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\begin{document}

\chapter{Convection and the Environment}

We first discuss the two classical necessary conditions long thought
to be needed for the production of deep convection, conditional instability
and a triggering mechanism. We then discuss more recently developed
criteria, which attempt to establish both necessary and sufficient
conditions for the development of convection.


\section{Conditional instability}

%
\begin{figure}
\begin{centering}
\includegraphics[scale=0.6]{sounding}
\par\end{centering}

\caption{\label{cap:sounding} Schematic of a typical conditionally unstable
atmospheric sounding. LCL is lifting condensation level, LFC is level
of free convection, and LNB is level of neutral buoyancy. Plotted
are the moist entropy and the saturated moist entropy as a function
of pressure.}

\end{figure}


Figure \ref{cap:sounding} shows a schematic of a conditionally unstable
sounding expressed in terms of the moist entropy and the saturated
moist entropy plotted as a function of height. \emph{Conditional instability}
means that finite energy must be expended to lift a parcel to the
point where it is positively buoyant.  The decrease in both saturated
and unsaturated moist entropy up to some level, followed by an increase
with height is universally found in conditionally unstable soundings.

The saturated moist entropy is the moist entropy with the vapor mixing
ratio replaced by the saturated mixing ratio. The moist entropy is
conserved in both moist and dry adiabatic processes, so the moist
entropy of a surface parcel lifted through the troposphere follows
the vertical trajectory shown in figure \ref{cap:sounding}. The saturated
moist entropy of the surface parcel initially follows a line of constant
potential temperature, which low in the troposphere slants sharply
to the left, as shown. However, when the saturated entropy line intersects
the moist entropy line, the parcel becomes saturated. This level is
called the \emph{lifting condensation level} (LCL). Subsequently the
saturated moist entropy equals the moist entropy, so both follow the
vertical trajectory of the lifted surface parcel.

Neglecting virtual temperature effects, the difference between the
saturated moist entropy of a parcel and its surrounding environment
is a measure of the buoyancy of the parcel relative to the environment.
Thus, when the saturated moist entropy of the parcel exceeds the saturated
moist entropy of the environment, the buoyancy of the parcel, which
typically is negative initially, becomes positive. This level is called
the \emph{level of free convection} (LFC). The parcel will eventually
reach its \emph{level of neutral buoyancy} (LNB), after which the
buoyancy becomes negative.

The buoyancy of a lifted surface parcel is initially negative relative
to the environment (with the exception of a possible shallow region
of positive buoyancy over sun-heated land) in almost all cases. Thus,
some energy source is required to lift the parcel initially to its
level of free convection. The required energy per unit parcel mass
is called the \emph{convective inhibition} (CIN). We now derive an
equation for CIN. Realizing that the net buoyancy force consists of
the downward force of gravity on the parcel plus the upward pressure
force which equals the weight of the displaced parcel of environmental
air $M_{e}$, the work needed to lift the parcel from its initial
level ($z=z_{I}$) to the level of free convection ($z=z_{LFC}$)
is the CIN times the mass of the parcel $M_{p}$:\begin{equation}
\mathrm{CIN}\times M_{p}=\int_{z_{I}}^{z_{LFC}}g(M_{p}-M_{e})dz,\label{eq:cin1}\end{equation}
where $g$ is the acceleration of gravity. Dividing by $M_{p}$ results
in\begin{equation}
\mathrm{CIN}=\int_{z_{I}}^{z_{LFC}}g(1-\rho_{e}/\rho_{p})dz,\label{eq:cin2}\end{equation}
where $\rho_{p}$ and $\rho_{e}$ are the densities of the parcel
and the environment.

It is more convenient to express equation (\ref{eq:cin2}) in the
form of a pressure integral, converting from geometrical height to
environmental pressure $p$ using the hydrostatic equation $dp=-g\rho_{e}dz$:\begin{equation}
\mathrm{CIN}=\int_{p_{LFC}}^{p_{I}}(1/\rho_{e}-1/\rho_{p})dp,\label{eq:cin3}\end{equation}
where $p_{I}$ and $p_{LFC}$ are the pressures at the initial level
and the level of free convection. Using the equation of state for
an air parcel with vapor mixing ratio $r_{V}$ and condensate mixing
ratio $r_{C}$,\begin{equation}
\frac{1}{\rho}=\frac{R_{D}T}{p}(1+0.61r_{V}-r_{C}),\label{eq:cin4}\end{equation}
we finally get\begin{equation}
\mathrm{CIN}=R_{D}\int_{}^{}[T_{e}(1+0.61r_{Ve})-T_{p}(1+0.61r_{Vp}-r_{Cp})]d\ln p,\label{eq:cin5}\end{equation}
where a subscripted $e$ indicates environment and a $p$ indicates
parcel values. We have assumed that no condensate exists in the environment.
The \emph{virtual temperature} is defined\begin{equation}
T_{V}=T(1+0.61r_{V}-r_{C}),\label{eq:virtualtemperature}\end{equation}
so the integrand is simply the difference between the virtual temperatures
of the environment and the parcel.

