application of normal mode functions for the improved

15◦S7.5◦S

0◦7.5◦N15◦N

(a) -20 days

15◦S7.5◦S

0◦7.5◦N15◦N

(b) -15 days

15◦S7.5◦S

0◦7.5◦N15◦N

(c) -10 days

15◦S7.5◦S

0◦7.5◦N15◦N

(d) -5 days

15◦S7.5◦S

0◦7.5◦N15◦N

(e) 0 days

15◦S7.5◦S

0◦7.5◦N15◦N

(f) 5 days

15◦S7.5◦S

0◦7.5◦N15◦N

(g) 10 days

15◦S7.5◦S

0◦7.5◦N15◦N

(h) 15 days

15◦S7.5◦S

0◦7.5◦N15◦N

0◦ 90◦E 180◦ 90◦W0◦

(i) 20 days

CSIRO

Application of normal mode functions for theimproved balance in the CAFE data assimilation

system and characterisation of modes of variabilityVassili Kitsios & Terry O’Kane

2nd International Conference on Seasonal to Decadal Prediction (S2D), 17th-21st, Boulder, CO, USA

OCEANS AND ATMOSPHEREwww.csiro.au

Application of normal mode functions for the improved balance in the CAFE data assimilation system and characterisation of modes of variability : Slide 2 of 19

Approach

• Will focus on the characterisation of the Madden Julian Oscillation (MJO)via a Normal Mode Functions (NMF) decomposition of JRA55, and dis-cuss implications for Normal Mode Inialisation (NMI).• MJO is a mode of variability resulting from coupled tropical deep convec-

tion and atmospheric dynamics.• Use key MJO properties to identify representatives NMFs:

– Eastward propagating– Dominant variance over intra-seasonal timescales: 30-90 days.– Tropics centric dynamics.– Horizontal velocity field has a dominant longitudinal wave of k = 1.– Dominant component of the zonal velocity is symmetric about the

equator.• Using one such mode we produce phase and conditional averages of:

– velocity potential to illustrate atmospheric dynamics– outgoing longwave radiation to illustrate convection

• Acknowledge Zagar for sharing the NCAR NMF code, MODES.


What are Normal Mode Functions ?

• Decompose 3D (λ, φ, σ) velocity (u, v) and geopotential height (h) fieldsinto horizonal and vertical scales, and mode type, using the eigensolu-tion of the linearised primitive equations on a sphere.• Each scale decomposed into: Balanced Component (BAL) ; Eastward

Inertial Gravity Wave (EIG) ; Westward Inertial Gravity Wave (WIG).• Vertical Structure Functions (VSF), Gm in σ coordinates.

• Longitudinal (λ) waves eikλ

• Meridional (φ) Horizonal Structure Functions (HSF) of wind and height(U, V,H) for EIG, WIG and BAL modes.

[u(σ, λ, φ, t)v(σ, λ, φ, t)h(σ, λ, φ, t)

]=

M∑

m=1

Gm(σ)

[ √gDm 0 00

√gDm 0

0 0 gDm

]K∑

k=−Keikλ

N∑

n=0

EIG,WIG,BAL∑

p

[U(φ)−iV (φ)H(φ)

]p

knm

χpknm(t)

• Complex coefficients, χpknm(t), represent contributions of each compo-nent.


Vertical Structure Functions

• VSF given by solution of the VSF eigenvalue problem (EVP).• VSF EVP requires only a time and horizontally averaged static stability

profile in σ coordinates.• Equivalent heights Dm (eigenvalues) are indicative of vertical scale.• VSFGm(σ) (eigenvectors) havem−1 zero crossings. G1 is the barotropic,G2 the first baroclinic, G3 the second baroclinic, ...• For large m, the VSF represent boundary layer processes.

101 102 103 104 105

Γ0 [K]

100

101

102

103

σ×p 0

[hPa]

(a) static stability ≡ Γ0

0 10 20 30 40vertical mode number

10−1

100

101

102

103

104

Dm[m

etres]

(b) equivalent height ≡ Dm

−0.5 0.0 0.5amplitude [unitless]

100

101

102

103

σ×p 0

[hPa]

(c) vertical structure functions

m=1

m=2

m=8

m=15

m=28


Horizontal Structure Functions

• HSF given by solution of a HSF EVP for each equivalent height (Dm).• Eigenvectors give meridional dependence, with mode types (EIG, WIG,

BAL) defined by symmetry properties.• Frequency ν (eigenvalue) is indicative of temporal scale.

