Icarus 207 (2010) 932–937

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Icarus

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Threshold of wave generation on Titan’s lakes and seas: Effect of viscosityand implications for Cassini observations

Ralph D. Lorenz a,*, Claire Newman b,c, Jonathan I. Lunine d

a Space Department, Johns Hopkins University Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723, USAb Ashima Research, 600 S. Lake Ave., Suite 303, Pasadena, CA 91106, USAc GPS, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USAd Department of Physics, University of Rome Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Rome, Italy

a r t i c l e i n f o

Article history:Received 10 October 2009Revised 27 November 2009Accepted 2 December 2009Available online 11 December 2009

Keywords:TitanMeteorologyGeological processes

0019-1035/$ - see front matter Published by Elsevierdoi:10.1016/j.icarus.2009.12.004

* Corresponding authorE-mail address: ralph.lorenz@jhuapl.edu (R.D. Lore

a b s t r a c t

Motivated by radar and near-infrared data indicating that Titan’s polar lakes are extremely smooth, weconsider the conditions under which a lake surface will be ruffled by wind to form capillary waves.We evaluate laboratory data on wind generation and derive, without scaling for surface tension effects,a threshold for pure methane/ethane of�0.5–1 m/s. However, we compute the physical properties of pre-dicted Titan lake compositions using the National Institute for Standards Technology (NIST) code andnote that dissolved amounts of C3 and C4 compounds are likely to make Titan lakes much more viscousthan pure ethane or methane, even without allowing for suspended particulates which would increasethe viscosity further. Wind tunnel experiments show a strong dependence of capillary wave growth onliquid viscosity, and this effect may explain the apparent absence so far of waves, contrary to prior expec-tations that generation of gravity waves by wind should be easy on Titan. On the other hand, we note thatwinds over Titan lakes predicted with the TitanWRF Global Circulation Model indicate radar observationsso far have in any case been when winds have been low (�0.5–0.7 m/s), possibly below the wave gener-ation threshold, while peak winds during summer may reach 1–2 m/s. Thus observations of Titan’s north-ern lakes during the coming years by the Cassini Solstice mission offer the highest probability ofobserving wind-roughening of lake surfaces, while observations of Ontario Lacus in the south will likelycontinue to show it to be flat and smooth.

Published by Elsevier Inc.

1. Introduction

In our Solar System at present, only Earth and Titan are knownhave standing bodies of meteorologically emplaced liquid on theirsurface. Some early consideration of waves on Titan (Srokosz et al.,1992; Ghafoor et al., 2000; Ori et al., 1998) focussed on gravitywaves, and noted that in Titan’s low gravity, such waves wouldbe large. Elachi et al. (1991) noted that capillary waves would beof a comparable scale to those on Earth, and might substantially af-fect the backscatter of liquid surfaces imaged by radar. The Huy-gens probe was designed with the prospect of splashing downinto a sea of liquid hydrocarbons (e.g. Lorenz, 1994), and wasequipped with tilt sensors expressly for measuring wave motion(e.g. Zarnecki et al., 1997).

Ghafoor et al. (2000) scaled empirical terrestrial wave spectra toTitan gravity, and suggested that a 1 m/s wind would generatewaves with a significant wave height of 20 cm (comparable withthose generated by winds of 3 m/s on Earth). However, Lorenz

Inc.

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et al. (2005) pointed out that the different air density and liquidproperties (density, surface tension and viscosity) would all influ-ence the wave generation process. In particular, they showed inwind tunnel experiments that waves form more easily on a kero-sene surface than a water one.

These considerations remained somewhat academic for sometime, since the Huygens probe landed on a solid surface, albeitone that was both damp (Lorenz et al., 2006) and soft (Zarneckiet al., 2005). However, as Cassini data on Titan’s polar regionsemerged, bodies of standing liquid were eventually identified(Stofan et al., 2007; Brown et al., 2008; Turtle et al., 2009). Thesebodies, including over 100 small lakes (e.g. Hayes et al., 2008)and two large seas (Ligeia Mare, about 500 km across, and KrakenMare, about 1000 km across) are almost exclusively in the north-ern hemisphere (Lopes et al., 2007; Aharonson et al., 2009). Onlya couple of lakes, notably 250 � 70 km Ontario Lacus (Turtleet al., 2009; Brown et al., 2008; Barnes et al., 2009; Lorenz et al.,2009; Wall et al., in press) are present in the south. Why thereshould be such a dichotomy between the hemispheres is notknown, although a compelling idea (Aharonson et al., 2009) is thatthe astronomical configuration of the seasons, with southern