A similar integral yielding the energy released per unit mass in a
parcel ascending from the level of free convection to the level of
neutral buoyancy is called the \emph{convective available potential
energy} (CAPE):\begin{equation}
\mathrm{CAPE}=R_{D}\int_{\ln p_{LNB}}^{\ln p_{LFC}}[T_{p}(1+0.61r_{Vp}-r_{Cp})-T_{e}(1+0.61r_{Ve})]d\ln p.\label{eq:cape1}\end{equation}
The order of environmental and parcel quantities is reversed in the
integrand compared to the expression for CIN, since CAPE is energy
released by the parcel ascent, whereas CIN is the external energy
required to lift the parcel. 

Generally the negative parcel buoyancy resulting in positive $\mathrm{CIN}$
for daytime convective environments comes from a stable layer above
the lifting condensation level where the parcel is saturated. The
specific moist entropy $s$ of the parcel in this case equals the
saturated moist entropy $s_{s}$, which is a function only of the
temperature and the pressure as long as the condensed water content
is small enough to be neglected. At a given pressure level the difference
between the temperature of the parcel and the environment can therefore
be related to the difference between the moist entropy of the parcel
and the saturated moist entropy of the environment,\begin{equation}
s_{p}-s_{se}=s_{sp}-s_{se}=\frac{\partial s_{s}}{\partial T}(T_{p}-T_{e}),\label{eq:cin6}\end{equation}
where the partial derivative of the saturated moist entropy is taken
at constant pressure. Thus, to the extent that the difference between
real and virtual temperatures can be ignored, the part of the $\mathrm{CIN}$
above the lifting condensation level can be written\begin{equation}
\mathrm{CIN}_{moist}\approx R_{D}\int_{\ln p_{LFC}}^{\ln p_{LCL}}\left(\frac{\partial s_{s}}{\partial T}\right)^{-1}\left[s_{se}(p)-s_{p}\right]d\ln p.\label{eq:cin7}\end{equation}
Similarly, the $\mathrm{CAPE}$ can be approximated\begin{equation}
\mathrm{CAPE}_{moist}\approx R_{D}\int_{\ln p_{LNB}}^{\ln p_{LFC}}\left(\frac{\partial s_{s}}{\partial T}\right)^{-1}\left[s_{p}-s_{se}(p)\right]d\ln p.\label{eq:cape2}\end{equation}



\section{Control of CAPE and CIN}

CAPE and CIN are functions of the characteristics of the convective
parcel itself and the environment through which it ascends. Convective
parcels are largely produced in the planetary boundary layer, so the
characteristics of this layer and the factors controlling it such
as surface fluxes and entrainment from the free troposphere are crucial.


\subsection{Planetary boundary layer}

Almost all deep convection originates in a layer in the troposphere
called the \emph{planetary boundary layer} (PBL). This is the layer
adjacent to the surface in which turbulence acts to mix the layer.
Though in practice this mixing is never carried to completion, a useful
idealized model of the PBL assumes that the mixing keeps this layer
homogenized. Atmospheric motions can distort this layer, increasing
or decreasing its thickness and horizontal area. In addition, the
PBL generally entrains air from the quiescent free troposphere overlying
it.

\begin{center}
%
\begin{figure}
\includegraphics[scale=0.6]{pblevolve}

\caption{\label{cap:pblevolve} The left panel illustrates the evolution of
a PBL control volume to stretching or shrinking in the $x$, $y$
(into page), and $p$ directions plus addition of mass via entrainment
from above. The right panel shows the assumed form of any intensive
variable $\chi$ as a function of pressure $p$.}

\end{figure}

\par\end{center}

Figure \ref{cap:pblevolve} shows how a well-mixed intensive variable
$\chi$ in a control volume might respond to deformations and entrainment
which take place over a time interval $\Delta t$, with values at
the end of this interval represented by a superscripted asterisk.
We assume that the horizontal area of the control volume changes from
$xy$ to $x^{*}y^{*}$. Since the pressure is the weight per unit
area of the overlying air, the mass in the initial control volume,
with base pressure $p_{S}$ and top pressure $p_{T}$ is $xy(p_{S}-p_{T})/g$,
where $g$ is the acceleration of gravity. With no entrainment, the
control volume neither gains nor loses mass. If $p_{S}^{*}$ is the
base pressure after the elapse of the interval and $p_{O}$ is the
pressure the top would have had at this point in the absence of entrainment,
then conservation of mass tells us that\begin{equation}
xy(p_{S}-p_{T})/g=x^{*}y^{*}(p_{S}^{*}-p_{O})/g.\label{eq:pblmasscons}\end{equation}
Because of entrainment, the actual top of the PBL is $p_{T}^{*}=p_{O}-\Delta p$
at the end of the interval, meaning that the mass in the control volume
has increased by approximately $xy\Delta p/g$ during the interval.