−30 −20 −10 0 10 20 30longitudinal wavenumber (k)

100

101

102

103

104

1/νq knm[days]

(a) timescales for m = 28

BAL

EIG

WIG

MRG

KW

0 10 20 30 40vertical mode number (m)

100

101

102

103

1/νq knm[days]

90days

60days

30days

(b) timescales

BAL(1,1,m)

WIG(1,0,m)

EIG(1,0,m)

−1 0 1amplitude

−75

−50

−25

0

25

50

75

latitude(φ)

(c) U qknm(φ)

EIG(1,0,28)

EIG(1,0,8)

BAL(1,1,15)

BAL(1,1,8)

• Recall for MJO: k = 1; tropics centric; U is symmetric• Only the EIG n = 0, WIG n = 0, and BAL n = 1 are tropics centric with

the appropriate symmetries.• EIG m = 23− 32 ; BAL m = 11− 22 have intra-seasonal timescales.


Energy Contribution χpknm(t)χp∗knm(t) of NMFs in JRA55

• BAL dominates for low k, IG dominate for high k• BAL dominates for all vertical modes m• For (k, n) = (1, 1) BAL HSF eigenvectors have MJO-like properties:

– BAL (k, n,m) = (1, 1, 2) and (1, 1, 2) local peaks in energy, but toofast

– BAL (k, n,m) = (1, 1, 15) has HSF eigenvalue of 46 days.

• For (k, n) = (1, 0) EIG HSF eigenvectors also have MJO-like properties:

– EIG (k, n,m) = (1, 0, 8) has most energy, but timescale too fast

– EIG (k, n,m) = (1, 0, 28) has HSF eigenvalue of 56 days.

100 101 102

zonal wavenumber (k)

10−4

10−3

10−2

10−1

100

101

102

103

104

nonzonal

energy

[J/kg]

(a)

BAL

EIG

WIG

total

0 10 20 30vertical mode (m)

(b)


(c) (k, n) = (1, 1)


(d) (k, n) = (1, 0)


Cross-spectral Analysis of Candidate NMFs

• All candidate modes are tropics centric and have the appropriate symme-tries.• Only EIG (k, n,m) = (1, 0, 28) has an intra-seasonal timescale, and

propagates eastward, but has low energy.• Cross-spectral analysis identifies only slow intra-seasonal timescales are

coherent between EIG (k, n,m) = (1, 0, 28) and the more energeticmodes.• Fast Kelvin wave removed from energetic EIG (k, n,m) = (1, 0, 8).

2 4 6 8 10k

0.0

0.1

0.2

0.3

0.4

0.5

f[1/days]

30days

(a) PSD of EIG(k,0,8) [J/kg]

10−4

10−3

10−2

10−1

100

101

2 4 6 8 10k

(b) PSD of EIG(k,0,28) [J/kg]

10−4

10−3

10−2

10−1

2 4 6 8 10k

(c) coherence

0.0

0.2

0.4

0.6

0.8

1.0

2 4 6 8 10k

(d) coherence output [J/kg]

10−4

10−3

10−2

10−1

100

101

5 10 15 20 25 30 35m

0.0

0.1

0.2

0.3

0.4

0.5

f[1/days]

(e) coherence output of EIG(1,0,m) [J/kg]

10−4

10−3

10−2

10−1

100

101

102

5 10 15 20 25 30 35m

(f) coherence output of BAL(1,1,m) [J/kg]

10−4

10−3

10−2

10−1

100

101

102


Cross-spectral Analysis of Candidate NMFs

• All candidate modes are tropics centric and have the appropriate symme-tries.• Only EIG (k, n,m) = (1, 0, 28) has an intra-seasonal timescale, and

propagates eastward, but has low energy.• Cross-spectral analysis identifies only slow intra-seasonal timescales are

coherent between EIG (k, n,m) = (1, 0, 28) and the more energeticmodes.• Repeated for all vertical scales (m) with k = 1 for BAL and EIG.

2 4 6 8 10k

0.0

0.1

0.2

0.3

0.4

0.5

f[1/days]

30days


10−4

10−3

10−2

10−1

100

101

2 4 6 8 10k


10−4

10−3

10−2

10−1

2 4 6 8 10k

(c) coherence

0.0

0.2

0.4

0.6

0.8

1.0

2 4 6 8 10k


10−4

10−3

10−2

10−1

100

101

5 10 15 20 25 30 35m

0.0

0.1

0.2

0.3

0.4

0.5

f[1/days]


10−4

10−3

10−2

10−1

100

101

102

5 10 15 20 25 30 35m


10−4

10−3

10−2

10−1

100

101

102

• Other candidate modes highlighted in coherence output clusters.


Phase Average on Basis of EIG (k, n,m) = (1, 0, 28)

• Phase angle calculated from complex χpnmk. Dates associated with eachphase angle octant averaged. All modes contribute to phase averages.

• Velocity potential is a propagating longitudinal wave, with a vertical signchange representing upper level divergence and lower level convergence.


