EFS 4
/
1
4.
Equatorial Waves and Tropical Dynamics
EFS 4
/
2
T
ropical Meteorology
Dynamics of equatorial regions very different to midlatitudes.
Low latitudes can i
nfluence
higher latitudes through
‘teleconnections’.
E.g.
: influence of the El Nino Southern Oscillation (ENSO)
Earth is spherical and rotates!
Major difference between midlatiudes and tropics:
Coriolis parameter
2 sin
f
small in tropics
/
f y
largest at equator
70%
17%
7%
10
º
4
º
45
º
EFS 4
/
3
Energy sources
lateral
forcing from midlatitudes
baro
tropic
energy conversions
systematic
interaction with convective heating
Governing equations
In log

pressure co

ordinates
(
0
ln
p
z H
p
,
Dz
w
Dt
with
h
D
w
Dt t
z
u
)
:
Momentum
h h h
f
t
z
u u k u
(
4
.
1
)
Hydrostatic
RT
H
z
(
4
.
2
)
Continuity
0
u v w w
x y H
z
(
4
.
3
)
Thermodynamic
2
h
p
w N H J
T
t R c
u
(
4
.
4
)
EFS 4
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4
Scaling analysis
3
2.5 10 m
D
vertical depth
6
10 m
L
horizontal length
1
5ms
U
horizontal velocity
5
~10 s
time
3
~ 8 10 m
H
scale height
2 1
~1.2 10 s
N
buoyancy frequency
h
h h h h
f
t
z
u
u u u k u
U
2
U
L
WU
D
fU
L
M
idlatitudes
:
4 1
~10 s
f
and
Ro ~/
U fL
small so
Coriolis term
balances the pressure gradient
term.
T
ropics
:
5 1
~10 s
f
. I
f assume
vertical velocity
W
small
,
Coriolis term
4 2
~10 ms
fU
usually
dominant on LHS
,
hence
2 2
~100m s
.
EFS 4
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5
F
rom hydrostatic equation
~ ~1K
H
T
R D
.
G
eopotential &
temperature fluctuations
order magnitude smaller
than those with
same scale at midlatitudes.
Small size of
f
in tropics:
leads to
small geopotential (& hence temperature)
fluctuations
compared to midlatitudes.
Temperature fluctuations associated with
available potential
energy
:

in tropics disturbances must grow by means other than
baroclinic energy conversions.
C
onsider th
ermodynamic equation
&
assume flow adiabatic (
J
=0)
2
0
h
T w N H
T
t R
u
T
UT
L
2
WN H
R
Local time rate of chan
ge of temperature dominates
horizontal
advection and hence
3 1
2
~ ~ 2.5 10 ms
RT
W
HN
–
vertical
velocity
small
as we assumed.
EFS 4
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6
Balances in continuity equation:
6 1
~ ~ ~ 5 10 s
u v U
x y L
,
6 1
~ ~10 s
w W
D
z
and
7 1
~ 3 10 s
W
H
.
Considerabl
e
cancellation between
divergence
terms
2 2
~ ~
u v W
x y D
N D
.
Clouds
Latent heat is
dominant source of energy of tropical circulation
with radiative heating playing an important but secondary role.
Motions on
scale of individual
convective clouds and
mesoscale
and synoptic systems in which they are sometimes or
ganised are
an important part of determining the strength and locatio
n of
convection that drives
lar
ge

scale flow.
D
istribution of radiative heating and coolin
g are strongly
influenced by
distribution and physical characteristics of clouds.
Inclusion of t
his level of resolution in models is not very practical,
however the average effect of convective and radiative processes
must be parameterised
.
EFS 4
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7
Equatorial waves
Equatorial waves exist for a range of spatial and temporal
scales
and are
trapped near equato
r
.
P
ropagate in the zonal and vertical directions.
Coriolis force changes sign at the equator
.
Diabatic heating by organised tropical convection
can excite
atmospheric equatorial wavses, whereas wind stresses can excite
oceanic equatorial waves.
Atmosph
eric equatorial wave propagation can cause the
effects of
convective storms
to be communicated over
large longitudinal
distances
, thus
producing remote responses to locali
sed heat
sources.
B
y influencing the pattern of low