Fig. 1. Wave height in water measured as a function of windspeed by Kahma andDonelan (1988) compared with measurements at a shorter fetch by Lorenz et al.(2005). The lower viscosity of kerosene appears to roughly compensate for theshorter fetch. Windspeed thresholds are from the literature summarized by Kahmaand Donelan (1988).

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summer shorter but hotter than the northern summer (as is thecase on Mars), drives volatiles such as methane and ethane tothe north pole.

A distinctive feature of the appearance of Titan’s northern lakes(Stofan et al., 2007) is their extremely low radar reflectivity. Thiswas noted to require a rather smooth surface, although quantita-tive constraints were not determined. Similarly, Brown et al.(2008) suggest that the near-infrared reflectivity of Ontario Lacusin the south (observed on Cassini encounter T38, December2007) is such as to require it to be very smooth (they extrapolateobservations to indicate that the 5-lm reflectivity at zero airmass‘approaches’ zero, which would imply a lack of scattering facets atthat scale, but do not determine a quantitative roughness value).More recently, a detailed analysis of the amplitude histograms ofradar altimeter data from T49 (December 2008) showing strikingspecular reflections (Wye et al., 2009) indicates that Ontario Lacusis exceptionally smooth, with a root-mean-square surface heightvariation of <3 mm over �100 m scattering areas. This suggeststhat even capillary waves are not present, which raises the ques-tion of what conditions this might imply about the lake and/orthe winds above it.

Recent imaging radar observations (Wall et al., in press) havehowever identified morphological evidence of wave action onone shore of Ontario Lacus. So, pre-Cassini expectations were thatwaves should be easy to generate on Titan, and there is some mor-phological evidence that some time in the past waves have oc-curred, yet in the handful of observations of lakes and seas onTitan so far, all the indications are that waves are not present. Thispaper proposes some solutions to this apparent paradox.

2. Wave generation threshold

For the present we confine ourselves to the qualitative questionof whether there are ‘perceptable’ waves or not – a fuller quantita-tive analysis of wave growth will require consideration of the dom-inant wavelength and quite possibly other considerations such asthe depth of the liquid and the strength of the wind-blown cur-rents in the liquid. In fact, there is not a particularly clean answerto this question even for waves on Earth, which have received sci-entific attention for over a century and a half. Several theoreticalmodels exist for the generation of waves (notably the shear model(Miles, 1996) and the ‘sheltering’ model (Jeffreys, 1925)), but noneof them are analytically simple or comprehensively accurate.

Threshold windspeeds are summarized in a paper by Kahmaand Donelan (1988) – we summarize their findings in Fig. 1. Empir-ical field observations on Earth dating back to Russell in Scotland in1844 suggest thresholds ranging between 0.4 m/s and 2 m/s, thescatter resulting somewhat on the subjective assessment of whenorganized waves appear. Laboratory thresholds are generally abouttwice as high, presumably in part due to the lower turbulence inwind tunnel flows.

Reviewing the data, then, a terrestrial threshold speed range of1–2 m/s is consistent with theoretical, field and laboratory studies.All else being equal, the same wind stress would be generated onTitan by winds of 0.5–1 m/s, since the atmosphere is four timesdenser; this should be the wave generation threshold for a liquidthat has the same properties at Titan surface conditions (notably,90–94 K) as water does on Earth.

Energy is provided to a wave from the freestream wind by acombination of shear friction and differential pressure on the up-wind and downwind faces of the wave. Larger waves will extractmore energy from the wind, but energy is lost due to viscosity (par-ticularly when the wave amplitude becomes comparable with thedepth) and to nonlinear effects which transfer energy to longer-wavelength gravity waves. Thus as the waves grow, their dissipa-

tion increases too, and ultimately an approximate steady-state re-sults where the wave height becomes constant (in the oceancontext, this is a ‘fully-developed’ sea).

