Influence of Robin Boundary Condition on Thermal Rotating Convection

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Differences between FF and FT cases with respect to Ekman number

We determine the critical Rayleigh number and azimuthal wave number (when it is relevant) at the linear onset of convection for different rotation rates (Ek = +∞ to 10−6), in both FF and FT end-member cases. We checked that the symmetric modes (with respect to the equatorial plane) are always the preferred unstable modes at the linear onset of convection (i.e.their RaC is the smallest).
In fig. 2.2 we plot RaC as a function of Ek, for different values of Pr, in order to illustrate the existence of two regimes (expected from previous studies, e.g. Gastine et al. (2016)): in the first one (Ek → +∞) rotation does not affect RaC while this number increases when Ek decreases in the second regime. For small enough Ek, the expected power law RaC ∝ Ek is observed (Chandrasekhar, 1953). Note that, although RaC is a relevant criterion to discriminate FF and FT setups at high values of the Ekman number, it is no longer the case when rotation impacts notably the critical Rayleigh number. The effect of the Prandtl number is contrasted: the general behaviour of RaC as a function of Ek is not affected by the value of Pr but its influence on the exact value of RaC depends on the Ekman number. For Ek ≥ 10−1, Pr has no visible effect since cases are differentiated only by their boundary conditions. This is in line with the classical behaviour of non-rotating Rayleigh-Bénard convection for which the onset of convection is known to be independent of the value of the Prandtl number (e.g. Chandrasekhar, 1961). On the other hand, for Ek ≤ 10−3, the Pr = 0.3 cases become distinct from the cases with larger values of Pr (Pr = 3 or 30).
Let us now turn to the azimuthal structure of the flow at the onset of convection. Fig. 2.3 shows the critical value of the azimuthal number m for the most unstable mode for both boundary conditions as a function of the Ekman number. We observe no difference between FF and FT configurations for high rotation rates: in the limit of small values of Ek, m scales as Ek−1/3, as expected (Busse, 1970). For large values of Ek, the critical value of m tends toward a constant. In the case of Bi = 0, the transition to a constant value of m happens sharply at around Ek = 2 × 10−3. This particularity may be compared to the transition described by Falsaperla et al. (2010) at Ek ‘ 10−2.
We should also mention that in the non-rotating case, Ek = ∞, the m number is not relevant. Indeed in a non rotating setup, the most unstable mode is degenerated in terms of m number. Nevertheless, in that case, the critical degree l can be determined and, with our chosen setup, is equal to 5 for a FF boundary condition and 6 for a FT one.

Transition from FF to FT owing to Robin boundary condition

We determine the critical Rayleigh number for different values of the Biot number with various Ekman numbers in the same range as formerly. Increasing Bi from 0 to +1 tunes the boundary condition from a fixed flux upper boundary condition to a fixed temperature one (FF to FT transition). We can reasonably expect that the amplitude of this transition is limited by the extremal cases described in section 2.6.1. Since the convective setup can evolve continuously from FF to FT configuration thanks to the Robin boundary condition, we expect (as the simplest hypothesis) significant influence of this condition on the onset only in the first regime (high Ekman numbers), in which a large RaC difference is observed.
In order to determine the evolution of the critical Rayleigh number with respect to the Biot number, we plot on fig. 2.5, for a given Ekman number, its normalised values, gRaC = RaC(Bi) RaC(Bi = 1). (2.13).
A continuous and monotonous transition can be observed in terms of the critical Rayleigh number. Without rotation, the effect is the largest, with a minimum ratio of 0.5 (RaC(FF) = RaC(FT)/2). As expected from the previous section, the amplitude of the transition is reduced when the Ekman number is decreased. Finally, most of the transition occurs for Bi 2 [0.03, 30].

Possible dependence on the Prandtl number

We have shown in fig. 2.4, that the relative difference of RaC measured by the quantity , decreases when Ek decreases. This general behaviour is not substantially affected by the variation of Pr in the range 0.3 Pr 30 that we investigated. In order to discuss the robustness of our observations concerning the effect of Bi against variations of the Prandtl number, we studied the FF to FT transition in a nonrotating case for Pr = 0.3 and 30. A representation of the transition in terms of critical Rayleigh number is plotted in fig. 2.8. We can note that the transitional regime is almost independent from Pr. In particular, the range of Biot number over which the transition from FF to FT occurs is 0.3 Bi 30.

