2. The Interaction of Waves and
Convection in the Tropics
1
Summary
Interest in tropical waves and their interaction with convection has been rekindled in
recent years by the discovery, using satellite infrared data to track high cloud, that
such waves closely display the dispersive properties of linear, inviscid wave theory
for an atmosphere with a resting basic state and equivalent depths between 10 and
100 m. While several current approaches focus on internal modes in the atmosphere,
this is inconsistent with the absence of internal modes in the atmosphere which is
characterized by a single isolated eigenmode and a continuous spectrum. It will be
shown, using a triggering type approach to convection, that the observed properties of
waves are consistent with a continuous spectrum. The approach assumes that the total
convection is determined by mean evaporation, but that the convection is patterned by
zero averaged perturbations to triggering energy following the recent approach of
Mapes. The observed convection associated with the migrating semidiurnal tide is
used to calibrate the timescale for the convective response to patterning. This time
scale is more representative of tropical mesoscale systems than of convection itself.
It is shown that this time scale leads to the observed phase lead of low level
convergence in tropical waves vis a vis the convective heating. Finally, it is shown
that this phase is sensitive to the equivalent depth which it is suggested is the basis for
the selection of equivalent depth. Reasonable simulations of observed waves are
readily obtained.
2
Waves in a stratified spherical fluid
Linearized equations of motion lead to a separable second
order partial differential equation (in altitude and latitude)
with solutions of the form:
w( ,  , z, t )  e x / 2ei (t s )  yn ( x)n ( )
where
x
z
0
dz
H
Lhn , ,s,,a n   0 (Laplace's Tidal Equation)
M hxn , H ,  yn   F (Vertical Structure Equation)
and hn is the separation constant
(known as equivalent depth).
3
The vertical structure equation is characteristically of
the form
d 2 yn
2


