Magnetohydrodynamic Modeling of the Solar Corona and Heliosphere

In this chapter, we review the status of the art of the three-dimensional (3D) time-dependent magnetohydrodynamic (MHD) models of solar wind. We first highlight the influence of adverse space weather conditions by presenting the Halloween Sun-Earth events and their associated effects in the terrestrial space and point out the importance of 3D physics-based MHD models in space weather forecast. Then we summarize three well-known frameworks, including the architectures, the characteristics of the numerical schemes, and the applications in modeling the background solar wind and solar disturbances’ propagation in interplanetary space. The three architectural MHD model frameworks are CORona-HELiosphere (CORHEL), Space Weather Modeling Framework (SWMF) and Space Weather Integrated Model (SWIM). These models are developed respectively by the Center for Integrated Space Weather Modeling (CISM), the Center for Space Environment Modeling (CSEM), and the Solar-Interplanetary-GeoMAgnetic (SIGMA) Weather Group of the State Key Laboratory for Space Weather, National Space Science Center, Chinese Academy of Sciences. Next we give a brief description of hybrid empirical-MHD models for solar-interplanetary modeling, each of which is comprised of a theoretic, empirical or kinematic solar coronal model and a 3D MHD heliospheric model. Then concise summarizations are presented for other miscellaneous models for solar-interplanetary modeling. For completeness, 3D MHD models in studying the interaction between the solar wind and the magnetosphere and the ionosphere are also presented in this chapter. Finally, brief concluding remarks are given in the last section.

This chapter is devoted to the description of finite volume method (FVM). FVM is a discretization technique for partial differential equations, especially those that arise from physical conservation laws. FVM uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations. Currently, FVM is a common practice for discretizing computational fluid/MHD dynamics equations. Here we devote our attention to the cell-centered FVM in three-dimensional computational domain. We begin with a general introduction to the first order finite volume methods, followed by a description of high-resolution formulations for conservative laws. It is mentioned that all the formulations are aimed to be applicable at least to the hexahedral cell, particularly quadrilaterally-faced hexahedron (cuboid) with 6 faces, 12 edges, and 8 vertices.

A second-order Godunov-type finite volume method (FVM) to advance the equations of single-fluid solar wind plasma magnetohydrodynamics (MHD) in time has been implemented into a numerical code. This code operates on a three-dimensional (3D) spherical shell with both non-staggered and staggered grids on the overlapping grid system with hexahedral cells of quadrilateral frustum type. By merging geometrical factors in spherical coordinates into the reformulation of fluxes, flux evaluation is made easy to achieve, and thus many numerical schemes with the total variation diminishing (TVD) slope limiters and approximate Roe solvers intended for Cartesian case can follow in the present context of spherical grid described here. At the same time, alternative strategies to ensure a solenoidal magnetic field, such as projection Poisson (PP) solver, hyperbolic divergence cleaning (HDC) method derived from generalized Lagrange multiplier (GLM) formulation of MHD system and constrained transport (CT) method, are employed. In this chapter, an FVM is described exemplarily on a six-component composite grid system by using a minmod limiter for oscillation control. Additionally, an implicit dual time-stepping technique is demonstrated to model the steady state solar wind ambient. Being of second order in space and time, this model is written in FORTRAN language with Message Passing Interface (MPI) parallelization, and validated in modeling the large-scale structure of solar wind from the Sun to Earth process (hereafter called Sun-to-Earth Process MHD model, also STEP-MHD model for brief). To demonstrate the suitability of our code for the simulation of solar wind ambient from the Sun to Earth, selected results from Carrington rotations (CR) during different solar activity phases are presented to show its capability of producing structured solar wind in agreement with observations.

This chapter describes the 3D Solar-Interplanetary space-time Conservation Element and Solution Element (SIP-CESE) MHD model and its application on the solar wind study. SIP-CESE MHD model is established on pentahedral cells with each cell a triangular prism composed of two triangular bases and three rectangular sides. The basic principle as well as the implementation detail of the CESE numerical scheme are presented. Two examples are given to illustrate the validity and capacity of modeling solar wind: (i) two-dimensional coronal dynamical structure with multipole magnetic fields and (ii) three-dimensional coronal dynamical structure, using measured solar surface magnetic fields and the empirical values of the plasma properties on the solar surface as the initial conditions for the set of MHD equations, and then the relaxation method to achieve a quasi-steady state. These examples show that the MHD model possesses the ability to model the Sun-Earth environment. Finally, the evolution of the large-scale coronal magnetic structure during solar cycle 23 is investigated by the SIP-CESE MHD model.

This chapter introduces the implementation of the SIP-CESE MHD model in six component overset grid system with the aim of mitigating the problem of singularity and mesh convergence near the poles. The SIP-CESE MHD model introduced in Chap. 4 is applied to solar wind simulation with this grid system. New numerical features are also added, including: (1) The \(\nabla \cdot \mathbf {B}\) constraint error cleaning procedure via an easy-to-use fast multigrid Poisson solver, (2) The Courant-number-insensitive method that reduces the numerical viscosity without generating any instability, (3) The time-integration by multiple time stepping, (4) The time-dependent boundary condition at the subsonic region by limiting the mass flux escaping through the solar surface. In order to produce fast and slow plasma streams of the solar wind, the volumetric heating and momentum source terms are included by considering the topological effect of the magnetic field expansion factor \(f_s\) and the minimum angular distance \(\theta _b\) (at the photosphere) between an open field foot point and its nearest coronal hole boundary. The simulation result for the 3D steady-state background solar wind during Carrington rotation (CR) 1911 from the Sun to Earth is presented. The good agreements between numerical result and observations, such as the Large Angle and Spectrometric Coronagraph (LASCO) aboard the Solar Heliospheric Observatory satellite (SOHO) in the corona and Wind observations at 1 AU, demonstrated the efficiency, accuracy and the ability to reasonably produce the structured solar wind of the model.

