Dynamical modelling of river deltas on Titan and Earth

Planetary and Space Science 105 (2015) 65–79

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Dynamical modelling of river deltas on Titan and Earth Piotr P. Witek n, Leszek Czechowski Institute of Geophysics, Faculty of Physics, University of Warsaw, ul. Pasteura 7, 02-093 Warszawa, Poland

art ic l e i nf o

a b s t r a c t

Article history: Received 20 March 2014 Received in revised form 8 October 2014 Accepted 6 November 2014 Available online 15 November 2014

The surface of Titan hosts a unique Earth-like environment with lakes and rivers, and active ‘hydrologic’ cycle of methane. We investigate sediment transport in Titanian rivers and deposition in Titanian lakes with particular attention to formation of river deltas. The obtained results are compared with analogous terrestrial processes. The numerical model based on Navier–Stokes equations for depth-integrated two dimensional turbulent flow and additional equations for bed-load and suspended-load sediment transport was used in our research. It is found that transport of icy grains in Titanian rivers is more effective than silicate grains of the same size in terrestrial rivers for the same assumed total discharge. This effect is explained theoretically using dimensionless form of equations or comparing forces acting on the grains. Our calculations confirm previous results (Burr et al., 2006. Icarus. 181, 235–242). We calculate also models with organic sediments of different densities, namely 1500 and 800 kg m
3. We found substantial differences between materials of varying densities on Titan, but they are less pronounced than differences between Titan and Earth. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Titan Hydrology River deltas

1. Introduction Titan and Earth are the only celestial bodies in the Solar System where current p–T conditions allow for permanent presence of the liquid on their surfaces as well as for hydrologic/hydrocarbon cycles. The precipitation is observed on both bodies as well as the river and stream valleys formed by surface runoff. The processes in these rivers are the subject of investigation of ExRiMoG group (Extraterrstrial Rivers’ Modeling Group). In the present paper we consider the evolution of deltas on Titan and Earth. We use numerical model based on the physical equations of fluid dynamics. Such models are widely used for terrestrial rivers but according to our best knowledge it is the first paper where dynamical numerical model is used for modelling of Titan’s deltas. The scope of the research is limited to flow with moderate discharge. It is an extension of research presented by Witek and Czechowski (2013). Terrestrial deltas are the subject of numerous papers and textbooks; e.g. Edmonds and Slingerland (2007), Julien (2010), Melosh (2011), Robert (2003). Some of their results are presented in the present section. The papers that concern extraterrestrial deltas are discussed in Section 4.5. The paper is organized as follows. In the rest of Introduction we present basic data concerning Titan, its rivers and lakes, as well as, the basic classification of rivers’ deltas. Section 2 presents equations, boundary conditions and parameters of our model. The results are

n

Corresponding author. Tel.: þ 48 509849765. E-mail address: [email protected] (P.P. Witek).

http://dx.doi.org/10.1016/j.pss.2014.11.005 0032-0633/& 2014 Elsevier Ltd. All rights reserved.

given in Section 3. Section 4 contains discussion of our results, their comparison with some other research, explanations of some observed processes and short discussion concerning possible dimensionless system of equations. Conclusions are in the last section. 1.1. Surface of Titan Titan’s low density (1880 kg m
3) is typical for icy moons and indicates a large fraction of water ice in composition of the satellite. Titan is large enough to be internally differentiated, with an external shell thought to be composed mostly of the lowpressure form of water ice (ice I). The surface is covered by a thin layer of unknown materials, most likely a mixture of precipitated organic materials derived from atmospheric photochemistry and crystalline ice (Sotin et al., 2009). The number of impact craters observed on Titan is very low (around 60, according to Neish and Lorenz, 2014), suggesting a high rate of resurfacing (Jaumann et al., 2009). The data concerning topography are sparse; the maximum known altitude difference is only about 2.5 km on the scale of the whole moon (Lorenz et al., 2013). Lakes of liquid hydrocarbons and numerous landforms created by fluvial activity have been observed by the Cassini-Hyugens mission on the surface of Titan (Stofan et al., 2007; Langhans et al., 2012). Channels thought to be river valleys are present on the surface; some of them form dendritic networks. Some channels are radar-dark so they could be presently filled with liquid. Other channels are radar-bright, indicating a rough, dry surface, most probably grains of solid material (Lorenz et al., 2008). Several dry valleys have alluvial fans at the terminus. Many dark channels