We assume that the initial value of $\chi$ in the control volume
is $\chi_{B}$ and its final value is $\chi_{B}^{*}$, while the value
in the free troposphere just above the control volume is $\chi_{E}$.
If $\chi$ has an internal source per unit mass per unit time of $S_{\chi}$
and a surface flux per unit area per unit time $F_{\chi}$, then we
can compute $\chi_{B}^{*}$ assuming that it is the mass-weighted
average of the pre-existing value in the control volume and the value
in the entrained air plus modifications due to the surface flux and
the source term:\begin{equation}
\frac{x^{*}y^{*}(p_{S}^{*}-p_{T}^{*})\chi_{B}^{*}}{g}=\frac{xy(p_{S}-p_{T})(\chi_{B}+S_{\chi}\Delta t)}{g}+\frac{xy\Delta p\chi_{E}}{g}+xyF_{\chi}\Delta t.\label{eq:pblbal}\end{equation}
Letting $\Delta t\rightarrow0$ and invoking equation (\ref{eq:pblmasscons})
results in the governing equation for $\chi_{B}$,\begin{equation}
\frac{d\chi_{B}}{dt}=\frac{gF_{\chi}}{p_{S}-p_{T}}+\frac{\chi_{E}-\chi_{B}}{p_{S}-p_{T}}\omega_{E}+S_{\chi},\label{eq:pblgov}\end{equation}
where\begin{equation}
\omega_{E}=\lim_{\Delta t\rightarrow0}\frac{\Delta p}{\Delta t}\label{eq:pblentrainvel}\end{equation}
is the \emph{entrainment velocity}.

Defining the area $A=xy$ and $A^{*}=x^{*}y^{*}$, we note that $p_{T}^{*}=p_{O}-\Delta p$,
and that $p_{S}^{*}-p_{O}=(p_{S}-p_{T})A/A^{*}\approx(p_{S}-p_{T})(1-\Delta A/A)$,
where $\Delta A=A^{*}-A$. From this we can determine the thickening
rate of the PBL in pressure coordinates,\begin{equation}
\frac{d(p_{S}-p_{T})}{dt}=-(p_{S}-p_{T})\frac{d\ln A}{dt}+\omega_{E}.\label{eq:pblriserate}\end{equation}
If we define the horizontal flow velocity in the PBL as $\mathbf{v}_{B}$,
then it is easily demonstrated that\begin{equation}
\frac{d\ln A}{dt}=\nabla_{h}\cdot\mathbf{v}_{B}\label{eq:pbldivergence}\end{equation}
where $\nabla_{h}$ is the two-dimensional horizontal gradient operator.

%
\begin{figure}
\begin{centering}
\includegraphics[scale=0.6]{pblexample1}
\par\end{centering}

\caption{\label{cap:pblexample1} Growing cloud-free PBL resulting from surface
dry entropy and water vapor fluxes. The dry entropy of the PBL, $s_{DB}$,
matches the dry entropy of the environment, $s_{DE}$, at PBL top
$p_{T}$.}

\end{figure}


We now analyze an example to illustrate how these equations are applied.
For this example we assume that $d\ln A/dt=0$. Figure \ref{cap:pblexample1}
illustrates a cloud-free PBL in an environment with a linear environmental
profile of dry entropy $s_{DE}=\Gamma(p_{S}-p_{T})$ (which is conserved
in the non-condensing case) and a constant environmental mixing ratio
$r_{TE}$ of total cloud water. Constant surface fluxes of dry entropy
$F_{s}$ and water vapor $F_{r}$ are specified and possible source
terms (such as radiation for entropy) are ignored. As is observed
in developing boundary layers in light winds, the value of dry entropy
in the PBL closely matches the environmental value at PBL top. Under
these conditions equation (\ref{eq:pblgov}) reduces to\begin{equation}
\frac{ds_{DB}}{dt}=\frac{gF_{s}}{p_{S}-p_{T}},\label{eq:pblex1ent}\end{equation}
for the dry entropy and to\begin{equation}
\frac{dr_{TB}}{dt}=\frac{gF_{r}}{p_{S}-p_{T}}+\frac{r_{TE}-r_{TB}}{p_{S}-p_{T}}\omega_{E}\label{eq:pblex1moist}\end{equation}
for the mixing ratio.

The condition on dry entropy at the top of the PBL leads to\begin{equation}
s_{DB}=\Gamma(p_{S}-p_{T}).\label{eq:pblex1topcond}\end{equation}
Using this to eliminate $p_{S}-p_{T}$ in equation (\ref{eq:pblex1ent})
leads to a simple differential equation for $s_{DB}$ which has the
solution\begin{equation}
s_{DB}=(2gF_{s}\Gamma t)^{1/2}.\label{eq:pblex1entsoln}\end{equation}
Thus, the entropy of the PBL increases with the square root of time,
as does the thickness of the PBL.

