it is not imposed 
externally. The forecaster, trying to predict the evolution of individual 
clouds, might profit from looking at the CIn, but those who are trying to 
understand the large-scale factors that determine the mesoscale structure would 
be looking at part of the outcome, not part of the cause. 

\section{The physics of convective quasi-equilibrium}

Part of the difficulty some have in accepting the quasi-equilibrium postulate 
may have to do with problems visualizing how it may work in nature. In the case 
of dry boundary layer convection, it is relatively easy to understand the 
process. Suppose, for example, that the rate of radiative cooling is increased 
in some individual atmospheric layer above the surface. At first, this layer 
may be expected to cool. But as soon as it does so, it is more unstable with 
respect to the air just below it, and less unstable with respect to the air 
just above it. This provides not only for an increase in the convective heat 
flux from the lower layer, but a decrease of the flux to the higher layer; both 
act to increase the convergence of the convective heat flux, thus warming the 
layer. 

It is more difficult to imagine what happens in a moist convecting layer. Start 
with a state of pure radiative convective equilibrium, and to make life simple, 
specify the radiative cooling profile. Now suppose we increase the rate of 
cooling in some atmospheric layer above the subcloud layer. If this layer 
happens to be {\em just} above the subcloud layer, then it is not difficult to 
see that the convective flux from the boundary layer will increase, just as in 
the dry case, and there will be a compensating warming. But what happens if the 
extra cooling is introduced to a layer far removed from the subcloud layer? The 
subcloud layer simply cannot know directly about this development and there is 
little or no basis for thinking that there will be a compensating increase in 
mass flux out of the subcloud layer. Even if there were, this would entail an 
extra warming not only in the layer to which we added the cooling, but to all 
layers below that layer. The warming of these other layers, to which we did not 
add extra cooling, would quickly stabilize the lower atmosphere and cut off the 
convection. 

There are two ways that nature resolves this paradox, as becomes evident on 
examining the response of explicit ensembles to changes in imposed cooling 
rates. First, the mass flux can increase in the individual layer to which we 
add extra cooling {\em without} increasing the mass flux out of the boundary 
layer. This occurs because of entrainment. While the exact physics of 
entrainment into cumulus clouds are not well understood, it is becoming 
increasingly clear that the rate of entrainment is sensitive to the vertical 
gradient 
of the buoyancy of the clouds (Bretherton and Smolarkiewicz, 
1989). Cooling an individual layer will have the effect of increasing the 
buoyancy of clouds rising into that layer. This increases the upward 
acceleration of air in the clouds and leads to greater entrainment just below 
the layer of extra cooling. This in turn increases the mass flux in the layer. 
The increased compensating subsidence outside the cloud warms the layer, 
opposing the initial added cooling. The physics is very different from what 
happens in the dry case, but the effect is the same. 

The second response to the presence of a layer of extra cooling is entailed in 
the precipitation physics. Adding cooling to the system means that, to reach 
equilibrium, there must be an increase in precipitation. How this happens is 
complex, but it is crucial to recognize that any increase in precipitation will 
also, in general, increase the magnitude of any unsaturated downdrafts driven 
by evaporation of precipitation. This will generally occur below the layer into 
which extra cooling has been added. Since no cooling has been added there, the 
increased downdraft mass flux must be compensated by an increased updraft mass 
flux. One may think of it this way: the upward mass flux compensates not just 
the imposed radiative cooling, but also the (interactive) evaporative cooling.  
So there {\em can} be an increase in {\em updraft} mass flux out of the 
subcloud layer. This can help warm the layer to which the extra cooling has 
been added. 

Entrainment and adjustments of the unsaturated downdraft are together very 
effective in compensating for changes in the imposed forcing. To illustrate 
this, Figure 6 shows the imposed radiative cooling profiles and equilibrium 
convective heating profiles for a variety of experiments using a single-column 
model with the convective scheme of Emanuel (1998). This is not explicitly a 
quasi-equilibrium scheme. Instead, it calculates the cloud base updraft mass flux 
based 
on an assumption of quasi-equilibrium of subcloud layer air with respect to the 
air {\em just above} the subcloud layer, as advocated by Raymond (1995). But, 
unlike the general quasi-equilibrium closure of Arakawa and Schubert (1974), 
the mass flux above cloud base is not calculated explicitly from a 
quasi-equilibrium assumption; rather, the rate of entrainment into clouds is 
allowed to respond to vertical variations of cloud buoyancy. 