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terminate in lakes, especially in the polar regions of Titan where large clusters of lakes exist. It is observed that appearance and disappearance of clouds in the atmosphere is associated with changes in brightness of the surface (Turtle et al., 2011a). This indicates the presence of active volatile cycle analogous to terrestrial hydrologic cycle (Lunine and Lorenz, 2009). 1.2. Hydrocarbon cycle The atmosphere of Titan consists mostly of molecular nitrogen (95% in the troposphere) with addition of methane and noble gases. Various organic compounds are synthesized from methane and nitrogen by set of complex photochemical reactions in the upper atmosphere (Lavvas et al., 2008). Under Titanian conditions, methane is able to form clouds and rains down to the surface. The rains have a seasonal character (Turtle et al., 2011a). When Titan (and Saturn) is near the solstice the rains occur in the summer hemisphere, and closer to equinox the rains may fall in the tropics (Lunine and Lorenz, 2009). In general, transport of methane from the atmosphere to the surface could take form of the steady drizzle and average precipitation rates are comparable to terrestrial arid and semi-arid regions (Tokano et al., 2006). The precipitation could also have form of sudden massive storm events. The resulting flash floods could cause large-scale changes in the appearance of the surface (Turtle et al., 2011a). Note that the most dramatic changes of terrestrial rivers occur during high discharge period (e.g. floods). Bankfull discharge is dominant for alluvial channels (e.g. Julien, 2002). Therefore, one can expect that the flash floods are likely also the dominant driver of large scale processes on Titan. Such events are however known to be relatively infrequent. We plan to calculate the effects these processes in the next papers but first we investigate the evolution for typical discharge. Therefore, in the present research we consider moderate discharge, an order of magnitude smaller than bankfull discharge for our channel. 1.3. Titanian lakes Numerous hydrocarbon lakes exist in the polar regions of Titan and few lakes have been postulated in the equatorial region as well (Griffith et al., 2012). Some of these features are radar-dark; they are thought to be filled with liquid and they are called ‘lakes’ (lacus). Other are radar-bright, which suggests that the rough lake bed is exposed. The largest cluster of lakes exists near the north pole of Titan. Three of these lakes are large enough to be classified as ‘seas’ (maria). Several large rivers enter the seas, terminating in estuaries. On the southern hemisphere of Titan the largest hydrocarbon lake is Ontario Lacus, the fifth largest body of liquids identified on the moon. It is situated in 300 m deep depression with flat floor, but the thickness of the liquid layer is not expected to exceed a few meters (Lorenz et al., 2010a). The currently observed lake may be a remnant of much larger sea of the size of Ligea Mare (Stofan et al., 2012). Observations have shown the bright annulus surrounding the dark lake, probably composed of evaporites (Barnes et al., 2011). The annuli of that type develop over terrestrial lakes under the conditions of prolonged drought (Barnes et al., 2009). Observations have also shown the reduction of visible area of the lake, interpreted as the recession of the southern shoreline (Turtle et al., 2011b). If this interpretation is correct, then the evaporation rate corresponds to fall of the lake level with the rate of 1 m per year (Hayes et al., 2011). Other interpretations of this phenomenon (without assumption of the fall of the lake level) have been also proposed (Cornet et al., 2012). Over 100 km-long river valley connects to Ontario Lacus at the south-western shore. At the mouth of the river there exists a

unique lobate structure entering the lake (Fig. 2). It is interpreted as a protruding delta, the first form of this kind observed on the surface of Titan. The delta has two lobes; their positions relative to the visible river channel suggest switching of the distributary channels (Wall et al., 2010). The resolution of the images is too low to observe the feeder channels. They are supposed to flow inside the lobes. The valley is about 1 km wide, but the width of the actual river channel is unknown (Wall et al., 2010). The tides of low amplitude are predicted in Ontario Lacus (Barnes et al., 2009; Tokano, 2010), however the apparent delta on the coast of the lake does not appear to show signs of tidal modification. The delta front may be modified by wave action, but the structure is generally dominated by the river processes (Wall et al., 2010). Therefore, in the present research we adopt the model with no tides or waves. Observations with Visual and Infrared Mapping Spectrometer (VIMS) have revealed the presence of ethane in Ontario Lacus (Brown et al., 2008). The compostion of Titan lakes in equilibrium with the atmosphere is predicted to be ethane-rich (Cordier et al., 2009). However, the northern seas could be out of equilibrium with the atmosphere, much as on the Earth (Lorenz, 2014). Currently no river delta has been identified on the shores of northern seas. Some of the river valleys seem flooded—e.g. southern coasts of Ligeia Mare have a ria morphology (Lorenz et al., 2012). Terrain in the vicinity of this sea is hilly, southern banks have steep slopes. The floor of this mare can be seen by Cassini radar through the liquid, which is now thought to be up to 160 m deep (Mastrogiuseppe et al., 2014; Wye et al., 2010). According to General Circulation Models (GCM) the precipitation in such high latitude prevails over evaporation of methane and Ligea Mare is thought to be methane-rich. This result is consistent with the dielectric constant (ε  1.7) suggested by interpretation of microwave radiometry data (Lorenz, 2014). Several channels appear to begin in the surrounding terrain and continue on the sea floor. These observations could indicate geologically recent rise of the liquid level: some valleys were formed when the surface of the lake was lower (Lorenz et al., 2012). Therefore, some depositional landforms created in the past may be currently drowned. The other factor responsible for the formation of channels visible on the lake floor could be turbidity currents. 1.4. Titanian rivers Valleys with morphology suggesting fluvial origin are distributed around the whole globe of Titan. They are visible in polar areas containing lakes, but they occur also in dry equatorial regions. Areas close to the north pole of Titan are believed to be humid, based on large number of lakes, whereas equatorial regions are arid or semi-arid. In the southern hemisphere the number of lakes is much lower, suggesting climate changes possibly due to asymmetric forcing (Aharonson et al., 2009; Lunine and Lorenz, 2009). The channels can be radar-bright, indicating centimetrescale rough terrain, probably an empty river bed covered with pebbles similar to those observed by the Huygens probe. The radar-dark channels may be either filled with liquid or deposits of particles of sizes smaller than the radar wavelength (2.17 cm). It should be noted, that Cassini spacecraft instruments have limited resolution ( 300 m for RADAR and  500 m for VIMS (Visual and Infrared Mapping Spectrometer) and small valleys are not visible in SAR (Synthetic Aperture Radar) images. For example, the Huygens probe imaged a small dendritic network of channels, during its descent through the atmosphere, invisible for Cassini instruments (Tomasko et al., 2005; Grotzinger et al., 2013). It should be noted that images acquired by Cassini show river valleys, but the actual river channel may occupy only a tiny fraction of the valley. The valleys on the Earth are well known to