yn  F
2
dx
where,
dT0
1 1æ
gö
1
÷
ç
çç
l =
+ ÷
÷
÷
÷ 4H 2
hn T0 çè dz c p ø
2
Essentially, the equivalent depth is a measure of the
vertical wavenumber, and depending on the relation
of hn to zonal wavenumber, frequency, etc., the wave
will essentially be an internal gravity or Rossby wave
or some combination of the two.
4
V
e
r
t
i
c
a
l
W
a
v
e
l
e
n
g
t
h
v
.
E
q
u
i
v
a
l
e
n
t
D
e
p
t
h
f
o
r
T
r
o
p
o
s
p
h
e
r
e
2
3
1
0
6
5
4
3
2
EquivalentDph(m)
2
1
0
6
5
4
3
2
1
1
0
6
5
4
3
2
0
1
0
0
1
0
23
4
5
6
7
8
1
1
0
23
4
5
6
7
8
2
V
e
r
t
i
c
a
l
W
a
v
e
l
e
n
g
t
h
(
k
m
)
1
0
2
5
Free v. forced waves
Forced waves: We are given  and s. Laplace’s Tidal
Equation is solved for hn and n, where hn is an eigenvalue
and n is an eigenfunction. The forcing is expanded in these
eigenfunctions (known as Hough Functions), and the vertical
structure equation is solved for the response to each
component of the forcing.
Free waves: The vertical structure equation is solved in
the absence of forcing. The eigenvalues are the
equivalent depths of the fluid system. For shallow
water, the only eigenvalue is the depth of the fluid. For a
stably stratified liquid with a top, the equivalent depths
correspond to an infinite set of vertical modes. For the
atmosphere, there is generally only a single eigenvalue,
6
corresponding to a Lamb mode with an equivalent depth of
about 10 km. There is also a continuous spectrum. For
each equivalent depth and zonal wavenumber, Laplace’s
Tidal Equation is solved for the eigenfrequencies, n, and
the associated Hough Functions. For each Hough
Function, one obtains a relation between frequency and
wavenumber.
The name ‘equivalent depth’ was chosen by analogy with
the shallow water case where the equivalent depth was the
actual depth of the fluid.
Despite the fact that the atmosphere generally is found to
have only a single equivalent depth (10 km), many tropical
waves are observed to behave as though they had a
relatively unique equivalent depth of around 12-60m.
7
Wheeler-Kiladis (1999) analysis of OLR
The new ‘equivalent depth’ is generally assumed to
arise from the interaction of waves with convection.
Note that MJO is not characterized by a particular
equivalent depth.
8
It should be noted that the Wheeler-Kiladis results
implicitly demonstrate the relevance of simple linearized
tropical wave theory.
It should also be noted that recent attempts to interpret the
Wheeler-Kiladis results in terms of internal normal modes
(Mapes, 2000, Majda and Schefter, 2001, Emanuel et al,
1994) are inconsistent with the spectral properties of the
vertical structure equation for realistic atmospheres
without lids.
9
Various approaches to interaction between large scale
motion system and convection.
1. Charney and Eliassen (1964) and early Wave-CISK (Yamasaki,
1969, Lindzen, 1974): Effective cumulus heating drives large
scale motion which includes low level convergence (due to
Ekman pumping in CISK, and vertically propagating wave in
Wave-CISK) which lifts air to Lifting Condensation Level (ca
500m). Problems: Eliassen critique: Trade wind boundary layer is
turbulent so that there is no need to lift air to LCL.
N.B. Early wave-CISK studies always got equivalent depths of
about 10 m corresponding to vertical wavelengths of about 2
km (corresponding to a quarter wavelength equal to the height
of the lifting condensation level).
10
2. Moisture budget (Tiedke parameterization, Lindzen (1981),
later Wave-CISK: Stevens and Lindzen (1978) ): Convection
is a response to moisture budget below about 2 km (Trade
Inversion). Problem ( Stevens and Lindzen (1978)) : WaveCISK doesn't provide instability. (Note, however, observed M c
equals M c , suggesting that observed waves equilibrate by
exhausting moisture supply.)
N.B. An equivalent
depth of about 30m
corresponds to a vertical
wavelength of about 8
km (and a quarter
wavelength to the height
of the trade inversion).
Cumulonimbus
Trade Inv.
Trade Wind
Boundary L.
Mixed Layer
LCL11
3. Wave triggering: In an unstable environment any
perturbation will trigger patterning of existing convection.
E
M c     V
q
In general, convective elements are randomly distributed.
In mass budget approach, Mc responds to directly determined
convergence below trade inversion.
In triggering approach, convection automatically provides
self-consistent low level convergence, but perturbations to
convergence determine pattern of convection. Of course, if
the perturbation provides more convergence, the convection
will not have to provide as much. (Basically analogous to
Benard convection.) Note, that the present work uses w as
a measure of triggering, but, in fact, any variable could
be used without fundamentally altering the formalism. 12
In the present approach, waves do not change the total amount
of convective activity. Rather, the low level fields of the wave
perturbation biases the random breakdowns of CIE produced
by boundary layer turbulence so as to pattern the convection
which would occur anyway. Thus, the mean amount of
convection is essentially determined by the mean evaporation.
While there is evidence that squall systems play an important
role in the convection itself, other systems ranging from
gravity waves to easterly and Kelvin waves to the Hadley and
Walker circulations serve primarily to pattern the convection
that would otherwise exist. Moreover, all the sources of
patterning can simultaneously coexist.
13
Squalls modulated by
Easterly Wave
L
Low level general circulation
associated with Hadley and Walker
circulations (modulates mean
convection)
H
ITCZ
Easterly Wave (modulates
convection organized by mean
circulation)
As noted by Reed & Recker (1971) and many since, the
amplitude in precipitation of tropical waves tends to be
approximately the mean precipitation -- suggesting a
reorganization of existing precipitation rather than the
production of additional precipitation.
14
Cumulus Heating
Cumulus convection does not directly heat the ambient
atmosphere. However, that portion of the mean vertical
velocity which is carried in cumulus towers also does not
contribute to adiabatic cooling. Thus, we must subtract this
part from the adiabatic cooling: i.e., the adiabatic cooling
term becomes

