4. Equatorial Waves and Tropical Dynamics

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Nov 16, 2013 (3 years and 7 months ago)

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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
/
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
/
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
/
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
/
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
/
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
/
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
/
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
/
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
/
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
/
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
/
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
/
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
/
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
/
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
/
19



EFS 4
/
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)