P.P. Witek, L. Czechowski / Planetary and Space Science 105 (2015) 65–79

be wider and deeper than the rivers flowing inside them; analogous situation is expected on Titan where fluvial valleys have width of up to several kilometres (Jaumann et al., 2008). 1.5. Source of sediments Although the surface of Titan is extensively dissected by fluvial channels, laboratory experiments have shown that the water ice is resistant to abrasion in Titan’s conditions (similarly to terrestrial mudstones and sandstones) and due to higher buoyancy of icy sediments in liquid methane compared to quartz in water, the incision rates can be much lower (Collins, 2005; Sklar et al., 2012). This result suggests that the prior fracturing of the bedrock is needed to create regolith (Burr et al., 2013a). The surface of Titan is likely to be porous (Kossacki and Lorenz, 1996) and saturated with hydrocarbons (Czechowski and Kossacki, 2012; Neish and Lorenz, 2014), with mechanical properties different from uniform block of ice. The analysis of the rectangular morphology of river networks on Titan suggests that rivers often use previously existing surface fractures, including faults (Burr et al., 2013b). Sources of sediments on Titan require further research, but the presence of rounded cobbles at Huygens landing site (Tomasko et al., 2005; Grotzinger et al., 2013) and equatorial dunes supports the idea that various processes of sediment transport operate on the surface. 1.6. River deltas A delta can be defined as a deposit built by a feeder system into or against a standing body of liquid (e.g. Nemec, 2009). The flow of a river or a stream results in erosion of its bed and banks. The eroded material is transported downstream. When the flow enters the basin, it slows down and its ability to transport sediments is diminished. Eventually the material is deposited on the floor. On the Earth, various types of deltas are observed. Galloway (1975) proposed a triangle plot for the classification of deltas (Fig. 1). Usually in lakes the wave action and tides are not very important and the deltas are dominated by the flow of the feeder river. Their lobate or elongate shapes are created when the river builds out into the lake (or sea) from the coastline. Prominent examples include Selenga delta in Lake Baikal and Mississippi delta in the Gulf of Mexico. Wave deltas form when waves push the sediments back to the coast or carry them away, creating shore-parallel spits and bars. The Nile and the Rhône rivers have such deltas. Tidal deltas are characterized by channels perpendicular to the coast, created by bidirectional flows: examples include the Ganges and Brahmaputra deltas. There are also mixed, intermediate types. The Niger delta, for example, is influenced by both wave and tidal actions. Orton and Reading (1993) pointed out the importance of the grain size for the shape and development of river deltas. Sand and gravel tend to build fan or braid deltas, whereas mud and fine sand is responsible for the creation of elongate (‘bird’s foot delta’) and lobate deltas. The liquid flow entering the lake is best described as free turbulent jet (Anderson and Anderson, 2010). The interactions between the entering jet, the lake fluid, and the delta front, influence sediment deposition: the jet may be inertia-, friction-, or buoyancy-dominated, resulting in different flow regimes and morphologies. The buoyantly supported jet with density lower than the basin water is called hypopycnal. This form is common for fine sediments transported into a sea, due to the salinity of seawater. In terrestrial freshwater lakes the river waters, containing sediments, are generally denser than lake waters and tend to flow close to the bottom (hyperpycnal flow). In case of equal densities the flow is called homopycnal (Leeder, 2011; Orton and Reading, 1993). A high proportion of suspended load in total sediment load results in a relatively small mouth bar deposited from bed load

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and an extensive gentle-sloped delta front and prodelta deposits formed by hypopycnal flows. A high proportion of bed load results in steep-sloped delta with a higher proportion of mouth bar gravels and sands. A typical delta forms over the subaqueous part and some deltas may lack (as yet) the sub-aerial part (Nemec, 2009). However, in the case of the relatively rapid rise of the liquid level in the reservoir, the formerly sub-aerial delta plain may become submerged.

2. Numerical model 2.1. Equations The dynamical analysis of flow, transport and deposition in terrestrial and Titanian rivers is performed using the package CCHE modified for the specific conditions on Titan—Table 1. The first set consists of equations governing the flow. It consists of the Navier–Stokes equations for depth-integrated two dimensional, turbulent flow and the depth-integrated continuity equation (Jia and Wang, 2001):
  ∂u ∂u ∂u ∂Z 1 ∂ðhτxx Þ ∂ hτxy τ þu þv ¼
g þ þ
bx ; ð1Þ ∂t ∂x ∂y ∂x h ∂x ∂y hρ 
 
τby ∂ hτyy ∂v ∂v ∂v ∂Z 1 ∂ hτyx þ
; þu þv ¼
g þ ∂x ∂y ∂t ∂x ∂y ∂y h hρ

ð2Þ

∂Z ∂uh ∂vh þ þ ¼ 0; ∂t ∂x ∂y

ð3Þ

where τxy, τxx, τyx and τyy are depth integrated Reynolds stresses, given by approximation based on Boussinesq’s assumption
 τij ¼ νt ui;j þ uj;i ; vt is the eddy viscosity coefficient, calculated using k–ε turbulence model (Wu, 2001); u and v are depth-integrated velocity components in x and y directions, respectively; t is the time; g is the gravitational acceleration; Z is the water surface; h is the local water depth. The shear stresses on the bed and flow interface, τbx and τby, are calculated using Manning’s coefficient (n):

τbx ¼ τby ¼

1 1=3

h

1 1=3

h

ρgn2 uU;

ð4Þ

ρgn2 yU;

ð5Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where U ¼ u2 þ v2 . The second set of equations describes transport of sediments. The transport is modelled using three-dimensional convection– diffusion equation. Sediments are divided into seven classes of different grain sizes (see Table 4), and each class treated with the following convection–diffusion equation:     ∂ck ∂ðuck Þ ∂ðvck Þ ∂ðwck Þ ∂ðωsk ck Þ ∂ ∂c ∂ ∂c þ þ ¼ þ þ εs k þ εs k ∂x ∂y ∂z ∂z ∂x ∂y ∂t ∂x ∂y   ∂ ∂ck ; ð6Þ εs þ ∂z ∂z where ck is the concentration of kth size class of sediment (k ¼1, …, 7); ωsk is the settling velocity (calculated with the Zhang’s formula, see e.g. Cheng, 1997) of kth size class of sediment; εs is the eddy diffusivity. For calculating the bed change the overall mass balance equation is adopted:  
 ∂ qbky þ qsky ∂ðhC tk Þ ∂ qbkx þqskx 0 ∂zbk ¼ 0; ð7Þ þ þ þ ð1
p Þ ∂y ∂x ∂t ∂t where qskx and qsky are the suspended load transport rates in x and y directions; qbkx and qbky are the components of bed load

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Fig. 1. Classification of river deltas according to processes dominant in shaping of delta plains, after Galloway (1975).

