Equatorial waves and superrotation in thestratosphere of a Titan general circulation model
Neil Lewis1, Peter Read1, Nick Lombardo2, Juan Lora2
1Atmospheric, Oceanic and Planetary Physics, University of Oxford
2Department of Earth and Planetary Sciences, Yale University

FDEPS Seminar: 1 December 2023

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

(Sharkey et al., 2021)

1 Key features of Titan
Parameter

Titan

EarthRadius (km)

2575

6371

Rotation period (Earth days)

16

1

Surface gravity (m s-2)

1.35

9.81

Surface pressure (bar)1.5

194% N2
5% CH478% N2
21% O2~.1% H2O

Insolation (W m-2)

14.9

1368

Obliquity (deg)

27

23.45

Eccentricity

0.029

0.0167

Length of year (Earth years)

29

1

Atmospheric composition

Late NHsummerSpringequinox

Early NHsummer

NH summer
solstice(Hörst, 2017)(Fulchignoni et al., 2005)

Key features of Titan• Complicated radiative budget
and T profileStratosphere• Greenhouse-like troposphere• Heated via surface• Anti-greenhouse stratosphere• Heated directly via hazes• Complicated chemistry andthermodynamics

Troposphere• Optically thick photochemicalhazes in stratosphere• Condensible constituents (CH4)with clouds, rain, lakes….

• Strong seasonal variability
30/11/2023

FDEPS2023

5

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

2 Equatorial superrotation – what is it?
Super-rotation is a phenomenon in atmospheric dynamics where the axial
angular momentum of an atmosphere in-some-way exceeds that of theunderlying planet.
A maximum off of the equator will typically be inertially unstable, so superrotation is invariably manifest as a zonal jet centred on the equator.
Hide’s theorem says that in axisymmetric, inviscid atmosphere

Dm
=�
Dt

where m……………………………….
= a cos � (Ωa cos � + u), and the overline denotes a zonal average.

Friction + overturning will lead to m = 0. Therefore, some process that can transport m up-gradientis required. This can be achieved by eddies.

2 Equatorial superrotation – results from idealised modeling
Idealised ‘Earth-like’ models show that superrotation emerges naturally forplanets with low rotation rate / radius.

(Mitchell and Vallis, 2010)

Smaller / slower rotating

2 Equatorial superrotation – results from idealised modeling
Idealised ‘Earth-like’ models show that superrotation emerges naturally forplanets with low rotation rate / radius.
Barotropic, ageostrophic instability generates a‘Rossby—Kelvin wave’ that induces
equatorward momentum flux during spin-up:(Wang and Mitchell, 2014)

Superrotation maintained by ‘equatorial
Rossby waves’, generated bybaroclinic/barotropic instability:(Dias-Pinto and Mitchell,2014)

Equatorial super-rotation –what about Titan?
• Situation on Titan is complicatedby seasonal cycle andlatitudinally asymmetric shear

From Lebonnois et al. [2012]𝑢! (𝐿! )Equatorial super-rotation –what about Titan?

𝑣̅ 𝑢!

• Situation on Titan is complicatedby seasonal cycle andlatitudinally asymmetric shear• Leads to a seasonal cycle in the
momentum budget𝑢" 𝑣′From Lebonnois et al. [2012]Equatorial super-rotation – Lower troposphere
Stratospherew < 4 cycles/T day 𝑢′(𝜔 ) ~6 cycles/T day
what about Titan?
#• Situation on Titan is complicatedby seasonal cycle andlatitudinally asymmetric shear• Leads to a seasonal cycle in the
momentum budget• What are the eddies like?
• Which contribute most to AM
transport?
• where do they come from?

Stratosphere: ~11 cycles/T day• Regular barotropic instability?

From Lebonnois et al. [2012]

Newman et al. [2011]

Equatorial super-rotation –what about Titan?
• Situation on Titan is complicatedby seasonal cycle andlatitudinally asymmetric shear• Leads to a seasonal cycle in the
momentum budget• What are the eddies like?
• Which contribute most to AM
transport?
• where do they come from?

Stratosphere: ~11 cycles/T day• Regular barotropic instability?

From Lebonnois et al. [2012]Equatorial super-rotation –what about Titan?
• Situation on Titan is complicatedby seasonal cycle andlatitudinally asymmetric shear• Leads to a seasonal cycle in the
momentum budget• What are the eddies like?
• Which contribute most to AM
transport?
• where do they come from?

LS = 295ow < 4/Tday
w = 6/Tday
w =11/Tday

MMC

Stratosphere: ~11 cycles/T day• Regular barotropic instability?

From Lebonnois et al. [2012]Equatorial super-rotation –what about Titan?
Contours of• Situation on Titan is complicatedby seasonal cycle andlatitudinally asymmetric shear• Leads to a seasonal cycle in themomentum budget• What are the eddies like?
• Which contribute most to AM
transport?
• where do they come from?