In the field, it is readily observed that patches of capillary waves(‘catspaw’) form quickly with fetches of less than a few meters.Laboratory experiments with fetches of less than several metersmay underestimate wave height (or equivalently, overestimatethe threshold for a given wave height) because waves have nothad time to grow fully. This discrepancy can be noted in experi-ments where the wave height is measured at several fetches simul-taneously. It is also seen in Fig. 1 where the experiments of Kahmaand Donelan (1988) with a fetch of 4.7 m are compared with thoseof Lorenz et al. (2005) with a fetch of only 1 m – the required wind-speeds in the latter are double the former. The experiments of Lor-enz et al. (2005) also included generation of waves in kerosene,which has (see Table 1) some physical properties somewhat closerto candidate Titan liquids than does water. They found that thethreshold for wave generation in kerosene was somewhat lowerthan for water – indeed, in Fig. 1 we see that the waveheights fora fetch of �1 m in kerosene are comparable with those for a fetchof 4.7 m in water. Curiously, Kahma and Donelan (1988) note thatin at least some circumstances higher surface tension can lead tolarger waves, all else being equal. Apparently some combinationof kerosene’s lower density and lower viscosity than water over-whelms the effect of its lower surface tension, such that in Lorenz’sexperiments waves were easier to generate.

It is clear that the parameter space of density, surface tensionand viscosity has not been adequately explored experimentallyto fully separate these effects, and thus future wind tunnel exper-iments in this direction are urged. The higher air density is, how-ever, a somewhat straightforward factor to include and is thebasis for suggesting a 0.5–1 m/s threshold for the present by scal-ing the recommended terrestrial value (note that since we discusscapillary waves, the different gravity on Titan is not a factor –though that is not the case in considering the ultimate size of largegravity waves once capillary waves form and begin to transfer en-ergy from wind to lake).

Note that the effect of viscosity is in part moderated by thedepth of the liquid – viscous shear increases substantially if thedepth is shallow. This is an important point to bear in mind in windtunnel experiments, but can also be an important effect in the field

Table 1Physical properties of candidate Titan lake compositions and comparison fluids. Top section denotes assumed conditions and composition, X denoting mole fraction of therespective component. The Cordier (simp) compositions denote a simplified variant of the Cordier et al. (2009) composition where all compounds heavier than propane arelumped with propane, to determine the effect of the butane fraction.

Context Pure CH4 Pure CH4 Rain Rain Pure C2H6 Pure C2H6 Cordier Cordier Cordier(simp)

Cordier(simp)

Water Liquidnitrogen

Toluene

T (K) 94 92 90 94 92 94 90 94 90 94 290 77 293X[C2H6] 100 100 74 74 74 74X[N2] 25 25 0.5 0.5 0.5 0.5X[CH4] 100 100 75 75 10 10 10 10X[C3H8] 7 7 15.5 15.5X[C4H10] 8.5 8.5

Density (kg/m3) 454 456 531 518 650 647 662 658 654 650 1000 800 867Specific heat Cp (kJ/kg K) 3.29 3.29 3.21 3.25 2.25 2.27 2.35 2.4 2.39 2.42 4.2 2.05 1.67Sound speed (m/s) 1574 1589 1294 1260 1985 1975 2026 1905 2028 1931 1482 945 1345Viscosity (lPa s) 208 222 167 151 1246 1141 1736 1423 1528 1263 1000 155 585k (W/m/k) 0.229 0.232 0.192 0.18 0.252 0.251 0.248 0.245 0.245 0.244 0.6 0.13 0.134Refractive index 1.287 1.272 1.38 1.33 1.497Dielectric constant 1.65 1.61 1.917 1.4 2.4Surface tension (N/m) 1.80E�02 1.80E�02 0.073 0.03 0.028

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in localities where lakebeds have very shallow slopes (see Fig. 2).Since slopes at Ontario Lacus are only of the order of �10�4 (e.g.Lorenz et al., 2009), centimeter depths likely to provide substantialdamping may extend to kilometers from the shoreline – an effectthat may manifest itself as a specularly-reflecting rim around anotherwise wind-roughened lake.