Table 1 Gravitational accelerations and Manning roughness coefficient used in our models. Manning roughness coefficient is after Geleynse et al. (2010).

Fig. 2. Ontario Lacus on Titan with two-lobed delta at southwestern shore indicated with the arrow. The eastern lobe may be abandoned or fed by the channel that is not visible at this resolution.

transport rate in x and y directions. p0 is the porosity of bed material; ∂zb =∂t is the total bed deformation rate; Ctk is the depthaveraged concentration of total load (suspended load þbed load). 2.2. Material properties Several kinds of liquid can be found on Titan. Methane can exist near the surface in both gaseous and liquid forms. Condensation of

Name [unit]

Value

Gravitational acceleration on Earth [m s
2] Gravitational acceleration on Titan [m s
2] Manning roughness coefficient [m
1/ 3 s]

9.817 1.352 0.03

methane into droplets leads to the rainfall. Nitrogen is soluble in liquid methane in Titanian conditions, therefore the rain most likely consists of mixture of these substances (Graves et al., 2008). Chemical composition of Titan’s lakes can be much more complicated; spectroscopic observations have revealed the presence of ethane and probably propane and butane in Ontario Lacus (Brown et al., 2008). Furthermore, their compositions change with time due to evaporation of methane (see e.g. Hayes et al., 2011) and retention of less volatile organic species. In our calculations we used three mixtures of liquids with the properties given in Table 2 and three solids given in Table 3. We consider water ice grains as the natural choice for sediment, because they are composed of the most abundant substance in the lithosphere. In fact, rounded cobbles on Huygens landing site are probably composed of ice (Tomasko et al., 2005). Another possible choice is the mixture of solid organic compounds, precipitated from the atmosphere. Since the composition of such

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Table 2 Physical properties of water (for terrestrial conditions) and three considered Titanian liquids at 94 K (after Cordier et al., 2009; Lorenz et al., 2010b). Name

Density [kg m
3]

Kinematic viscosity [m2 s
1]

Water ‘Rain’: 75% methane, 25% nitrogen 100% methane ‘Lake’: 74% ethane, 10% methane, 8,5% butane, 7% propane, 0.5% nitrogen

999.84 518 454 658

1.52  10
6 2.92  10
7 4.58  10
7 2.16  10
6

Table 3 Physical properties of considered sediments for both planetary bodies. Densities for ice and dense organic matter in Titanian conditions are after Burr et al. (2006) and Perron et al. (2006). Specific gravity for Titanian sediments is calculated assuming liquid identified as ‘rain’ in Table 2. Name

Quartz (for Earth) Water ice (pure) Organic matter A mixture of water ice and low density hydrocarbons

Density [kg m
3]

Specific gravity

2650 940 1500 800

2.650 1.815 2.896 1.544

mixture is hard to predict, its properties span wide range. We adopt the value used by Burr et al. (2006) for dense organic matter. Most of Titan’s surface is covered with layer of hydrocarbons and icy sediments may contain some admixture of hydrocarbons. Therefore, sediments density could be different than for pure ice; higher or lower depending on composition and density of hydrocarbons. To explore the other possibility we include in our calculations the sediments with density lower than water ice (0.8 g cm
3); this corresponds to either organic sludge (Lorenz et al., 2003) or ice-organic mixture. The density of Titan tholin analogues created in laboratory can be as low as 0.4 g cm
3 (Hörst and Tolbert, 2013), but the sediments of such low density would float and would not form the river delta. Specific gravity is the dimensionless ratio of the sediment density to the liquid density. It is an important factor when considering buoyancy. The concentration of sediments is scaled for each Titanian case to keep the volume of transported sediments constant. Since the grain size distribution in Titan’s rivers is unknown, our chosen distribution of fractions is based on the terrestrial example. The authors of CCHE package (Jia and Wang, 2001; Zhang, 2006) provide an example of fractions in a part of Vistula river in Poland; our distribution is based on that example but it is scaled down. Note, that our model is designed to reproduce the formation of deltaic depositional system at the terminus of a long river, such as the one in the Ontario Lacus region, which is over 100 km in length. The mean sediment radius decreases downstream the river; this is the reason why we use very fine sediments in our simulations. 2.3. Geometry and boundary conditions The geometry of our model is artificial and simple. The considered area consists of the short (terminal) part of the river channel perpendicular to the shore and of the rectangular part of the lake (Fig. 3). It is similar to geometry of model presented by Geleynse et al. (2010) but we used a different shape of the mouth of the river, i.e. we cut off the corners at the river’s mouth to produce triangular shape. Moreover our model is substantially smaller (width of the river is  75 m, compared to  550 m in Geleynse et al., 2010). The channel and the reservoir have constant bed slope of 0.0005 rad. Obviously, the channel lies higher than

Table 4 Fractions of grains and their initial distribution in our model. Diameter [mm]

Boundary condition for suspended load [%]

Boundary condition for bed load [%]

Initial bed composition [%]