( w  M c )
z

The term M c
constitutes an effective cumulus
z
heating term.
The patterning of the convection gives rise to a
contribution to the effective cumulus heating in the form
of the pattern. The contribution of the patterning to the
mean is, however, zero.
15
Modes of cumulus-large-scale flow interaction
(regardless of interaction):
a) Self-excitation where, for example, motion systems
forced by effective cumulus heating provide low level
convergence that, in turn, triggers the convective pattern
that forces the wave.
b) Motion systems that have their origin in processes
separate from effective cumulus heating provide low level
convergence with the resulting effective cumulus heating
modifying the motion system.
 Examples of (a) are the tropical waves analyzed by
Wheeler and Kiladis (1999) (except for the MJO).
 Examples of (b) are the solar semidiurnal tide, and,
possibly, the MJO.
We will begin with the second case.
16
Madden-Julian Oscillation (MJO)
Baroclinic instability of subtropical jet leads to wave 1-2 40 day
westerly waves which are strongly coherent with the tropical
MJO (Straus and Lindzen, 2001).
Unfortunately, it is not obvious how such instabilities would
effectively penetrate to near equatorial regions.
17
From Straus and Lindzen, 2000
18
2
l :
qy
u- c
Unfortunately,
prevailing easterlies
prevent propagation.
Tunneling, however,
remains a possibility.
Also, in reanalyses, qy is
negative below 900 mb.
19
Case of the solar semidiurnal tide:
Forcing is primarily due to insolation absorption by ozone
(Butler and Small, 1963) and water vapor (Siebert, 1961).
Amplitude okay, but phase is about one hour off (0900, 2100
By contrast, the diurnal component of the
instead of 1000, 2200).
rainfall does not have a uniform phase in
local time.
Lindzen (1978) and Hamilton (1981) showed that observed
semidiurnal component of rainfall provided additional forcing
that would correct discrepancy. However, tidal convergence
was one order of magnitude less than needed. This implies that
triggering rather than direct forcing of convective pattern is
involved.
Moreover, tidal component of rainfall is only a fifth of mean
rainfall. The reason, we suggest, is that tidal time scale (12
hours/2p) is short compared with convective response to
20
triggering.
Unperturbed Convection
Equilibrated Perturbation
Cumulonimbus
Trade Wind
Boundary L.
Mixed Layer
W
M c = M c ; M 'c = 0
M c = M c + M 'c sin kx; M 'c = M c
W
21
The following equation roughly describes how we expect M c
to behave:
1 d
 
 1 M c  M c sgn(sin t )

 a dt 
For convenience, we will let ,
above equation becomes
t  x
so that the
 d
 

1

 M c  M c sgn(sin x)
 a dx 
Although M c will, of course, be distorted from a sine
wave, its impact on the wave will be associated with its
projection on the sinusoidal w component.
22
(
a
/

