Lectures 4 & 5: Interaction of convection and large
scales, and tropical dynamics
Adam Sobel
Columbia University
ahs129@columbia.edu

December 21, 2013

Adam Sobel (Columbia)

Lecture 4

December 21, 2013

1 / 71

Overview
1

Large-scale heat and moisture budgets

2

Ascending and descending motion, rainy and dry weather,radiative-convective equilibrium

3

Tropical wave dynamics

4

Free wave solutions

5

Steady, forced-dissipative (Gill) problem

6

Interactive convection - quasi-equilibrium theory

7

The moisture mode

8

More on convective parameterization

9

Conserved variable budgets, gross moist stability

10

Observed relationships between moisture and other fields
Adam Sobel (Columbia)

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December 21, 2013

2 / 71Equations

Following Yanai et al., we write the prognostic equations for dry static
energy and specific humidity in flux form in pressure coordinates:∂s∂+ ∇ · sv +(ωs) = QR ,
∂t
∂p
∂q
∂
+ ∇ · qv +(ωq) = e − c,
∂t
∂p

(1)
(2)

Here the dry static energy s = cp T + gz, q is specific humidity, c and e are
the condensation and evaporation rates of water (in units of kg m−3 s −1 ),and QR is the radiative heating rate (J kg −1 s −1 , K s −1 if you divide by
cp ). The divergence operator ∇· represents a horizontal divergence only.

Adam Sobel (Columbia)

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Now consider ensemble averages — or ”grid box averages”, under thecommon assumption that on the scales we are interested in there aremany clouds in a ”grid box” — and deviations from those averages,
obeying standard Reynolds averaging rulesa = a + a0
ab = ab + a0 b 0(3)
Then apply the averaging operator to the equations, and rearrange:∂s∂+ ∇ · sv +(ω s) = Q1 ,
∂t
∂p
∂q
∂
+ ∇ · qv +(ω q) = −Q2 /lv ,
∂t
∂p
where
∂ 0 0
Q1 = QR + lv (c − e) −ωs,∂p∂
Q2 = lv (c − e) + lv ω 0 q 0 .
∂pAdam Sobel (Columbia)

Lecture 4

December 21, 2013

(4)
(5)

(6)
(7)
4 / 71

For convenience we also define
Qc = Q1 − QR = lv (c − e) −

∂ 0 0
ωs
∂p

(8)

as the ”convective heating”. Neglecting ∇ · s 0 v0 and ∇ · q 0 v0 , (4) and (5)become∂s∂+∇·s v+(ω s) = Q1 ,
∂t
∂p
∂q
∂
+∇·q v+(ω q) = −Q2 /lv ,
∂t
∂p

(9)
(10)

the difference being that the average is applied separately to v, s, and q
on the LHS rather than to quadratic products vs, vq.
Expand the terms on the LHS using the chain rule and then use the formof mass continuity that is valid in pressure coordinates under thehydrostatic approximation,∇·v+
Adam Sobel (Columbia)

∂ω
= 0,
∂p

Lecture 4(11)
December 21, 20135 / 71to obtain the advective forms∂s∂s+ v · ∇s + ω= Q1 = Qc + QR ,
∂t
∂p
∂q
∂q
+ v · ∇q + ω= −Q2 /lv ,
∂t
∂p

(12)
(13)

Qc and Q2 are the effective large-scale heat source and large-scalemoisture sink due to convection.
A closed theory, or numerical model, must represent Qc and Q2 asfunctions of resolved variables. This is known as parameterization.
The simplest parameterization is convective adjustment: convection
maintains a moist adiabatic profile, while satisfying all conservation laws in
the vertical integral. But there are many others, much more complex...

Adam Sobel (Columbia)

Lecture 4

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What profiles of MSE, saturation MSE, Q1 , Q2 look like in deep
convective regions of the tropics. (Yanai et al. 1973)

Note Q1 peaks higher than Q2 ; most condensation occurs lower, where q
is higher; turbulence transports condensation-warmed air upwards.