0.01 0.1 0.25 0.5 2 4 16

35 25 15 15 10 0 0

10 10 20 25 25 9 1

10 10 20 25 25 9 1

the bottom of the lake (Fig. 4). The discharge at the inflow is constant, set to 10 m3 s
1. It corresponds to unit discharge of 0.13 m2 s
1. This value is in the range given by Jaumann et al. (2008). The liquid can freely flow through three sides of the rectangular area to simulate larger reservoir; the boundaries have constant level of liquid, 0.5 m above the reference level. The initial water surface level is constant, equal to 0.5 m for whole considered region (the river and the lake). The chosen value of Manning roughness coefficient corresponds to coarse sand or gravel bed on the Earth. 2.4. Modelling procedure Our calculations are performed in two stages. In the first the velocity field is calculated using Eqs. (1)–(5) over two hours until stabilization. This stage removes short-term transient effects which may arise from artificial initial conditions (i.e. flow starts from zero velocity, initial level of the liquid is constant, etc). It improves the reliability of modelling, because transient effects of the flow field could influence the deposition. The second stage is the simulation of sediment transport. The velocity field is recalculated every 20 time steps (40 s) to account for bed changes. The sediment transport is calculated using the system (6) and (7).

3. Results The model gives a few physical fields: velocity field, bed elevation, distribution of D50 (median grain size), suspended load concentration, shear stress, eddy viscosity and bed composition. Because of the boundary conditions the level of liquid in the lake is almost constant and the depth change is a result of sedimentation. We restrained our calculations to one liquid, the Titanian ‘rain’, because calculations with other liquids give very similar results in terms of flow fields (see also Misiura and Czechowski, 2013a,b). This behaviour is mostly a result of turbulent nature of the flow. The values of eddy viscosity are often a few orders of magnitude larger than kinematic viscosity, reducing the differences in the flow fields. Note however, that transport and sedimentation are not the same for these liquids for many reasons, e.g. different settling velocities of the sediment grains. Fortunately, different settling velocities could be modelled also by different density of

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Fig. 3. Initial bed elevation in the considered models.

Fig. 4. Initial vertical cross-section and boundary conditions for the considered models.

the grains. In present model we choose the most likely liquid and we consider three densities of grains. Liquid contained in the lake in our model has the same density as the river. Figs. 5 and 6 present bed elevation for terrestrial deltas after 60 d 8 h and after 98 d 23 h of evolution. Corresponding results for icy sediments in Titanian conditions are given in Figs. 7 and 8, respectively. Figs. 9 and 10 present vertical sections along chosen lines. Note that the distribution of sediments at the mouth of the river delta (in our models as well as in real deltas) is often not symmetric, although boundary conditions are assumed to be symmetric. It is a result of the fact that the ideal symmetric system of the flow and sedimentation is usually not stable. Even very small disturbance of symmetry will grow leading to finite asymmetry. In our model the small asymmetry of the grid serves as this disturbance. Consider small breaching of the sandy bar by water stream. The stream leads to erosion that widens the breaching etc. Eventually asymmetric lateral channel is formed. Formation of deltaic deposits in our model begins with creation of subaqueous part, initially the mouth bar grows in front of river channel (see Figs. 5–8 and cross sections in Figs. 9 and 10). Accumulation of

sediments leads to branching of the flow, which is visible in the velocity field (Fig. 11). The new velocity field leads to creation of additional depositional bars, and the process continues leading to formation of mostly subaqueous prodelta. When sufficient amount of sediments is deposited in front of the river channel, parts of our ‘delta’ become sub-aerial, completely blocking the flow in some directions. New areas of deposition emerge in front of deeper channels formed in the delta. For conditions chosen in our model the terrestrial delta consists of cone-shaped alluvial deposits only. In Fig. 5 almost the entire delta is subaqueous. Deposited sediments force the flow to branch into several streams, indicating that ‘delta’ will continue to evolve. Fig. 6 shows the later stage of evolution; the delta is now partially sub-aerial. For Titan we present calculations with three different densities of the sediments: 940 kg m
3 for water ice and 1500 kg m
3 (dense organic matter) and 800 kg m
3 (a mixture of water ice and low density organic matter). The results for Titanian conditions are different compared to terrestrial, although the same discharge and the same total volume of sediments are assumed. Note the dramatic difference between the shapes of both sedimentary

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Fig. 5. Bed elevation after 60 days 8 h—Earth. Vertical black line marks the cross sections given in Figs. 9 and 11. In this model, deposited sediments force the flow to branch into several streams, indicating that ‘delta’ will continue to evolve.

Fig. 6. Bed elevation after 98 days 23 h—Earth. Deposited sediments have created wide sub-aerial delta. Note different color scale comparing to Fig. 5.

structures. For the Earth the sediments form wide cone-shaped deposits. On Titan the material is deposited further from the mouth of the river forming well developed elongate delta with three bars. Generally, our results indicate that the transport of material on Titan is more effective compared to Earth. Further evolution of this model is presented in Fig. 8. The cross section is given in Fig. 9. The sediments initially have been

deposited along the slightly asymmetric, narrow central bar. Later the flow evolved and created two more depositional bars. In terrestrial conditions, the jet rapidly decelerates and expands laterally, therefore the wide ‘middle-ground’ bar is created. The overall picture resembles the case with a friction-dominated jet (Leeder, 2011). In Titanian conditions, the jet is probably inertiadominated, and the sediments are transported by the flow deep

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Fig. 7. Bed elevation after 60 days 9 h—Titan, icy sediments. Coarser sediments have been deposited along the slightly asymmetric, narrow bar and the finest sediments are dispersed over large area and therefore they are less visible in this image. Initially, they created the central bar but later they form two additional depositional bars. Vertical black line marks the cross section given in Fig. 10.

Fig. 8. Bed elevation after 98 days 23 h—Titan; icy sediments. New depositional bars are visible and large portion of the delta became sub-aerial. Compare with Fig. 7.

into the lake, creating the bar with steep front; its lunate shape is recognizable in Fig. 7 (see also e.g. Leeder, 2011). This fact is related to lower gravity on Titan that results in smaller friction between the fluid and the lake bed. Effects of different viscosity of the liquids are less important here.

