Effects of a long lived global magma ocean on mantle dynamics of the early Moon

2021 ◽  
Author(s):  
Adrien Morison ◽  
Stephane Labrosse ◽  
Daniela Bolrao ◽  
Antoine Rozel ◽  
Maxim Ballmer ◽  
...  
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Fractional Crystallization ◽  
Thermal Evolution ◽  
Classical Case ◽  
Magma Ocean ◽  
Viscous Forces ◽  
Solid Part ◽  
Convection Patterns ◽  
Magma Oceans

<p>The light plagioclase-enriched crust as well as the KREEP layer at the surface of the Moon are believed to be remnants of the bottom-up crystallization of a global Lunar Magma Ocean.  In such a setup, the primitive Lunar solid mantle is coated by a liquid magma ocean of similar composition. We propose here to study the dynamic and evolution of the primitive Lunar solid mantle, accounting for the presence of the Lunar Magma Ocean.</p><p>We solve numerically the equations of solid-state convection in the solid part of the mantle.  This model is coupled to 1D models of crystallization of the magma oceans to self-consistently compute the thickening of the solid part as heat is evacuated from the mantle.  We take into account fractional crystallization at the freezing front.</p><p>Moreover, the boundaries between the solid and the magma oceans are phase-change interfaces.  Convecting matter in the solid arriving near the boundary or getting away from it forms a topography which can be erased by melting or freezing.  Hence, provided the melting and freezing occurs rapidly compared to the time needed to build the topographies by viscous forces, dynamical exchange of matter can occur between the solid mantle and the magma oceans.  We take this effect into account in our model with a boundary condition applied to the solid.</p><p>We find that the boundary condition allowing matter to cross the interfaces between the solid and the magma oceans greatly affects the convection patterns in the solid as well as its heat flux.  Larger-scale convection patterns are selected compared to the classical case with non-penetrative boundary conditions; and the heat transfert in the solid is more efficient with these boundary conditions.  This affects the long term thermal evolution of the mantle as well as the shape of chemical heterogeneities that can be built by fractional crystallization of magma oceans.</p>

Characteristics of an hybrid atmosphere with disk-captured and degassing contributions over a rocky planet’s magma ocean. A modeling approach.

2020 ◽  
Author(s):  
Athanasia Nikolaou ◽  
Lorenzo Mugnai ◽  
Oliver Herbort ◽  
Enzo Pascale ◽  
Peter Woitke
Keyword(s):  
Thermal Evolution ◽  
Chemical Characterization ◽  
Protoplanetary Disk ◽  
International Institute ◽  
Atmospheric Conditions ◽  
Magma Ocean ◽  
Silicate Mineral ◽  
Chemical Abundances ◽  
Magma Oceans ◽  
The University

<p>Motivation:<br />   Early during their formation the planets capture an amount of atmosphere from the protoplanetary disk (Ikoma et al. 2018, Odert et al. 2018, Lammer et al. 2020, Kimura and Ikoma 2020). An additional proportion of their atmosphere is provided during the magma ocean stage by interior degassing. The latter mechanism is assumed to be the main provider of the final atmospheric mass. Its composition is compromised by the source silicate mineral and its chemical characterization (Gaillard and Scaillet 2014, Herbort et al. 2020).<br />   Numerous studies support the degassing of the oxidized gas species H<sub>2</sub>O and CO<sub>2</sub> as main contributions from the magma ocean phase (Abe and Matsui 1988, Abe 1993, Elkins-Tanton 2008, Schaefer et al. 2012, Lebrun et al. 2013, Lupu et al. 2014, Gaillard and Scaillet 2014, Salvador et al. 2017, Nikolaou et al. 2019). Previous work has also shown that H<sub>2</sub>O, in particular, plays a crucial role (Hamano et al. 2013, Katyal et al. 2019, Turbet et al. 2019) in thermal blanketing. H<sub>2</sub>O possibly leads to “long-term” (Hamano et al 2013) or “conditionally continuous” (Nikolaou et al. 2019) magma oceans that effectively cease to cool. Water also ties directly to the availability of hydrogen that drives hydrodynamic escape (Airapetian et al. 2017, Lammer et al. 2018). CO<sub>2 </sub>factors into both above processes, as well (Wordsworth and Pierrehumbert 2013, Odert et al. 2018). Constraining the H<sub>2</sub>O and CO<sub>2</sub> abundances early after formation is indispensible to the planet’s thermal evolution and extensive modeling effort has been devoted to it. Their constraint would in particular help revisit which magma ocean types among transient-conditionally continuous-permanent (Nikolaou et al. 2019) are detectable in future exoplanetary missions (ARIEL, Tinetti et al. 2018; PLATO, Rauer et al. 2014).<br /> </p> <p>Method:<br />   In this work we focus on the combination of degassed and disk-captured atmosphere under the assumption of chemical equilibrium. Using simulations from the 1D Convective Ocean of Magma Radiative Atmosphere and Degassing model (Nikolaou et al. 2019) we obtain the thermal evolution and degassing tracks of a rocky planet. In order to evaluate the chemical abundances under equilibrium conditions we employ the thermodynamical model GGchem (Woitke et al. 2018).<br />   We explore the atmospheric conditions during the lifetime of a magma ocean under varying mineral compositions and protoplanetary disk contributions. We discuss the results in the context of the likely magma ocean types.<br /> <br />A.N. and P.W. wish to thank the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) of the University of Vienna, Thematic Programme on “Astrophysical Origins: Pathways from Star Formation to Habitable Planets” 2019, which enabled this collaboration.</p>