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Behaviour of purely FT or FF setups Global diagnostics

First of all, we study end-member configurations in terms of Bi values, that is Bi = 0 or Bi = 1, for rotating (Ek = 10−4) or non-rotating cases, for Pr = 0.3 or 3. The Rayleigh number has been set between Ra RaC and Ra/RaC 103. As expected from numerous previous studies (e.g. Gastine et al., 2016; Long et al., 2020), several successive convection regimes are observable when Ra increases above RaC.
We first consider Nu = f(Ra) as represented on fig. 2.10. We observe the well known convective behaviour (e.g. King et al. (2012) or Gastine et al. (2016)): an increase of Nu with Ra starting from Nu = 1 for Ra RaC. Rotating cases show a higher RaC and a smaller Nu at a given Ra compared to the non-rotating cases. Since the slope of Nu = f(Ra, Ek 6= 1) is also steeper (it can approach 1/2 for Ra 107) rotating and non-rotating cases show increasingly similar Nu when Ra is increased. We do not reach large enough values of Ra to see the convergence of moderately rotating and nonrotating cases that has been documented (Gastine et al., 2015, 2016; King et al., 2012).
Indeed the convective Rossby number, Roc = (RaEk2 Pr )1/2, used to compare a priori the Coriolis force to the buoyancy force Gilman (1977)) of Ek = 10−4 cases is always below 10 (see appendix A.2.2). Nevertheless, for weakly rotating cases, this convergence is well observed for Ra > 106. The comparison of FT and FF cases on the basis of Nu values at given Ra shows that the two configurations almost always show distinct behaviour. Generally speaking Nu(FF) < Nu(FT). The only exception is noted for the Ek = 10−4 cases when Ra RaC which shows no real difference between FF and FT configurations.
This agrees with the fact that the RaC of these cases were found with the linear stability analysis to be very similar. Indeed, it is expected (Busse and Or, 1986; Gillet and Jones, 2006) that, near the threshold, the behaviour of the system is controlled by Ra RaC −1. In the non-rotating case, for large enough Ra, it is known that the Nusselt number scales as Ra where is expected to be similar to 1/3 (1/3 in Plumley and Julien (2019) for example or 2/7 are both frequently used, see for example Gastine et al. (2015) or Iyer et al. (2020) for FT configuration and Long et al. (2020), King et al. (2012) or Johnston and Doering (2009) for symmetric FF one). Here, we observe = 2/7. This behaviour is well observed in our data set for the FT configuration. Nevertheless, a different exponent (2/9 here, as justified below) seems to be necessary to model the FF data. All these general observations can be applied to both Pr = 0.3 and Pr = 3 cases.

Table of contents :

1 Introduction 
1.1 Une brève histoire du concept d’océan de magma
1.1.1 Premiers modèles physiques de la naissance du système solaire
1.1.2 Le refroidissement de la Terre selon Buffon
1.1.3 Une Terre primitive liquide ?
1.1.4 L’apport des roches lunaires
1.1.5 Impact géant et océan de magma lunaire
1.1.6 Quelques modélisations géophysiques de l’océan de magma terrestre
1.1.7 Des témoignages géochimiques plus ambigus
1.1.8 Bilan
1.2 Enjeux, motivations et méthodes
1.2.1 Des océans de magma bien mélangés : le modèle standard
1.2.2 Impact de la cristallisation fractionnée et motivation de notre
1.2.3 Le choix d’une approche hydrodynamique
1.2.4 L’adimensionnement du problème, une démarche préalable indispensable
1.2.5 Bilan
2 Influence of Robin Boundary Condition on Thermal Rotating Convection
2.1 Résumé
2.2 Abstract
2.3 Erratum
2.4 Introduction
2.5 Methods
2.5.1 Physical System
2.5.2 Numerical Tools
2.6 Linear Stability Analysis
2.6.1 Differences between FF and FT cases with respect to Ekman number
2.6.2 Transition from FF to FT owing to Robin boundary condition .
2.6.3 Possible dependence on the Prandtl number
2.7 Non Linear Simulations
2.7.1 Behaviour of purely FT or FF setups
2.7.2 FF to FT transition
2.8 Discussion
3 Thermo-Solutal Convection in Magma Oceans 
3.1 Résumé
3.2 Abstract
3.3 Introduction
3.3.1 The Concept of Magma Ocean
3.3.2 Terrestrial Case
3.3.3 Thermal Convection & Partial Crystallization
3.3.4 Thermo-solutal Convection
3.4 Methods
3.4.1 Physical System
3.4.2 Reference Fields
3.4.3 Equations of Conservation
3.4.4 Boundary Conditions
3.4.5 Parameters
3.4.6 Numerical Tools & Diagnoses
3.5 Heat and Mass Transport Regimes Observed
3.5.1 Fully Convective Regime
3.5.2 Erosion Regime
3.5.3 Enduring Stratification Regime
3.6 Regimes Diagram
3.6.1 Dependence with other parameters
3.6.2 Dependence on initial conditions
3.7 Discussion
4 Conclusion 
4.1 Résumé
4.2 Perspectives


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