Fig. 11 presents evolution of the velocity of the flow in terrestrial conditions. Initial velocity (symmetric solid profile) is higher near the centre than on the sides; the same is observed in simulations of Titanian environment. It is a simple result of geometry (the distance to the mouth of the river). In the following

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Fig. 9. Evolution of the lake bed on Earth, cross section along the line given in Fig. 5. Vertical axis: bed elevation in meters. Horizontal axis: distance (in meters) from the upper end of the line. The legend gives the times (in simulated days) corresponding to each profile. The sediments form wide cone-shaped deposits, reflecting high deceleration of the flow in terrestrial conditions. Compare with three bars forming in model of Titan—Fig. 10.

Fig. 10. Evolution of the lake bed on Titan, cross section along the line given in Fig. 7. Vertical axis: bed elevation in meters. Horizontal axis: distance (in meters) from the upper end of the line. Compare with Fig. 9 for terrestrial conditions. Note that initially (t¼ 22 d) only the central bar formed; later (t¼ 43 d) two lateral bars formed, but the sediments were still deposited on the central bar, causing it to widen. Small irregularities grew in size with time. The deposits rose up to the liquid level (0.5 m).

Fig. 11. Evolution of the velocity magnitude on Earth, cross section along the line given in Fig. 5. Horizontal axis: distance (in meters) from the upper end of the line. High velocity near the centre is a result of geometry (the distance to the mouth of the river is shorter than on the sides). The decrease of the velocity in the center is a result of formation of the central bar. Shifting of the peak to the left at t ¼106 d is a result of redirection of the flow due to formation of large subaerial deposits.

stages of evolution the velocity in the central parts of terrestrial delta decreases as a result of formation of a tiny ‘islands’ located on the central bar. The islands divide the stream and two lateral maxima of velocity are observed (line corresponding to 84 days in Fig. 11). Figs. 12–14 present results for sediments density 1500 kg m
3. In Fig. 12 the lateral bars are still forming, which means that the evolution is slower due to reduced mobility of sediments. The overall distribution of sediments in Fig. 13 is similar to distribution in Fig. 7. Note substantial asymmetry of elevation of sediments in Fig. 14; the elevation of deposits on the left hand side of the river is considerably lower comparing to right hand side. Figs. 15 and 16 present results for sediments density 800 kg m
3. The overall distribution of sediments is different, more similar to distribution in Fig. 8. Note that the asymmetry of central part of the

deposits is much lower. Note also the difference in evolution compared to the previous case: the areas between the bars are gradually filled with sediments. Our calculations show substantial differences for three different densities of sediments. Denser sediments tend to form narrower deposits, while sediments of lower density are distributed over larger area. These differences are also less pronounced than differences between Titan and Earth. The dramatic difference between terrestrial and Titanian deltas could be explained as a result of more effective sediment transport on Titan. Simple explanation of this result is presented below in Sections 4.4 and 4.5. Note also, that the effectiveness of sediment transport is a function of the sediment grain size (see Orton and Reading, 1993). Because the disturbances and instabilities of different origin are common features of the flow, typical deltas (both real and modelled) are usually asymmetric. The averaging of several models (with

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Fig. 12. Bed elevation after 63 days 22 h—Titan, dense organic sediments (1500 kg m
3). In this simulation the volume of the sediments is the same as on Earth. The lateral bars are barely recognizable; they started forming much later compared to the case with icy sediments (see Fig. 7).

Fig. 13. Bed elevation after 99 days 3 h—Titan, dense organic sediments (1500 kg m
3). The geometry of the delta is similar to Fig. 8, however much smaller portion of the delta is subaqueous.

different disturbances) could remove chaotic effects of the disturbances. However, numerical models are deterministic and the repetitions of the run give the same (asymmetric) results. In our models the most pronounced effect of disturbances is strong asymmetry of deposits and channels with respect to the axis of symmetry of the domain. This effect of asymmetry of the grid could be removed by

symmetrization of the results with respect to this axis. Fig. 17 gives results of such procedure for the model of Titan with icy sediments, for cross sections located 251 m (upper) and 295 m from the shore (lower). Note small difference between original topography and topography after symmetrization for the upper cross section; for other cross sections the difference is substantially higher.

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Fig. 14. Evolution of the lake bed on Titan—dense organic sediments. Cross section of the deposits is similar to Fig. 9, but the asymmetry is more pronounced here. The lateral bars are recognizable at ca. 85 d, but later the sediments are deposited further away from them.

Fig. 15. Bed elevation after 98 days 21 h—Titan, low density organic sediments (800 kg m
3). In this simulation the volume of the sediments is the same as in other models. The geometry of the delta is similar to Fig. 8.

Fig. 16. Evolution of the lake bed on Titan—low density organic sediments. Cross section of the deposits is similar to Fig. 9.

Some phenomena occurring in short time scales, can result in large scale changes in the sediment pattern (e.g. Geleynse et al., 2010; Melosh, 2011). Note, that our model is not designed to reproduce a major shift in river course, which can occur during large flood (avulsion). Floods on Titan may result from extreme precipitation events, but the discharges are expected to be much smaller than those from glacial floods on Earth and Mars (Burr, 2010).

4. Discussion and interpretations 4.1. Problems of modelling Numerical modelling offers special opportunity for comparison of processes in Titanian and terrestrial rivers. At least basic assumptions concerning terrestrial rivers could be applied

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Fig. 17. Bed topography along cross sections located 251 m (upper) and 295 m from the shore (lower) for model of Titan with icy sediments. Dashed line gives the original result. Solid line represents the same results after symmetrization. Note that the differences between two lines in the upper panel are very limited, while in the lower panel they are much more pronounced.

