Thermo-hydrodynamics of comet 67P/Churyumov–Gerasimenko’s atmosphere 

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Sublimation patterns on an ice bed

Different patterns on a sublimating surface, such as penitentes and ice waves, have been observed on Earth [60, 61], as in Fig. 1.7. Some of them also form (or are expected to form) on other planetary bodies [62–64]. These patterns are always found in specific environments, where the partial pressure of vapour (corresponding to the ice substance) in the atmosphere is low, sublimation therefore significantly contributes to ablation and takes a part in the development of patterns on the ice surface.
It is already known that penitentes form in sublimation conditions by differential ablation due to self-illumination, vapor diffusion and heat conduction [66, 67]. Regarding ice waves, the emergence and evolution can be described using a similar hydrodynamics as granular (snow or sand) dunes and ripples. They differ by the presence of particle transport in one case and the sublimation at the interface in the other. Recent studies have shows that complex interactions between sublimation-related mass transfer and turbulent flow in the lower-atmosphere lead to the development of stunning spiral-shaped topographic ice waves at the surface of the North Polar cap of Mars, and it is the periodic spatial variations in sublimation-related ablation rates that are responsible for the development of these topographic waves [68, 69]. So far, the genesis of these patterns is still not completely known, and further work is needed to understand the role of sublimation in the development of such wavy patterns, the relevant dynamical mechanisms controlling the mass balance, the dynamics at different scales and so on.
Another group of similar phenomena are dissolution patterns, such as scallops and icicle ripples, which always develops in limestone caves and in caves in ice, as well as other precipitation and dissolution interfaces [70–73]. These patterns result from the interaction of a soluble surface and an adjacent turbulent flow, and this is beyond the discussion in this thesis.

Fluid flow over the rippled patterns

All the patterns mentioned above are resulted from the interaction between substrates and fluid flows over the interfaces. A good understanding of the flow field near the patterns plays a key role in explaining the emergence and evolution of these patterns. As a general description, we briefly introduce here a two-dimensional incompressible flow over the interface, with x, z and ξ denote the flow direction, vertical direction and the interface profile, respectively. Following the standard separation between average quantities and fluctuating ones (denoted with a prime), the mean velocity field ui is governed by Reynolds averaged Navier Stokes equations: ∂i ui = 0, (1.8) ρ(∂t ui + uj ∂j ui ) = ∂j τi j − ∂i p, (1.9).
where τi j contains the deviatoric part of the Reynolds stress tensor −ρu′iu′j . The trace of the Reynolds stress tensor is included inside p. We use here a Prandtl-like first order turbulence closure in which the distance to the bed z − ξ determines the mixing length ℓ, and the mixing frequency is given by the strain rate modulus |γ˙| = γ˙i j γ˙i j , where we have introduced the strain rate tensor γ˙i j = ∂i uj + ∂j ui .
In the general case, we can write the stress tensor components as the sum of the viscous and turbulent contributions: τ xz τ x x τzz.
= ρ ℓ2|γ˙| + ν γ˙xz .
= ρ ℓ2|γ˙| + ν γ˙x x .
= ρ ℓ2|γ˙| + ν γ˙zz .

Theoretical modelling

The purpose of this section is to provide a brief but self-contained summary of the theoretical framework within which we analyze our experimental data. The theoretical description of the flow over a flexible sheet has been treated in a general way several decades ago, as e.g. summarized by Paidoussis [1], see chapter 10 and references therein. Here, we restrict this analysis to a two-dimensional linear perturbation theory. Furthermore, we hypothesize that the air flow can be decomposed into a turbulent inner boundary layer and an outer laminar flow which can be described as an incompressible perfect fluid. As we only need the pressure field, which is almost constant across the inner layer according to the boundary layer theory, we will simply describe the outer layer. Under these simplifying assumptions, we are able to derive analytical scaling laws for the frequency and the wavenumber of the most unstable mode in the asymptotic limit of either very flexible or very rigid sheets, and which were not available in the literature.

Governing equations

We consider a flexible sheet of infinite span and length submitted to an air flow along the x-axis. For later rescalings, we denote by V the characteristic velocity of the wind, i.e. the average air velocity at a given and fixed altitude zw (in the experiment zw ≃ 7.5 cm). Assuming that the motion of the sheet is independent of the coordinate y, we denote ζ(x, t) as its deflection with respect to the reference line z = 0. In the limit of small deflections with respect to a flat reference state, the sheet obeys the linearized Euler-Bernoulli beam equation: m ∂ 2ζ + D ∂ 4ζ + δp = 0, (2.1) ∂ t2 ∂ x4. with δp the air pressure jump across the sheet. m and D are respectively the mass per unit surface and the bending rigidity (Table 2.1).
Neglecting viscous stress in the outer layer and assuming incompressibility (recall that velocities are on the order of a few m/s, i.e. corresponding to very low Mach numbers), mass and momentum conservations for the flow field are therefore expressed by Euler equations: ∂ u ∇ · u = 0, 1 ∇p, (2.2) + (u · ∇) u = − (2.3) ∂ t ρ.
where u and p are the velocity and pressure fields, and ρ is the air density. Finally, the fluid velocity on the sheet should be equal to the sheet velocity, in order to ensure the impermeability of the sheet: dζ u(z = ζ) · n = dt , (2.4).