3. Titan lake properties

We now turn to the composition and properties of Titan lakes.The most probable composition in equilibrium with the presentatmosphere is ethane-rich (see below), although a methane-richseasonal layer cannot be ruled out (except at Ontario, where theBrown et al. (2008) observation implies at least some ethane). Pure

Fig. 2. Aerial photo by the first author of Lake Eyre in South Australia in July 2009.The lake is exceptionally shallow (meters deep yet hundreds of kilometer across)and its margins where the depth is only millimeters form a band where capillarywaves are damped – on typical beaches this band is only a few centimeter wide, butEyre is shallow enough that this damped area is several tens of meters across. Thismargin appears bright because the geometry here permits a specular reflection ofthe bright sky towards the Sun, whereas the main part of the lake to the left is roughand reflects the overall darker sky.

methane, which would provide an end-member case for manyphysical properties, can be discounted as its existence would rep-resent an extreme thermodynamic disequilibrium. A methane–nitrogen composition is considered as a concession to the possibil-ity of a seasonal layer or recent rainstorm. Thus, pure substancedata in Table 1 are provided for context and information only(see also Lorenz et al., 2003). A 25% nitrogen/75% methane liquidis roughly in thermodynamic equilibrium with the ambient atmo-sphere of �5% methane.

However, some amount of higher hydrocarbons is also pro-duced by atmospheric photochemistry and will be deposited inthe lakes. The amounts produced differ between various photo-chemical models (see Lorenz and Lunine (1996) and also discus-sion of solutes in Raulin (1987)) and may or may not reachsaturation in the liquid. More recent work by Cordier et al.(2009) suggests values that range from �7–8% for C3H8 (propane),2–3% for HCN (hydrogen cyanide), 1% for C2H2 (acetylene) andsome C4 hydrocarbons, to 0.2% for C6H6 (benzene), 0.1% for CH3CNand 0.03% for CO2. Although density and surface tension are notstrongly sensitive to these minor constituents, viscosity can be sig-nificantly affected, as is well-known in the petrochemicalsindustry.

We have calculated the relevant properties in Table 1 using theNIST-14 Mixture Property Database (Friend, 1992). This programuses a Peng–Robinson equation of state for coexisting-phase com-position calculations. The NIST Extended Corresponding Statesmodel with exact shape factors and van der Waals one-fluid mix-ing rules are used for single phase properties.

A result of particular note is that ethane-rich compositions aresignificantly more (�5�) viscous than is pure methane or methanenitrogen. Further, the ‘doped’ Cordier compositions with propaneand butane have viscosities higher still – in some cases exceedingan order of magnitude higher than pure methane. Fig. 3 shows theeffect of such a viscosity increase (generated in an experiment byKahma and Donelan (1998) by exploiting the temperature depen-dence of the viscosity of water). This viscosity increase drops thewave height by a factor of 5 or so (or equivalently, increases thewave generation threshold).

The effective viscosity of Titan lakes may be enhanced substan-tially further above the ‘clean’ liquid values above, since solidmaterial (such as tholin haze) may be very fine-grained, and willthus have a very small sedimentation velocity. Lunine (1992) notedthat the terminal velocity of micron-sized particles (resembling thehaze) in Titan seas would be only of the order of centimeter peryear – and thus easily stirred by tidal, wind-driven or thermal

Fig. 3. Height of capillary waves in water at various temperatures as measured byKahma and Donelan (1988) in a wind tunnel with freestream flowspeed of 3.3 m/sand a fetch of 4.5 m. The data have been plotted against tabulated viscosity of waterat the relevant temperatures – the strong dependence of wave height on viscosity isevident, and the predicted viscosity of Titan lakes containing heavy compounds(Table 1) is higher than the maximum for water. The dynamic pressure of terrestrialair at 3.3 m/s corresponds to Titan winds of 1.6 m/s.

Fig. 4. Near-surface windspeeds (calculated in the TitanWRF model at �90 maltitude) over Ontario Lacus as a function of solar longitude Ls. Winds are strongest,peaking at 1.5–2 m/s in early southern summer (Ls � 200–240�). Cassini observa-tions have been and will be exclusively when low winds (<0.5 m/s) are predicted.The VIMS T38 observation in December 2007 was at Ls = 340�.