)
=
5
S
o
l
u
t
i
o
n
F
o
u
r
i
e
r
P
r
o
j
e
c
t
i
o
n
0
.
4
(
a
/

)
=
0
.
1
6
S
o
l
u
t
i
o
n
F
o
u
r
i
e
r
P
r
o
j
e
c
t
i
o
n
1
M'c/(meanM c)
M'c/(meanM c)
0
.
2
0
0
.
0
1
0
.
2
0
1
2
t
i
m
e
s
c
a
l
e
d
b
y
p
e
r
i
o
d
3
0
1
2
3
t
i
m
e
s
c
a
l
e
d
b
y
p
e
r
i
o
d
The quantity a/ determines the amplitude and phase of M c¢.
For semidiurnal tide, we know that M c¢/ M c » 0.2 . This,
in turn, implies a/  0.16, or a-1 11.94 hours. The phase
lag is about 81.8o, which is what is needed to correct the
tide. For most tropical waves, adjustment time is relatively
short.
23
Self-excited tropical waves.
Self excitation with triggering only requires that triggering be
in phase approximately with heating required to force wave.
Procedure: Using characteristic heating profile, calculate low
level w as a function of equivalent depth, and see how phase
varies with equivalent depth.
N.B. In contrast to most earlier studies, we are not
associating observed waves with instabilities. Rather, we
are looking for consistency between triggering and wave
forcing. Waves for which these are inconsistent will self
destruct (though the rate of self-destruction depends on the
degree of inconsistency).
24
In point of fact, the calculation of waves is subject to
considerable complexity and uncertainty -- especially due to
such factors as cumulus friction and trade wind boundary
layer turbulence. However, there is a simple test of
triggering which bypasses these complexities.
The patterning time of about 11.9 hours implies that the low
level convergence should lead the upper level heating by
this amount in order for the two to be compatible.
25
From Straub &
Kiladis, 2002.
26
From Straub &
Kiladis, 2002.
27
1
5
0
P
a
t
t
e
r
n
i
n
g
a
d
j
u
s
t
m
e
n
t
t
i
m
e
=
1
1
.
9
h
o
u
r
s
PhaseLd(gres)
1
0
0
5
0
0
3
8
1
3
1
8
P
e
r
i
o
d
(
d
a
y
s
)
28
The following simplified approach illustrates how one
ought to approach the problem of determining the preferred
equivalent depth. The approach is probably too simple to
take the results as being anything other than a very rough
approach. They are, nonetheless, remarkably successful.
29
We consider the linearized equation for vertical structure of the
‘vertical velocity' in log p coordinates, w*, weighted by exp( x / 2) ,
w (where x  ln( p / ps ) , p=pressure, and ps=surface pressure):
d 2w
R
2
 w
exp( x / 2)Q( x)
2
dx
gh
where  2 
S 1

h 4


and S  R  dT   T 
g  dx

T
h is the equivalent depth,
is the basic state temperature, g is
the acceleration of gravity,   R / c p , R = the gas constant for
air, cp is the heat capacity of air at constant pressure, and Q(x) is
the vertical distribution of effective convective heating.
30