It is evident in Figure 6 that even bizarre profiles of imposed radiative 
cooling are compensated by the net convective heating profiles, demonstrating 
the efficacy of the adjustment process. Figure 7 shows that the resulting 
temperature profiles are all very close to a moist adiabatic profile. Thus the 
assumption that convection relaxes the temperature profile of a convecting 
layer back towards a moist adiabat is well verified in this model. David Neelin 
discusses in this volume the profound implications that this has for 
understanding tropical dynamics. 

\section{Non-equilibrium thinking}

Most students of meteorology are conditioned to think of convection in 
non-equilibrium terms, being first introduced to the concept of conditional 
instability through the illustration of highly metastable soundings from places 
like Oklahoma. Instability accumulates under some ``lid" and is released 
suddenly when convective temperature is attained or when some mesoscale process 
locally removes the potential barrier to convection. This may very well be an 
appropriate mode of thinking about the type of convection that often results in 
severe thunderstorms. But it is probably inappropriate for thinking about many 
tropical circulation systems. 

No where is the disparity between equilibrium and nonequilibrium thinking more 
on display than in discussions about hurricanes. As reviewed very thoroughly by 
Yanai (1964), most of the earliest attempts to model hurricanes, beginning in 
the early 1960's, focused on finding a particular mode by which stored 
conditional instability is released. As earlier theoretical studies had predicted, 
conditional instability is released at the scale of individual clouds. All 
attempts to run numerical simulations of hurricanes as modes of release of 
conditional instability failed to produce a hurricane-scale vortex. Earlier 
theoretical work by Riehl (1950) and Kleinschmidt (1951) had shown that the 
warmth of the eyewall could only be explained by the enormous enthalpy transfer 
from ocean to atmosphere that occurs in the high wind region of the storm. 
Although the principals involved in this work were undoubtedly aware of this 
earlier theoretical work, they evidently considered the heat transfer to be a 
secondary issue. 

The failure of these earliest attempts at numerical simulation formed a large 
part of the motivation behind the development of the theory of Conditional 
Instability of the Second Kind (CISK) by Charney and Eliassen (1964) and Ooyama 
(1964). The history of the development of CISK is reviewed very nicely in the 
chapter by Kasahara in this volume. The idea of CISK was stated very 
beautifully by Charney and Eliassen (1964):
\vspace{0.2in}

\begin{minipage}{4.5in}
{\em ``...we should look upon the pre-hurricane depression and the cumulus cell 
not as competing for the same energy, for in this competition the cumulus cell 
must win; rather we should consider the two as supporting one another - the 
cumulus cell by supplying the heat energy for driving the depression, and the 
depression by producing the low-level convergence of moisture into the cumulus 
cell."}
\end{minipage}
\vspace{0.2in}

In my view, a fatal flaw was introduced into thinking about tropical 
circulations by this enormously influential work. It is the idea that latent 
heat release can ever be an energy source for equilibrium circulations, an idea 
disproved in section II above. This flaw was exacerbated by later work that 
also introduced the incorrect notion that the vertical profile of convective 
heating is an internal property of the convective clouds that can, to a first 
approximation, be specified independently of the environment. 

The Charney and Eliassen work attempted to demonstrate CISK by posing a balanced 
model in which, as in the case of unbalanced models, the latent heat release is 
taken to be proportional to vertical velocity but, unlike unbalanced models, the 
vertical velocity was constrained to be that associated with Ekman pumping. Thus 
constrained, the model dutifully produced a linear instability with tropical 
cyclone-like characteristics, but even in this case the most rapidly growing 
modes were of small scale. 

The difference between nonequilibrium (CISK) thinking and equilibrium thinking, 
in the case of a tropical cyclone, is illustrated in Figure 8. In nonequilibrium 
thinking, the ambient atmosphere has a reservoir of usable potential energy for 
convection. The tropical cyclone is a means of releasing that instability on a 
large scale. In equilibrium thinking, the storm passes through an infinite 
sequence of convective equilibrium states, and the warmth of the eyewall is a 
consequence of the energy flux from the ocean. In reality, there is always some 
stored potential energy to balance dissipation in clouds (see Emanuel and Bister, 
1996), and there is never perfect equilibrium in an evolving system. 
Nevertheless, as subsequent work (Emanuel, 1989) showed, approximating the 
evolution as passing through an infinite sequence of equilibrium states yields a 
realistic numerical simulation.

