Identifying the MJO Skeleton in Observational Data

. Identifying the MJO Skeleton in Observational Data Sam Stechmann, Wisconsin Andrew Majda, NYU World Weather Open Science Conference August 20, 2014 Montreal, Canada

Theoretical prediction of MJO structure low level pressure contours 2 y (1000 km) 0 2 0 10 20 30 suppressed convection (A < 0) enhanced convection (A > 0) Ingredients: Kelvin wave structure on equator Rossby gyre structure off equator Observed MJO structure Hendon & Salby 1994

Main Question: Can the MJO be isolated in observational data without the use of temporal filtering or EOFs? Instead: use theoretical prediction of MJO structure.

Outline 1. Modeling the MJO Skeleton Provides theoretical prediction of MJO structure 2. Identifying the MJO Skeleton in Observational Data 3. OLR and diabatic heating

MJO Skeleton Model Majda & Stechmann 2009 u t yv = p x yu = p y 0 = p z +θ u x +v y +w z = 0 θ t +w = Ha s θ Linearized primitive equations Equatorial long-wave scaling Coriolis term: equatorial β-plane approx. + q t Qw = Ha+s q a t = Γqa Key mechanism: q a interaction Minimal number of parameters: s θ, Q,Γ Dynamic equation for convective activity q: lower tropospheric moisture anomaly a: amplitude of convective activity envelope Minimal number of parameters: s θ, Q,Γ Minimal nonlinear oscillator model

MJO Skeleton Model (vertical truncation) u t yv θ x = 0 yu θ y = 0 θ t u x v y = Ha s θ q t + Q(u x +v y ) = Ha+s q a t = Γqa Truncate at first vertical baroclinic mode (sinz, cosz) Shallow water equations (long-wave) Matsuno Gill-like model without dissipative mechanisms but with lower tropospheric moisture anomaly, q envelope of convection/wave activity, a, provides dynamic planetary-scale heating Conserved energy: [ 1 t 2 u2 + 1 2 θ2 + 1 Q 21 Q even with source terms s θ = s q = S ( θ+ q Q ) 2 + H ] Γ Q a S Γ Q loga x (uθ) y (vθ) = 0

MJO Skeleton Model (vertical and meridional truncation) K t +K x = 1 2 HA Q t + 1 2 QKx 1 6 2 R t 1 3 R x = 2 2 3 QR x = HA ( 1+ 1 6 Q A t = ΓQ(Ā+A) ) HA Meridional structures: K: Kelvin wave structure R: first symmetric equatorial Rossby wave structure Q: exp( y 2 /2) A: exp( y 2 /2) 3D 1D model

Outline 1. Modeling the MJO Skeleton Provides theoretical prediction of MJO structure 2. Identifying the MJO Skeleton in Observational Data 3. OLR and diabatic heating

Overview of Procedure 1. Vertical (z) projection (using 850 hpa and 200 hpa data) 2. Meridional (y) projection (using parabolic cylinder functions) 3. Zonal (x) projection (using Fourier modes k = 1 to k max ) 4. Define K,R,Q,A 5. Construct the MJO eigenmode e MJO Plus 2 other standard steps: Remove the seasonal cycle Remove ENSO variability (120-day running time average) Question: Does this procedure isolate the MJO in observational data? Note: No temporal bandpass filter or EOFs are used

Vertical projection (using 850 hpa and 200 hpa data) Isolate the 1st baroclinic mode: u = u(850 hpa) u(200 hpa) 2 2 Z(850 hpa) Z(200 hpa) θ = 2 2 Z = geopotential height Z θ from hydrostatic balance: p z θ Z p θ

Meridional y projection Expand using meridional basis functions: Parabolic cylinder functions φ m (y): 0.8 0.4 u(y) = m=0 u m φ m (y) φ m (y) 0 0.4 φ 0 (y) φ 1 (y) φ 2 (y) 0.8 φ m (y) H m (y) e y2 /2 60S 30S 0 30N 60N latitude φ 0 (y) 1 e y2 /2 φ 1 (y) y e y2 /2 φ 2 (y) (2y 2 1)e y2 /2 Average from 15 S 15 N Anti-symmetric Needed for Rossby gyres

Meridional y projection Example with u(x,y) p=200 hpa : Raw data Mode 0 Modes 0+1+2 90 N uwnd at 200 hpa on day 1 70 90 N mode 0 70 90 N mode 0+1+2 70 60 N 60 50 60 N 60 50 60 N 60 50 latitude (deg) 30 N 0 30 S 40 30 20 10 latitude (deg) 30 N 0 30 S 40 30 20 10 latitude (deg) 30 N 0 30 S 40 30 20 10 60 S 0 10 60 S 0 10 60 S 0 10 90 S 0 60 E 120 E 180 120 W 60 W longitude (deg) 20 90 S 0 60 E 120 E 180 120 W 60 W longitude (deg) 20 90 S 0 60 E 120 E 180 120 W 60 W longitude (deg) 20 u(y) u 0 φ 0 (y) u 0 φ 0 (y)+u 1 φ 1 (y)+u 2 φ 2 (y)

