an analytical framework for understanding tropical
TRANSCRIPT
An Analytical Framework for UnderstandingTropical Meridional Modes
Cristian Martinez-Villalobos and Daniel J. Vimont
1Atmospheric and Oceanic Sciences DepartmentUniversity of Wisconsin- Madison
Jan 25, 20172017 AMS Meeting, Seattle
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
The Atlantic and Pacific Meridional Modes
Figure: Pacific Meridional Mode (PMM) and Atlantic Meridional Mode (AMM).Chiang and Vimont 2004
AMM: Equatorially antisymmetric dipoleAMM: Flow from cold to warm hemispherePMM: Analogous to AMMPMM: Important Equatorially Symmetric PartWind-Evaporation-SST (WES) feedback.
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
The WES feedback (Xie and Philander 1994)
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
In addition... Equatorially symmetric case.
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
Equatorially Antisymmetric and Symmetric patternssupported by the WES feedback
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
Take home messages I
Both equatorially symmetric and antisymmetric structures mayarise through the WES feedback interaction.Symmetric and antisymmetric modes will interact differently withother components of the climate system (e.g. ENSO)What are the differences in the dynamics of equatoriallysymmetric and antisymmetric modes sustained by the WESfeedback?
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
WES feedback simple model
Slab Ocean
∂T∂t
= αu − εT T
Mean easterly trades (α > 0)u > 0 ->Relaxation oftrades, positive SSTtendencyεT Thermal relaxation time
Gill-Matsuno Atmosphere
εu − yv = −∂φ∂x
yu = −∂φ∂y
εφ+∂u∂x
+∂v∂y
= −KqT
Long wave approximation
Effective coupling of the system Kqα. (Martinez-Villalobos andVimont 2016)Atmosphere determined by SSTsKq , α homogeneous coupling
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
Separation onto Parabolic Cylinder Function Space
T (t , y , x) =∞∑
m=0
Tm(t)ψm(y)exp(ikx)
m even -> equatoriallysymmetric SST structuresm odd -> equatoriallyantisymmetric SSTstructureslow m -> loadingspreferentially close to theequatorhigh m -> peak farther fromequator
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
SST evolution under the WES feedback in paraboliccylinder function space
∂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. (1)
Interaction between contiguous m modes mediated by equatorialatmospheric Rossby and Kelvin waves.Equatorial Symmetry preserving evolution.Equatorward PropagationDynamics of antisymmetric and symmetric modes the same form ≥ 2 modes.Difference only on most equatorially confined structures.
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
SST evolution under the WES feedback in paraboliccylinder function space
∂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 (2)
m = 1
∂T1
∂t=
12
Kqα[(−3
−5ε+ ikT1 −
√6
−5ε+ ikT3]− εT T1 (3)
Key role of atmospheric Kelvin wave in T0 equationNo analogous term in T1 evolution
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
Equatorially symmetric and antisymmetric example
Figure: Left Eq symmetric structure evolution. Right Eq antisymmetricstructure evolution. k = 2π
120o
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
Take home messages II
Both equatorially symmetric and antisymmetric modes emergenaturally from the same framework.WES feedback interaction recast as the interaction betweendifferent SST modes mediated by Equatorial atmospheric waves.Both symmetric and antisymmetric modes propagateequatorwardEquatorially symmetric and antisymmetric modes dynamics differonly for the most equatorially confined SST patterns.(Atmospheric Kelvin Wave Role).
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
Zonally homogeneous SST patterns (k = 0)
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)
h(m) ->WES feedback propagationg(m) ->WES feedback growthg > 0 WES feedback positive, g < 0 WES feedback negativef (m, σ) = g(m)σ − 2-> Balance between WES feedback andthermal damping
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
Symmetric and antisymmetric propagation
Equatorward propagationa. Decay until ψ1(y)projection.b. Decayc. Similar to b at thebeginningc. No Kelvin -> Growth
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
Zonally inhomogeneous SSTs (k 6= 0)
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)
Kelvin wave reduceddamping compared to k = 0.More growth of equatoriallysymmetric modes. Figure: a. (b). Kelvin (first Rossby)
wave response to T = ψ0(y). c TotalresponseMartinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
Take home messages III
WES feedback is positive for all Tm modes, with the exception ofthe most equatorially confined SST mode (T0 Kelvin waveaffect)WES feedback favors growth of equatorially antisymmetric SSTstructuresFinite zonal wavelength There are regimes in the parameterspace where equatorially symmetric structures can grow throughthe WES feedback.
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes
Relevant Publications
Vimont, D. J. (2010). Transient growth of thermodynamicallycoupled variations in the tropics under an equatorially symmetricmean state. Journal of Climate, 23(21):5771-5789.Martinez-Villalobos, C. and Vimont, D. J. (2016). The Role of theMean State in Meridional Mode Structure and Growth. Journal ofClimate, 29(10):3907-3921.Martinez-Villalobos, C. and Vimont, D. J. (2017). An analyticalframework for understanding Tropical Meridional Modes. Journalof Climate, In Press.
Martinez-Villalobos and Vimont An Analytical Framework for Understanding Tropical Meridional Modes