an analytical framework for understanding meridional mode ... · cristian martinez-villalobos...

An analytical framework for understandingMeridional Mode dynamics

Cristian Martinez-Villalobos Daniel J. Vimont

1Atmospheric and Oceanic Sciences DepartmentUniversity of Wisconsin- Madison

Oct 18, 2016Meridional Mode Workshop

Madison, WI

Martinez-Villalobos, Vimont An analytical framework for understanding Meridional Mode dynamics

PMM and AMM patterns

Figure: Left: Pacific Meridional Mode (PMM). Right: Atlantic MeridionalMode (AMM). (Chiang and Vimont 2004)

AMM Equatorially AntisymmetricAMM Mass flow from cold to warm hemispherePMM Important Equatorially Symmetric Part.


Thermodynamic Slab Ocean coupled to Gill-Matsunoatmospheric model

∂T∂t

= αu − εT T (1)

Mean easterly trades (α > 0)u > 0 ->Relaxation oftrades, positive SSTtendencyεT Thermal relaxation time

εu − yv = −∂φ∂x

yu = −∂φ∂y

εφ+∂u∂x

+∂v∂y

= −KqT (2)

Long wave approximation

Effective coupling of the system Kqα. (Martinez-Villalobos andVimont 2016)Atmosphere determined by SSTsKq , α homogeneous coupling


Separation into Parabolic Cylinder functions

T (t , y , x) =∞∑

m=0

Tm(t)ψm(y)exp(ikx)

(3)

m even -> equatoriallysymmetric SST structuresm odd -> equatoriallyantisymmetric SSTstructures

Figure: First 4 Parabolic Cylinderfunctions

low m -> loadingspreferentially close to theequatorhigh m -> peak farther fromequator


Equation in ψm(y) space

u can be written in terms of atmospheric Kelvin wave (q0) andRossby waves m (qm+1)

q0 = −KqT0

ε+ ik

qm+1 =Kq(

√m(m + 1)Tm−1 + mTm+1)

−(2m + 1)ε+ ik(4)

∂Tm

∂t=

12

Kqα[

√(m − 1)m

−(2m − 1)ε+ ikTm−2 + (

m − 1−(2m − 1)ε+ ik

− m + 2−(2m + 3)ε+ ik

)Tm

−√

(m + 1)(m + 2)−(2m + 3)ε+ ik

Tm+2]− εT Tm. (5)

Tm evolution affected by Tm−2 and Tm+2

Interaction mediated by atmospheric Rossby waves m − 1 (orKelvin if T0 eqn) and m + 1


Equatorially symmetric and antisymmetric example

Figure: Left Eq symmetric structure evolution. RightEq antisymmetric structure evolution. k = 2π

120o

Couplinghomogenous

Symmetric andantisymmetricindependent

WES feedbackcan sustainequatoriallysymmetric modes

Eq antisymmetricmore growthpotential


Near equator symmetric and antisymmetric equation

∂Tm∂t equation same for m ≥ 2

m = 0

∂T0

∂t=

12

Kqα[(−1ε+ ik

− 2−3ε+ ik

)T0 −√

2−3ε+ ik

T2]− εT T0 (6)

m = 1

∂T1

∂t=

12

Kqα[(−3

−5ε+ ikT1 −

√6

−5ε+ ikT3]− εT T1 (7)

Key role of atmospheric Kelvin wave in T0 equationNo analogous term in T1 evolution


Zonally homogeneous SST patterns

k → 0σ =

KqαεT ε

Ratio of coupling to damping ("stability parameter")

∂Tm

∂t=

12εT (−h(m−1)σTm−2+(g(m)σ−2)Tm +h(m+1)σTm+2) (8)

h(m) ->Exchange/interaction terms mediated by Rossby wave mg(m) ->WES feedback self interaction.g > 0 WES feedback positive, g < 0 WES feedback negativef (m, σ) = g(m)σ − 2-> Balance between WES feedback andthermal dampingf > 0 potential for growth of Tm

f < 0 -> Tm decays


h and f functions

m

0 1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6 a. h(m) vs m

m

0 1 2 3 4 5 6 7 8 9

-5

0

5

b. f(m,σ ) vs m

σ=0.5σ0

σ=σ0

σ=2σ0

Figure: a Exchange function. b Growthfunction

h flat functionWES feedbackpositive for m ≥ 1WES feedbacknegative for m = 0No Kelvin waveWES (red dots)WES feedbackpeaks for m = 1 andthen decreasesPotential for growthdepends on thestability parameter


SST growth

∂|Tm|2

∂t= εT (−h(m − 1)σTm−2Tm + f (m, σ)|Tm|2 + h(m + 1)σTm+2Tm)

∂|Tm+2|2

∂t= εT (−h(m + 1)σTmTm+2 + f (m + 2, σ)|Tm+2|2 + h(m + 3)σTm+4Tm+2).

(9)

∂ < T 2 >

∂t∝

∑m=0

f (m, σ)|Tm|2 (10)

Interaction (∝ h) shifts variance aroundEquatorward propagationNon-normal interactionf key to short term growth prospect


Symmetric and antisymmetric propagation

Equatorward propagationa. Decay until ψ1(y)projection.b. Decayc. Similar to b at thebeginningc. No Kelvin -> Growth


Non-normality

0 100 200 300

Gro

wth

0

1

2

3

4

a. Antisymmetric (T1) growth

Non-normalNormal

Days0 100 200 300

Gro

wth

0

0.5

1

1.5

b. Symmetric (T0) growth

Non-normalNormal

Interaction between modes -> Non-normalityh = 0

f > 0 unchecked growthf < 0 exponential decay

Interaction -> Potential short term growth, but seeding futuredecay


Zonally inhomogeneous case

ν = kε

ν

0 1 2 3 4 5

Gro

wth

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6 Maximum growth vs ν

µant

µsym

µsym

(NK)

Symmetric andantisymmetricgrowth more similarKelvin wavereduced dampingcompared to k = 0Kelvin wave lessimportant inresponse


Figure: a. 180 day symmetric initialcondition. b 180 day evolution (noKelvin). c Growth comparison.

Figure: a. (b). Kelvin (first Rossby)wave response to T = ψ0(y). c Totalresponse


Conclusions

WES feedback -> Interaction between SST modes mediated byRossby and Kelvin wavesInteraction non-normalEquatorward propagation arises naturally as a result of thedifferent modes interactionHomogeneous coupling -> Symmetric and antisymmetriccoupled structures evolve independentlyLarge scale variability -> WES feedback positive for all SSTmodes except for circulation generated by T = ψ0(y)Large scale variability -> Kelvin wave acts as an additionaldamping to the symmetric modeAntisymmetric structures preferentially excited by WES feedbackRegimes in parameter space where equatorially symmetricmodes coupled by WES feedback are sustained


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