Five years after the CISK papers were published, Ooyama (1969) presented the 
first genuinely successful numerical simulation of a tropical cyclone. It 
captured the essential physics of the intensification process, and documented the 
sensitive dependence of the vortex evolution on the exchange coefficients of 
enthalpy and momentum at the sea surface. It confirmed the deductions of Riehl 
(1950) and others that surface enthalpy fluxes are essential; when they are 
excluded from the model very little happens. 

One might have thought that Ooyama's and subsequent numerical simulations would 
finally lay to rest the notion that the controlling process in tropical cyclone 
intensification is some kind of cooperation between the cyclone circulation and 
cumulus clouds. That this has not happened is a testament to the attractiveness 
of nonequilibrium thinking and, I think, the eloquence of the Charney and 
Eliassen paper. The Ooyama simulation left an opening for CISK adherents: It 
began with a highly unstable vertical sounding and used a cumulus 
parameterization that could not switch on unless there was moisture convergence 
in the boundary layer. Together, this meant that the resting state was indeed 
linearly unstable to CISK, as shown by Ooyama in the same paper. But it was also 
clear from the nonlinear simulations that, even with the loaded gun, CISK could 
account for no more than a trivial initial intensification. In a later paper, 
Ooyama (1982) downplayed the significance of the linear instability, and Klaus and 
Reeder 
(1997) showed that Ooyama's original model, when initialized with a neutral 
sounding, 
is metastable in the same sense as the model of Emanuel (1989) which also begins 
with a neutral sounding. The latter model showed that a perfectly acceptable 
numerical simulation of a tropical cyclone results even under the extreme 
assumption that the model is always precisely neutral to cumulus convection. 

Craig and Gray (1996), using a cloud-resolving, nonhydrostatic primitive equation 
model of 
a tropical cyclone, showed that, at all stages of intensification, the rate of 
development is directly proportional to the surface enthalpy exchange coefficient 
and inversely proportional to the drag coefficient, in direct contradiction to 
the predictions of CISK. 

The most recent variation of nonequilibrium thinking holds that tropical cyclones 
cannot develop until the rotation is strong enough that the local deformation 
radius is of the same order as the scale of convective clouds. This notion was 
introduced by Ooyama (1982) and furthered by Hack and Schubert (1986). The general 
idea is that an isolated 
heat source cannot produce finite warming in a nonrotating, infinite fluid, 
because the warming is spread over an infinite region. Warming only commences 
when the deformation radius becomes appreciably small. 

The problem with this idea is that it postulates a single, noninteractive cloud 
whereas equilibrium thinking demands that we consider an ensemble of clouds always 
in equilibrium with the forcing. A large-scale circulation with a large 
deformation radius can be efficiently warmed by an array of clouds and there is 
no physical reason to postulate a single cloud. On the contrary, doing so is 
unphysical. In a nonrotating unbounded domain initially in radiative-convective 
equilibrium, a sudden increase in the surface fluxes will be accompanied by a 
rapid increase in the domain temperature at all levels up to the tropopause, in 
spite of the total absence of rotation. There are many dynamical reasons why the 
rate of intensification of tropical cyclones may depend on their intensity, but 
the ratio of the cloud scale to the local deformation radius is not one of them, 
as demonstrated by tropical cyclone models that constrain the free atmospheric 
temperature to always lie on a moist adiabat determined by the thermodynamic 
properties of boundary layer air. 

Non-equilibrium thinkers continue to hold that tropical cyclones are powered by 
latent heat release in cumulus clouds, though surface energy fluxes are of course 
necessary to maintain the reservoir of conditional instability. This is analogous 
to claiming that cars are powered by their drive trains, though engines are of 
course necessary to keep torque on them. In both cases the awkwardness arises 
from a failure to separate time scales....in the case of a car, the time scale 
for transmission of torque through the drive train is tiny compared to the time 
scale determined by the power of the engine and the inertia of the car; in the 
case of the hurricane, the time scale for transmission of enthalpy by convection, 
of order a few hours, is small compared to the time scale determined by the rate 
of surface enthalpy flux and the inertia of the cyclone. No numerical simulation 
has ever demonstrated a critical role for convective time scales in the evolution 
of the vortex. 

