Free and convectively coupled equatorial waves diagnosis using 3-D Normal Modes

Free and convectively coupled equatorial waves diagnosis using 3-D Normal Modes Carlos A. F. Marques and J.M. Castanheira CESAM & Department of Physics University of Aveiro Portugal MODES Workshop - Boulder 26-28 Aug. 215

Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. (215) DOI:1.12/qj.2563 Convectively coupled equatorial-wave diagnosis using three-dimensional normal modes José M. Castanheira* and Carlos A. F. Marques CESAM & Department of Physics, University of Aveiro, Portugal *Correspondence to: J. M. Castanheira, CESAM & Department of Physics, University of Aveiro, Campus de Santiago, 381-193 Aveiro, Portugal. E-mail: jcast@ua.pt A new methodology for the diagnosis of convectively coupled equatorial waves (CCEWs) is presented. It is based on a pre-filtering of the geopotential and horizontal wind, using three-dimensional (3D) normal mode functions of the adiabatic linearized equations of a resting atmosphere, followed by a space time spectral analysis to identify the spectral regions of coherence. The methodology permits a direct detection of various types of equatorial wave, compares the dispersion characteristics of the coupled waves with the theoretical dispersion curves and allows an identification of which vertical modes are more involved in the convection. Moreover, the proposed methodology is able to show the existence of free dry waves and moist coupled waves with a common vertical structure, which is in conformity with the

Outline Observation of Convectively Coupled Equatorial Waves (CCEW) Theory of equatorial waves Data and Method Results Coherence spectra Vertical decomposition Power spectra Separating stratosphere and troposphere QBO effects on the vertical propagation of equatorial waves Conclusions

CCEW Sep 1 Daily OLR from NOAA, [1 o S to 2.5 o N], 1992 1993 29 28 Oct 1 27 26 25 Nov 1 24 23 22 21 Dec 1 2 Wm 2 19 18 Jan 1 17 16 15 14 Feb 1 13 12 11 Mar 1 6 E 12 E 18 12 W 6 W Longitude

Equatorial waves Linearized primitive equations: u t f v + 1 φ a cos θ λ = (1) v t + f u + 1 φ a θ = (2) V + 1 ρ z (ρ w) = (3) ( ) 1 φ + wn 2 ρ t z = (4)

Equatorial waves Linearized primitive equations: u t f v + 1 φ a cos θ λ = (1) v t + f u + 1 φ a θ = (2) V + 1 ρ z (ρ w) = (3) ( ) 1 φ + wn 2 ρ t z = (4) Assuming N constant and combining (3) and (4) to eliminate w:

Equatorial waves Linearized primitive equations: u t f v + 1 φ a cos θ λ = (5) v t + f u + 1 φ a θ = (6) [ ( )] 1 φ ρ N 2 V t ρ z z = (7)

Equatorial waves Linearized primitive equations: u t f v + 1 φ a cos θ λ = (5) v t + f u + 1 φ a θ = (6) [ ( )] 1 φ ρ N 2 V t ρ z z = (7) Considering solutions with separable vertical structure: [ ] [u, v, φ] = G(z) ũ(t, θ, λ), ṽ(t, θ, λ), φ(t, θ, λ)

Equatorial waves One obtains the horizontal structure equations: ũ t f ṽ + 1 φ a cos θ λ = (8) ṽ t + f ũ + 1 φ a θ = (9) φ t + gh e Ṽ = (1)

Equatorial waves One obtains the horizontal structure equations: ũ t f ṽ + 1 φ a cos θ λ = (8) ṽ t + f ũ + 1 φ a θ = (9) φ t + gh e Ṽ = (1) and the vertical structure equation: ( ) 1 G ρ + N2 G = (11) ρ z z gh e where 1/gh e is the separation constant.