The entrainment velocity equation in this case can be obtained from
equation (\ref{eq:pblriserate}):\begin{equation}
\omega_{E}=\frac{d(p_{S}-p_{T})}{dt}=\left(\frac{gF_{s}}{2\Gamma t}\right)^{1/2}=\frac{p_{S}-p_{T}}{2t}.\label{eq:pblex1entrainvel}\end{equation}


The moisture equation (\ref{eq:pblex1moist}) is slightly more difficult
to solve. However, the substitution $r_{TB}-r_{TE}=\eta/t^{1/2}$
leads to a solution in terms of $\eta$, which upon re-expression
in terms of $r_{TB}$ is\begin{equation}
r_{TB}=r_{TE}-\frac{(r_{0}-r_{TE})\Delta p_{0}}{p_{S}-p_{T}}+\frac{\Gamma F_{r}(p_{S}-p_{T})}{2F_{s}}\left(1-\frac{\Delta p_{0}^{2}}{(p_{S}-p_{T})^{2}}\right)\label{eq:pblex1moistsoln1}\end{equation}
where $r_{0}$ is the total water mixing ratio in the PBL at the time\begin{equation}
t_{0}=\frac{\Delta p_{0}^{2}\Gamma}{2gF_{s}},\label{eq:pblex1starttime}\end{equation}
 and where $\Delta p_{0}=p_{S}-p_{T}$ is the PBL thickness at this
time. Defining the dimensionless PBL thickness $\sigma=(p_{S}-p_{T})/\Delta p_{0}$,
the PBL mixing ratio equation simplifies to\begin{equation}
r_{TB}=\frac{r_{0}+(\sigma-1)r_{TE}}{\sigma}+\left(\frac{F_{r}\Gamma\Delta p_{0}}{2F_{s}}\right)\left(\frac{\sigma^{2}-1}{\sigma}\right).\label{eq:pblex1mostsoln2}\end{equation}
Thus, the PBL mixing ratio is controlled by competing mechanisms which
respectively tend to increase and decrease the mixing ratio as the
PBL thickens by entrainment. The first term in this equation tends
to decrease the mixing ratio with increasing thickness as long as
$r_{TE}<r_{0}$, i. e., when the initial boundary layer mixing ratio
exceeds that of the environment. This decrease is due simply to the
entrainment of drier air from above the PBL. The second term increases
with time at a rate which is proportional to the surface moisture
flux. Thus, the surface moisture flux must exceed a certain critical
value for the mixing ratio of the PBL to increase with time. Over
land, the moisture flux depends strongly on the wetness of the soil
and on the transpiration rate of vegetation.

Recall that a key parameter in the development of deep convection
is the PBL moist entropy. We can combine equation (\ref{eq:pblex1topcond}),
written $s_{DB}=\Gamma\Delta p_{0}\sigma$, and equation (\ref{eq:pblex1mostsoln2})
to get the simplified form of the moist entropy $s=s_{D}+Lr_{V}/T_{F}$
in the PBL, assuming that the PBL remains unsaturated so that $r_{V}=r_{T}$:\begin{equation}
s_{B}=\Gamma\Delta p_{0}\left[\sigma+\frac{1}{2\mathrm{B}}\left(\frac{\sigma^{2}-1}{\sigma}\right)\right]+\frac{L[r_{0}+(\sigma-1)r_{TE}]}{\sigma T_{F}},\label{eq:pblex1moistent}\end{equation}
where the dimensionless quantity\begin{equation}
\mathrm{B}=\frac{T_{F}F_{s}}{LF_{r}}\label{eq:pblbowenratio}\end{equation}
is called the \emph{Bowen ratio}. The Bowen ratio is the ratio of
sensible to latent heat transfer to the atmosphere from the surface.

Entrainment of dry air from above the PBL tends to decrease the PBL
moist entropy in the above formulation. This is manifested in the
second term on the right side of equation (\ref{eq:pblex1moistent}),
which asymptotically drives the mixing ratio to the free tropospheric
value for large $\sigma$. Surface moisture fluxes can counteract
this if they are large enough relative to the surface heat fluxes.
This eventuality is manifested by a small Bowen ratio. Since the development
of deep convection requires $s_{B}$ to exceed a certain threshold,
the importance of surface moisture fluxes (i. e., evaporation from
the surface) cannot be overemphasized. This can lead to positive feedback:
if the ground is already moist, more convection and precipitation
is likely to occur. However, if the ground is dry, deep convection
and precipitation are inhibited.