Zonal x projection Use zonal Fourier modes k = 1 to k max For visual clarity, k max = 3 here Procedure can be done for any desired wavenumbers (e.g., k = 1 to 8)

Change variables from (u,θ) to (K,R) Define K,R u(y) = u 0 φ 0 (y)+u 1 φ 1 (y)+u 2 φ 2 (y)+ θ(y) = θ 0 φ 0 (y)+θ 1 φ 1 (y)+θ 2 φ 2 (y)+ K = 1 2 u 0 1 2 θ 0, R = R 1 = 1 2 u 0 1 2 θ 0 +u 2 θ 2, K,R ingredients: Both u and θ Both equatorial and off-equatorial

K and R: Case Study: TOGA COARE a K [nondim] b R [nondim] Jun Jun May May Apr Apr Mar Mar Feb Feb Jan Jan Dec Dec Nov Nov Oct Oct Sep Sep Aug Aug Jul 0 60E 120E 180 120W 60W 0.1 0.05 0 0.05 0.1 K strongest during January Always eastward-propagating Propagates both fast, slow Jul 0 60E 120E 180 120W 60W 0.2 0 0.2 R strong during January R eastward-propagating in January Propagates both eastward, westward

How to define Q and A? Q is lower-tropospheric specific humidity: Q = 1 4 shum(925 hpa)+ 1 2 shum(850 hpa)+ 1 shum(700 hpa) 4 A OLR as first guess: HA = H OLR OLR H OLR = 0.056 K d 1 W m 2 Estimates of H OLR : Christy 1991; Yanai & Tomita 1998; Stechmann & Ogrosky 2014, submitted

Q and A: Case Study: TOGA COARE c Q [nondim] d A [nondim] Jun Jun May May Apr Apr Mar Mar Feb Feb Jan Jan Dec Dec Nov Nov Oct Oct Sep Sep Aug Aug Jul 0 60E 120E 180 120W 60W 0.2 0.1 0 0.1 0.2 Q is somewhat noisy Some propagating signals Some standing signals Jul 0 60E 120E 180 120W 60W 0.2 0 0.2 A strongest during Dec Jan Feb A eastward-propagating in DJF Propagates both eastward, westward

Construct MJO eigenmode e MJO Construct MJO using the following ingredients/proportions of K, R, Q, A: 1 0.8 k=1 k=2 k=3 MJO Amplitude 0.6 0.4 0.2 0 K R Q A Work with each zonal wavenumber independently: k = 1,2,3, MJO eigenmode evolution: MJOS(x,t) e MJO = amplitude structure MJOS(x, t) = MJO Skeleton signal

Case Study: TOGA COARE MJOS(x, t) MJOS(x,t) 1992 1993 Jun May Apr Mar Feb Jan Dec Nov Oct Sep Aug Jul 0 60E 120E 180 120W 60W (Note: here time goes down) (Straub 2013) 0.2 0 0.2 MJOS(x, t) isolates MJO without the use of temporal filtering or EOFs

Case Study: TOGA COARE MJOS(x, t) MJOS(x,t) 1992 1993 Jun May Apr Mar Feb Jan Dec Nov Oct Sep Aug Jul 0 60E 120E 180 120W 60W (Straub 2013) 0.2 0 0.2 MJOS(x, t) provides details of spatial variations

Outline 1. Modeling the MJO Skeleton Provides theoretical prediction of MJO structure 2. Identifying the MJO Skeleton in Observational Data 3. OLR and diabatic heating

The Walker circulation, diabatic heating, and outgoing longwave radiation Stechmann & Ogrosky 2014, submitted to Geophysical Research Letters OLR Diabatic Heating?

Matsuno Gill-like model, without damping: dk dx = H OLR 2 OLR, H OLR = 0.056 K d 1 (W m 2 ) 1 Schematic diagram of overturning circulation z (a) Observed winds of steady Kelvin wave 30N EQ 30S (b) 0.1 0.05 Amplitude of steady Kelvin wave obs model K(x) 0 0.05 (c) 0.1 0 60E 120E 180 120W 60W 0 Obs vs Model: Pattern Correlation = 0.98

OLR Diabatic Heating on shorter time scales? Average over 3 months: El Niño 1997 JJA La Niña 1998 99 DJF 30N 30N EQ EQ 30S (a) 30S (b) 4 2 0 2 4 4 2 0 2 4 K(x) 0.1 0.05 0 0.05 0.1 (c) obs model 0 90E 180 90W 0 0.1 0.05 0 0.05 0.1 (d) 0 90E 180 90W 0 Pattern correlations: 0.94 (left) and 0.91 (right)

Summary Identifying the MJO Skeleton Eigenmode in Observational Data Features of MJO Eigenmode procedure: Flexibility Any spatial pattern is possible (not just the k 1 EOF) Contributions included from many variables, latitudes u,z,q,olr Both equatorial and off-equatorial fields Rossby gyres Theory-based rather than empirical Theoretical weights (ingredients) of K, R, Q, A MJOS(x, t) Required parameters are physically based (e.g., H OLR, Q, etc.) (can estimate them independently, off-line?) Manuscript submitted to Monthly Weather Review