Equatorial waves For a wave with vertical wave number m: [ ] [u, v, φ] = û(θ), ˆv(θ), ˆφ(θ) e z/2h e imz e i(kλ+ωt)

Equatorial waves For a wave with vertical wave number m: [ ] [u, v, φ] = û(θ), ˆv(θ), ˆφ(θ) e z/2h e imz e i(kλ+ωt) the equivalent depth is given by h e = N 2 g ( m 2 + 1 4H 2 ) (12)

Equatorial waves Using the equatorial β-plane approximation: ũ t βy ṽ + φ x ṽ t + βy ũ + φ y ( φ u t + gh e x + v ) y = (13) = (14) = (15)

Equatorial waves Using the equatorial β-plane approximation: ũ t βy ṽ + φ x ṽ t + βy ũ + φ y ( φ u t + gh e x + v ) y = (13) = (14) = (15) A complete set of zonal propagating wave solutions of these system of shallow water equations was found by Matsuno (1966).

3-D normal mode basis We solved the equations over the sphere in isobaric coordinates, with the vertical structure equation given by: ( ) 1 G + 1 G = p S p gh e

3-D normal mode basis We solved the equations over the sphere in isobaric coordinates, with the vertical structure equation given by: ( ) 1 G + 1 G = p S p gh e The VSE was solved numerically with a spectral method as in Kasahara (1984) and Castanheira et al. (1999).

1 m= m=1 m=2 Vertical Structure Functions 5 1 45 4 m=3 m=4 m=5 5 45 4 Pressure (hpa) 1 1 1 2 5 hpa 15 hpa 35 3 25 2 15 1 1 1 2 5 hpa 15 hpa 35 3 25 2 15 Altitude (km) 1 5 hpa 5 1 3 85 hpa 6 3 3 6 1 5 hpa 5 1 3 85 hpa 6 3 3 6 1 m=6 m=7 m=8 5 45 4 1 m=9 m=1 m=11 5 45 4 Pressure (hpa) 1 1 1 2 5 hpa 15 hpa 35 3 25 2 15 1 1 1 2 5 hpa 15 hpa 35 3 25 2 15 Altitude (km) 1 5 hpa 5 1 3 85 hpa 6 3 3 6 1 5 hpa 5 1 3 85 hpa 6 3 3 6

3-D normal mode basis The horizontal structure equations over the sphere ũ t f ṽ + 1 φ a cos θ λ = (16) ṽ t + f ũ + 1 φ a θ = (17) φ t + gh e Ṽ = (18) were solved using the methodology of Swarztrauber and Kasahara (1985).

Shallow water waves over the sphere (h 1 = 5591 m) 9 a) Kelvin b) Equatorial Rossby (n r = 1) 6 3 Latitude 3 6 9 6 E 12 E 18 12 W 6 W 6 E 12 E 18 12 W 6 W 9 c) Mixed Rossby-Gravity d) Eastward Inertio-Gravity (n e = ) 6 3 Latitude 3 6 9 6 E 12 E 18 12 W 6 W 6 E 12 E 18 12 W 6 W Longitude Longitude

Shallow water waves over the sphere (h 4 = 474 m) 9 a) Kelvin b) Equatorial Rossby (n r = 1) 6 3 Latitude 3 6 9 6 E 12 E 18 12 W 6 W 6 E 12 E 18 12 W 6 W 9 c) Mixed Rossby-Gravity d) Eastward Inertio-Gravity (n e = ) 6 3 Latitude 3 6 9 6 E 12 E 18 12 W 6 W 6 E 12 E 18 12 W 6 W Longitude Longitude

.8 Dispersion curves for equatorial waves 1.25.75 1.33.7 1.42.65 1.54.6 1.67.55 WIG 2 1.82.5 2..45.4.35.3 WIG 1 EIG 2.5 1 m.25 34 m 22 m 14 m 4..2 53 m 5..15 MRG 96 m KELVIN.1 1. ER.5 2.

Data Outgoing Longwave Radiation (OLR) from the National Oceanic and Atmospheric Administration (NOAA) for the period 1979-212. Horizontal wind (u, v) and geopotencial (φ) from the ERA-Interim reanalysis (1979-212).

