The North pole of Titan seen by Cassini. © NASA

Tides in the lakes of Titan

Hi there! The satellite of Saturn Titan has hydrocarbon seas, i.e. lakes made of liquid ethane and methane. When you have a sea, or a lake, you may have tides, and this is what this study is about. I present you A numerical study of tides in Titan’s northern seas, Kraken and Ligeia Maria, by David Vincent, Özgür Karatekin, Jonathan Lambrechts, Ralph D. Lorenz, Véronique Dehant, and Éric Deleersnijder, which has recently been accepted for publication in Icarus.

The lakes of Titan

The presence of hydrocarbons in such a thick atmosphere as the one of Titan has suggested since the spacecraft Voyager 1 than methane and ethane could exist in the liquid state on the surface of Titan. There could even be a cycle of methane, as there is a hydrological cycle on Earth, in which the liquid methane on the surface feeds the clouds of gaseous methane in the atmosphere, and conversely.

The spacecraft Cassini has detected dark smooth features, which revealed to be these hydrocarbon seas. Here is a list of the largest ones:

Location Diameter
Kraken Mare 68.0°N 310.0°W 1,170 km
Ligeia Mare 79.0°N 248.0°W 500 km
Punga Mare 85.1°N 339.7°W 380 km
Jingpo Lacus 73.0°N 336.0°W 240 km
Ontario Lacus 72.0°S 183.0°W 235 km
Mackay Lacus 78.32°N 97.53°W 180 km
Bolsena Lacus 75.75°N 10.28°W 101 km

I present you only the detected lakes with a diameter larger than 100 km, but some have been detected with a diameter as small as 6 km. It appears that these lakes are located at high latitudes, i.e. in the polar regions. Moreover, there is an obvious North-South asymmetry, i.e. there are much more lakes in the Northern hemisphere than in the Southern one. This could be due to the circulation of clouds of Titan: they would form near the equator, from the evaporation of liquid hydrocarbons, and migrate to the poles, where they would precipitate (i.e. rain) into lakes. Let us now focus on the largest two seas, i.e. Kraken and Ligeia Maria.

Kraken and Ligeia Maria

Kraken and Ligeia Maria are two adjacent seas, which are connected by a strait, named Trevize Fretum, which permit liquid exchanges. Kraken is composed of two basins, named Kraken 1 (north) and Kraken 2 (south), which are connected by a strait named Seldon Fretum, which dimensions are similar to the strait of Gibraltar, between Morocco and Spain.

Kraken and Ligeia Maria. © NASA
Kraken and Ligeia Maria. © NASA

Alike the Moon and Sun which raise tides on our seas, Saturn raises tides on the lakes. These tides cannot be measured yet, but they can be simulated, and this is what the authors did. In a previous study, they had simulated the tides on Ontario Lacus.

They honestly admit that the tides on Kraken and Ligeia Maria have already been simulated by other authors. Here, they use a more efficient technique, i.e. which uses less computational resources, and get consistent results.

Numerical modeling with SLIM

Computational fluid dynamics, often referred as CFD, is far from an easy task. The reason is that the dynamics of fluids in ruled by non-linear partial derivative equations like the famous Navier-Stokes, i.e. equations which depend on several variables, like the time, the temperature, the location (i.e. where are you exactly on the lake?), etc. Moreover, they depend on several parameters, some of them being barely constrained. We accurately know the gravitational tidal torque due to Saturn, however we have many uncertainties on the elasticity of the crust of Titan, on the geometry of the coast, on the bathymetry, i.e. the bottom of the seas. So, several sets of parameters have to be considered, for which numerical simulations should be run.

It is classical to use a finite element method for problems of CFD (Computational Fluid Dynamics, remember?). This consists to model the seas not as continuous domains, but as a mesh of finite elements, here triangular, on which the equations are defined.
The structure of the mesh is critical. A first, maybe intuitive, approach would be to consider finite elements of equal size, but it appears that this way of integrating the equations is computationally expensive and could be optimized. Actually, the behavior of the fluid is very sensitive to the location close to the coasts, but much less in the middle of the seas. In other words, the mesh needs to be tighter at the coasts. The authors built an appropriate mesh, which is unstructured and follow the so-called Galerkin method, which adapts the mesh to the equations.

The authors then integrated the equations with their homemade SLIM software, for Second-generation Louvain-la-Neuve Ice-ocean Model. The city of Louvain-la-Neuve hosts the French speaking Belgian University Université Catholique de Louvain, where most of this study has been conducted. The model SLIM has been originally built for hydrology, to model the behavior of fluids on Earth, and its simulations have been successfully confronted to terrain measurements. It thus makes sense to use it for modeling the behavior of liquid hydrocarbons on Titan.

In this study, the authors used the 2-dimensional shallow water equations, which are depth-integrated. In other words, they directly simulated the surface rather than the whole volume of the seas, which of course requires much less computation time.
Let us now see their results.

Low diurnal tides

The authors simulated the tides over 150 Titan days. A Titan day is 15.95 days long, which is the orbital period of Titan around Saturn. During this period, the distance Titan-Saturn varies between 1,186,680 and 1,257,060 km because the orbit of Titan is eccentric, and so does the intensity of the tidal torque. This intensity also varies because of the obliquity of Titan, i.e. the tilt of its rotation axis, which is 0.3°. Because of these two quantities, we have a period of variation of 15.95 days, and its harmonics, i.e. half the period, a third of the period, etc.

It appeared from the simulations that the 15.95-d response is by far the dominant one, except at some specific locations where the tides cancel out (amphidromic points). The highest tides are 0.29 m and 0.14 m in Kraken and Ligeia, respectively.

Higher responses could have been expected in case of resonances between eigenmodes of the fluids, i.e. natural frequencies of oscillations, and the excitation frequencies due to the gravitational action of Saturn. It actually appeared that the eigenmodes, which have been computed by SLIM, have much shorter periods than the Titan day, which prevents any significant resonance. The author did not consider the whole motion of Titan around Saturn, in particular the neglected planetary perturbations, which would have induced additional exciting modes. Anyway, the corresponding periods would have been much longer than the Titan day, and would not have excited any resonance. They would just have given the annual variations of tides, with a period of 29.4 years, which is the orbital period of Saturn around the Sun.

Fluid exchanges between the lakes

SLIM permits to trace fluid particles, which reveals the fluid exchanges between the basins. Because of their narrow geometry, the straits are places where the currents are the strongest, i.e. 0.3 m/s in Seldon Fretum.
The volumetric exchanges are 3 times stronger between Kraken 1 and Kraken 2 than between Kraken and Ligeia. These exchanges behave as an oscillator, i.e. they are periodic with respect to the Titan day. As a consequence, there is a strong correlation between the volume of Kraken 1, and the one of Kraken 2. Anyway, these exchanges are weak with respect to the volume of the basins.

The attenuation is critical

The authors studied the influence of the response with respect to different parameters: the bathymetry of the seas (i.e., the geometry of the bottom), the influence of bottom friction, the depth of Trevize Fretum, and the attenuation factor γ2, which represents the viscoelastic response of the surface of Titan to the tidal excitation. It appears that γ2 plays a key role. Actually, the maximum tidal range is an increasing function of the attenuation, and in Seldon and Trevize Fretum, the maximum velocities behave as a square root of γ2. It thus affects the fluid exchanges. Moreover, these exchanges are also affected by the depth of Trevize Fretum, which is barely constrained.

Another mission to Titan is needed to better constrain these parameters!

The study and its authors

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook. And let me wish you a healthy and happy year 2018.

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