Adam Sobel (Columbia)

Lecture 4

December 21, 20137 / 71Vertically integrated budgets
If we integrate the moisture equation in the vertical over the depth of theconvective layer (whose boundaries in pressure are p0 , pT ), use massconservation, assume steady state, and assume that ωq = 0 at p0 and
pT , we get
Z p0
Z p0
g −1
Q2 dp = Lv (P − E ) = −g −1 Lv∇ · (vq) dp,
(14)
pT

pT

where E = −g −1 ω 0 q 0 |p0 ≈ ρ0 w 0 q 0 |z=0 is the surface water vapor flux
(evaporation). Similarly,Z
1 p0
Qc dp = Lv P + H
g pT
where H is the sensible heat flux (usually H  E over ocean).

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A comment on tropical atmospheric dynamics:
We can choose to describe the problem of understanding the tropicalatmospheric circulation — both weather and climate — as being
composed of two parts:
1

Given large-scale fields, find Q1 and Q2 ,2

Use Q1 and Q2 and the governing equations to evolve the large-scale
fields in time.

Part of this problem is finding the large-scale dynamical response of thedry variables — winds, temperatures, pressures — to Q1 . I claim that thispart is relatively straightforward and well-understood at this point. It isclassic geophysical fluid dynamics.
Finding Q1 and Q2 , and understanding how they depend on the large
scales, remains the greater challenge.

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Large-scale budget regimes
In the discussion that follows we make the further assumptions:1

The atmosphere is in statistically steady state, so we can neglect thetime derivative terms

2

We neglect the horizontal advection terms v · ∇s, v · ∇q. For dry
static energy, in the tropics, this is a good approximation, as
horizontal temperature gradients are very small. For moisture it issometimes an ok approximation, sometimes not so good,quantitatively, but is adequate for purposes of our discussion here.

With these approximations our equations becomeωAdam Sobel (Columbia)∂s= Q1 = Qc + QR ,∂p∂qω= −Q2 /lv ,∂p

Lecture 4

(15)
(16)

December 21, 201310 / 71

Some notes

ω

∂s
∂q
= Q1 = Qc + QR , ω= −Q2 /lv ,
∂p
∂p

(17)

Keep in mind that as QR is generally negative in the troposphere (with avalue on the order of 1 K d −1 , while Qc and Q2 are generally either
positive or zero.

Adam Sobel (Columbia)

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Now we consider the different kinds of balances that can occur in theseequations.

Adam Sobel (Columbia)

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Radiative-convective equilibrium, 1

We define a state of radiative-convective equilibrium (RCE) as one inwhich ω = 0, as appropriate for, say, an average over the entire planet. In
that case (15) implies
Qc = −QR ,(18)convective heating balances radiative cooling. Eq. (16) implies
Q2 = 0.

(19)This does not mean that precipitation is zero. Rather, we can easily showthat in RCE,P = E,
precipitation equals surface evaporation, and also

Adam Sobel (Columbia)

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Radiative-convective equilibrium, 2
lv P + H = −g

−1

Z p0
QR dp,
pT

which says that the total condensation heating plus surface sensible heat
flux (H) equals the radiative cooling. Going back to the definition of Q2 ,we conclude that the condensation is not itself zero (unless surface
evaporation is, which is unlikely over most surfaces, and certainly overocean) but that it is balanced by upward turbulent moisture transport,e −c =

∂ 0 0
ωq.
∂p

(20)

Moisture is evaporated at the surface, transported upward by turbulence(which generally means, in clouds, above the shallow subcloud layer
adjacent to the surface), and condenses there. This condensation would
tend to increase dry static energy due to the enthalpy of vaporization, butradiative cooling compensates, preventing this warming from occurring.
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Radiative-convective equilibrium, 3There is no large-scale vertical motion, and thus by the mass conservation
equation (11), no divergent horizontal velocity. Thus there is nolarge-scale circulation whatsoever (apart from the somewhat artificial
possibility of a purely rotational, nondivergent one which would, under ourapproximations, transport no energy or moisture).