Method: filtering the dynamical fields Projection of the horizontal wind (u, v) and geopotential (φ) onto the normal modes of the linearized primitive equations on the sphere. u v φ = m,k,n w mkn (t) G m (p) e ikλ C m U(θ) iv (θ) Z(θ) mkn Reconstitution of the horizontal wind (u, v) and geopotential (φ) with a given subset of modes (filtering): u n v n φ n = m,k w mkn (t) G m (p) e ikλ C m U(θ) iv (θ) Z(θ) mkn

Method: space-time cross-spectral analysis (Hayashi, 1982) Considering a space-time series Y (λ, t) = k [C k (t) cos(kλ) + S k (t) sin(kλ)]

Method: space-time cross-spectral analysis (Hayashi, 1982) Considering a space-time series Y (λ, t) = k [C k (t) cos(kλ) + S k (t) sin(kλ)] The space-time power spectra is given by 4 P k,±ω (Y ) = P ω (C k ) + P ω (S k ) ± 2 Q ω (C k, S k ), where P ω and Q ω are the time power and quadrature spectra, respectively.

Method: space-time cross-spectral analysis (Hayashi, 1982) Spectral coherence and phase difference between two fields Y (λ, t) and Y (λ, t) are given by Coh 2 k,±ω (Y, Y ) = K 2 k,±ω (Y, Y ) + Q 2 k,±ω (Y, Y ) P k,±ω (Y )P k,±ω (Y ) and Ph k,±ω (Y, Y ) = tan 1 [ Q k,±ω (Y, Y )/K k,±ω (Y, Y ) ] where

Method: space-time cross-spectral analysis (Hayashi, 1982) where 4 K k,±ω (Y, Y ) = K ω (C k, C k ) + K ω (S k, S k ) ±Q ω (C k, S k ) Q ω (S k, C k ) and 4 Q k,±ω (Y, Y ) = ±Q ω (C k, C k ) ± Q ω (S k, S k ) K ω (C k, S k ) + K ω (S k, C k ) are cospectra and quadrature spectra, respectively.

Testing the methodology Reconstructing the circulation field u v φ = m,k,n w mkn (t) G m (p) e ikλ C m U(θ) iv (θ) Z(θ) mkn Symmetric and antisymmetric components of a variable Y Y S (θ) = Y A (θ) = Y (θ) + Y ( θ) 2 Y (θ) Y ( θ) 2

Squared coherence between symmetric components of zonal wind and OLR.5 a) Original u (15 hpa) 96 m b) Reconstructed u (15 hpa) 2..45 53 m.4 34 m 2.5.35 22 m.5.3.25.2 KELVIN 14 m 1 m 4. 5..4.15.1 96 m ER 1..5 2..3.5 c) Original u (85 hpa) d) Reconstructed u (85 hpa) 2..45.4 2.5.2.35.3.25.2 4. 5..1.15.1 1..5 2.

.5.45.4 Squared coherence between anti symmetric components of zonal wind and OLR a) Original u (15 hpa) 34 m b) Reconstructed u (15 hpa) 53 m 22 m 96 m EIG 14 m 1 m 2. 2.5.35.5.3.25.2 4. 5..4.15 MRG.1 1..5 2..3.5 c) Original u (85 hpa) d) Reconstructed u (85 hpa) 2..45.4 2.5.2.35.3.25.2 4. 5..1.15.1 1..5 2.

Selecting waves Fixing a given pair of zonal, k and meridional n indices, waves of a given type are selected u kn v kn φ kn = m w mkn (t) G m (p) e ikλ C m U(θ) iv (θ) Z(θ) mkn

Results Space-time coherence spectra for eastward inertio-gravity (EIG), mixed Rossby-gravity (MRG), Kelvin (Kel) and Equatorial Rossby (ER) waves.