level moisture
convergence,
atmo
spheric equatorial waves can partly control the
spatial and
temporal distribution of convective heating
.
Oceanic equat
orial wave propagation,
can cause local wind stress
anomalies to remotely influence the
thermocline depth and the
SST.
EFS 4
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8
Fastest moving wa
ves are eastward propagating
Kelvin Waves
,
that have no north

south component and move at a constant
velocity, independent of the east

west wavenumber. Typical
Kelvin waves for the Pacific might move at about 3 ms

1
and take
about
2 months to cross the Pac
ific
.
The other important waves are
equatorial Rossby waves
. These
are
more slowly moving
(the fastest Rossby wave is about a third
the speed of the Kelvin wave) and have phase velocities that
propagate westward.
It is the generation of Kelvin and Rossby
waves by the atmosphere
that gives rise to the Pacific climate oscillation known as
El Nino
Southern Oscillation (ENSO)
.
EFS 4
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9
Equatorial
β

plane
Focus on troposphere.
Introduce theory in simplest context.
Following Matsuno (1966)
.
Use shallow water model
.
Equatorial
β

plane
(
cos 1
,
sin/
y a
,
f y
).
Linearised shallow water eq
uations for perturbations on a
motionless basic state of mean depth
e
h
are:
u
yv
t x
(
4
.
5
)
v
yu
t y
(
4
.
6
)
0
e
u v
gh
t x y
(
4
.
7
)
where
gh
is the geopotential disturbance.
EFS 4
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10
Seek soluti
ons in form of zonally propagating waves
, i.e. assume
wavelike solutions but retain y

variation
:
ˆ
,,Re ( ),( ),( ) exp[ ( )]
ˆ ˆ
u v u y v y y i kx t
Substituting into
(
4
.
5
)
to
(
4
.
7
)
gives set of three equations for
ˆ
,,
ˆ ˆ
u v
. Re

arranged to second order diff equ
for
ˆ
v
only:
2 2 2 2
2
2
ˆ
0
ˆ
e e
d v k y
k v
gh gh
dy
And we require
ˆ
v
to
decay to zero
at large
y
.
This is the
Schrödinger equation
with a simple harmonic
potential energy.
One solution is
0
ˆ
v
. Other solutions exist only
for a given
k
if
takes a particular value.
Convenient to non

dimensionalise and set
2
2
1
2 2
e
e
gh
k
k
gh
and
2
Y/2
( )e
ˆ
v F Y
where
1/4
1/2
e
gh
Y y
EFS 4
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11
Then
can be re

written as
the
Hermite differential equa
tion
2 2 0
F YF F
.
S
olutions which satisfy the boundary conditions are
F cH
,
where
n
for
0,1,2...
n
and
( )
n
H Y
is a Hermite polynomial.
The first few He
rmite polynomials are:
0
1
H
,
1
( ) 2
H Y Y
,
2
2
( ) 4 2
H Y Y
,
3
3
( ) 8 12
H Y Y Y
.
Clearly the solutions are ‘trapped’ near the equator by the
exponential.
EFS 4
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12
We have found
horizontal dispersion
rela
tion
, i.e.:
2
2
2 1
e
e
gh
k
k n
gh
,
0,1,2...
n
(
4
.
8
)
Since this equation
is cubic in
, we have three roots for
when
n
and
k
are specified.
We can now make various approximations.
At
low frequencies
,
2
/
e
gh
is smaller
the other terms and we
have
Rossby
2
(2 1)/
e
k
k n gh
. This corresponds to
equatorial Rossby waves
. These waves are westward

propagating
only as
is of opposite sign to
k
.
At
high frequencies
, the term
/
k
is small and we have
2
(2 1)
IG e e
n gh k gh
. These are the eastward and
westward propagating
inertio

gravity waves
.
EFS 4
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13
For
case
0
n
, from
(
4
.
8
)
,
0
2
1 4
1
2
n e
e
k gh
k gh
.
This corresponds to an eastward

propagating
inertio

gravity
wave
and a westward
Rossby

gravity
wave (sometimes called a Yanai
wave).
Has properties of Rossby and gravity waves in different
limits.
For
case
0
ˆ
v
.
T
he equations for
ˆ
,,
ˆ ˆ
u v
derived from
(
4
.
5
)
to
(
4
.
7
)
give dispersion relation for fast