 x  x1  
S ( x)  S1  ( S2  S1 ) 1  tanh 
  
 1  



 x  x3  
 x  x2  
( S3  S2 ) 1  tanh 
 
   ( S4  S3 ) 1  tanh 


 2 
 3 


 x  xc 
Q  e sin  p

x

x
c 
 T
bx
4
0
Altiude(komtrs)
3
0
2
0
1
0
0
1
7
01
9
02
1
02
3
02
5
02
7
02
9
03
1
0
T
e
m
p
e
r
a
t
u
r
e
(
K
)
31
5
0
5
0
S
t
a
n
d
a
r
d
B
a
s
i
c
S
t
a
t
e
a
n
d
H
e
a
t
i
n
g
B
a
s
i
c
S
t
a
t
e
a
n
d
H
e
a
t
i
n
g
f
r
o
m
S
L
S
3
0
0
Phase(dgr)
Phase(dgr)
1
0
1
0
5
0
3
0
5
0
1
0
0
1
0
3
0
5
0
h
(
m
e
t
e
r
s
)
7
0
9
0
1
0
3
0
5
0
7
0
9
0
h
(
m
e
t
e
r
s
)
N.B. Given the oversimplified physics, it doesn’t pay to take the
precise phases too seriously. More important is the fact that
phase varies substantially with h, allowing for the possibility of
selection. Nonetheless, in these cases the proper phase seems to
32
occur for h’s that are too large.
Mapes (Mapes, B.E. (2000) Convective Inhibition, SubgridScale Triggering Energy, and Stratiform Instability in a
Toy Tropical Wave Model. J. Atmos. Sci., 57, 1515-1535)
pointed out the ubiquitous importance of shallow heating from
cumulus congestus that is shallower, and somewhat smaller than
cumulonimbus heating, and precedes it by several hours.
33
The addition of a relatively small amount of shallow heating at
low levels does, indeed, correct the phase appropriately.
Interestingly, the effect does not depend much on the phase
lead of the shallow convection.
For example, we might add the following to our Q
æ x - xc ÷
ö
ç
÷
Qcongestus = Ke sin çç
÷
÷
x
x
è tc
cø
ij
where xtc = 0.75
For K, we will try 0.2. For φ, we will try 0 and π/6.
34
1
5
0
S
t
a
n
d
a
r
d
B
a
s
i
c
S
t
a
t
e
1
0
0
B
a
s
i
c
S
t
a
t
e
f
r
o
m
S
L
S
i
n
g
l
e
S
o
u
r
c
e
D
o
u
b
l
e
S
o
u
r
c
e
D
o
u
b
l
e
S
o
u
r
c
e
P
L
1
0
0
5
0
S
i
n
g
l
e
C
o
n
g
C
o
n
g
P
L
Phase(dgr)
Phase(dgr)
5
0
0
0
5
0
5
0
1
0
0
3
4
5
6
7
8
3
4
5
6
7
8
3
0 2
1 2
2 2
1
0
1
0
1
0
E
q
u
i
v
a
l
e
n
t
D
e
p
t
h
(
m
e
t
e
r
s
)
4
5
6
7
8
1
1
0
2 3
4
5
6
7
8
2
1
0
E
q
u
i
v
a
l
e
n
t
D
e
p
t
h
(
m
e
t
e
r
s
)
2 3
We see that for widely different basic states, and small
congestus source, equivalent depths between around 12-60m
will either be self-sustaining or slow to self-destruct, while
outside this range, self-destruction will be more rapid.
35
The vertical structure of waves for a single h displays the
characteristics that are observed.
2
.
5
2
.
5
2
.
0
2
.
0
h
=
1
8
m
F
o
r
c
i
n
g
i
s
s
u
m
o
f
d
e
e
p
c
u
m
u
l
u
s
h
e
a
t
i
n
g
,
a
n
d
s
m
a
l
l
c
u
m
u
l
u
s
c
o
n
g
e
s
t
u
s
h
e
a
t
i
n
g
w
i
t
h
1
.
5
1
.
5
h
=
1
8
m
Altiudenscalhigts
Altiudenscalhigts
3
0
d
e
g
r
e
e
p
h
a
s
e
l
e
a
d
.
1
.
0
F
o
r
c
i
n
g
i
s
s
u
m
o
f
d
e
e
p
c
u
m
u
l
u
s
h
e
a
t
i
n
g
,
1
.
0
a
n
d
s
m
a
l
l
c
u
m
u
l
u
s
c
o
n
g
e
s
t
u
s
h
e
a
t
i
n
g
w
i
t
h
3
0
d
e
g
r
e
e
p
h
a
s
e
l
e
a
d
.
0
.
5
0
.
0
0
.
0
0
0
.
5
0
.
0
5
0
.
1
0
0
.
1
5
A
m
p
l
i
t
u
d
e
o
f
w
(
a
r
b
i
t
r
a
r
y
u
n
i
t
s
)
0
.
0
0
.
2
0
2
0
0
1
0
0
0
1
0
0
2
0
0
P
h
a
s
e
o
f
w
(
d
e
g
r
e
e
s
)
The relatively small stratospheric leakage may be an artifact of
the choice for the vertical structure of the thermal forcing.
However, the fact that the solution follows the heating in the
heating region, and looks like a vertically propagating wave
above this region is almost certainly robust and corresponds to 36
observations.
Even for the present basic state, there is more leakage for longer
vertical wavelengths or, equivalently, larger equivalent depths.
2
.
5
2
.
5
h
=
5
2
m
S
o
u
r
c
e
s
a
r
e
d
e
e
p
c
o
n
v
e
c
t
i
o
n
a
n
d
c
o
n
g
e
s
t
u
s
w
i
t
h
3
0
d
e
g
r
e
e
p
h
a
s
e
l
e
a
d
2
.
0
1
.
0
1
.
0
0
.
5
0
.
0
0
.
0
h
=
5
2
m
S
o
u
r
c
e
s
a
r
e
d
e
e
p
c
u
m
u
l
u
s
a
n
d
c
o
n
g
e
s
t
u
s
w
i
t
h
3
0
d
e
g
r
e
e
p
h
a
s
e
l
e
a
d
1
.
5
Altiudenscalhigts
Altiudenscalhigts
1
.
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37
d 2 yn
2