Adam Sobel (Columbia)

Lecture 4

December 21, 201315 / 71Nonprecipitating, descending region, 1
Now let’s consider a region in which there is no deep convection, thus nosignificant condensation of water in the free troposphere.
We then expect that turbulence will be limited to a shallow boundary layer
near the surface, and it is only in this layer that the turbulent transportterms in (6) and (10) are significantly different from zero. Thus in the freetroposphere, as we have already assumed c = e = 0, we have
Qc = Q2 = 0 as well.
Assuming that QR is not zero, but rather is negative, there is no way we
can have a steady balance in the temperature equation unless we haveω∂s/∂p < 0 as well. As stable stratification to dry motions implies (andwe observe) ∂s/∂p < 0, this implies ω > 0, descent.

Adam Sobel (Columbia)

Lecture 4

December 21, 201316 / 71Nonprecipitating, descending region, 2
The balance is (overbars understood; everything that follows is in terms of
averaged variables unless primes are explicitly written):ω∂s= QR .
∂p(21)

In the moisture budget, we must then have
ω

∂q
=0
∂p

(22)

in the free troposphere. Since we have already found that ω is not zero
this implies that
∂q
= 0,
(23)
∂p
which means q is constant with respect to pressure.
Adam Sobel (Columbia)

Lecture 4

December 21, 201317 / 71Nonprecipitating, descending region, 3

But up near the tropopause the temperature becomes very small, thus by
Clausius-Clapeyron so does the saturation vapor pressure and presumably
the vapor pressure itself. We can consider the value of q to be set up
there, say q = qtp , but compared to the saturation values lower down thisis not much different from q ≈ 0.
Thus we expect a very dry free troposphere in these descending regions.
In reality the degree of dryness will often be somewhat reduced byhorizontal advection from moister regions, which we have neglected.

Adam Sobel (Columbia)

Lecture 4

December 21, 201318 / 71Nonprecipitating, descending region, 4

In the boundary layer we have
Q2 = lv

∂ 0 0
ωq,
∂p

thus our moisture budget isω∂q∂= − ω0q0,
∂p
∂p

(24)

which says that the turbulent transport of moisture up from the surfacebalances large-scale advective drying.

Adam Sobel (Columbia)

Lecture 4

December 21, 201319 / 71Nonprecipitating, descending region, 5
The dry static energy budget in the boundary layer isω∂s∂= − ω 0 s 0 + QR ,
∂p
∂p

(25)

or putting terms which will be of similar sign together,
QR =

∂ 0 0
∂s
ω s +ω ,
∂p
∂p

(26)

which says that the turbulent sensible heat transport (which if we getmore specific will include a contribution connected directly to the surface
sensible heat flux as well as a contribution due to turbulent entrainment ofpotentially warm free tropospheric air) plus large-scale advective(adiabatic) warming together balance radiative cooling.

Adam Sobel (Columbia)

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Strongly precipitating, ascending region, 1

Now we consider a region in which Qc + QR > 0. This implies that ω < 0,as our temperature equation (15) can be writtenω=

Qc + QR
,.
∂s/∂p

(27)

The vertical integral of Q2 is constrained once we know Qc and thesurface fluxes H and E , as we can easily derive
Z p0
Z p0
Q2 dp =
Qc dp − H − lv E .
pTAdam Sobel (Columbia)

pT

Lecture 4

December 21, 201321 / 71

Strongly precipitating, ascending region, 2
The vertical structure of Q2 is not so clearly constrained, but if we know
the profiles of q and ω we can infer what it must be from the moisturebudget (16).
Further, from (14) we can deduce that in this regime P > E . If ω < 0throughout the troposphere (as is generally true in deep convective
regimes), and furthermore if we assume that the convective layer isbounded in the vertical by points at which ω = 0, it follows that ∂ω/∂p is
positive in the lower troposphere and negative in the upper.
From that, the fact that q invariably decreases with height (increases withpressure), and the neglect of horizontal gradients, it follows that P > E inthis regime, implying net import of water vapor, or ”moisture
convergence”.

Adam Sobel (Columbia)

Lecture 4

December 21, 201322 / 71

Strongly precipitating, ascending region, 3Note that in this regime we expect the turbulent transport terms to beimportant throughout the convecting layer, which will be the entire
troposphere or a large fraction of it. So the distinction between theboundary layer and the free troposphere is less clear here, though often
there is still a thin layer well-mixed in s and q just near the surface which
we might consider to be the boundary layer.

