Wind Driven Capillary-Gravity Waves on Titan's Lakes: Hard to Detect Or

Wind driven capillary-gravity waves on Titan's Lakes: Hard to Detect or Non-Existent? A. G. Hayes, 1 R. D. Lorenz 2 M. A. Donelan, 3 T. Schneider, 4 M. P. Lamb, 4 J. M. Mitchell, 5 M. Manga, 6 J. I. Lunine, 1 H. L. Tolman, 7 O. Aharonson,4 and the Cassini RADAR Team 1Astronomy Department, Cornell University 2Applied Physics Laboratory, Johns Hopkins University 3Rosenstiel School of Marine and Atmospheric Science, University of Miami 4Division of Geological and Planetary Sciences, California Institute of Technology 5Department of Earth and Space Sciences, UCLA 1Department of Earth and Planetary Science, University of California at Berkeley 7Marine Modeling and Analysis Branch, NOAA/NCEP/EMC Abstract Background Modeling Instrument Response Saturn's moon Titan has lakes and seas of liquid hydrocarbon and a dense atmosphere, an Titan is the only extraterrestrial body currently known to support standing bodies of liquid on its surface and, along with Specular observations, such as altimetry (e.g Wye et al., 2009) or glint (e.g. Barnes et al., environment conducive to generating wind-waves. Regardless, Cassini observations thus Earth and Mars, is one of only three places in the solar system that we know to possess or have possessed an active 2011) measurements, will be sensitive to even the smallest gravity-capillary waves. far show no indication of wave activity. In an attempt to understand this apparent hydrologic cycle. The Cassini spacecraft, a joint endeavor between NASA and ESA, was launched in 1997 and reached However, it is unclear whether or not wave fields will be observable to RADAR at off-nadir discrepancy, we apply wind-wave generation models to the Titan environment. Results Saturn in 2004. In order to penetrate Titan' atmosphere, Cassini is equipped with a Ku-band radar capable of obtaining geometries. At moderate incidence angles (10◦ − 60◦) microwave backscatter is suggest threshold wind speeds of 0.4-0.7 m/s (at 10 m) for likely lake compositions, images of the surface at a scale of ∼ 300 m. Cassini's instrument suite also includes the Imaging Science Subsystem dominated by Bragg scatter. In a broad spectrum wave field, however, the smaller depending on liquid viscosity. While available observations have been acquired near (ISS) and Visual and Infrared Mapping Spectrometer (VIMS). These instruments have unveiled a world at once strange Bragg-scattering waves are tilted by their larger counterparts. In order to accurately Equinox, when predicted wind-speeds are below this threshold, future observations will be and familiar to our own, with extensive equatorial dunes, well-organized channel networks draining from mountains into estimate the radar backscatter, a composite model is required in which the effect of low acquired during Northern Spring/Summer, when wind-speeds above the threshold are large basins, and perhaps most astonishingly, large seas and lakes filled with liquid hydrocarbons (Figure 2 and 3). frequency waves is taken into account (e.g. Donelan & Pierson, 1987). TextTextText 180° 0° 10° predicted by Global Circulation Models. 190° 170° 0125 250 500 750 1,000 350° 10° ◦ ◦ 160° Titan’s 20° 200° km 340° Between 10 and 60 incidence, the decrease in dielectric constant () between water 150°Polar Distribution of Hydrologic Features 30° 210° 330° 140° 40° 220° 0125 250 500 750 1,000 km 320° and liquid hydrocarbon results in an 8-15 dB decrease in the Fresnel reflectivity at Wave Generation on Hydrocarbon Lakes 130° 50° 230° 310° Fluvial Valleys 120° 60° horizontal polarization. For Bragg scatter on Titan, this decrease is partially offset by the 240° Empty Lakes 300°300° 110° 70°70° Filled Seas/Lakes 290° increase in spectral power at the Bragg frequency resulting from the increased Wind blowing over a liquid surface generates waves that grow in amplitude with increasing 250° 290° Dark Lacustrine 100° 80° Features 80° 80° 0° 8 280° wind speed. As wind is turbulent, there are always pressure differences available to initiate 260° SAR Incidence 2280° atmosphere-to-fluid density ratio and reduced wave phase speed relative to Earth . 