Section 5: Kelvin waves

1. Introduction2. Shallow water theory3. Observation4. Representation in GCM5. 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)

v

Equatorial β-plan

1cos ay /sin

yf

he

z

xEq.

h

f is the coriolis parameter β is the Rossby parameter

u

y

Linearized Shallow Water Equations for perturbations on a motionless

basic state of mean depth he

0

0

0

yv

xugh

t

yuy

tv

xvy

tu

e

(1.1)

(1.2)

(1.3)

hg is the geopotential disturbancewhere

Momentum:

Continuity:

Seek solutions in form of zonally propagating waves, i.e., assume wevalike solution but retain y-variation:

tkxiyyvyuvu exp)(ˆ),(ˆ),(ˆRe,,

Substituting this into (1.1-1.3) gives:

0

0

0

yvuikghi

yuyvi

ikvyui

e

(2.1)

(2.2)

(2.3)

Eliminating u’ from (2.1) and (2.2) gives:

0)( 222

y

ykivy

Elimination of Φ between (3) and (4) and assuming ω2 ≠ ghek2 gives:

0ˆˆ 222

2

2

2

v

ghykk

ghdyvd

ee

Requires v̂ to decay to zero at large |y| (motion near the equator)

and from (2.1) and (2.3) gives:

0)( 22

vykyvghikgh ee

(3)

(4)

(5)

Schrödinger equation with simple harmonic potential energy, solutions are:

0ˆ v

Other solutions exist only for given k if ω takes particular value.

Non-dimensionalize and set

y

ghY

eYFv

ghkkgh

e

Y

e

e

4/1

2/1

2/

22

2

)(ˆ

21

2

(5) can be re-written as Hermite polynomial equation

022 FFYF

Solutions that satisfy the boundary conditions are:

cHF where n for ,...2,1,0n

and )( yHn Is a Hermite polynomial

Horizontal dispersion relation:

...2,1,0,1222

nnkk

ghgh

e

e

ω is cubic 3 roots for ω when k and n are specified.At low frequencies: equatorial Rossby waveAt high frequencies: Inertio-gravity waveFor n = 0 : eastward inertio-gravity waves and Yanai wave.

(6)

Theoretical Dispersion Relationships for Shallow Water Modes on Eq. Plane

Freq

uenc

y ω

Zonal Wavenumber k

Matsuno, 1966

Theoretical Dispersion Relationships for Shallow Water Modes on Eq. Plane

Freq

uenc

y ω

Zonal Wavenumber k

Westward Eastward

Matsuno, 1966

Theoretical Dispersion Relationships for Shallow Water Modes on Eq. Plane

Kelvin

Eastward Inertio-Gravity

Equatorial Rossby

Freq

uenc

y ω

Zonal Wavenumber k

Mixed Rossby-gravity (Yanai)

n =

-1

n = 0

n = 1n = 3

n = 1

n = 2

n = 3

n = 4

Westward Inertio-Gravity

Matsuno, 1966

For the Kelvin wave case, v’ = 0 (2.1-2.3) become:

dispersion relation given by (7.1) and (7.3):

kgheKelvin With meridional structure of zonal wind:

eghyuu

2expˆ

2

0

Zonal velocity and geopotential perturbations vary in latitude as Gaussian functions centered on the equator

0

0

0

uikghiy

uy

ikui

e

(7.1)

(7.2)

(7.3)

(8)

(9)

Kelvin Wave Theoretical Structure

Wind, Pressure (contours), Divergence, blue negative

Zonal phase speed

kcp

Zonal component of group velocity

kcg

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

2/1

eghL

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 modelMultilevel 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