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Comet 67P/Churyumov–Gerasimenko and the Rosetta mission

67P/Churyumov–Gerasimenko (abbreviated as 67P) is a Jupiter-family comet, originally from the Kuiper belt. It was first observed on photographic plates in 1969 by Soviet astronomers Klim Ivanovych Churyumov and Svetlana Ivanovna Gerasimenko, after whom it is named. As in Fig. 3.1a, 67P has two lobes, a small one and a large one, with a thick neck connecting in between. The current orbit is shown in Fig. 3.1b, and the latest fly-by over the perihelion was on 13 August 2015. There are 19 distinct regions on 67P, with each named after an Egyptian deity [106]. Parameters of 67P and its current orbit are listed in Table 3.1.
67P was the destination of the European Space Agency (ESA) Rosetta mission (Fig. 3.1b). Rosetta is a space probe launched on 2 March 2004 from the Guiana Space Centre in French Guiana. On 6 August 2014, it reached 67P, performed a series of manoeuvres and entered orbit on 10 September 2014. Rosetta’s lander, Philae (Fig. 3.1c), touched down on its surface on 12 November 2014, becoming the first spacecraft to land on a comet nucleus. Along with Philae, Rosetta is performing detailed investigations of 67P, and the mission continues to return data from the spacecraft in orbit and from the lander in the comet’s surface as of 2015. On 30 September 2016, the Rosetta spacecraft ended its mission by landing on the comet in its Ma’at region.

Table of contents :

List of figures
List of tables
1 General introduction 
1.1 Flag flapping instability induced by wind
1.2 Granular patterns on an erodible bed
1.3 Sublimation patterns on an ice bed
1.4 Fluid flow over the rippled patterns
1.5 Outline of the thesis
I Travelling waves on highly flexible substrates 
2 Paper waves in the wind 
2.1 Introduction
2.2 Experimental study
2.2.1 Experimental setup
2.2.2 Experimental data
2.3 Theoretical modelling
2.3.1 Governing equations
2.3.2 Linearized problem
2.3.3 Dispersion relation
2.3.4 Asymptotic analysis and scaling laws
2.4 Comparisons with experiments
2.4.1 Selection of angular frequency and wavenumber
2.4.2 Finite amplitude effects
2.5 Results and Discussions
II Giant ripples on comet 67P/Churyumov-Gerasimenko 
3 Introduction 
3.1 Comet 67P/Churyumov–Gerasimenko and the Rosetta mission
3.2 Unexpected bedforms on the comet
3.3 Outline of the part
4 Thermo-hydrodynamics of comet 67P/Churyumov–Gerasimenko’s atmosphere 
4.1 Gravity
4.2 Thermal process of the comet’s nucleus
4.2.1 Thermal diffusion
4.2.2 Ice sublimation
4.3 Hydrodynamics of the comet’s atmosphere
4.3.1 Outer layer flow
4.3.2 Turbulent boundary layer
4.3.3 Porous sub-surface layer
4.4 Results and discussions
4.4.1 Temperatures, density and pressure
4.4.2 Vapour flux
4.4.3 Wind
4.5 A brief summary
5 Sediment transport 
5.1 Grain size
5.2 Transport threshold
5.2.1 Threshold velocity ut
5.2.2 Cohesion
5.2.3 A comparison of ut and u∗
5.2.4 Dependence of ut on d
5.3 Transport mode and saturated transport
5.3.1 Transport mode
5.3.2 Saturated transport flux qsat
5.3.3 Saturation length Lsat
5.4 A brief summary
6 The nature of the bedforms 
6.1 Dispersion relation
6.2 Most unstable mode
6.2.1 Wave length selection
6.2.2 Bedform growth and propagation
6.3 Conclusions
III Sublimation dunes on Pluto 
7 Introduction 
7.1 Pluto and the New Horizons mission
7.2 Rythmic patterns on the surface of Pluto
8 Pluto’s atmosphere 
8.1 A general description
8.2 Thermo-hydrodynamics of Pluto’s atmosphere
8.2.1 Thermal processes
8.2.2 Hydrodynamical description
8.3 A brief summary
9 Physical model of sublimation dunes 
9.1 Governing equations
9.1.1 Thermal processes
9.1.2 Hydrodynamics
9.1.3 Sublimation interface
9.1.4 Self-illumination on a modulated surface
9.2 Linearised problem
9.2.1 Base state
9.2.2 First order fields
9.2.3 Base state in a dimensionless form
9.2.4 Interfacial equations
9.2.5 Linearised system
9.3 Dispersion relation
9.3.1 Instability due to heat diffusion and convection
9.3.2 Instability due to solar radiation
9.3.3 Application to Pluto
9.4 Conclusions
IV Aeolian sand ripples 
10 Aeolian sand ripples instability 
10.1 Introduction
10.2 A simplified transport model
10.2.1 Hop length modulation
10.2.2 Flux modulation
10.3 A simplified model for bed evolution
10.4 Discussions
11 Conclusions and perspectives 
References

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