Adam Sobel (Columbia)

Lecture 4

December 21, 201323 / 71

The weak temperature gradientapproximation and the shallow-to-deep
cumulus transition
Adam Sobel
FDEPS, Kyoto

Climatology: Free tropospheric temperature is homogeneous. SST
and precipitation have much more structure, and resemble each other.

At first order, we have a monotonic relationship between SST and
precipitation.

Waliser et al. 1993

The monotonic relationship between SST and precipitation will be
explained using two principles:
1. Tropospheric temperature is horizontally uniform, and
2. Deep convection is a response to instability.

Why is the free tropospheric temperature so uniform?

In geostrophic adjustment in a compressible fluid, a localized temperature
anomaly will spread out until it reaches the deformation radius.

x

T

time

Ld
t=0

t >Ld/cAt the equator, the effective deformation radius is ~1500 km in latitude,
and the entire circumference of the earth in longitude. Thatʼs whytemperature is uniform in the tropics.

The equation for potential temperature is∂tθ + uh·rhθ + wS = Qc + QR,
with Qc convective heating, QR radiative heating, S static
stability (proportional to ∂zθ)
Since θ is held nearly constant (at const. z) by large-scale
adjustment, the dominant balance is
wS ¼ Qc + QR

Letʼs build our understanding of tropical dynamics startingfrom a single column∂tT + wS = Qc + QR,∂tq + w∂zq = Qq
with S / ∂zθ,

Qq=(e-c) + ∂z wʼqʼ

where we have neglected horizontal advection terms uh·rh(θ,q).
A single column model is one which solves theseequations, with no horizontal dimension.

The convective heating & moistening are parameterized as functionals of
T(z), q(z), the temperature and moisture profiles, generally minimizing
some measure of conditional instability.
We divide z into a boundary layer (b) and free troposphere (f). Write
Qc (T,q) = Qc(Tb,qb,qf,Tf)
Qq (T,q) = Qq(Tb,qb,qf,Tf)
Tb and qb are controlled by the boundary conditions (e.g., SST). Tf
must be the same everywhere. For an area small compared to theentire tropics, Tf is an external parameter, not part of the solution.
It is determined by convective adjustment to the boundary conditions
elsewhere in the tropics - to the tropical mean SST, under thesimplest assumption.

freetroposphereboundarylayer
rest of tropics

location of interest

∂tT + wS = Qc(Tb,qb,qf,Tf) + QR(Tb,qb,qf,Tf),∂tq + w∂zq = Qq(Tb,qb,qf,Tf)

If instead we model the whole tropics (or whole planet),
we get radiative-convective equilibrium, w=0. It would be
the observed state if the whole globe had uniform SST.
In that case Tf must be part of the solution.
The temperature profile adjusts to neutrality to buoyant
ascent given Tb, qb, which depend on the SST.

Manabe and Strickler 1964,
J. Atmos. Sci. 21, 361-385.

RCE temperature as a function of latitude and pressure, calculated with
the Emanuel single column model. Uniform CO2, ozone, surface wind speed.

In RCE, P=E, and atmos. temperature increases ~linearly with SST (slope>1)

In the real tropical atmosphere, free tropospheric temperature doesnʼtadjust to the local SST. Precipitation does.

This is a shallow-to-deep convection transition as air moves overincreasing SST!

Stevens (2005, Ann. Rev. Earth. Planet. Sci.)

SST increasing along low-level flowIn the tropics, the dominant balance in the T equation in thefree troposphere is not
Qc + QR ¼ 0, as in RCE but instead
Qc + QR ¼ wS,
with Tf itself held nearly constant at const. z by geostrophic
adjustment.
RCE may be a useful theoretical construct, but it is not a good
approximation to the state of the atmosphere locally. It does
hold approximately in the tropical mean.