Squared coherence between u EIG and anti symmetric OLR.5.45 a) 5 hpa 34 m 53 m 22 m 96 m EIG 14 m b) 15 hpa 2..4 1 m 2.5.35.5.3.25.2 4. 5..4.15 MRG.1 1..5 2..3.5 c) 5 hpa d) 85 hpa 2..45.4 2.5.2.35.3.25.2 4. 5..1.15.1 1..5 2.

Squared coherence between u MRG and anti symmetric OLR.5.45 a) 5 hpa 34 m 53 m 22 m 96 m EIG 14 m b) 15 hpa 2..4 1 m 2.5.35.5.3.25.2 4. 5..4.15 MRG.1 1..5 2..3.5 c) 5 hpa d) 85 hpa 2..45.4 2.5.2.35.3.25.2 4. 5..1.15.1 1..5 2.

Squared coherence between u KEL and symmetric OLR.5 a) 5 hpa 96 m b) 15 hpa 2..45 53 m.4 34 m 2.5.35 22 m.5.3.25.2 KELVIN 14 m 1 m 4. 5..4.15.1 96 m ER 1..5 2..3.5 c) 5 hpa d) 85 hpa 2..45.4 2.5.2.35.3.25.2 4. 5..1.15.1 1..5 2.

Squared coherence between u ER and symmetric OLR.5 a) 5 hpa 96 m b) 15 hpa 2..45 53 m.4 34 m 2.5.35 22 m.5.3.25.2 988 m KELVIN 14 m 1 m 4. 5..4.15.1 96 m ER 1..5 2..3.5 c) 5 hpa d) 85 hpa 2..45.4 2.5.2.35.3.25.2 4. 5..1.15.1 1..5 2.

Vertical mode decomposition of waves Fixing the vertical, zonal,and meridional indices, mkn, respectively, we obtain a vertical modal decomposition of the waves u mkn v mkn φ mkn = w mkn (t) G m (p) e ikλ C U(θ) iv (θ) Z(θ) Now for, the cosine and sine coefficients are given by mkn Ck mn R [w mkn (t)] Sk mn I [w mkn (t)] and we may analyze the coherence of the 3-D waves with the OLR and interpret their power spectra as the total (Kinetic + Available Potential) energy spectra.

Squared coherence between the Kelvin baroclinic modes and symmetric OLR.5.45.4 a) Kelvin (m=1) 5591 m b) Kelvin (m=2) 2241 m c) Kelvin (m=3) 937 m 2. 2.5.35.3.25.2.15 5591 m KELVIN 2241 m 937 m 4. 5..1.5 ER 1. 2..5.45.4 d) Kelvin (m=4) 474 m e) Kelvin (m=5) 279 m f) Kelvin (m=6) 17 m 2. 2.5.35.3.25.2.15.1.5 474 m 279 m 17 m 4. 5. 1. 2..1.2.3.4.5

Squared coherence between the Kelvin baroclinic modes and symmetric OLR.5.45.4 a) Kelvin (m=7) 116 m b) Kelvin (m=8) 78 m c) Kelvin (m=9) 2. 2.5.35.3.25.2.15 ER KELVIN 4. 5..1.5 116 m 78 m 1. 2..5.45.4 d) Kelvin (m=1) 37 m e) Kelvin (m=11) 26 m f) Kelvin (m=12) 2. 2.5.35.3.25.2.15 4. 5..1.5 1. 2..1.2.3.4.5

1 2 m= m=1 m=2 Vertical Structure Functions 1 2 16 m=3 m=4 15 hpa 14 m=5 15 hpa 16 14 12 12 Pressure (hpa) 5 hpa 1 8 6 5 hpa 1 8 6 Altitude (km) 4 4 2 85 hpa 1 3 3 1.5 1.5 3 2 85 hpa 1 3 3 1.5 1.5 3 1 2 m=6 m=7 m=8 15 hpa 16 14 1 2 m=9 m=1 m=11 15 hpa 16 14 12 12 Pressure (hpa) 5 hpa 1 8 6 5 hpa 1 8 6 Altitude (km) 4 4 2 85 hpa 1 3 3 1.5 1.5 3 2 85 hpa 1 3 3 1.5 1.5 3