Towards 3D modelling of convection in planetesimals and meteorite parent bodies

2019 ◽  
Vol 490 (1) ◽  
pp. L47-L51 ◽  
Author(s):  
Wladimir Neumann
Keyword(s):  
Solid State ◽  
Thermal Evolution ◽  
Asteroid Belt ◽  
Magma Ocean ◽  
Early Solar System ◽  
Conduction Model ◽  
Time Requirements ◽  
Convection Model ◽  
Magma Oceans ◽  
Free Material

ABSTRACT Observations of asteroid belt members, investigations of meteorites and thermal evolution models converge on the paradigm of the ubiquity of melting processes in the planetesimals of the early Solar system. At least partial melting of planetesimals that fulfilled size and accretion time requirements to surpass the solidus temperatures of metal and silicates led to the weakening of the rock due to the interstitial melt. A decrease of the viscosity relative to melt-free material facilitates solid-state convection on partially molten bodies. Additional melting can produce liquid-like layers with suspended particles, i.e. magma oceans. Thermal evolution models indicate that partially molten layers can occur in the interior of undifferentiated bodies and in silicate mantles of differentiated ones. They can exist before a magma ocean forms or after it solidifies and above a whole-mantle magma ocean or below a shallow magma ocean. Thus, convection is likely. Attempts to model and to quantify the effects of convection in planetesimals remain rare. This study discusses the possibility of solid-state convection in partially molten planetesimals, presents a first-order comparison of a 3D mantle convection model with a conduction model taking a Vesta-sized body as an example and illustrates the importance of convection for meteorite parent bodies.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

scholarly journals Stiffness Estimation and Equivalence of Boundary Conditions in FEM Models

2021 ◽  
Vol 11 (4) ◽  
pp. 1482
Author(s):  
Róbert Huňady ◽  
Pavol Lengvarský ◽  
Peter Pavelka ◽  
Adam Kaľavský ◽  
Jakub Mlotek
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Boundary Conditions ◽  
Model Updating ◽  
Element Model ◽  
Computational Time ◽  
Stiffness Coefficients ◽  
First Case ◽  
Numerical Approaches

The paper deals with methods of equivalence of boundary conditions in finite element models that are based on finite element model updating technique. The proposed methods are based on the determination of the stiffness parameters in the section plate or region, where the boundary condition or the removed part of the model is replaced by the bushing connector. Two methods for determining its elastic properties are described. In the first case, the stiffness coefficients are determined by a series of static finite element analyses that are used to obtain the response of the removed part to the six basic types of loads. The second method is a combination of experimental and numerical approaches. The natural frequencies obtained by the measurement are used in finite element (FE) optimization, in which the response of the model is tuned by changing the stiffness coefficients of the bushing. Both methods provide a good estimate of the stiffness at the region where the model is replaced by an equivalent boundary condition. This increases the accuracy of the numerical model and also saves computational time and capacity due to element reduction.

scholarly journals Bootstrapping boundary-localized interactions

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Connor Behan ◽  
Lorenzo Di Pietro ◽  
Edoardo Lauria ◽  
Balt C. van Rees
Keyword(s):  
Boundary Condition ◽  
Scalar Field ◽  
Boundary Conditions ◽  
Parameter Space ◽  
Three Dimensional ◽  
Conformal Boundary ◽  
Dimensional Boundary ◽  
Dirichlet And Neumann Conditions ◽  
Boundary Fields ◽  
Boundary Dynamics