\subsection{Surface fluxes}

Scaling arguments originating with G. I. Taylor give us equations
for surface fluxes of entropy, moisture, and momentum. For an intensive
variable $\chi$, with PBL value $\chi_{B}$, the surface-air flux
is approximately\begin{equation}
F_{\chi}=\rho_{s}C_{E}(\chi_{ss}-\chi_{B})U_{eff},\label{eq:bulkseaairflux1}\end{equation}
where $\rho_{s}$ is the air density at the sea surface, $C_{E}\approx1\times10^{-3}$
is an exchange coefficient, and $\chi_{ss}$ is the value of $\chi$
for air in immediate contact with the surface. The quantity $U_{eff}$
is the \emph{effective surface wind speed},\begin{equation}
U_{eff}=(U_{B}^{2}+W^{2})^{1/2},\label{eq:effectivewind}\end{equation}
where $U_{B}$ is the mean PBL wind speed and $W$ is a parameter
which accounts for the averaged effect of PBL wind variability, which
is important in low wind situations. Typically we find that $W\approx3\sunit{m}\sunit{s}^{-1}$.
The exchange coefficient $C_{E}$ is a function of the atmospheric
static stability near the surface, with instability leading to larger
values of $C_{E}$. If strong stability exists in this layer, then
$C_{E}$ becomes small, effectively shutting off the fluxes.

Surface fluxes of three variables are of particular importance, entropy,
water vapor, and momentum. Over the ocean, the values of $\chi_{ss}$
are well-defined for each of these variables. For entropy and water
vapor, $\chi_{ss}$ equals respectively the saturated moist entropy
and saturation mixing ratio at the temperature and pressure of the
sea surface. For momentum, it equals the velocity of the sea surface,
generally close to zero. Over land as well as over the ocean, the
momentum flux depends on a parameter called the \emph{roughness length}
(see Stull, 1988), which has an effect on the value of $C_{E}$.

Determination of the entropy and moisture fluxes over land is more
difficult than over the ocean. The quantity $\chi_{ss}$ cannot be
specified as easily as it can over the ocean. Instead, the usual approach
is to compute explicitly the budget of heat and moisture in the soil
as a function of depth. A byproduct of this approach is the surface
fluxes of these quantities. A complicating factor over land is the
role of vegetation in transporting moisture from beneath the land
surface into the atmosphere. This leads to a large role for biological
mechanisms in the surface moisture flux over land.


\subsection{Geographically forced convection}

Geographical inhomogeneities such as topography and variability in
surface properties can cause certain regions to be more favorable
to the initiation of deep convection. Obvious cases of this convective
focusing are the initiation of convection due to the elevated heating
of the morning sun on high terrain, and differential solar heating
at the land-sea boundary. The net effect in these cases is to modify
locally the depth, entropy, and mixing ratio of the PBL, leading to
strong local variations in CAPE and CIN. Typically, deep convection
forms in regions in which the PBL has become locally thickened, as
this thickening has the effect of reducing the convective inhibition.
The thickened region of the PBL may also exhibit larger moist entropy,
which also reduces CIN and increases CAPE as well.


\subsection{Large-scale forcing}

Geophysical fluid dynamics GFD tells us how the large-scale flow works
in the earth's atmosphere. Consideration of this topic is beyond the
scope of the course, but the essential point is that GFD exerts a
strong control over the wind and temperature profiles in the free
troposphere. Thus, the free tropospheric contribution to CAPE and
CIN is largely determined by GFD. The interaction between the large-scale
flows governed by GFD and the collective behavior of convection is
a fascinating topic which we don't have time to discuss here. Large-scale
flows also govern (together with convection itself) the distribution
of moisture in the atmosphere. The distribution of moisture is of
great import to convection.

One aspect of large-scale forcing can be discussed here, namely the
effect of large-scale pressure gradients imposed from aloft on the
PBL. One of the problems posed below addresses the issue of Ekman
balance in the PBL -- a situation in which a local balance exists
between pressure gradient, Coriolis, and surface drag forces. The
actual PBL flow is often not too far from that prescribed by Ekman
balance, though recent work suggests that incorporation of the momentum
fluxes from air entrained from the free troposphere is necessary for
reasonable accuracy in some instances. Convergence in the Ekman balance
flow results in thickening of the PBL and possible destabilization
to deep convection.


\section{Development of deep convection}


\subsection{Transition to deep convection}

According to the classical picture of convective initiation, convection
occurs when sufficient convective available potential energy exists
and when the convective inhibition is locally reduced to the point
at which a convective parcel can be lifted to the level of free convection
by mechanical processes acting in the PBL. Though it is clear that
the combination of these two conditions generally produces moist convection,
it is not obvious that the resulting convection will be either deep
or precipitating, and in most cases it is neither.