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currents. If the burden of suspended sediment is high, the bulk vis-cosity could be increased, damping waves further than would bepredicted from the liquid composition alone.

In sum, given the range of plausible liquid properties on Titan(Table 1), the wave generation threshold could vary by a factor of2 or more – see Fig. 3, and thus it is harder than on Earth to suggesta specific value. A strong possibility exists, however, that some Ti-tan lakes may be somewhat ‘tarry’ and thus reluctant to formwaves.

Fig. 5. Near-surface (90 m) windspeed predicted by the TitanWRF GCM for agridcell roughly in the center of the large northern hemisphere sea Kraken Mare. Itis seen that the SAR observations during the nominal mission (T25–T30) took placein the somewhat calm late winter. The upcoming T64 observation is in early spring(Ls � 4�) where winds are picking up. The most likely chance to observe wavesappears to be on the T86 SAR and T91 altimetry observations, while winds areexpected to decline by the summer solstice, at which (T126) Cassini may make afinal observation.

4. Titan lake surface wind predictions

We now compare the threshold with estimates of near-surfacewinds over Titan lakes, for which there are at present no directobservations. We thus use models. TitanWRF is a global model ofTitan’s atmosphere from the surface to about 400 km, adaptedfrom the limited area WRF (Weather Research and Forecasting)model that is used extensively in terrestrial weather prediction.Wind predictions for the lowest grid cell in the TitanWRF GCM(�90 m altitude) were extracted over two major lake targets, On-tario Lacus in the south and Kraken Mare in the north, and areshown in Figs. 4 and 5. In particular, the winds during epochs ofCassini RADAR observations are noted. TitanWRF predictions forlow level winds can vary depending on the parameters used in agiven simulation (e.g. the amount of horizontal diffusion appliedwithin the model). However, the typical wind speeds and the basicvariation in zonal wind with height agree with the measurementsmade by the Huygens Doppler Wind Experiment (as shown in, e.g.Folkner et al. (2006)). Specifically, the features of the Huygenswind profile, namely low near-surface winds, then easterlies peak-ing at a few kilometers, then westerlies aloft, are reproduced cor-rectly, although there are some significant differences in detail.At this location and time, TitanWRF somewhat underestimatesthe observed near-surface westerlies below 1 km from ��0.7 to+0.1 m/s (compared to an observed range of��1.0–1.2 m/s). Titan-WRF also predicts strong easterlies, peaking at >3 m/s between �6and 11 km (Huygens observed similar peak wind speeds but at�3 km), with westerlies not predicted until above �15 km(whereas Huygens observed westerlies of a few meter per second

at this altitude, suggesting the cross-over from easterlies towesterlies occurred lower). The gross behavior of the model, interms of seasonal variation of zonal winds, agrees with other mod-els, none of which reproduces all present observations (a notablechallenge is to reproduce the winds implied by the equatorialdunes (Lorenz and Radebaugh, 2009)).

Some support for the TitanWRF predictions over the lakes is gi-ven by an independent GCM by Tokano (2009). He predicts frictionspeeds (a more direct measure of wind stress) of 0.07 m/s at ahypothetical 75�N lake. Adopting a drag coefficient of 0.002, thisfriction speed corresponds to a freestream speed of �1.6 m/s,rather commensurate with the TitanWRF results. A slight differ-ence in the models is that the peak windspeed occurs early in

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the summer in TitanWRF, but at or just after the solstice in Tokano(2009).

We can see that if the wave threshold is 0.7 m/s or less, waveswill not have occurred anywhere observed on Titan yet (which isconsistent with their nondetection, although a robust upper limiton wave roughness has not been established for the T25–T30SAR observations), and waves on Ontario Lacus will not be ob-served by Cassini at all. On the other hand, waves can be expectedto occur during upcoming northern sea observations if the thresh-old windspeed is �1 m/s, notably on T64 in December 2009, and onT86 and T91 early in the next decade. If the threshold is more than1.5–2 m/s, waves are not likely to occur at all in the present cli-matic regime. Note that evidence of wave action at Ontario maymean either that winds were higher in an earlier season in thepresent Titan (as suggested in Fig. 4), or that wind stress was high-er still in some historical climate regime, or that the wind thresh-old was lower in the past due to a more methane-rich compositionthan in the present epoch. Or some combination of these factorsmay have applied.