In studying the transition from SCu to Cu, a constant vertical motion profile
was assumed. This was ok because both types of shallow cloud layers occur
under essentially the same overlying free tropospheric state, dry and warmwith large-scale descent.
This will not work for the shallow to deep transition, because in that case
The large-scale vertical motion must change sign during the transition (ifSubstantial precipitation rates are to be achieved, representative of the real
deep convective regions)We use the WTG system:∂tTf=0,wfS = Qc(Tb,qb,qf;Tf) + QR(Tb,qb,qf;Tf),∂tTb + wbS = Qc(Tb,qb,qf;Tf) + QR(Tb,qb,qf;Tf ,Ts),∂tqf + w∂zqf = Qq (Tb,qb,qf;Tf)∂tqb + w∂zqb = Qq (Tb,qb,qf;Tf ,Ts)
solution variables (fns of z,t)externally specified; Ts = SST
For many purposes it is ok to assume constantrelative humidity in boundary layer, and surface
air temperature diagnostically related (and close to)surface temperature, thus
qb¼ qb(Tb)¼ qb(Tb(Ts))

WTG/RCE Calculations with Emanuelsingle-column models(Renno et al. 1994, JGR, 99, 14429-14441; Bony andEmanuel)
•
•
•
•
•
•
•

Emanuel convective parameterization
Bony-Emanuel cloud scheme (in some calculations)
Goddard/French radiative schemes
Fixed SST
CO2, Ozone at reasonable present-day values
Surface wind speed=7 m/s
No horizontal advection of moisture (for WTG, meansthat horizontal moisture gradient is assumed to vanish)precipitation

Precipitation in WTG simulations with Emanuelʼs model. Dots are fromobserved January climatology of SST and Precip over oceans 20S-20N.

SST
Sobel and Bretherton 2000

Deep convection is controlled by stability of the sounding (e.g.,
Arakawa and Schubert 1974). The stability here is determined by
the free tropospheric temperature and the SST.

moist adiabat rising from sfc

stable, no rain

unstable, rainzfree tropospheric T(z)same in both places

low SSThigh SSTRelative humidity and pressure vertical velocity (ω) profiles as functions of
SST in the same set of simulations

The tropospheric temperature is horizontally uniform throughout
the tropics. Its value is set by a tropical mean adjustment to themean SST in deep convective regions.
Near-surface air adjusts to the local surface temperature, while
free tropospheric air cannot, since it has to be the same as elsewhere.
Over a relatively cold surface, the sounding is stable.
Over a relatively warm surface, the sounding is unstable.
This gives us the zero-order relationship between SST and
precipitation.
NB the relationship is between time-mean CAPE and precip; on
sub-monthly time scales, it doesnʼt work.

Precipitation in RCE and WTG.

2011

Ramsay and Sobel 2010

Similar calculations with cloud-resolving model, in 2D and 3D(Wang and Sobel 2011, J. Geophys. Res.)Although SST controls the transition in these simulations, it is mediated by water
vapor (Wang and Sobel 2011, J. Geophys. Res.; see also Wang and Sobel 2012)

ENSO Teleconnections

work with
http://www.rpdc.tas.gov.au/soer/image/377/index.php
John Chiang(Chiang and Sobel 2002, J. Climate)97-98 El Nino. Early event. Large warm SST anomalies in central/east Pacific,
weak anomalies elsewhere. Negative precip anomalies in warm pool.

97-98 El Nino. Late event. Tropical Indian and Atlantic warm up. Negative
precip anomalies go away.

Typical El Nino precip anomalies are negative throughout the tropics,
except the eastern/central Pacific. This suggests a global-scale explanation.

Ropelewski and Halpert 1987

Interannual tropospheric temperature anomalies are very homogeneous
within the tropics,

Yulaeva and Wallace 1994

and related to tropical mean SST
anomalies

Sobel, Held, and Bretherton 2002

We user our single-column modeling strategy to understand tropical
ENSO teleconnections, now with interactive slab ocean

Free
tropo
PBL

Central & E. Pacific
west

east

“remote tropics”(atmospheric column)∂tTf=0,wfS = Qc(Tb,qb,qf;Tf) + QR(Tb,qb,qf;Tf),∂tTb + wbS = Qc(Tb,qb,qf;Tf) + QR(Tb,qb,qf;Tf ,Ts),∂tqf + w∂zqf = Qq (Tb,qb,qf;Tf)∂tqb + w∂zqb = Qq (Tb,qb,qf;Tf ,Ts)C∂tTs=net surface heat flux