One final point to be made about non-equilibrium thinking concerns ordinary 
conditional instability itself. In non-equilibrium thinking, the degree of 
conditional instability is thought of as an external condition that determines 
various properties of the convection. In equilibrium thinking, by contrast, the 
degree of conditional instability is determined by the convection itself, 
together with the forcing.  Quasi-equilibrium does not imply actual invariance of 
CAPE, any more than quasi-geostrophy implies invariance of velocities. In the 
quasi-equilibrium view, both the intensity of the convection and CAPE are 
determined by the forcing and, to second order, the time rate of change of the 
forcing. CAPE is not a predictor, though it can be an indicator. 

\section{Equilibrium thinking}

The underlying proposition in quasi-equilibrium thinking is that convection 
rapidly adjusts the temperature profile back toward a moist adiabat in a way that 
preserves the vertically integrated enthalpy. To a first order of approximation, 
convection keeps the atmospheric temperature profile on a moist adiabat tied to 
the subcloud layer entropy. This strongly constrains the vertical structure of 
the horizontal and vertical velocities as well as the temperature perturbations 
associated with large-scale disturbances in convecting atmospheres, as discussed 
by Neelin and Yu (1994) and Emanuel et al. (1994). It also means that, in 
convecting atmospheres, the problem of predicting the evolution in three 
dimensions of atmospheric variables reduces largely to the problem of predicting 
the evolution of subcloud layer entropy. 

For all disturbances with time scales appreciably greater than the time scale of 
convective adjustment, the vertical structure of the disturbance is completely 
determined by the shape, in $T-p$ space, of a moist adiabat (Neelin and Yu, 1994). 
Moreover, the static stability felt by such disturbances is not related to the 
degree of conditional instability but rather to a ``gross moist stability" that 
depends on the shape of the vertical moist static energy profile (Neelin and Yu, 
ibid.). Calculations of the distribution and magnitude of this gross stability have 
been presented by Neelin (1998). The exact magnitude of this stability measure, on 
the other hand, depends not only on the shape of the moist static energy profile but 
on the relationship between static energy and moisture fluctuations. This 
relationship, in turn, depends on the details of cloud microphysical processes, 
which are poorly understood. 

One of the most basic issues we may address in quasi-equilibrium
thinking is what happens when an internal gravity wave passes through
a background atmosphere in radiative-convective equilibrium. (Note
that the equivalent question in nonequilibrium thinking is 
what happens when an internal wave passes through a cloudless
background atmosphere but which contains stored CAPE.) To assert that the
convection remains close to a state of statistical equilibrium with
the large scale, we must assert that the horizontal wavelength is
much larger than the intercloud spacing of the background state and
that the wave period turns out to be much longer than the convective
adjustment time scale.

Note first that, as of this writing, the answer to the question posed
above has not been obtained by direct numerical simulation with
cloud-resolving models.  Such models, at least in three dimensions, are
not quite capable of simultaneously containing a reasonably large
gravity wave and resolving individual clouds.

The first and most basic suggestion is how the radiative-convective
equilibrium atmosphere responds to large-scale ascent and descent.
To begin with, we make use of the fact that, so long as all
convective kinetic energy  is locally dissipated, convection does
not alter the mass integral of the system moist state energy, $h$:

\beqd
h= C_p T + L_v q +gz.
\eeqd

Thus, if we neglect perturbations to the radiative heating, the
equation for the vertically integrated moist static energy
perturbation from the mean state is

\beq
{\partial[h^\prime]\over \partial t} = - [\omega^\prime {\partial
\overline h\over \partial \rho}],
\lab{12}
\eeq

\noindent where the brackets denote an integral over the mass of the convecting
layer, and the overbar signifies the background state.  As discussed
earlier, the vertical structure of $\omega^\prime$ is determined by the
condition that the temperature profile remains moist adiabatic.
Thus, the sign of the response of $h^\prime$ to ascent or descent
depends on the convolution of the vertical structure function of
$\omega$ with ${\partial \overline h/\partial p}$, as pointed out by Neelin and
Yu (1994).  It is straightforward to calculate this from atmospheric
soundings, and this has been done (Neelin, 1998).  The conclusion is
that this measure of stratification is stable throughout the
atmosphere, meaning that upward motion will be associated with
decreasing $h^\prime$.