Abstract We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a ‘shadow pair’ of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a ‘kink’ in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

scholarly journals An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

2020 ◽  
Vol 54 (4) ◽  
pp. 1373-1413 ◽  
Author(s):  
Huaiqian You ◽  
XinYang Lu ◽  
Nathaniel Task ◽  
Yue Yu
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Nonlocal Diffusion ◽  
Order Convergence ◽  
Flux Boundary Condition ◽  
Nonlocal Interactions ◽  
Local Case ◽  
Neumann Type ◽  
Type Boundary

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

Chaotic Vibration of a Two-dimensional Non-strictly Hyperbolic Equation

2018 ◽  
Vol 61 (4) ◽  
pp. 768-786 ◽  
Author(s):  
Liangliang Li ◽  
Jing Tian ◽  
Goong Chen
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Method Of Characteristics ◽  
Nonlinear Boundary ◽  
Chaotic Oscillation ◽  
Nonlinear Boundary Conditions ◽  
Rigorous Proof ◽  
Two Dimensional ◽  
Chaotic Vibration

AbstractThe study of chaotic vibration for multidimensional PDEs due to nonlinear boundary conditions is challenging. In this paper, we mainly investigate the chaotic oscillation of a two-dimensional non-strictly hyperbolic equation due to an energy-injecting boundary condition and a distributed self-regulating boundary condition. By using the method of characteristics, we give a rigorous proof of the onset of the chaotic vibration phenomenon of the zD non-strictly hyperbolic equation. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs. Numerical simulations are also provided.

NEW MODELS IN MICROPOLAR FLUID AND THEIR APPLICATION TO LUBRICATION

2005 ◽  
Vol 15 (03) ◽  
pp. 343-374 ◽  
Author(s):  
GUY BAYADA ◽  
NADIA BENHABOUCHA ◽  
MICHÈLE CHAMBAT
Keyword(s):  
Boundary Condition ◽  
Friction Coefficient ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Micropolar Fluid ◽  
Reynolds Equation ◽  
Slip Boundary Condition ◽  
Slip Boundary ◽  
New Models ◽  
Uniqueness Of The Solution

A thin micropolar fluid with new boundary conditions at the fluid-solid interface, linking the velocity and the microrotation by introducing a so-called "boundary viscosity" is presented. The existence and uniqueness of the solution is proved and, by way of asymptotic analysis, a generalized micropolar Reynolds equation is derived. Numerical results show the influence of the new boundary conditions for the load and the friction coefficient. Comparisons are made with other works retaining a no slip boundary condition.

Application of Generalized Differential Quadrature Method to the Bending of Thick Laminated Plates with Various Boundary Conditions

2006 ◽  
Vol 5-6 ◽  
pp. 407-414 ◽  
Author(s):  
Mohammad Mohammadi Aghdam ◽  
M.R.N. Farahani ◽  
M. Dashty ◽  
S.M. Rezaei Niya
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Simple Procedure ◽  
Shear Deformation Theory ◽  
Laminated Plates ◽  
Differential Quadrature ◽  
Governing Equations ◽  
First Order ◽  
Various Boundary Conditions ◽  
Generalized Differential

Bending analysis of thick laminated rectangular plates with various boundary conditions is presented using Generalized Differential Quadrature (GDQ) method. Based on the Reissner first order shear deformation theory, the governing equations include a system of eight first order partial differential equations in terms of unknown displacements, forces and moments. Presence of all plate variables in the governing equations provide a simple procedure to satisfy different boundary condition during application of GDQ method to obtain accurate results with relatively small number of grid points even for plates with free edges .Illustrative examples including various combinations of clamped, simply supported and free boundary condition are given to demonstrate the accuracy and convergence of the presented GDQ technique. Results are compared with other analytical and finite element predictions and show reasonably good agreement.

Conduction of heat in a solid with periodic boundary conditions, with an application to the surface temperature of the moon

1953 ◽  
Vol 49 (2) ◽  
pp. 355-359 ◽  
Author(s):  
J. C. Jaeger
Keyword(s):  
Boundary Condition ◽  
Thermal Properties ◽  
Circular Cylinder ◽  
Boundary Conditions ◽  
Surface Temperature ◽  
Numerical Method ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
The Moon ◽  
Non Linear

The object of this note is to indicate a numerical method for finding periodic solutions of a number of important problems in conduction of heat in which the boundary conditions are periodic in the time and may be linear or non-linear. One example is that of a circular cylinder which is heated by friction along the generators through a rotating arc of its circumference, the remainder of the surface being kept at constant temperature; here the boundary conditions are linear but mixed. Another example, which will be discussed in detail below, is that of the variation of the surface temperature of the moon during a lunation; in this case the boundary condition is non-linear. In all cases the thermal properties of the solid will be assumed to be independent of temperature. Only the semi-infinite solid will be considered here, though the method applies equally well to other cases.

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