Table 5 Minimum, typical and maximum ratios of settling velocities and dimensionless numbers Bk for each grain size class for three different compositions. Composition

‘Rain’ with dense organic sediments

Diameter [mm] ωE/ωT 0.01 1.2 0.1 1.2 0.25 1.6 0.5 2.0 2 2.4 4 2.5 16 2.5

BE/BT 0.5 0.6 .07 .09 1.1 1.1 1.1

‘Rain’ with icy sediments

‘Lake’ with low density ice-organic mixture

ωE/ωT 2.8 2.8 3.1 3.4 3.8 3.8 3.8

ωE/ωT 79.0 78.3 69.7 46.6 12.4 9.0 7.6

BE/BT 1.3 1.3 1.4 1.6 1.7 1.7 1.7

BE/BT 35.7 35.4 31.5 21.1 5.6 4.1 3.5

also for Titanian ones. In both cases we are modelling turbulent flows in open channels. The properties of liquids and sediments are different but the differences are not dramatic comparing to other natural flows, e.g. flows in atmosphere or in lava channels. Various methods are used in practice of modelling (see e.g. Jaumann et al., 2009; Melosh, 2011; Robert, 2003). Laboratory models give reliable results concerning local processes that can be reproduced on the same scale in the laboratory. However, there are problems with large-scale processes because the sizes of laboratory model are much smaller than those of natural rivers. Dimensional analysis is another method. It is based on Buckingham π theorem (e.g. Malvern, 1969, p. 462). This method requires only knowledge of units used for given physical quantities but its results are limited (see e.g. Robert, 2003). The best theoretical method of modelling is based on dimensionless form of equations. It requires the knowledge of physical system of equations that describes the considered physical processes. The method is very popular in fluid dynamics, especially if fluids with different properties are considered. For terrestrial rivers with the fresh water as the liquid and quartz grains as the solids the advantage of the method is not obvious. However, considering rivers on different celestial bodies (Earth, Mars, Titan) we have different liquids, solids and values of other parameters (e.g. gravity). Therefore,

we believe that at least some problems will be better described using dimensionless form of the equations.

4.2. Dimensionless form of equations The dimensionless forms of equation have some advantages over a dimensional form. Usually some reduction of the number of parameters could be achieved (e.g. Czechowski and Kossacki, 2012). Moreover solution of dimensionless system of equations could be valid for whole class of physical situations, i.e. the solution is valid if the corresponding dimensionless numbers have the same values. The way to obtain dimensionless form is not unique. To define natural units of the length, mass, and time one can use different physical quantities. In typical approach dimensionless numbers like Froude and Reynolds numbers are introduced (e.g. Malvern, 1969, Section 7.4). Note however, that in definition of the Reynolds number: Re¼ Uh/ν, v is the coefficient of kinematic viscosity. This coefficient is of secondary importance for turbulent flow typical in rivers. The eddy viscosity is often a few orders of magnitude larger than the kinematic viscosity, but it depends on the local flow. Therefore, we try to find methods that do not use the viscosity at all. We propose here the method based on the global parameters of our model: discharge at the inflow Q [m3 s
1], gravity g [m s
2] and fluid density ρ [kg m
3] (see also Czechowski, 2014). Using these parameters we can define three natural units as follows: length: L¼ Q2/5 g
1/5, time: T ¼Q1/5 g
3/5, mass: M¼ ρ Q6/5 g
3/5. After this transformation the right hand side of Eq. (6) takes the form:
        ∂ ω0sk ∂ ∂ ∂ 0 ∂ 0 ∂ 0 ∂ þC ε ε ε ð8Þ þ þ ck ; Bk ∂z0 ∂x0 s ∂x0 ∂y s ∂y0 ∂z0 s ∂z0 where Bk ¼ ωsk Q
1/5 g
2/5 and C¼ ε0 Q
3/5 g
1/5 are dimensionless numbers. The ε0 is the reference value of the eddy diffusivity. C characterizes the role of diffusion of sediments described by ck while Bk characterizes the role of settling of sediments.

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4.3. Comparison of B and C Compare now values of two numbers that determine transport of sediments, i.e. Bk and C. Assuming the same value of the reference eddy diffusivity ε0 for terrestrial and Titanian rivers, one can find that ratio CE/CT is 0.7 where indices E and T denote values for the Earth and Titan, respectively. Because diffusion is more important for suspended load then one can expect that suspended load is more efficient in Titanian rivers than in terrestrial ones. This fact is really observed in our calculations (see Section 3). The value of the number Bk depends on the size of grain labelled by k via the settling velocity. Note that Bk can be expressed also by other dimensionless numbers. For uniform or near-uniform flow it can be expressed simply as: Bk ¼kk A
1/2 S
1/2, where kk is the dimensionless ratio of settling velocity and friction velocity (see: Bagnold, 1966; Burr et al., 2006), A¼ Q2/5 g
1/5 h
1 is a dimensionless number arising in transformation of equations of motion, and S is the bed slope. The ratios of BE/BT and the ratios of settling velocities for each sediment size class in three cases (minimum and maximum values and for typical composition) are given in Table 5. For the liquid with composition of Titanian ‘rain’ and small icy grains the ratio has value  1.3 and it increases to  1.7 for the largest grains considered here. Unfortunately, we have no direct interpretation for the ratio BE/BT. 4.4. Comparison of forces Some characteristics of described processes could be explained by simple consideration of basic forces acting on the grains. The flowing fluid acts on sediment grains by the drag force FD. It is the main driving force of sediments transport. The negative buoyancy Fg is responsible for impeding the motion of loose sediments (see e.g. Fig. 9.4 in Melosh, 2011). Let E be the ratio of these forces, i.e.: E ¼Fg/FD. Of course, the low value of E is necessary for effective sediment transport while large value means that transport is difficult or even impossible. Fg acting on a spherical grain is: Fg ¼(4/3) π r3g (ρg
ρf) g, where rg is the radius of grain, ρg is density of grain, ρf is density of fluid, and g is the gravity. Depending on the Reynolds number Re the drag force is given by: FD ¼ (1/2) CD π r2g ρf v2 (for high Re) or FD ¼ 6 π η rg v (for low Re)—see also: Julien (2010). CD is between 0.4–0.5 for Re in the range 1000–200 000, and 0.1–0.2 for Re4400 000. Let ET and EE be the ratio for Titan and Earth, respectively. For turbulent flow the ratio ET/EE is 0.075; for laminar flow the ratio ET/EE is 0.39. Generally the river is assumed to be turbulent (e.g. Julien, 2010, Ch. 6) and equations for turbulent motion is used to describe flow in a river (e.g. Jia and Wang, 2001). The flow of fluid transporting sediments in suspension is strongly turbulent, but close to the bottom turbulence could be less important and viscous forces could be dominant (e.g. Melosh, 2011, p. 358). Anyway the ratio ET/EE o1 suggest that sediments transport in Titanian rivers is more effective than in terrestrial ones. This result is in agreement with Burr et al. (2006) as well as with our calculations. It is more difficult to determine the roles of the bed load and the suspended load. The low value of the ratio ET/EE for turbulent flow (ET/EE o0.1) suggests that on Titan the suspended load transport is more effective. Our model as well as the model of Misiura and Czechowski (2013a,b) confirmed this conclusion. For laminar flow ET is also lower than EE, so one could expect that bed load transport for Titan is also more effective. However, our numerical models indicate that this is not so evident. The bed load transport on Titan could be of the same order as for terrestrial rivers. Results of Misiura and Czechowski (2013a,b) suggest that the bed