Now if $h^\prime$ decreases, then either $T^\prime$ decreases, $q^\prime$
decreases, or both.  In all of the observational studies of which I
am aware, ascent is associated with increasing $q^\prime$, implying
that $T^\prime$ must decrease with $h^\prime$.  Moreover, even if the
extreme assumption is made that the relative humidity is invariant
with large-scale vertical motion, it is easy to show that $T^\prime$
must have the same sign as $h^\prime$:

\begin{eqnarray}
h^\prime &=& c_pT^\prime + L_vq^\prime \nonumber \\
         &=& c_p T^\prime + L_v {\cal H} ({\partial q^*\over \partial
T})_p  T^\prime \label{13},
\end{eqnarray}

\noindent where $\cal H$ is the relative humidity.  Since $(\partial q^*/\partial
T)_p$ is positive, $T^\prime$ and $h^\prime$ must have the same sign when
$\cal H$ is constant.  For $T^\prime$ to have the opposite sign as
$h^\prime$, $\cal H$ would have to have a rather strong negative
correlation with ascent, which is certainly not observed.

Thus, although the exact value of the response of temperature to
ascent depends on the details of how clouds humidify the atmosphere,
there is little question that ascent causes cooling, just as in a
dry, stable atmosphere.   The magnitude of this cooling is less, and
sometimes much less, than in a nonconvecting atmosphere.

It pays to consider the same process from another angle, introduced
by the author (Emanuel, 1989).  It begins with the observation that
the temperature of convecting atmospheres lies close to an adiabat
originating in the subcloud layer.  Thus, in strict statistical
equilibrium as defined by Emanuel et al. (1994), the temperature of
the troposphere is tied to the moist static energy of the subcloud
layer.  Now, if large-scale ascent occurs, there will be convergence
in the subcloud layer but this does {\em not} change the 
moist static energy, which is, after all, a conserved variable.  But
to keep the deep troposphere approximately moist adiabatic, there
must be enhanced convection to counter the adiabatic cooling
associated with the large-scale ascent.  Enhanced convection will be
associated also with enhanced downdrafts, which import low static
energy into the subcloud layer.  The reduction of subcloud-layer
moist static energy will then be associated with a net cooling of the
lower troposphere.  This argument is fully equivalent to the
preceding one, but makes the importance of downdrafts more
explicit.  Reducing the relative magnitude of the response of the 
convective downdraft moist static energy flux reduces the effective
stratification.   When this flux vanishes, as in a saturated
atmosphere, so does the effective stratification.

Thus we may expect a large-scale gravity wave, propagating through a
radiative-convective equilibrium background state, to propagate much
more slowly than an equivalent wave propagating through a cloudless
atmosphere with the same stratification.

But there is, after all, a difference.  In the case of a dry
atmosphere, there is no time lag between vertical displacement and
buoyancy.  On the other hand, it takes time for convection to respond
to changes in its environment.  This time lag is probably on the
order of hours, but its effect on the large-scale dynamics is
important.  Consider again the problem of an internal gravity wave
passing through a background state in radiative-convective
equilibrium.  As described before, upward motion will be associated
with cooling, and, ordinarily, the lowest temperatures will occur 1/4
cycle after the strongest upward motion.  Now take into account the
small lag in the response of the convection.  Now the convection,
rather than being precisely in phase with the wave ascent, will
slightly lag it, thus shifting slightly toward the cold phase of the
disturbance.  This will cause the convective heating to have a
negative correlation with temperature, thereby draining
perturbation potential energy from the wave.  This effect is called
{\em moist convective damping}.  This was shown by the author (Emanuel,
1993) to damp linear equatorial waves of all kinds in a simple model
with a quasi-equilibrium-type convective closure.  Neelin and Yu
(1994) showed it more generally to be true of linear disturbances in
a vertically continuous atmosphere using the Betts (1986) convection
scheme, and Brown and Bretherton (1995) demonstrated it in a linear
model using the Emanuel (1991) convection scheme.  Here I demonstrate
that it applies to a quasi-linear, two-column model.  The model is linear in the
sense that it ignores wave-wave interaction, and retains strictly
sinusoidal spatial distribution of the wave variables, but it allows
for arbitrary amplitude of the wave.  Nonlinear terms in the momentum
equation are ignored, but are retained in the thermodynamic equation.
The linearized inviscid, hydrostatic vorticity equation in a
nonrotating atmosphere may be written

\beq
{\partial ^2\over \partial z^2}  {\partial w\over \partial t} =
\nabla_2^2 B,
\lab{14}
\eeq

\noindent where $w$ is the vertical velocity, $B$ is the buoyancy and $\nabla_2^2$ 
is
the horizontal Laplacian operator.  Assuming horizontally sinusoidal
disturbances with combined horizontal wavenumber $K$, \eq{14} becomes