Squared coherence between the MRG baroclinic modes and anti symmetric OLR.5.45 a) MRG (m=1) 5591 m b) MRG (m=2) 2241 m c) MRG (m=3) 937 m 2..4 EIG 2.5.35.3.25.2.15 MRG 4. 5..1.5 1. 2..5.45 d) MRG (m=4) 474 m e) MRG (m=5) 279 m f) MRG (m=6) 17 m 2..4 2.5.35.3.25.2.15 4. 5..1.5 1. 2..1.2.3.4.5

Squared coherence between the MRG baroclinic modes and anti symmetric OLR.5.45 a) MRG (m=7) 116 m b) MRG (m=8) 78 m c) MRG (m=9) 2..4 EIG 2.5.35.3.25.2.15 MRG 4. 5..1.5 1. 2..5.45.4 d) MRG (m=1) 37 m e) MRG (m=11) 26 m f) MRG (m=12) 2. 2.5.35.3.25.2.15 4. 5..1.5 1. 2..1.2.3.4.5

Squared coherence between the EIG baroclinic modes and anti symmetric OLR.5.45 a) EIG (m=1) 5591 m b) EIG (m=2) 2241 m c) EIG (m=3) 937 m 2..4 EIG 2.5.35.3.25.2.15 MRG 4. 5..1.5 1. 2..5.45 d) EIG (m=4) 474 m e) EIG (m=5) 279 m f) EIG (m=6) 17 m 2..4 2.5.35.3.25.2.15 4. 5..1.5 1. 2..1.2.3.4.5

Squared coherence between the EIG baroclinic modes and anti symmetric OLR.5.45 a) EIG (m=7) 116 m b) EIG (m=8) 78 m c) EIG (m=9) 2..4 EIG 2.5.35.3.25.2.15 MRG 4. 5..1.5 1. 2..5.45.4 d) EIG (m=1) 37 m e) EIG (m=11) 26 m f) EIG (m=12) 2. 2.5.35.3.25.2.15 4. 5..1.5 1. 2..1.2.3.4.5

Squared coherence between the ER baroclinic modes and symmetric OLR.5.45.4 a) ER (m=1) 5591 m b) ER (m=2) 2241 m c) ER (m=3) 937 m 2. 2.5.35.3.25.2.15 5591 m KELVIN 2241 m 937 m 4. 5..1.5 ER 1. 2..5.45.4 d) ER (m=4) 474 m e) ER (m=5) 279 m f) ER (m=6) 17 m 2. 2.5.35.3.25.2.15.1.5 474 m 279 m 17 m 4. 5. 1. 2..1.2.3.4.5

Squared coherence between the ER baroclinic modes and symmetric OLR.5.45.4 a) ER (m=7) 116 m b) ER (m=8) 78 m c) ER (m=9) 2. 2.5.35.3.25.2.15 ER KELVIN 4. 5..1.5 116 m 78 m 1. 2..5.45.4 d) ER (m=1) 37 m e) ER (m=11) 26 m f) ER (m=12) 2. 2.5.35.3.25.2.15 4. 5..1.5 1. 2..1.2.3.4.5

Normal Mode energy spectrum divided by the background red noise.5.45.4 a) Kelvin (m=1) 5591 m b) Kelvin (m=2) c) Kelvin (m=3) 2241 m 937 m 2. 2.5.35.3.25.2.15 5591 m KELVIN 2241 m 937 m 4. 5..1.5 ER 1. 2..5.45.4 d) Kelvin (m=4) 474 m e) Kelvin (m=5) 279 m f) Kelvin (m=6) 17 m 2. 2.5.35.3.25.2.15.1.5 474 m 279 m 17 m 4. 5. 1. 2. 1 1.5 2 2.5 3 3.5 4 4.5 5