Phase Average on Basis of EIG (k, n,m) = (1, 0, 28)

• Phase angle calculated from complex χpnmk. Dates associated with eachphase angle octant averaged. All modes contribute to phase averages.

• OLR has a dipole pattern over the maritime continent, tracking with veloc-ity potential of like sign.


Large and Persistent MJO Events

• Start and end of each event defined as discontinuities in phase angle ofEIG (k, n,m) = (1, 0, 28).• Persistent events have a continuous phase for longer than 270◦.• Large events have a magnitude in the upper quartile.• Composite average magnitude is greater than background for 20 days be-

fore and after day 0.• Composite average phase angle indicates eastward propagation.

• Dates associated with each phase shift identified and averaged to producecomposite fields of velocity potential and outgoing longwave radiation.


Velocity Potential at 200hPa - Wave Like

15◦S7.5◦S

0◦7.5◦N15◦N

(a) -20 days

15◦S7.5◦S

0◦7.5◦N15◦N

(b) -15 days

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0◦7.5◦N15◦N

(c) -10 days

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0◦7.5◦N15◦N

(d) -5 days

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0◦7.5◦N15◦N

(e) 0 days

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0◦7.5◦N15◦N

(f) 5 days

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0◦7.5◦N15◦N

(g) 10 days

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0◦7.5◦N15◦N

(h) 15 days

15◦S7.5◦S

0◦7.5◦N15◦N

0◦ 90◦E 180◦ 90◦W0◦

(i) 20 days


Outgoing Longwave Radiation at 200hPa - Dipole Like

15◦S7.5◦S

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(a) -20 days

15◦S7.5◦S

0◦7.5◦N15◦N

(b) -15 days

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0◦7.5◦N15◦N

(c) -10 days

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0◦7.5◦N15◦N

(d) -5 days

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0◦7.5◦N15◦N

(e) 0 days

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0◦7.5◦N15◦N

(f) 5 days

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0◦7.5◦N15◦N

(g) 10 days

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0◦7.5◦N15◦N

(h) 15 days

15◦S7.5◦S

0◦7.5◦N15◦N

0◦ 90◦E 180◦ 90◦W0◦

(i) 20 days


Instantaneous Comparison of NMFs and WH

• Wheeler & Hendon index based on first two PCs of meridionally averagedu at 200hPa, 850hPa and OLR within 15◦S and 15◦N .• Compare to tropics centric NMFs, with MJO-like symmetries:

– BAL(k, n,m) = (1, 1, 8) : energetic, but westward, timescale too fast.

– EIG(k, n,m) = (1, 0, 8) : energetic, but timescale too fast.

– BAL(k, n,m) = (1, 1, 15) : intra-seasonal timescale, but westward.

– EIG(k, n,m) = (1, 0, 28) : intra-seasonal timescale, eastward, notenergetic.

• Correlation of BAL (k, n,m) = (1, 1, 8) with WH when filtered to retaintemporal scales coherent with EIG (k, n,m) = (1, 0, 28) is 0.78.

2001 2003 2005 2007 2009 2011 2013 20150.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

√〈W

H〉 M

A

0.0

0.5

1.0

1.5

2.0

2.5

3.0

√〈E

q knm〉 M

A

(a) 91 day moving average

BAL(1,1,8) unfiltered BAL(1,1,8) EIG(1,0,8) BAL(1,1,15) EIG(1,0,28) unfiltered WH

2012-0

2-05

2012-0

2-19

2012-0

3-04

2012-0

3-18

2012-0

4-01

2012-0

4-15

−150

−100

−50

0

50

100

150

phaseangle

(b) phase angle

−2 −1 0 1 2RMM1

−2

−1

0

1

2

RMM2

(c) phase space WH

−2 −1 0 1 2 3real(χq

knm)√Dmg/2

−2

−1

0

1

2

imaginary(χq knm)√

Dmg/2

(d) phase space BAL(1,1,8)


Instantaneous Comparison of NMFs and WH

• Wheeler & Hendon index based on first two PCs of meridionally averagedu at 200hPa, 850hPa and OLR within 15◦S and 15◦N .• In comparision to WH, coherence filtered BAL(k, n,m) = (1, 1, 8) mode

exhibits:– high correlation in magnitude over the entire time series (1958-2016)– consistent phase propagation for a specific event (Jan 2012)– consistent spiralling in for a specific event in phase space

2001 2003 2005 2007 2009 2011 2013 20150.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

√〈W

H〉 M

A

0.0

0.5

1.0

1.5

2.0

2.5

3.0

√〈E

q knm〉 M

A

(a) 91 day moving average

BAL(1,1,8) unfiltered BAL(1,1,8) EIG(1,0,8) BAL(1,1,15) EIG(1,0,28) unfiltered WH