\beq
{\partial ^2\over \partial z^2}  {\partial w\over \partial t} = -
K^2B.
\lab{15}
\eeq 

\noindent This equation can be solved in a single column, if $K$ is specified.
One problem that arises in doing this, however, is that to define the
buoyancy, B, one must define a background state.  This is easy enough
for infinitesimal disturbances, but for real disturbances the
background state may itself change.  To avoid this difficulty, we
solve \eq{15} in a two-column model, where each column is out of phase
with the other ($w_1 = -w_2$) and define $B$ as

\beq
B_i = {1\over 2} {g\over \overline T_v} (T_{vi} -T_{vj}),
\lab{16}
\eeq

\noindent where the subscript $i$ denotes the column in question and the
subscript $j$ denotes the other column, while $\overline T_v \equiv
{1\over 2} (T_{vi} + T_{vj})$.  Each column is identical to the single
column model described in section III, except that vertical advection
terms are added to the heat and moisture equations, and \eq{15} is also
solved in each column.  Moreover, to guarantee global energy
conservation, heat and moisture are advected from one column to the other
according to the horizontal circulation implied by mass continuity
and $w$.

The two-column model is initialized with an arbitrary perturbation to
the buoyancy.  Figure 9 shows the time evolution of the vertical
velocity at a particular level in one of the two columns, for
wavenumber 1.  One observes a decaying oscillation with a period of
roughly 40 days.  Examination of the behavior of the system for other
specified horizontal wavenumbers, $K$, shows the same general behavior,
with the period decreasing and the rate of decay increasing with
increasing $K$.

This is completely consistent with earlier results by Emanuel (1993),
Neelin and Yu (1994), and Brown and Bretherton (1995).  We can say
with some certainty that internal gravity waves propagating through a
background state in radiative-convective equilibrium will be damped.

This finding is at odds with CISK models, dating back at least to
Lindzen (1974).  What is different about these models?  There are two
key elements of CISK models that allow them to produce unstable
internal waves.  The first is the partial or complete decoupling of
convective heating from buoyancy which occurs when the rate of
heating is related to moisture convergence.  This permits buoyancy to
accumulate in one phase of a wave and be reduced in another and so is
explicitly contrary to the quasi-equilibrium hypothesis.  The
second is, in the case of some models, a specification of the
vertical profile of convective heating that is independent of the
vertical profile of cloud buoyancy.  This also allows for in-phase
fluctuations of buoyancy and heating.

Other aspects of quasi-equilibrium thinking are discussed in Emanuel
et al. (1994).  The two most important conclusions that emerge from a
quasi-equilibrium point of view are:


\begin{itemize}

\item Large-scale disturbances in convecting atmosphere ``feel" a reduced,
but still positive, effective static stability

\item Such disturbances are also damped in proportion to their frequency.
\end{itemize}

If this mode of thinking is correct, then we must turn away from
convection per se as an ``explanation" for large-scale disturbances
and think of it instead as a means of rapidly redistributing enthalpy
in the vertical.  Potential candidates for the sources of large-scale
disturbances in the tropics include

\begin{itemize}

\item Horizontal gradients of the radiative-convective equilibrium
temperature.  These are responsible for the Walker-Hadley and monsoon
circulations, for example.  (Held and Hou, 1980; Plumb and Hou,
1992) 

\item Wind-induced surface heat exchange (WISHE).  The feedback between
wind and surface enthalpy flux is responsible for tropical cyclones
and many play a role in other tropical phenomena.

\item Transmission of wave energy from outside the tropics.  There are now
well-documented instances of this phenomenon (Kiladis, 1998).

\item Dynamical instabilities.  The instability of the African easterly jet
in summer is the source of easterly waves, for example.

\end{itemize}

\section{Summary}

Until quite recently, quasi-equilibrium has been thought of primarily as a closure 
for convective parameterizations; its effect on the way we think about convection 
has been relatively slow to come about. In this paper I have reviewed and in some 
small ways extended Emanuel et al.'s (1994) exploration of the full implications of 
quasi-equilibrium. The main structure of quasi-equilibrium thinking emphasizes the 
following points:

\begin{itemize}

\item Latent heating is a concept that applies to the dynamics of individual clouds. 
In contrast, it plays {\it no role} in the energetics of cumulus ensembles. 