Normal Mode energy spectrum divided by the background red noise.5.45.4 a) Kelvin (m=7) 116 m b) Kelvin (m=8) 78 m c) Kelvin (m=9) 2. 2.5.35.3.25.2.15 ER KELVIN 4. 5..1.5 116 m 78 m 1. 2..5.45.4 d) Kelvin (m=1) 37 m e) Kelvin (m=11) 26 m f) Kelvin (m=12) 2. 2.5.35.3.25.2.15 4. 5..1.5 1. 2. 1 1.5 2 2.5 3 3.5 4 4.5 5

Normal Mode energy spectrum divided by the background red noise.5.45 a) MRG (m=1) 5591 m b) MRG (m=2) 2241 m c) MRG (m=3) 937 m 2..4 EIG 2.5.35.3.25.2.15 MRG 4. 5..1 1..5 2..5.45 d) MRG (m=4) 474 m e) MRG (m=5) 279 m f) MRG (m=6) 17 m 2..4 2.5.35.3.25.2.15 4. 5..1 1..5 2. 1 1.5 2 2.5 3 3.5 4 4.5 5

Normal Mode energy spectrum divided by the background red noise.5.45 a) MRG (m=7) 116 m b) MRG (m=8) 78 m c) MRG (m=9) 2..4 EIG 2.5.35.3.25.2.15.1.5 MRG 4. 5. 1. 2..5.45.4 d) MRG (m=1) 37 m e) MRG (m=11) 26 m f) MRG (m=12) 2. 2.5.35.3.25.2.15 4. 5..1 1..5 2. 1 1.5 2 2.5 3 3.5 4 4.5 5

Normal Mode energy spectrum divided by the background red noise.5.45 a) EIG (m=1) 5591 m b) EIG (m=2) 2241 m c) EIG (m=3) 937 m 2..4 EIG 2.5.35.3.25.2.15 MRG 4. 5..1 1..5 2..5.45 d) EIG (m=4) 474 m e) EIG (m=5) 279 m f) EIG (m=6) 17 m 2..4 2.5.35.3.25.2.15 4. 5..1 1..5 2. 1 1.5 2 2.5 3 3.5 4 4.5 5

Normal Mode energy spectrum divided by the background red noise.5.45 a) EIG (m=7) 116 m b) EIG (m=8) 78 m c) EIG (m=9) 2..4 EIG 2.5.35.3.25.2.15.1.5 MRG 4. 5. 1. 2..5.45.4 d) EIG (m=1) 37 m e) EIG (m=11) 26 m f) EIG (m=12) 2. 2.5.35.3.25.2.15 4. 5..1 1..5 2. 1 1.5 2 2.5 3 3.5 4 4.5 5

Normal Mode energy spectrum divided by the background red noise.5.45.4 a) ER (m=1) 5591 m b) ER (m=2) c) ER (m=3) 2241 m 937 m 2. 2.5.35.3.25.2.15 5591 m KELVIN 2241 m 937 m 4. 5..1.5 ER 1. 2..5.45.4 d) ER (m=4) 474 m e) ER (m=5) 279 m f) ER (m=6) 17 m 2. 2.5.35.3.25.2.15.1.5 474 m 279 m 17 m 4. 5. 1. 2. 1 1.5 2 2.5 3 3.5 4 4.5 5

Normal Mode energy spectrum divided by the background red noise.5.45.4 a) ER (m=7) 116 m b) ER (m=8) 78 m c) ER (m=9) 2. 2.5.35.3.25.2.15 ER KELVIN 4. 5..1.5 116 m 78 m 1. 2..5.45.4 d) ER (m=1) 37 m e) ER (m=11) 26 m f) ER (m=12) 2. 2.5.35.3.25.2.15 4. 5..1.5 1. 2. 1 1.5 2 2.5 3 3.5 4 4.5 5