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load transport for mid-course of the river on Earth could be even higher than for Titan (at least for some range of sizes of the grains). The higher bed load in terrestrial rivers could be explained as an effect of larger shear stresses on the bed of terrestrial rivers (the shear is proportional to the product ρg, see e.g. Melosh, 2011, p. 403). Unfortunately, the simple considerations presented here are not able to give final conclusions. The problem needs further research.

4.5. Comparison with other models There are many other attempts to model formation and evolution of river deltas, but the investigations have been so far focused on Earth. This is highly understandable given enormous data sets concerning deltas on Earth, including field studies that are impossible (presently) for other planetary bodies. Several papers are also dedicated to study ancient Martian deltas (e.g. Kleinhans et al., 2010; Kraal et al., 2008). The recent high resolution images of Mars indicate a large number of deltas. Note that most information concerning Martian deltas comes from the remote sensing. There are a few examples of well-defined layering of deltaic deposits, most notably in Eberswalde crater where stratigraphy can be constructed (Pondrelli et al., 2008). Similar knowledge is unavailable for Titan, although it can be speculated that one delta lobe in Ontario Lacus is abandoned (Wall et al., 2010). The models of formation and evolution of deltas have come across difficulties arising from complexity of the process and various time scales involved in formation and reworking of the deltas. Recent applications of reduced complexity models have proven to be useful for simulations of terrestrial river deltas, especially in large time scales. However, these models rely mainly on phenomenological relations for erosion and sedimentation (e.g. Seybold et al., 2007, 2009). Application of these relations for other planetary bodies is not so straightforward. New successful models of river delta formation and evolution based on physics of flow and deposition processes have been developed recently. They are able to simulate not only development of river-dominated deltas, like our model but they could account also for effects of tides and waves (Geleynse et al., 2010, 2011). Nevertheless they have not been used for other planetary bodies besides the Earth. In the next stage of the research we plan to perform calculations of the models with values of some parameters (namely Q and n) additionally adapted for specific conditions on the celestial bodies, note that e.g. for free flowing river the total discharge Q depends not only on the slope of the bed but also on the gravity.

5. Conclusions

 We have confirmed that dynamical model of flow and sedi





mentation produce expected results for Earth and it is able to give estimated results for Titan. We observe that the first part of the delta visible above the liquid level is the mouth bar. It forces the flow to bifurcate, creating more depositional bars and driving the evolution of the delta. Sometimes the channel is blocked, abandoned and infilled. We confirmed the results of Misiura and Czechowski (2013a,b) that three different liquids considered for Titan give similar velocity fields. The main reason behind this fact is that the eddy viscosity is often a few orders of magnitude larger than kinematic viscosity. Transport of sediments in Titan’s rivers is more effective than in terrestrial rivers for the same discharge and the same total volume of sediments. Similar result for mid-course of the river was found by Misiura and Czechowski (2013a,b) and Burr et al. (2006).

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 The differences of sediment transport on Earth and Titan could







be explained theoretically by considering drag and buoyancy forces (see Section 4.4). The comparison of dimensionless number C for Titan and the Earth indicates that at least suspended load transport is more efficient in Titanian rivers than in terrestrial ones (see Section 4.3). The models show differences in evolution of deltaic complexes corresponding to different densities of sediments (compare Figs. 9, 14 and 16). The appearance of deltas formed in Titanian environment may reflect the composition (i.e. if the sediments are composed of pure ice, hydrocarbons or some mixture of both). The better understanding of delta formation processes needs further numerical simulations, experimental investigations and more observational data from space missions. Moreover, detailed comparisons with analogous terrestrial processes could be helpful. For example, we hope to estimate the effect of different grain sizes responsible for creation of different delta types under Titanian conditions. In general, higher buoyancy of icy material on Titan results in larger grain size than in analogous delta type in terrestrial conditions. Better understanding of these processes will give scientists an important tool for interpretation of data from spacecraft and to explain evolution of rivers on Titan. In-situ measurements on Titan are (and will be) extremely limited and the majority of the information comes from the remote sensing. Comparison of observations with modelling can help constrain parameters and develop methods of interpretation.

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