\item The state of radiative-convective equilibrium serves as the basic equilibrium 
state for quasi-equilibrium thinking in the same way that an east-west baroclinic 
flow serves as the basic state for quasi-geostrophic thinking about many 
mid-latitude flows. 

\item Disturbances with space and time scales much larger than convective 
overturning time scales and inter-cloud spacing characterizing the 
radiative-convective equilibrium state may be considered to be in quasi-equilibrium 
with the convective clouds.  

\item Such disturbances ``feel" an effective stratification that, while positive, is 
much less than typical dry stratifications. The stratification may also be related 
physically to drying of the subcloud layer by convective downdrafts. The effective 
stratification vanishes when the large scale becomes saturated, as happens in the 
core of tropical cyclones. 

\item Such disturbances are also damped in proportion to their frequency. This tend 
to filter high frequency disturbances and to damp most nascent tropical depressions.

\item Convection that is not close to being in equilibrium with explicitly simulated 
flows cannot be parameterized as a function of the explicitly resolved variables. 

\end{itemize}

A full appreciation of the consequences of quasi-equilibrium will no doubt lead to 
important advances in understanding and predicting large-scale disturbances in 
convecting atmospheres. 

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\end{thebibliography}

\vfill\eject

\centerline{\bf Figure Captions}

\vspace{0.4in}

\begin{enumerate}

\item Radiative-convective equilibrium over dry land. Solid arrows denote longwave 
radiative flux, which increases upward; dashed arrows denote turbulent convective 
heat flux, which decreases upward. There is no net flux divergence except at the 
surface, where it is balanced by absorption of solar radiation.

\item Radiative-convective equilibrium over a water surface. Arrows denote longwave 
radiative flux. 

\item Distribution of upward motion at 250 m in a three-dimensional numerical 
simulation of radiative-convective equilibrium over a water surface. The simulation 
has reached statistical equilibrium at this time. From Robe (1996).

\item As in Figure 3, but for a simulation with an imposed vertical wind shear from 
right to left, in the lowest 3 km. From Robe (1996).

\item Ratio of the variance to the domain-average of the precipitation in a 
three-dimensional numerical simulation of radiative-convective equilibrium over a 
water surface, as a function of space-time averaging. The ordinate is the length of 
time averaging while the abscissa is the length of averaging in space. This ratio 
asymptotes to $\sqrt{2}$ for short averaging intervals. From Islam et al. (1993).

\item The heat budget of a single-column model in radiative-convective equilibrium, 
showing the rate of heating as a function of pressure. In each case, the solid line 
denotes the convective heating rate, the dashed line the (imposed) radiative heating 
rate, and the thin dashed line the convergence of the dry tubulent heat flux.
\begin{description}
\item{a} Uniform radiative cooling in the troposphere.
\item{b} Same as (a) but with added cooling in the 400-500 mb layer.
\item{c} Same as (a) but with zero cooling in the 850-950 mb layer. 
\item{d} No cooling in the 500-950 mb layer. This shows that convection can 
penetrate even a deep layer of no large-scale destabilization. 
\end{description}

\item Departure of the ambient temperature from a reference moist adiabat for each 
of the experiments illustrated in Figure 6. A positive value means that the 
reference adiabat is warmer than the atmosphere. 

\item Two views of tropical cyclone physics. In each case, the dashed lines indicate 
surfaces of constant saturation $\theta_e$.
\begin{description}
\item{a} The CISK view. The frictionally induced inflow allows for convection in the 
core. The core region is warm because the convection is there and not in the outer 
region. The role of the ocean is to keep the subcloud layer ``stoked"; i.e., to 
prevent its $\theta_e$ from decreasing in the face of convective downdrafts. 
Intensification stops when the free troposphere saturation $\theta_e$ is the core 
increases to the initial value of $\theta_e$ in the boundary layer. 

\item{b} The WISHE view. The saturation $\theta_e$ is tied to the subcloud layer 
$\theta_e$. The core is warm because surface fluxes have increased $\theta_e$ there. 
The rate of intensification is limited by the surface fluxes, not by the convection, 
which is very fast by comparison. Intensification stops when the frictionally 
induced radial advection of low $\theta_e$ air balances the surface enthalpy flux. 
\end{description}

\item Variation with time of the pressure velocity ($\omega$) at 600 mb in a 
two-column model that has been perturbed away from radiative-convective equilibrium. 

\end{enumerate}

\end{document}             % End of document.