%$!&$' !

− 𝛽 & plot of 𝜕𝑀/𝜕𝑡

Stratosphere: ~11 cycles/T day• Regular barotropic instability?

From Newman et al. [2011]

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

3 Open questions + aim
Which (equatorial) wave modes are supported in Titan’s stratosphere,
and what is their spatial structure?
Which contribute to maintaining equatorial superrotation?
What generates them?
What relevance do the results from idealised modelling have?
- Theoretical understanding- Observed behaviour of Titan’s climate?

Our aim: to answer some of these!

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

4 Model configuration
Analyse equatorial waves and (Lombardo and Lora 2023)
eddy angular momentum fluxes inTAM
TAM = Titan Atmospheric ModelGFDL spectral core with T21
horizontal resolution (~4.5 degree)
50 levels in vertical 0 to ~450km
Realistic RT with seasonal cycle andabsorption of solar radiation due tohaze
Methane hydrological cycle
Data coverage from Lsol 180o – 360o, output is 6hrly average(NH autumn equinox -> NH winter -> NH spring equinox)

TAM Model configuration• GFDL spectral dynamical coreNH earlyWinter
NH Spring• Cf ISCA model• T21 horizontal resolution• ~4.5o resolution in lat x lon• 50 vertical levels• 0 < z < 450 km• Realistic radiative transfer• seasonal cycle,• solar absorption due to haze
• IR cooling due to hydrocarbons and CIA due toN2 etc.

• Mellor-Yamada boundary layer• Betts-Miller convection• CH4 hydrological cycle• Variable chemical composition [potentially]
Lombardo & Lora [2023]

NH earlySummerNHmidsummer

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

5 Basic circulation and eddy momentum fluxes
Key features:

Seasonal variation

Strongly stratified (N>0),
plus N depends strongly on
altitude / pressure:

Meridional shear

Seasonal cycle of u, Psi (TEM), and EP fluxes:

Vertical shear

5 Basic circulation and eddy momentum fluxes
Key features:

Seasonal cycle of u, Psi (TEM), and EP fluxes:

Nearly horizontal EP flux vectors where momentum
transport strongest – acceleration due to horizontal
component of divergence.
But! There is evidence of vertical wave propagation.
Caveat: direction unknown without knowing zonal direction of wave
propagation.

Acceleration in upper stratosphere strongest
with strong meridional shear.
Acceleration in lower stratosphere moreconsistent all year round.

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

6 Spectral analysis of equatorial waves
Overview:
6.1 Waves in NH late autumn / early winter (Lsol 226—234)- Upper stratosphere- Lower stratosphere- Vertical structure
6.2 Waves in NH winter (Lsol 266—274)
6.3 Waves in NH spring (Lsol 346—354)
6.4 Possible generation mechanisms

6 Spectral analysis of equatorial waves
Overview:
6.1 Waves in NH late autumn / early winter (Lsol 226—234)- Upper stratosphere- Lower stratosphere- Vertical structure
6.2 Waves in NH winter (Lsol 266—274)
6.3 Waves in NH spring (Lsol 346—354)
6.4 Possible generation mechanisms

6 Spectral analysis of equatorial waves – late autumn
EP flux filtered into contributions from prograde and retrograde
disturbances

Acceleration in upper
stratosphere due toprograde waves

Prograde: EP flux anti-parallel to direction ofwave propagation(Andrews et al. 1987, Imamura 2006)

Acceleration in lower
stratosphere due toretrograde waves

Retrograde: EP flux parallel to direction of wave
propagation

Both: waves propagate upwards

6 … analysis of equatorial waves – late autumn,
upper stratosphere
Spectral decomposition of eddy momentum flux and geopotential height

Prograde and retrograde waves present
at equator
Retrograde waves in mid-latitudes
Acceleration at equator due to modes
with low intrinsic frequency- mostly prograde
What do they look like?

6 … analysis of equatorial waves – late autumn,
upper stratosphere
Horizontal structure for selected waves

6. … analysis of equatorial waves – late autumn,
lower stratosphere

Acceleration in lower
stratosphere due toretrograde waves

6 … analysis of equatorial waves – late autumn,
lower stratosphere
Spectral decomposition of eddy momentum flux and geopotential height

Prograde and retrograde waves present
at equator, again!
Acceleration at equator due to modes
with low intrinsic frequency, again!
- BUT, mostly retrograde
Note also signature of eastward waves, and westward
wave with cp=-12 ms-1, similar to upper stratosphere
What do they look like?