2012-0

2-05

2012-0

2-19

2012-0

3-04

2012-0

3-18

2012-0

4-01

2012-0

4-15

−150

−100

−50

0

50

100

150

phaseangle

(b) phase angle

−2 −1 0 1 2RMM1

−2

−1

0

1

2

RMM2

(c) phase space WH

−2 −1 0 1 2 3real(χq

knm)√Dmg/2

−2

−1

0

1

2

imaginary(χq knm)√

Dmg/2

(d) phase space BAL(1,1,8)


Proposed MJO Skeleton

• Nonlinear interactions of these modes generates an energetic MJO ofeastward propagation and correct phase period.• All modes have MJO-like longitudinal and meridional structure.

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• Interaction strength inferred from cross-spectral analysis.• In the future will calculate the nonlinear transfer terms explicitly.


Concluding Remarks

• NMFs decompose a 3-D atmospheric geopotential height and horizon-tal velocity field into scale (zonal, meridional, vertical) and mode class(BAL, EIG, WIG).• MJO-like NMF modes and their interactions were isolated in the JRA-55.• A skeleton physical model of the MJO was proposed.

• Implications for Normal Modes Initialisation:– Since the IG waves have shorter timescales, they are potentially less

predictable over a multi-year period.– Naively one would think that filtering the IG waves would improve pre-

dictability.– However, the EIG waves (even of small vertical scale and low energy)

are shown here to be dynamically important for the MJO.


QuestionsVertical Structure Functions Horizontal Structure Functions

101 102 103 104 105

Γ0 [K]

100

101

102

103

σ×p 0

[hPa]

(a) static stability ≡ Γ0

0 10 20 30 40vertical mode number

10−1

100

101

102

103

104

Dm[m

etres]

(b) equivalent height ≡ Dm

−0.5 0.0 0.5amplitude [unitless]

100

101

102

103

σ×p 0

[hPa]

(c) vertical structure functions

m=1

m=2

m=8

m=15

m=28

−30 −20 −10 0 10 20 30longitudinal wavenumber (k)

100

101

102

103

104

1/νq knm[days]

(a) timescales for m = 28

BAL

EIG

WIG

MRG

KW

0 10 20 30 40vertical mode number (m)

100

101

102

103

1/νq knm[days]

90days

60days

30days

(b) timescales

BAL(1,1,m)

WIG(1,0,m)

EIG(1,0,m)

−1 0 1amplitude

−75

−50

−25

0

25

50

75

latitude(φ)

(c) U qknm(φ)

EIG(1,0,28)

EIG(1,0,8)

BAL(1,1,15)

BAL(1,1,8)

Cross-Spectral Analysis Phase Averages2 4 6 8 10k

0.0

0.1

0.2

0.3

0.4

0.5

f[1/days]

30days


10−4

10−3

10−2

10−1

100

101

2 4 6 8 10k


10−4

10−3

10−2

10−1

2 4 6 8 10k

(c) coherence

0.0

0.2

0.4

0.6

0.8

1.0

2 4 6 8 10k


10−4

10−3

10−2

10−1

100

101

5 10 15 20 25 30 35m

0.0

0.1

0.2

0.3

0.4

0.5

f[1/days]


10−4

10−3

10−2

10−1

100

101

102

5 10 15 20 25 30 35m


10−4

10−3

10−2

10−1

100

101

102

Composite Averages Skeleton Physical Model

15◦S7.5◦S

0◦7.5◦N15◦N

(a) -20 days

15◦S7.5◦S

0◦7.5◦N15◦N

(b) -15 days

15◦S7.5◦S

0◦7.5◦N15◦N

(c) -10 days

15◦S7.5◦S

0◦7.5◦N15◦N

(d) -5 days

15◦S7.5◦S

0◦7.5◦N15◦N

(e) 0 days

15◦S7.5◦S

0◦7.5◦N15◦N

(f) 5 days

15◦S7.5◦S

0◦7.5◦N15◦N

(g) 10 days

15◦S7.5◦S

0◦7.5◦N15◦N

(h) 15 days

15◦S7.5◦S

0◦7.5◦N15◦N

0◦ 90◦E 180◦ 90◦W0◦

(i) 20 days

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CSIRO

OCEANS AND ATMOSPHEREwww.csiro.au

CSIRO Oceans and Atmosphere

Vassili Kitsiost 0362325312e [email protected] research.csiro.au/dfp/

CSIRO Oceans and Atmosphere

Terry O’Kanete Terrence.O’[email protected] research.csiro.au/dfp/

mailto: [email protected]

mailto: Terrence.O'[email protected]

application of normal mode functions for the improved

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