Section 5: Kelvin waves 1.Introduction 2.Shallow water theory 3.Observation 4.Representation in GCM 5.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 m s -1. Dry Kelvin waves in the lower stratosphere have phase speed of 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) Equatorial β-plan hehe z x Eq. h f is the coriolis parameter β is the Rossby parameter y

Linearized Shallow Water Equations for perturbations on a motionless basic state of mean depth h e (1.1) (1.2) (1.3) is the geopotential disturbance where Momentum: Continuity:

Seek solutions in form of zonally propagating waves, i.e., assume wevalike solution but retain y-variation: Substituting this into ( ) gives: (2.1) (2.2) (2.3)

Eliminating u’ from (2.1) and (2.2) gives: Elimination of Φ between (3) and (4) and assuming ω 2 ≠ gh e k 2 gives: Requiresto decay to zero at large |y| (motion near the equator) and from (2.1) and (2.3) gives: (3) (4) (5)

Schrödinger equation with simple harmonic potential energy, solutions are: Other solutions exist only for given k if ω takes particular value. Non-dimensionalize and set

(5) can be re-written as Hermite polynomial equation Solutions that satisfy the boundary conditions are: where for and Is a Hermite polynomial

Horizontal dispersion relation: 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. (6)

Theoretical Dispersion Relationships for Shallow Water Modes on Eq.  Plane Frequency ω Zonal Wavenumber k Matsuno, 1966

Theoretical Dispersion Relationships for Shallow Water Modes on Eq.  Plane Frequency ω Zonal Wavenumber k WestwardEastward Matsuno, 1966

Theoretical Dispersion Relationships for Shallow Water Modes on Eq.  Plane Kelvin Eastward Inertio-Gravity Equatorial Rossby Frequency ω Zonal Wavenumber k Mixed Rossby-gravity (Yanai) n = -1 n = 0 n = 1 n = 3 n = 1 n = 2 n = 3 n = 4 Westward Inertio-Gravity Matsuno, 1966

For the Kelvin wave case, v’ = 0 ( ) become: dispersion relation given by (7.1) and (7.3): With meridional structure of zonal wind: Zonal velocity and geopotential perturbations vary in latitude as Gaussian functions centered on the equator (7.1) (7.2) (7.3) (8) (9)

Kelvin Wave Theoretical Structure Wind, Pressure (contours), Divergence, blue negative

Zonal phase speed Zonal component of group velocity Kelvin waves are non-dispersive with phase propagating relatively quickly to east with same speed as their group: m/s in troposphere correspond to h e = m m/s in ocean along the thermocline correspond to h e = m.

The horizontal scale of waves is given by equatorial Rossby radius for h e = m in troposphere, L = 6-13 o latitude. for h e = m in ocean, L = o 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