6 … analysis of equatorial waves – late autumn,
lower stratosphere
Horizontal structure for selected waves

Note:1) Re-appearance of MRG
2) Rossby—Kelvin -> pure Kelvin6 … analysis of equatorial waves – late autumn,
lower stratosphere
What about vertical structure?

Spectral decomposition of eddy
geopotential (at equator):

A number of waves have coherent verticalstructure.
Evidence for a critical region through which
disturbances can not propagate vertically
Prograde waves in lower stratosphereappear to be concentrated onto certain
pressure levels.
Roughly coincident with sharp gradient inbackground wind and also buoyancy freq.

6 … analysis of equatorial waves – late autumn,
lower stratosphere
What about vertical structure?

Spectral decomposition of eddy
geopotential (at equator):

What do they look like?

6 … analysis of equatorial waves – late autumn,
lower stratosphere
What do they look like?

Vertical propagation
away from a source
regionUnusual ‘verticallytrapped’ structure forsome prograde waves inlower stratosphere6 Spectral analysis of equatorial waves
Overview:
6.1 Waves in NH late autumn / early winter (Lsol 226—234)- Upper stratosphere- Lower stratosphere- Vertical structure
6.2 Waves in NH winter (Lsol 266—274)
6.3 Waves in NH spring (Lsol 346—354)
6.4 Possible generation mechanisms

6 Spectral analysis of equatorial waves
Overview:
6.1 Waves in NH late autumn / early winter (Lsol 226—234)- Upper stratosphere- Lower stratosphere- Vertical structure
6.2 Waves in NH winter (Lsol 266—274)
6.3 Waves in NH spring (Lsol 346—354)
6.4 Possible generation mechanisms

6 Spectral analysis of equatorial waves
Overview:
6.1 Waves in NH late autumn / early winter (Lsol 226—234)- Upper stratosphere- Lower stratosphere- Vertical structure
6.2 Waves in NH winter (Lsol 266—274)
6.3 Waves in NH spring (Lsol 346—354)
6.4 Possible generation mechanisms

6 Spectral analysis of equatorial waves
Overview:
6.1 Waves in NH late autumn / early winter (Lsol 226—234)- Upper stratosphere- Lower stratosphere- Vertical structure
6.2 Waves in NH winter (Lsol 266—274)
6.3 Waves in NH spring (Lsol 346—354)
6.4 Possible generation mechanisms

6 Spectral analysis of equatorial waves
Overview:
6.1 Waves in NH late autumn / early winter (Lsol 226—234)- Upper stratosphere- Lower stratosphere- Vertical structure
6.2 Waves in NH winter (Lsol 266—274)
6.3 Waves in NH spring (Lsol 346—354)
6.4 Possible generation mechanisms

6 Spectral analysis of equatorial waves
Overview:
6.1 Waves in NH late autumn / early winter (Lsol 226—234)- Upper stratosphere- Lower stratosphere- Vertical structure
6.2 Waves in NH winter (Lsol 266—274)
6.3 Waves in NH spring (Lsol 346—354)
6.4 Possible generation mechanisms

5 … analysis of equatorial waves – generation mechanisms(Wang and Mitchell, 2014)

Ageostrophic, shear instability?

Occurs when

Our work:≈1−2

5 … analysis of equatorial waves – generation mechanisms
What about for retrograde waves, that exist all year round?
Barotropic instability? (cf. Lebonnois et al., Newman et al.)Criterion met all year roundContents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

Contents
1 Key features of Titan
2 Equatorial superrotation
3 Open questions + aim
4 Model configuration
5 Basic circulation and eddy momentum fluxes
6 Spectral analysis of equatorial waves
7 Summary / discussion

7 Summary / discussion
Which (equatorial) wave modes are supported in Titan’s stratosphere,
and what is their spatial structure? - many: K, RK, ER, MRG (IG waves not shown)
Which contribute to maintaining equatorial superrotation?
- mostly RK in upper strat, ER in lower strat

What generates them?

- consistent with ageostrophic shear instability, and barotropic instability

What relevance do the results from idealised modelling have?
- some!
Despite differences, Titan resembles idealized model spin-up in upper
stratosphere, and equilibrated idealized model in lower
Qs: Are waves properly resolved in vertical? stratosphere….
Is the vertically trapped structure real?
What wave modes do we expect (e.g., from instabilityanalysis) from an asymmetric background with strong
vertical shear?

Should there be more vertical momentum
transport? (cf. the wind drop-off region on realnrut
Sa

nati
T
and

Equatorial superrotation – what about Titan?
Situation on Titan complicated by seasonal cycle and latitudinallyasymmetric shear
This leads to a seasonal cycle in momentum budget…
What are the eddies like?
(Lebonnois et al., 2012)(Newman et al., 2011)(Lebonnois et al., 2012)

These authors suggest regular barotropic instability as origin for the waves(Lebonnois et al., 2012)