5. Conclusions

We estimate that a wind of 0.5–1 m/s should generate capillarywaves on Titan in ‘clean’ methane–ethane–nitrogen lakes, and pre-vious work has suggested that such wind could build over fetchesof 20 km or more, leading to gravity waves with a 20 cm waveheight. However, we have noted the role of viscous damping inthe wave energy balance, and that the likely amount of heavierhydrocarbons dissolved in lakes may provide enough viscosity tosuppress wave growth, although further wind tunnel testing wouldbe desirable to understand this effect better. The presence of hea-vier solutes is most likely in southern hemisphere lakes such asOntario Lacus, which may have been evaporating in the presentepoch (Hayes et al., in preparation; Aharonson et al., 2009; see alsoBarnes et al., 2009).

Both the seasonal transport of methane over the last decade andthe longer-term transport of ethane to the northern hemisphere inthe present astronomical configuration, will tend to enrich thenorthern seas in more fluid components at the expense of OntarioLacus, which may consequently be more viscous and thus have ahigher wave generation threshold at present than northern lakes.We therefore expect future observations of Ontario by Cassini(which will perforce be by radar, since direct solar illumination isno longer available there now that autumnal equinox has passed)to continue to show no indication of waves.

On the other hand, northern seas are expected to be exposed tostronger winds for the next few years through northern summer.Although radar altimetry is a particularly sensitive indicator ofwave-roughening at the centimeter-scale via specular reflection(Wye et al., 2009), SAR imaging (T64, T86, T91) should also be ableto place constraints on roughness, subject to assumptions aboutvolume scattering, absorption coefficient and bottom reflection(Paillou et al., 2008; Notarnicola et al., 2009). In fact, radar back-scatter at incidence angles similar to those used by Cassini SAR isroutinely used to determine ocean surface winds on Earth – theseobservations (e.g. NASA’s Quickscat, or the Advanced Scatterome-ter ASCAT on ESA’s Metop satellite) measure each location fromtwo directions to determine wind direction as well as strength. IfTitan wind models can be adequately validated with other data,comparison of upcoming observations with those acquired whenwinds were lower in 2006–2007 (T25–T30) may allow a windthreshold to be inferred, and perhaps some constraints derivedon the lake viscosity. Additionally, as the Sun rises above the north-ern hemisphere seas, optical observations of the glitter patternaround the solar specular point may allow an independent con-

straint on sea surface roughness (Cox and Munk, 1954) to be deter-mined, albeit subject to assumptions about scattering in theatmosphere. The geometrical requirements for observing suchglints on high-latitude bodies of liquid meant that a previoussearch (West et al., 2005) for optical specular reflections wasunsuccessful. Finally, radio frequency experiments with similarscattering geometry to sunglint observations can be made as‘bistatic scattering’ observations with Cassini’s Radio Science Sub-system (RSS).

Finally, we note in passing that interest in extraterrestrial wavegeneration is not confined to Titan. The detection of optical glintsin lightcurves of extrasolar planets has been suggested as a meansof detecting oceans of liquid water and thereby inferring possiblehabitability of such planets (e.g. Williams and Gaidos, 2008; Cowanet al., 2009). The existence of a detectable glint depends, however,on the ocean roughness, which will depend on the atmosphericdensity as well as its windspeed. Furthermore, many extrasolarplanets and their satellites may be more Titan-like (Lunine, 2009)than Earth-like, considering the abundance of faint M-dwarfs inthe galaxy. Improved understanding of wave generation in general,which would benefit from wind tunnel experiments to explore theinfluence of various parameters, is therefore important, and forth-coming Cassini observations may serve to motivate such work.

Acknowledgments

R.L. acknowledges the support of the NASA Cassini Project atJ.P.L. via the Cassini RADAR investigation. J.L.’s work was supportedby the Italian program ‘‘Incentivazione alla mobilita’ di studiosistranieri e italiani residenti all’estero” and the Cassini program. Ja-son Soderblom and an anonymous referee are thanked for prompt,thorough and constructive reviews. Additionally, we thank LaurenWye for comments on the manuscript.

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