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[PPS07-P13] Improving the DISPH method for simulating planetary core formation
Keywords:planetary core formation, metal-silicate separation, SPH method, DISPH method, Rayleigh-Taylor instability
Introduction: No simulation methods of the planetary formation process have been established which can solve both accretion and differentiation processes simultaneously. In particular, only a few computational fluid dynamics simulations of global core-mantle differentiation processes (i.e., the planetary core formation) have been carried out.
The SPH (Smoothed Particle Hydrodynamics) method is widely used to simulate planetary accretion processes [e.g., Benz et al., 1986; Nakajima et al., 2021]. The popularity of the SPH method has led us to apply it to a global planetary core formation model. However, the standard SPH, which we shall refer to as SSPH, has a big problem that it cannot express physical phenomena correctly near material boundaries [Agertz et al., 2007]. To solve this problem, Saitoh and Makino (2013) and Hosono et al. (2013) proposed the Density-Independent SPH (DISPH) method. However, DISPH is still under development [Hosono et al., 2016; Takeyama et al., 2017; Hosono et al., 2019], and its treatments of boundaries and some thermodynamic quantities are not well suited for core-mantle differentiation processes.
This study attempts to improve the original DISPH in order to simulate global planetary core formation processes. Our aim is to address some problems of the original DISPH and to propose a way of solving them.
Method: The basic idea of SPH is that all the field quantities are smoothed with smooth functions centered on neighboring particles. However, in SSPH, unphysical phenomena occur near material boundaries due to the smoothing of the density, which should be discontinuous actually. As a solution to this problem, Saitoh and Makino (2013) proposed DISPH, which avoided using the density as a measure of the elemental volume. It succeeded in representing discontinuous material boundaries. On the other hand, we need to adjust DISPH for applying it to core-mantle differentiation processes. For example, we should rearrange (1) time-development equations of some thermodynamic quantities and (2) boundary treatments. The latter is important because the boundary treatments are known to have large influence on the calculation accuracy in SSPH [e.g., Bonet and Kulasegaram, 2002; Shao et al., 2012]. We have improved these points by (1) reconstructing the governing equations in the form appropriate for planetary interiors, and (2) applying a successful SSPH boundary treatment [Marrone et al., 2011; Asai et al., 2013] to DISPH. In addition, we propose a new treatment of the free-surface boundary for DISPH.
We use the improved DISPH to solve two-dimensional Rayleigh-Taylor Instability (RTI). First, we put heavy liquid metal on top of light liquid silicate (i.e., Magma Ocean). Then, we add a perturbation to the material boundary and run the code.
Result and Future Work: The figure shows the result for an initial perturbation of a horizontal wavenumber one. The color bar expresses the density. Heavier liquid metal sinks downward without mixing with lighter liquid silicate.
To summarize, we simulate two-dimensional RTI as an example of our improved DISPH using liquid metal and liquid silicate. In the near future, we will consider two-dimensional three-phase systems, including solid silicate, which is much viscous than the liquids. We will next extend them to three-dimensional systems of realistic global planetary core formation. In addition, we will develop a new algorithm to handle element-partitioning simultaneously.
The SPH (Smoothed Particle Hydrodynamics) method is widely used to simulate planetary accretion processes [e.g., Benz et al., 1986; Nakajima et al., 2021]. The popularity of the SPH method has led us to apply it to a global planetary core formation model. However, the standard SPH, which we shall refer to as SSPH, has a big problem that it cannot express physical phenomena correctly near material boundaries [Agertz et al., 2007]. To solve this problem, Saitoh and Makino (2013) and Hosono et al. (2013) proposed the Density-Independent SPH (DISPH) method. However, DISPH is still under development [Hosono et al., 2016; Takeyama et al., 2017; Hosono et al., 2019], and its treatments of boundaries and some thermodynamic quantities are not well suited for core-mantle differentiation processes.
This study attempts to improve the original DISPH in order to simulate global planetary core formation processes. Our aim is to address some problems of the original DISPH and to propose a way of solving them.
Method: The basic idea of SPH is that all the field quantities are smoothed with smooth functions centered on neighboring particles. However, in SSPH, unphysical phenomena occur near material boundaries due to the smoothing of the density, which should be discontinuous actually. As a solution to this problem, Saitoh and Makino (2013) proposed DISPH, which avoided using the density as a measure of the elemental volume. It succeeded in representing discontinuous material boundaries. On the other hand, we need to adjust DISPH for applying it to core-mantle differentiation processes. For example, we should rearrange (1) time-development equations of some thermodynamic quantities and (2) boundary treatments. The latter is important because the boundary treatments are known to have large influence on the calculation accuracy in SSPH [e.g., Bonet and Kulasegaram, 2002; Shao et al., 2012]. We have improved these points by (1) reconstructing the governing equations in the form appropriate for planetary interiors, and (2) applying a successful SSPH boundary treatment [Marrone et al., 2011; Asai et al., 2013] to DISPH. In addition, we propose a new treatment of the free-surface boundary for DISPH.
We use the improved DISPH to solve two-dimensional Rayleigh-Taylor Instability (RTI). First, we put heavy liquid metal on top of light liquid silicate (i.e., Magma Ocean). Then, we add a perturbation to the material boundary and run the code.
Result and Future Work: The figure shows the result for an initial perturbation of a horizontal wavenumber one. The color bar expresses the density. Heavier liquid metal sinks downward without mixing with lighter liquid silicate.
To summarize, we simulate two-dimensional RTI as an example of our improved DISPH using liquid metal and liquid silicate. In the near future, we will consider two-dimensional three-phase systems, including solid silicate, which is much viscous than the liquids. We will next extend them to three-dimensional systems of realistic global planetary core formation. In addition, we will develop a new algorithm to handle element-partitioning simultaneously.