%
\begin{figure}
\begin{centering}
\includegraphics[scale=0.6]{sound17}
\par\end{centering}

\caption{\label{cap:sound17} Actual sounding and cloud penetrations of a vigorous
convective cloud system in the tropical east Pacific (from Raymond
et al., 2003). The left solid line is the profile of environmental
moist entropy, while the right solid line is the saturated moist entropy
of the environment. The dashed lines are lines of constant potential
temperature or dry entropy and the dots show moist entropy values
inside the clouds. The vertical line shows the moist entropy of a
parcel lifted from the lower part of the subcloud layer.}

\end{figure}


The educated eye tells one that deep convection rarely if ever develops
directly out of the PBL as conventionally defined, but rather forms
from an extensive mass of pre-existing shallow cumulus clouds. These
clouds may start out as the well-mixed top of a PBL. However, as they
grow, the entrainment of free tropospheric air causes the mean conditions
in the cloud layer to deviate significantly from being well-mixed
with the underlying subcloud layer. The dynamics of this cloud layer
is like that of the \emph{mixing layer} envisioned by Frasier (1968)
or Raymond and Blyth (1986).

Figure \ref{cap:sound17} shows that values of moist entropy inside
a deep convective cloud over the tropical east Pacific do not approach
those of the sub-cloud layer, but are characteristic of parcels originating
from as high as $800\sunit{hPa}$. This result is typical of measurements
in many tropical oceanic clouds as well as those which form over land.

The energy for the mixing in the mixing layer comes from the fact
that the moist entropy decreases with height in this layer (see figure
\ref{cap:sounding}). If the mixing layer is saturated, then the reference
profile for static stability is one with constant moist entropy. If
the moist entropy decreases with height in this layer, then a vertically
displaced moist parcel will acquire a buoyancy which causes it to
accelerate away from its initial level. Numerical models of deep moist
convection show little evidence of entrainment of environmental air
by deep convective thermals at levels above the level of minimum environmental
moist entropy.

The mechanism by which deep convective thermals form out of cloudy
air in the mixing layer is shrouded in mystery. Such thermals must
typically be large enough to ascend to great altitude without experiencing
significant entrainment of environmental air. Is the development of
such a thermal out of a turbulent field simply the result of a statistical
accident? Does the successful thermal gain an advantage by commencing
the production of precipitation, leaving less liquid water to cause
evaporative cooling upon mixing with environmental air? Does the initiation
of ice nucleation lead to additional heating which is crucial to overcoming
the CIN? Many such questions remain to be answered.

Evidence is accumulating that deep convection is harder to produce
and sustain in environments with low rather than high relative humidity,
irrespective of the value of CAPE in the environmental sounding. This
is consistent with the idea that evaporative cooling upon mixing with
dry air is detrimental to the production of deep convection. However,
it is not known whether the mixing in question is of greatest importance
in the mixing layer, or whether the dry air saps the thermal primarily
after it leaves the mixing layer.


\subsection{Precipitation-induced downdrafts}

If precipitation falls out of a cloud into dry air, it will begin
to evaporate. This evaporation cools the air, resulting in the formation
of a precipitation-induced downdraft. (The weight of the precipitation
can by itself cause an air parcel to sink, but this negative buoyancy
is reversed as soon as the precipitation falls out of the parcel.)
In a layer in which the saturated moist entropy decreases with height,
the cooled parcel actually gains capacity to absorb evaporated water
as it descends to maintain buoyancy equilibrium with its surroundings.

%
\begin{figure}
\begin{centering}
\includegraphics[scale=0.6]{downdraftcool}
\par\end{centering}

\caption{\label{cap:downdraftcool} Cooling a parcel by evaporation in an environment
in which the saturated moist entropy ($s_{s}$) decreases (left panel)
or increases (right panel) with height. The trajectory segment BC
in each case represents the change of saturated moist entropy along
a line of constant potential temperature.}

\end{figure}


This phenomenon is shown in figure \ref{cap:downdraftcool}. Cooling
a parcel at constant height corresponds to moving it to the left in
the entropy-height plane, i. e., from point A to point B. The moist
entropy of the parcel remains unchanged during this cooling. Since
the parcel now has negative buoyancy, it sinks, with the saturated
moist entropy following a constant potential temperature line BC.
Sinking terminates when the saturated moist entropy of the parcel
matches that of the environment. At this level the parcel is again
in buoyancy equilibrium with the environment. The moist entropy of
the parcel doesn't change in these two processes, which means that
the trajectory of the parcel's moist entropy is vertical, DE.

Notice that the difference between moist entropy and saturated moist
entropy decreases when the saturated moist entropy increases with
height (right panel of figure \ref{cap:downdraftcool}), and increases
when the saturated moist entropy decreases with height (left panel).
The latter generally occurs in the lower part of the free troposphere.
An increase in this difference makes precipitation evaporate more
readily, whereas a decrease causes evaporation to be slower. Thus,
in the lower troposphere where saturated moist entropy decreases with
height, sinking due to evaporation of precipitation is a runaway process
which accelerates as the parcel sinks in the presence of precipitation,
while the reverse is true in the upper troposphere, where the saturated
moist entropy increases with height. In the latter case, if the moist
entropy of the parcel exceeds the saturated moist entropy of the environment
at some level, the parcel cannot sink beyond that level. This implies
that downdrafts reaching the surface originate primarily from the
level of minimum saturated moist entropy and below.