yn  F
2
dx
What is happening is that when the vertical scale of F is
larger than half the vertical wavelength given by , then the
particular solution dominates, and the deeper F is relative to
the half wavelength, the less is the leakage of the wave into
regions where F is small. However, what does leak out,
behaves like an homogeneous solution: ie, a radiating wave.
38
From Straub &
Kiladis, 2002.
39
As noted in Stevens, Lindzen and Shapiro (1977), solutions of simple inviscid
linear theory for tropical waves suffer from one major drawback: for observed
values of rainfall (i.e., effective cumulus heating), amplitudes of temperature
and horizontal velocity oscillations are too large. Models commonly replicate
tropical easterly waves, but with reduced wave components of rainfall. SLS
showed that these problems could readily be eliminated by the inclusion of a
simple model for cumulus momentum transport (Schneider and Lindzen,
1980). Since then, there has been much interest and controversy over the form
or even the existence of so-called ‘cumulus friction.' Sardeshmukh and
Hoskins (1987), for example, argued that there was no evidence for any such
phenomenon. Tung and Yanai (2002a,b), however, have recently presented
evidence to the contrary. Until these issues are resolved, the present approach
must be considered simply a suggestion for how simple tropical wave physics
can select the observed preferred equivalent depth.
The dispersion relation observed by Wheeler and Kiladis (1999) suggests that
such physics, nevertheless, remains relevant.
40
Despite uncertainty, the primary test of the patterning
hypothesis for equatorial waves was met.
The phase lead observed by Straub & Wheeler (2002) is
essentially what is called for by the patterning time derived
from the semidiurnal tide. This, in turn, implies quite a lot
about both tropical waves and convection: namely,
convection is triggered and waves are forced by the
resulting ‘effective cumulus heating.’
In addition, it was easy, within the context of classical wave
theory to simulate both the observed distribution of
equivalent depths and the relatively complex observed
vertical structure.
41
In summary:
1. Using the semidiurnal tide as an example of
patterning, one gets an estimate for the time scale for
patterning. This time scale is longer than the time
scales characteristic of convective elements. It is
more likely appropriate for squalls which organize most
convection.
(Note that, for simplicity, we have taken the adjustment rate
to be independent of the spatial scale of the perturbation.
More to the point, we don’t know what influence such spatial
scales should have.)
42
43
2. Patterning requires only a consistent phase in the
trade wind boundary layer. This talk has simply shown
that phase varies substantially with equivalent depth,
hovering around acceptable phases for the observed
equivalent depths -- especially when small heating
from cumulus congestus is included. This serves to
preferentially select these equivalent depths over
other equivalent depths which don't present consistent
phases. However, given the uncertainty as to how
triggering is actually to be formulated, this remains
speculative.
44
3. While the precise calculated phase should indeed
depend on aspects of the physics that were treated
casually at best, the observed phase lead for tropical
waves is, indeed, what is implied by the tidal estimate for
patterning time.
Note that the fact that the congestus clouds may play
an important role in the cumulonimbus response to
dynamic triggering, and that the inclusion of
congestus heating is important (in the present
formulation) in order to achieve consistency in phase
between triggering and cumulus heating at the
observed equivalent depth suggests that the
interaction of waves and convection may be more
subtle than anticipated.
45