moving eastward propagating
Kelvin wave
:
Kelvin
e
gh k
with a meridional structure of
2
0
exp
ˆ
2
e
y
u u
gh
.
Zonal
velocity and geopotential perturbations that vary in latitude
as Gaussian functions centred on the equator.
Velocity and pressure distributions in the horizontal plane for (a)
Kelvin waves, and
(b) Rossby

gravity waves (
Matsuno
1966).
EFS 4
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14
Z
onal
phase speed
is
/
p
c k
(i.e. determined by the position on
the graph wi
th respect to the origin).
Z
onal component of the
group velocity
is given by
/
g
c k
(i.e. the local slope of the curves).
EFS 4
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15
Zoology of
equatorial waves
Rossby waves
only propagate to the west, whereas their energy (as
inferred from their group velocity) may propagate to the east or
west.
Mixed Rossby

gravity waves
have westward and eastward energy
propagation.
Kelvin waves
are non

disper
sive with their phase propagating
relatively quickly to the east at the same speed as their group.
Eastward inertio

gravity waves
have phase and group velocities
to the east.
Westward inertio

gravity waves
have phase velocity to the west
and group velocity
also to the west except for very low zonal
wavenumbers.
EFS 4
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16
Phase speed
Inertio

gravity waves propagate much more quickly than Rossby
waves, while the Kelvin wave has phase speeds of intermediate
magnitude.
Typical values of the Kelvin wave speed are in the
range
1
10 50ms
e e
c gh
in the troposphere (corresponding to
10 250m
e
h
) with higher values in the middle atmosphere. For
internal ocean waves that propagate along the thermocline
1
0.5 3ms
e
c
(corresponding to
0.025 1m
e
h
)
.
Scale
The horizontal scale of the waves is determined by the equatorial
Rossby radius,
1/2
/
e
L gh
. For the troposphere, with the
value of
e
h
above, this gives 6

13
º
latitude. For intern
al modes in
the ocean, it gives 1.3

3.3
º
latitude.
EFS 4
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17
EFS 4
/
18
Model experiment
Multilevel primitive equation atmospheric model forced by
imposed heating over period of two days
. Heating representative
of latent heating in organized convection.
Kelvin
wave
Rossby
wave
Kelvin
wave
half

way around globe
Rossby
wave
dispersed and new
circulation cells
developed
EFS 4
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19
EFS 4
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20
Observations
First equatorial waves observed were Kelvin waves and Rossby

gravity waves from balloon soundings over Pacific. Propagating
vertically into stratosphere from tropospheric source (important
source of mo
mentum).
Oscillation 4

5 days: Rossby

gravity wave.
EFS 4
/
21
Wave

number Frequency Spectrum of Convectively

Coupled Equatorial Waves
Wave

number frequency spectral peaks of satellite

observed
outgoing long

wave radiation (OLR) between 15ºN and 15ºS. A)
antisymmetric component w.r.t. equator; B) symm
etric
component. Superimposed are dispersion curves of equatorial
waves (Wheeler and Kiladis, 1999; Wheeler et al, 2000).
EFS 4
/
22
Equatorial Rossby Wave n=1
Typical lower tropospheric
Rossby wave. Suppressed convection
in conjunction with equatorward flow. Modulates large

scale
tropical weather.
EFS 4
/
23
850mb ci
rculation pattern (contours of streamfunction and wind
vectors) and OLR anomalies (red and blue shading) of an observed
n = 1 equatorial Rossby wave.
Individual circulation dipoles
propagate to the west while new circulations develop in their wake
to their
east. Period of disturbance in this sequence is about 12
days. Enhanced convection (blue) in the poleward flow, and
suppressed convection (red) in the equatorward flow (Kiladis and
Wheeler, 1995)
EFS 4
/
24
Mixed Rossby

gravity wave
Streamfunction anomalies (contours and purple/orange shading),
wind vectors, and the OLR anomalies (blue is an indication of
enhanced convection and red is suppressed convection). Th
e
westward phase speed of these waves and a period of about 5 days
is readily apparent. Also of note is the assymetrical convection
patterns associated with these waves. The latitudinal coverage of
these plots is from 17.5°N to 17.5°S. (TOGA

COARE)
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