\subsection{Convective feedback on PBL}

So far we have discussed the possible production of deep convection
from the PBL. However, we have not yet discussed the feedbacks imposed
on the PBL by deep convection. The most important feedbacks appear
to be the extraction of updraft mass from the PBL and its partial
replacement by downdraft air. Equation (\ref{eq:pblmasscons}) representing
mass conservation in the PBL becomes\begin{equation}
xy(p_{S}-p_{T})/g-xy(M_{u}-M_{d})\Delta t=x^{*}y^{*}(p_{S}^{*}-p_{O})/g\label{eq:pblmasscons2}\end{equation}
when this is accounted for, where $M_{u}$ and $M_{d}$ are the mass
per unit area per unit time extracted from and returned to the PBL
by deep convection. Similarly, the governing equation (\ref{eq:pblgov})
for an intensive variable $\chi$ generalizes to\begin{equation}
\frac{d\chi_{B}}{dt}=\frac{gF_{\chi}}{p_{S}-p_{T}}+\frac{g(\chi_{d}-\chi_{B})M_{d}}{p_{S}-p_{T}}+\frac{\chi_{E}-\chi_{B}}{p_{S}-p_{T}}\omega_{E}+S_{\chi}\label{eq:pblgov2}\end{equation}
where $\chi_{d}$ is the value of $\chi$ in convective downdrafts,
while equation (\ref{eq:pblriserate}) becomes\begin{equation}
\frac{d(p_{S}-p_{T})}{dt}=-(p_{S}-p_{T})\frac{d\ln A}{dt}+g(M_{u}-M_{d})+\omega_{E}\label{eq:pblriserate2}\end{equation}
(see Raymond 1995).

The transports of entropy, moisture, and momentum into the PBL by
deep convection can have major effects on the PBL. This subject lies
at a sub-discipline boundary in the atmospheric sciences, and thus
has not received the attention it deserves.


\section{Precipitation and the environment}

%
\begin{figure}
\begin{centering}
\includegraphics{irsfrac}
\par\end{centering}

\caption{\label{fig:satfrac} Saturation fraction as a function of satellite
infrared brightness temperature and inferred precipitation rate in
the tropical east Pacific and the southwest Caribbean.}

\end{figure}


Just because there is moist convection does not necessarily mean that
rainfall occurs. The convection may occur in an environment which
is so dry that entrainment evaporates most of the condensed water.
Alternatively, precipitation which does form may fall into a deep,
dry boundary layer where it evaporates.

Recent work over the tropical oceans (Bretherton, Peters, and Back
2004) shows a high correlation between the rainfall rate averaged
over areas of order a few square degrees and the saturation fraction,
defined as\begin{equation}
\mathcal{F}\equiv\int_{0}^{p_{s}}r_{V}dp\left/\int_{0}^{p_{s}}r_{S}dp\right.\label{eq:defsatfrac}\end{equation}
where $p_{s}$ is the surface pressure and $r_{V}$ and $r_{S}$ are
the water vapor mixing ratio and the saturation mixing ratio. The
saturation fraction is a kind of column-integrated relative humidity.
Alternatively, it may be thought of as the ratio of precipitable water
to saturated precipitable water. Figure \ref{fig:satfrac} shows an
example of this relationship over the eastern tropical Pacific and
the southwest Caribbean sea. The applicability of this relationship
over land has yet to be determined.


\section{Form of convection}

A vast amount of work has been done in characterizing the morphology
of convective systems. This work is well described by Houze (1993)
and is treated very briefly here.


\subsection{Convective and stratiform rain}

A typical thunderstorm cell typically produces an initial gush of
heavy rain, with the intensity of the rain decreasing gradually over
an extended period. Multiple cells in an organized system can produce
an extended middle to upper tropospheric layer of stratiform cloud
which extends the lifetime and amount of this tail of light rain.
A formal distinction has been made between \emph{convective rain}
(the early period of heavy rain) and \emph{stratiform rain} (the extended
tail of light rain). The distinction is an empirical one, but it has
some physical basis. The heavy rain is typically the result of heavy
riming of ice crystals that fall back through the supercooled water
in the cell's updraft. The light stratiform rain generally consists
of ice crystals which have been blown out of the top of the updraft
into an upper level stratiform layer, where they aggregate into snow
flakes. They then fall out of the stratiform cloud, melting at the
freezing level and thus turning into small raindrops.

Radar observations of stratiform rain commonly show a \emph{bright
band} at the freezing level, i. e., a thin layer of elevated radar
reflectivity at this level. This is produced by melting snowflakes
which have yet to acquire the higher terminal velocity of raindrops,
but have developed higher reflectivity due to the partial melting
and development of a surface layer of liquid water. This bright band
does not exist in the region of convective rain, because the falling
precipitation particles are of higher density and fall speed, and
thus melts over a much broader vertical range.