270 90° 90° 90° 50° 90° ° 270° 270° and feed disturbances over the liquid (Phillips, 1957). These turbulent pressure 270° Nevertheless, the backscatter cross-sections (σo) expected from hydrocarbon wave fields 100° 100° 100° 280° 0° 260° 260° 260° ◦ 80° on Titan are low. For wind speeds below ∼ 1 m/s and incidence angles greater than 20 fluctuations account for linear growth (random phase) of micron-sized disturbances. Once 110° 110° 110° 290° 250° 250° 250° 70° 120° 120° these initial wavelets are generated, their presence can modify the air flow in the boundary at Great 120° predicted σo values are less than -25 dB, which is below the noise floor of most Cassini 300° 240° 240° Lakes 240° kes 60° Lakes 130° 130° 130° 310° 230° 230° layer above them and generate wave-induced pressure fluctuations that are in-phase with 230° SAR images (Figure 8). In scatterometry mode, however, the Cassini RADAR can measure 50° 140° 140° 140° 220° 220° 320° 220° 150° 150° 40° 150° σ values as low as -40 to -50 dB (West et al., 2008) which, according to the two-scale 210° 210° the wave slope leading to exponential growth. 210° o 330° 160° 160° 160° 200° 200° 30° 200° 170° 170° 170° 190° 190° 190° 0 ° 8 340° 1 0 ° 8 0 ° 1 1 8 0 ° 8 0 ° 1 8 20° 1 350° 0° 10° backscatter model, is sensitive enough to detect the presence or absence of waves within For deep water (coth(kd) ≈ 1), the phase speed of surface waves is given by the North Polar Region South Polar Region Figure 3: SAR image of Ligeia Mare in acquired between the ∼ 10 km real-apertures footprint. dispersion relation: Figure 2: Distribution of hydrologic features on Titan poleward of 55◦. Note the hemi- February and April of 2007. Note the absence of structure in 2 spherical dichotomy in lake density, which has been attributed to Saturn's eccentric orbit. A: Bragg Backscatter C: Two Component Model [Earth] D: Two Component Model [Titan] c ≈ γk=pl + g=k (1) −10 ° 0 0 the lake surface, consistent with a lack of wave activity. Earth: H O (0 C, ν = 1.85 cSt) 2 f An outline of the Earth's Great Lakes is provided for scale. ° Titan: Coordier et al. 2009 (ν =2.6 cSt) Earth: H 0 (30 C, ν = 0.84 cSt) ° f where k is the wavenumber, γ is the surface tension, p is the liquid density, and g is −15 2 f Earth: H O (0 C, ν =1.85 cSt) ν 2 f Titan: C H (94K, ν =1.7 cSt) l Titan: Cordier et al. 2009 ( = 2.6 cSt) ° 2 6 f f −5 Earth: H O (30 C, ν =0.84 cSt) −5 p4 −20 Titan: CH Rain (94K, ν = 0.31 cSt) 2 f Titan: CH Rain (94K, ν =0.31 cSt) gravity. The phase speed has a minimum value cmin = 4γg=pl that occurs at 4 f ° 4 f Observing Lake Surface Roughness U = 2 m/s θ=20 p −25 10 kmin = gpl/γ (see Table 1) . (dB) −10 −10 HH U = 1 m/s 10 o −30 ° σ −35 θ=30 Once initial wavelets are generated, a positive feedback between the waves and the I The first direct attempt to detect wave activity on Titan was performed by Wye et al. (2009), who showed that Ontario −15 −15 −40 logarithmic shear flow in the boundary can lead to exponential growth (Miles, 1957). Lacus had a maximum root-mean-square (RMS) roughness of 3 mm over the ∼100 m Fresnel zone of the real-aperture U = 8 m/s ° U = 4 m/s 10 10 ° −45 −20 θ=40 −20 Donelan & Pierson (1987) uses this model to predict the onset of observable wind-waves RADAR footprint during the December 2008 (T49) altimetry pass (Figure 4). θ=20 10 20 30 40 50 ° (dB) θ (°) (dB) =30 HH HH θ ° o in seawater, obtaining thresholds of 1:5 − 2 m/s, depending on temperature (i.e. o In July 2009, the VIMS instrument observed glints from the northern seas Kraken Mare and Jingpo Lacus (Stephan B: Specular Backscatter ° σ I σ −25 −25 −10 =50 viscosity). Donelan & Plant (2009) experimentally verified these predictions under varying et al., 2010) (Figure 5). Barnes et al. (2011) analyzed the glints and found that the RMS surface slope was at most θ θ=40 −15 Earth ° ◦ −30 ° θ=50 −30 environmental conditions (e.g., liquid density and viscosity). 0:15 over the ∼ 1 km specular footprint acquired during image acquisition. Titan ° −20 =60 θ θ=60 Both of these observations suggest a complete absence of wave activity, consistent with the calm −25 ° Donelan & Pierson (1987) relates the growth rate to the wind speed at a height above the (dB) U = 8 m/s −35 ° −35 10 θ=70 −30spec =70 surface winds predicted by available GCMs near equinox. o U = 4 m/s θ mean water level equal to one half the wavelength (λ) of the growing wind-wave (U ): σ λ/2 10 −35 −40 −40 U = 2 m/s ρa h i2 10 −40 U = 1 m/s (β =!) = 0:194 (U =c) cos(χ − χ¯) − 1 (2) 10 g Donelan λ/2 −45 −45 −45 ρl 10 20 30 40 50 0.4 0.6 1 1.6 2.5 4 6.3 0.4 0.6 1 1.6 2.5 4 6.3 10 θ ° U (m/s) U (m/s) where (χ − χ¯) is the deviation between the direction of wind and wave.

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