In this case, we force the SCM with time-varying tropospheric temperature,
representing that imposed on remote regions by SST in the Pacific.
Sea surface temperature
79-99 observed tropical
tropospheric temperature
perturbation

FORCING
X

precipitation

OUTPUT

The ocean surface warming is impressed from above by the free troposphere,
via the convective adjustment of the PBL to that, and then surface fluxes

Free troposphere

Tʼ

Planetary boundary(subcloud) layerθeʼ (equivalent

ocean

the “upped ante”(Neelin et al. 2003)

potential temp)

SSTʼ

Convection

Surface fluxes

The precipitation response results from a disequilibrium: the upper ocean
has not yet had a chance to adjust to the tropospheric warming. Adjustment
is slower for a deeper mixed layer. Precip response requires ocean “memory”.

Forcing

mld=1mModelprecipmld=40m
mld=160m

Reference1• Arakawa, A., and W. H. Schubert, 1974: Interaction of a Cumulus Cloud
Ensemble with the Large-Scale Environment, Part I. J. Atmos. Sci., 31, 674–
701
• Bony, S., and K. A. Emanuel, 2001: A Parameterization of the Cloudiness
Associated with Cumulus Convection; Evaluation Using TOGA COARE Data.
J. Atmos. Sci., 58, 3158–3183• Chiang, J. C. H., and A. H. Sobel, 2002: Tropical Tropospheric Temperature
Variations Caused by ENSO and Their Influence on the Remote Tropical
Climate. J. Clim., 15, 2616–2631• Manabe, S., and R. F. Strickler, 1964: Thermal Equilibrium of the
Atmosphere with a Convective Adjustment. J. Atmos. Sci., 21, 361–385• Neelin, J. D., 2003: Tropical drought regions in global warming and El Niño
teleconnections. Geophys. Res. Lett., 30, 2275
• Ramsay, H. a., and A. H. Sobel, 2011: Effects of Relative and Absolute Sea
Surface Temperature on Tropical Cyclone Potential Intensity Using a SingleColumn Model. J. Clim., 24, 183–193

Reference2• Rennó, N. O., K. A. Emanuel, and P. H. Stone, 1994: Radiative-convectivemodel with an explicit hydrologic cycle: 1. Formulation and sensitivity tomodel parameters. J. Geophys. Res., 99, 14429• Ropelewski, C. F., and M. S. Halpert, 1987: Global and Regional Scale
Precipitation Patterns Associated with the El Niño/Southern Oscillation.
Mon. Weather Rev., 115, 1606–1626• Sobel, A. H., and C. S. Bretherton, 2000: Modeling Tropical Precipitation ina Single Column. J. Clim., 13, 4378–4392• Sobel, A. H., I. M. Held, and C. S. Bretherton, 2002: The ENSO Signal in
Tropical Tropospheric Temperature. J. Clim., 15, 2702–2706• Stevens, B., 2005: Atmospheric Moist Convection. Annu. Rev. Earth Planet.
Sci., 33, 605–643• Waliser, D. E., N. E. Graham, and C. Gautier, 1993: Comparison of theHighly Reflective Cloud and Outgoing Longwave Radiation Datasets for Use
in Estimating Tropical Deep Convection. J. Clim., 6, 331–353

Reference3• Wang, S., and a. H. Sobel, 2011: Response of convection to relative sea
surface temperature: Cloud-resolving simulations in two and three
dimensions. J. Geophys. Res., 116, D11119• Wang, S., and a. H. Sobel, 2012: Impact of imposed drying on deep
convection in a cloud-resolving model. J. Geophys. Res., 117, D02112• Yanai, M., S. Esbensen, and J.-H. Chu, 1973: Determination of Bulk
Properties of Tropical Cloud Clusters from Large-Scale Heat and Moisture
Budgets. J. Atmos. Sci., 30, 611–627• Yulaeva, E., and J. M. Wallace, 1994: The Signature of ENSO in Global
Temperature and Precipitation Fields Derived from the Microwave
Sounding Unit. J. Clim., 7, 1719–1736