\subsection{Low shear convection}

Low-entropy downdrafts produced by evaporating precipitation spread
out in the form of gust fronts when they reach the surface. These
gust fronts propagate across the surface, sometimes lifting surrounding
PBL air to the level of free convection, thus producing new deep convective
cells. If the vertical shear of the horizontal wind is weak, this
production of cells and proliferation of gust fronts is somewhat random
and disorganized. It basically proceeds until the high-entropy PBL
air is exhausted, or at least until the swarm of gust fronts is unable
to lift the remaining conditionally unstable air to the level of free
convection. Such convection is thus \emph{autocatalytic} to a certain
degree, with the initial round of convective cells contributing to
the development of others.


\subsection{Shear and squall lines}

When significant wind shear exists in the troposphere, especially
in the lowest few kilometers, the lifting of undisturbed PBL air to
the level of free convection becomes most efficient on the down-shear
side of the downdraft pools of cool air. This can lead to the development
of squall lines which propagate downshear, resulting in highly organized,
intense convective systems. An extensive literature exists on the
morphology and dynamics of squall lines. See in particular the books
of Houze (1993) and Emanuel (1994) for more information.


\subsection{Supercell storms}

When the shear is strong enough and the convective available potential
energy is large, squall lines give way to a type of self-propagating
system called a \emph{supercell storm}. These systems develop cyclonic
rotation as the result of incorporating air into updrafts with significant
streamwise vorticity. The rotation acts dynamically to cause uplift
on one of the flanks of the storm, generally the right flank (facing
down-shear) in the northern hemisphere. Houze (1993) describes the
propagation mechanism of supercell storms.

These storms are the most violent local storms that occur, and frequently
have large hail and strong tornadoes associated with them.


\section{References}
\begin{description}
\item [{Emanuel,}] K. A., 1994: \emph{Atmospheric Convection}. Oxford University
Press, 580 pp. This is a popular text on convection and has extensive
coverage of atmospheric thermodynamics.
\item [{Houze,}] R. A. Jr., 1993: \emph{Cloud dynamics}. Academic Press,
San Diego, 570 pp. Robert Houze is the undisputed master of the morphology
of convective systems.
\item [{Raymond,}] D. J., 1995: Regulation of moist convection over the
west Pacific warm pool. \emph{J. Atmos. Sci.}, \textbf{52}, 3945-3959.
This paper outlines the theory of {}``boundary layer quasi-equilibrium''.
\item [{Stull,}] R. B., 1988: \emph{An introduction to boundary layer meteorology}.
Kluwer, Dordrecht, 666 pp. A reasonably up to date treatment of the
subject of boundary layers.
\end{description}

\section{Problems}
\begin{enumerate}
\item Using the simplified moist entropy $s=C_{PD}\ln(T/T_{F})-R_{D}\ln(p/p_{R})+Lr_{V}/T_{F}$
and ignoring virtual temperature effects, rewrite equation (\ref{eq:cin5})
for convective inhibition in terms of an integral over the difference
between the saturated moist entropies of the parcel and the environment.
You may assume that the temperature difference between the parcel
and environment is small.
\item Verify equation (\ref{eq:pbldivergence}).
\item Suppose a parcel with mixing ratio $18\sunit{g}\sunit{kg}^{-1}$ but
no liquid water is lifted to an environment where the mixing ratio
is $13\sunit{g}\sunit{kg}^{-1}$. At this level the temperatures of
the two parcels are the same at $295\sunit{K}$. How much greater
is the virtual temperature of the parcel than that of the environment?
\item Suppose the PBL is supplying deep convective updrafts with mass flux
$M_{u}$. Suppose further that the convection injects convective downdrafts
into the PBL in proportion to the updraft flux: $M_{d}=\alpha M_{u}$,
where $\alpha$ is a constant. The downdraft air has moist entropy
$s_{d}$ while the updraft is fed by the PBL, with moist entropy $s_{B}>s_{d}$.

\begin{enumerate}
\item Compute the value of $\nabla\cdot\mathbf{v}_{B}$ needed to keep $p_{S}-p_{T}$
constant, assuming that $\omega_{E}=0$.
\item If the surface flux of moist entropy $F_{s}$ and $s_{B}$ are constant
in time as well as $p_{S}-p_{T}$, determine $M_{u}$. How does this
mass flux depend on $\alpha$ and $s_{B}-s_{d}$?
\end{enumerate}
\item Ekman balance: Apply the PBL budget equation to the $x$ and $y$
components of the wind in the PBL. For simplicity assume that $\omega_{E}=0$
and apply the surface flux equation with $W=0$ in order to compute
surface drag. The source term for the wind contains the horizontal
pressure gradient and the Coriolis force per unit mass. For fixed
pressure gradient and Coriolis parameter, find the vector horizontal
wind, assuming that $p_{S}-p_{T}$ is given.
\end{enumerate}

\end{document}
