Transcript Section 5: Kelvin waves - University at Albany, SUNY

Section 5: Kelvin waves
1.
2.
3.
4.
5.
Introduction
Shallow water theory
Observation
Representation in GCM
Summary
5.1. Introduction
• Equatorial waves:
• Trapped near equator
• Propagate in zonal-vertical directions
• Coriolis force changes sign at the equator
• Can be oceanic or atmospheric.
• Diabatic heating by organized tropical convection can excite
atmospheric equatorial waves, wind stress can excite oceanic
equatorial waves.
• Atmospheric equatorial wave propagation is remote response to
localized heat source.
• Oceanic equatorial wave propagation can cause local wind stress
anomalies to remotely influence thermocline depth and SST.
• Described by the shallow water theory.
5.1. Introduction
• Kelvin waves were first identified by William Thomson
(Lord Kelvin) in the nineteenth century.
• Kelvin waves are large-scale waves whose structure
"traps" them so that they propagate along a physical
boundary such as a mountain range in the atmosphere
or a coastline in the ocean.
• In the tropics, each hemisphere can act as the barrier for
a Kelvin wave in the opposite atmosphere, resulting in
"equatorially-trapped" Kelvin waves.
• Kelvin waves are thought to be important for initiation of
the El Niño Southern Oscillation (ENSO) phenomenon
and for maintenance of the MJO.
5.1. Introduction
• Convectively-coupled atmospheric Kelvin waves have a
typical period of 6-7 days when measured at a fixed
point and phase speeds of 12-25 m s-1.
• Dry Kelvin waves in the lower stratosphere have phase
speed of 30-60 m s-1.
• Kelvin waves over the Indian Ocean generally propagate
more slowly (12–15 m s-1) than other regions.
• They are also slower, more frequent, and have higher
amplitude when they occur in the active convective stage
of the MJO.
5.2. Theory
• Shallow water model
• Matsuno (1966)
z
y
Equatorial β-plan
cos  1
he
f is the coriolis parameter
β is the Rossby parameter

v

u
sin     y / a
f  y
h
Eq.
x
Linearized Shallow Water Equations
for perturbations on a motionless
basic state of mean depth he
Momentum:
Continuity:
where
u
 
  yv  
0
t
x
v
 
  yu  
0
t
y
 u v 
 
 ghe 

  0
t
 x y 
   gh
(1.1)
(1.2)
(1.3)
is the geopotential disturbance
Seek solutions in form of zonally propagating
waves, i.e., assume wevalike solution but
retain y-variation:
u, v,  Reuˆ( y), vˆ( y),ˆ( y)expikx  t 
Substituting this into (1.1-1.3) gives:
 iu   yv  ik   0
 
 iv    yu 
0
y

v  
 i  ghe  iku 
  0
y 

(2.1)
(2.2)
(2.3)
Eliminating u’ from (2.1) and (2.2) gives:

 
(  y   )v  i ky     0
y 

2
2
2
(3)
and from (2.1) and (2.3) gives:
 v ky 
(  ghek )  ighe  
v   0
 y  
2
2
(4)
Elimination of Φ between (3) and (4) and assuming ω2 ≠ ghek2 gives:
d vˆ  
k  y 
2
vˆ  0
 
k 

2
dy  ghe

ghe 
2
Requires
2
vˆ
2
2
(5)
to decay to zero at large |y| (motion near the equator)
Schrödinger equation with simple harmonic potential energy, solutions are:
vˆ  0
Other solutions exist only for given k if ω takes
particular value.
Non-dimensionalize and set
 2
k  ghe 1
2

  
k 

  2
2
 ghe
vˆ  F (Y )e
Y
Y 2 / 2
 1/ 2
ghe 
1/ 4
y
(5) can be re-written as Hermite polynomial equation
F   2YF   2F  0
Solutions that satisfy the boundary conditions are:
F  cH
where
and
 n
Hn ( y)
for
n  0,1,2,...
Is a Hermite polynomial
Horizontal dispersion relation:
ghe   2
k 
2

  2n  1, n  0,1,2...
k 
  ghe

(6)
ω is cubic 3 roots for ω when k and n are specified.
At low frequencies: equatorial Rossby wave
At high frequencies: Inertio-gravity wave
For n = 0 : eastward inertio-gravity waves and Yanai
wave.
Frequency ω
Theoretical Dispersion Relationships for Shallow Water Modes on Eq.  Plane
Matsuno, 1966
Zonal Wavenumber k
Frequency ω
Theoretical Dispersion Relationships for Shallow Water Modes on Eq.  Plane
Westward Eastward
Matsuno, 1966
Zonal Wavenumber k
Theoretical Dispersion Relationships for Shallow Water Modes on Eq.  Plane
Eastward
Inertio-Gravity
Westward
Inertio-Gravity
Frequency ω
Kelvin
Mixed Rossby-gravity
(Yanai)
Matsuno, 1966
Equatorial Rossby
Zonal Wavenumber k
For the Kelvin wave case, v’ = 0 (2.1-2.3) become:
 iu  ik   0
 
 yu  
0
y
 i  ghe iku  0
(7.1)
(7.2)
(7.3)
dispersion relation given by (7.1) and (7.3):
Kelvin  ghe k
(8)
With meridional structure of zonal wind:
2


y
uˆ  u0 exp 
 2 gh
e





(9)
Zonal velocity and geopotential perturbations vary in latitude as Gaussian
functions centered on the equator
Kelvin Wave Theoretical Structure
Wind, Pressure (contours),
Divergence, blue negative
Zonal phase speed

cp 
k
Zonal component of group velocity

cg 
k
Kelvin waves are non-dispersive with phase
propagating relatively quickly to east with same speed
as their group:
10-50 m/s in troposphere correspond to he =10-250 m.
0.5-3 m/s in ocean along the thermocline correspond
to he =0.025-1 m.
The horizontal scale of waves is given by
equatorial Rossby radius
1/ 2
 ghe 
L  

  
for he = 10-250 m in troposphere, L = 6-13o latitude.
for he = 0.025-1 m in ocean, L = 1.3-3.3o latitude.
Model experiment: Gill model
Multilevel primitive
atmospheric model
forced by latent
heating in
organized
convection over 2
days.
imposed heating
Vectors: 200 hPa
horizontal wind
anomalies
Contours: surface
temperature
perturbations