Stratosphere : Normal Mode energy spectrum divided by the background red noise.5.45.4 a) Kelvin (m=1) 5591 m b) Kelvin (m=2) c) Kelvin (m=3) 2241 m 937 m 2. 2.5.35.3.25.2.15 5591 m KELVIN 2241 m 937 m 4. 5..1.5 ER 1. 2..5.45.4 d) Kelvin (m=4) 474 m e) Kelvin (m=5) 279 m f) Kelvin (m=6) 17 m 2. 2.5.35.3.25.2.15.1.5 474 m 279 m 17 m 4. 5. 1. 2. 1 1.5 2 2.5 3 3.5 4 4.5 5

Troposphere : Normal Mode energy spectrum divided by the background red noise.5.45.4 a) Kelvin (m=1) 5591 m b) Kelvin (m=2) c) Kelvin (m=3) 2241 m 937 m 2. 2.5.35.3.25.2.15 5591 m KELVIN 2241 m 937 m 4. 5..1.5 ER 1. 2..5.45.4 d) Kelvin (m=4) 474 m e) Kelvin (m=5) 279 m f) Kelvin (m=6) 17 m 2. 2.5.35.3.25.2.15.1.5 474 m 279 m 17 m 4. 5. 1. 2. 1 1.5 2 2.5 3 3.5 4 4.5 5

QBO effects The vertical propagation of equatorial waves was analysed for the two phases of the QBO. Considering the daily zonal mean 3-hPa zonal wind at the equator the QBO phases were defined as follows u( N, 3 hpa) > 5 m s 1 Westerly phase u( N, 3 hpa) < 5 m s 1 Easterly phase From the linear theory and using the slow-variation WKBJ approximation (Andrews et al., 1987), waves with zonal phase velocity c can propagate vertically only if the zonal winds satisfy c u(z) > for Kelvin waves β/k 2 < c u(z) < for MRG waves

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The ratios between the power spectra calculated for each QBO phase are represented in the next Figures F = PE QBO P W QBO

Ratio between the Normal Mode Power spectra for the two QBO phases (Easterly/Westerly).5.45.4 a) Kelvin (m=1) 5591 m b) Kelvin (m=2) 2241 m c) Kelvin (m=3) 937 m 2. 2.5.35.3.25.2.15 5591 m KELVIN 2241 m 937 m 4. 5..1.5 ER 1. 2..5.45.4 d) Kelvin (m=4) 474 m e) Kelvin (m=5) 279 m f) Kelvin (m=6) 17 m 2. 2.5.35.3.25.2.15.1.5 474 m 279 m 17 m 4. 5. 1. 2..3.5.7.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7

Ratio between the Normal Mode Power spectra for the two QBO phases (Easterly/Westerly).5.45.4 a) Kelvin (m=7) 116 m b) Kelvin (m=8) 78 m c) Kelvin (m=9) 2. 2.5.35.3.25.2.15 ER KELVIN 4. 5..1.5 116 m 78 m 1. 2..5.45.4 d) Kelvin (m=1) 37 m e) Kelvin (m=11) 26 m f) Kelvin (m=12) 2. 2.5.35.3.25.2.15 4. 5..1.5 1. 2..3.5.7.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7

Stratosphere : Ratio between the Normal Mode Power spectra for the two QBO phases (Easterly/Westerly).5 2. a) Kelvin (m=1) 5591 m b) Kelvin (m=2) c) Kelvin (m=3).45 2241 m 937 m.4 2.5.35.3.25 KELVIN 4. 5591 m.2 5. 2241 m 937 m.15.1.5 ER 1. 2..5.45.4 d) Kelvin (m=4) 474 m e) Kelvin (m=5) 279 m f) Kelvin (m=6) 17 m 2. 2.5.35.3.25.2.15.1.5 474 m 279 m 17 m 4. 5. 1. 2..3.5.7.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7

