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Asymptotic analysis and scaling laws
Scaling laws for the characteristics of the most unstable mode (¯km, ¯ωm, ¯σm) as well as for the cut-off wavenumber ¯kc can be analytically derived in the limits of asymptotically small and large ¯D . ¯kc and ¯km are calculated from (2.24) and (2.27), respectively. ¯ω m and ¯σm are obtained by introducing ¯km into Eqs. 2.25,2.26. Expanding these equations in the limit.
Selection of angular frequency and wavenumber
Considering the experimental parameters (see Tab. 2.1 and typical values in Section 2.2), the dimensionless rigidity lies in the range 10−3 – 10−2. For the analysis of the experimental data, we shall then make use of the scaling laws (2.28) and (2.30) obtained in the limit of small ¯D . Introducing back physical dimensions in these expressions, we obtain ωm ∼ 2ρ m V, (2.32) km ∼ ρ 2D 1/3 V2/3. (2.33)
The selected angular frequency purely results from the balance between dynamic pressure and inertia. The selected wavenumber results from the balance between dynamic pressure and elasticity. It is interesting to compare the phase velocity ωm/km with that of the elastic waves in the absence of wind flow. In the latter case, Eq. 2.1 tells us that the dispersion relation is simply ω = p D/mk2, which corresponds to a velocity, evaluated at the most unstable wavenumber, p D/mkm. This scales as km with V2/3, whereas ωm/km is here predicted to be proportional to V1/3. Our main goal is the experimental check of these scaling laws of ωm and km with the wind velocity V.
Finite amplitude effects
Although the results displayed in Fig. 2.7 show a good agreement of the selection of angular frequency and wavenumber in the experiments with the prediction of the linear stability analysis of the problem, we have also found some evidence for finite amplitude effects. Focusing on the paper material, we have systematically varied the sheet length. Data corresponding to different values of ΔL are displayed in Fig. 2.8, showing ω and k in the same rescaled way as in Fig. 2.7. The scaling law obeyed by the angular frequency is found to be independent of the sheet length, whereas that of the wavenumber shows small but systematic variations with ΔL. This shows the presence of non-linearities that are not described here, the linear regime corresponding to the limit of vanishing ΔL.
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.
Temperatures, density and pressure
The calculated results show both seasonal and diurnal time variations of the atmosphere characteristics. At perihelion, the vapor temperature peaks around 200K at the surface of the comet (Fig. 4.2), and the corresponding thermal velocity around 500 m/s, which is much larger than the escape velocity. The vapour mean free path ℓ is about 3 cm at the surface of the comet (Fig. 4.3). As ℓ is significantly smaller than the bedform emergent wavelength λ ≃ 5 m (Fig. 3.2a), hydrodynamics accurately describes the flow above relief.
The vapor density is around 10 times larger than the previous estimations due to the presence of the granular porous layer. As we can see, there exists an asymmetry between sunrise and sunset for temperature, density as well as the pressure (Fig. 4.4), which is simply results from thermal inertia, as some heat is accumulated in the superficial layer during in the morning and released in the afternoon.
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
Table of contents xi
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
Appendix A Measuring apparatus
List of symbols