Stratosphere : Ratio between the Normal Mode Power spectra for the two QBO phases (Easterly/Westerly).5 2. a) Kelvin (m=7) b) Kelvin (m=8) c) Kelvin (m=9).45 116 m 78 m.4 2.5.35.3.25 KELVIN 4..2 5..15 ER.1.5 116 m 78 m 1. 2..5.45.4 d) Kelvin (m=1) 37 m e) Kelvin (m=11) 26 m f) Kelvin (m=12) 2. 2.5.35.3.25.2.15 4. 5..1.5 1. 2..3.5.7.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7

The vertical wavenumber of Kelvin waves is given by m(z) = N c u

The vertical wavenumber of Kelvin waves is given by m(z) = N c u Therefore the westerly QBO phase is associated with smaller vertical wavelengths.

2 E QBO W QBO Stratospheric Kelvin waves 1.5 Energy (1 kj m 2 ) 1.5 1 2 3 4 5 6 7 8 9 1 11 12 13 vertical mode

2 E QBO W QBO Stratospheric Kelvin waves 1.5 Energy (1 kj m 2 ) 1.5 1 2 3 4 5 6 7 8 9 1 11 12 13 vertical mode Using typical values for the variables in the theoretical relationship h e = N 2 g ( m 2 + 1 4H 2 )

2 E QBO W QBO Stratospheric Kelvin waves 1.5 Energy (1 kj m 2 ) 1.5 1 2 3 4 5 6 7 8 9 1 11 12 13 vertical mode Using typical values for the variables in the theoretical relationship one obtains h e = N 2 g ( m 2 + 1 4H 2 ) h 7 = 116 m L z = 9.7 km h 9 = L z = 6.5 km

1 m= m=1 m=2 Vertical Structure Functions 5 1 45 m=3 m=4 m=5 5 45 4 4 Pressure (hpa) 1 1 35 3 1 1 35 3 Altitude (km) 5 hpa 25 5 hpa 25 2 1 2 6 3 3 6 15 hpa 2 1 2 6 3 3 6 15 hpa 1 m=6 m=7 m=8 5 hpa 5 45 1 m=9 m=1 5 hpa m=11 5 45 85 hpa 4 85 hpa 4 Pressure (hpa) 1 1 35 3 1 1 35 3 Altitude (km) 5 hpa 25 5 hpa 25 2 2 1 2 6 3 3 15 hpa 6 1 2 6 3 3 15 hpa 6

Stratosphere : Ratio between the Normal Mode Power spectra for the two QBO phases (Easterly/Westerly).5 a) MRG (m=1) b) MRG (m=2) c) MRG (m=3) 2241 m 937 m.45 5591 m 2..4 EIG 2.5.35.3.25.2.15 MRG 4. 5..1 1..5 2..5.45 d) MRG (m=4) 474 m e) MRG (m=5) 279 m f) MRG (m=6) 17 m 2..4 2.5.35.3.25.2.15 4. 5..1 1..5 2..3.5.7.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7

Stratosphere : Ratio between the Normal Mode Power spectra for the two QBO phases (Easterly/Westerly).5 a) MRG (m=7) 116 m b) MRG (m=8) c) MRG (m=9).45 78 m.4 EIG 2. 2.5.35.3.25.2.15.1.5 MRG 4. 5. 1. 2..5.45.4 d) MRG (m=1) 37 m e) MRG (m=11) 26 m f) MRG (m=12) 2. 2.5.35.3.25.2.15 4. 5..1 1..5 2..3.5.7.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7

Conclusions 3-D normal modes over the sphere are a useful tool for the study of equatorial waves:

Conclusions 3-D normal modes over the sphere are a useful tool for the study of equatorial waves: free and convectively coupled equatorial waves of different types are clearly identified;

Conclusions 3-D normal modes over the sphere are a useful tool for the study of equatorial waves: free and convectively coupled equatorial waves of different types are clearly identified; Doppler shifts and other wave propagation features predicted by the theory are clearly shown. Suggestion:The methodology can be made less constraining as in Yang et al. (23) or Gehne and Kleeman (212) but using the Hough vectors instead of parabolic cylinder functions.