Coronal-heliospheric space is characterized by disparate temporal and spatial scales as well as by different relevant physics in different domains such as the corona and the inner heliosphere. The scalable, massively parallel, block-based, adaptive-mesh refinement (AMR) allows resolving such disparate spatial and temporal scales throughout the computational domain with even less cells but can generate a necessary resolution. This chapter is devoted to the adaptive mesh refinement (AMR) implementation of Solar-Interplanetary space-time conservation element and solution element (CESE) magnetohydrodynamic model (SIP-CESE MHD model) with the aid of the parallel AMR package PARAMESH. Two AMR realization strategies are employed: one uses a solution adaptive technique directly for the CESE solver in the six-component grid system introduced in Chap. 5, while the other is implemented for the CESE solver of associated partial differential equations (PDEs) in the reference space of curvilinear coordinates transformed from the original governing partial differential equations in the physical space. Under these two AMR implementations, tests are carried out for the solar wind background study, and numerical results are compared with the observations in the solar corona and in interplanetary space from the Solar and Heliospheric Observatory (SOHO) and spacecraft data from OMNI.

Solar wind is rather than of steady state, but physically dynamic corresponding to the solar rotation, solar mass flow and solar magnetic field evolution. In fact, solar wind is constantly time varying. Data-driven modeling of solar wind here means to use continuously time-varying solar observations as input to drive models to produce solar wind background and reproduce eruptive process of solar active region, instead of using one instantaneous cadence of observation as input. The latter is in contrast called data-constrained modeling. The idea of data-driven modeling is to let continuously observed data and MHD evolve to produce what is happening next. This chapter describes how to use the global photospheric field maps generated by the solar surface flux Transport (SFT) model for long-interval synoptic data products or time interpolation for short-interval synoptic data products as input to drive the CESE-MHD coronal and solar wind model.

Solar corona is the source region of many explosive events, e.g., flares and CMEs, which play a leading role in causing adverse space weather. All the solar eruptions are energized from the magnetic fields, which is generated below the photosphere and transported into the upper atmosphere, i.e., the corona. Consequently, the coronal magnetic field holds the key to understanding the origin of the energetic events. Due to the absence of direct measurement, the three-dimensional magnetic field in the solar corona is usually “extrapolated” in numerical ways from the photosphere where a two-dimensional surface of magnetic vectors can be routinely measured. Up to the present, the nonlinear force-free field (NLFFF) model dominates the physical models for magnetic field extrapolation in the low corona. This chapter briefly reviews the present methods for NLFFF extrapolation with emphasis on the CESE–MHD–NLFFF code.

Solar corona is always in dynamical evolution as driven by the surface motions on the photosphere, which stress the coronal field by shearing, twisting and even braiding in a somewhat random way. Such processes inject continuously magnetic free energy and helicity into the corona, which, however, cannot be reserved forever. As a result, solar eruptions occur from time to time, like a volcanic eruption, which explosively releases the magnetic free energy into other forms like kinetic and thermal energies. As such dynamical process is not included in the force-free field extrapolation models, MHD model, in particular, constrained and even driven by the observed magnetograms, is required to follow the coronal dynamic evolution. Such data-driven MHD models have been recently developed and applied to study the solar active-region evolution as well as its eruption, which offers a new way for understanding sophisticatedly their underlying magnetic topology and mechanism. This chapter briefly presents data-driven CESE–MHD modeling of coronal magnetic evolutions and eruptions.

Coronal mass ejections (CMEs) consist of large-scale eruptions of magnetized plasma from the Sun, and they are considered to be the major drivers of adverse space weather disturbances that strongly affect our high-tech activities of humankind. Even the fastest CMEs require almost a day to arrive at Earth and initiate a geomagnetic storm. This allows, in principle, sufficient time to predict their impact. The geoeffectiveness of CMEs, that is, of the associated interplanetary CME (ICME) or magnetic cloud (MC), is primarily dependent on their Earth-side magnetic field direction (Bz), their velocity, and their associated ram pressure upon arrival at the magnetosphere. It is therefore highly expected to predict these parameters before an ICME (and the shock that potentially precedes it) arrives at Earth. A promising tool for such purpose is magnetohydrodynamic (MHD) numerical simulations. An accurate modeling of their onset and propagation up to 1 AU represents a key issue for more reliable space weather forecasts. To this end, a lot of CME-related models have been developed to describe their pre-eruption structures, their initiations, and their eruptions, and also the propagation of CMEs from the Sun to the Earth has been numerically investigated. In this chapter, after we briefly introduce the CME models so far, we numerically study the time-dependent evolution and propagation of the CME from the Sun to Earth using the 3D SIP-CESE MHD model introduced in Chap. 4, compare the simulation results with spacecraft observations and analyze in detail the CME’s propagation characteristics.

As we are approaching the end of this book, it is necessary to identify current challenges and potential directions to advance reliable predictive capability of the corona-heliosphere system, develop a research roadmap to tackle the scientific challenges of coupling observations and modeling with emerging data-science research to derive new knowledge from mass data, promote trans-disciplinary collaborations in the future, and find areas of convergence across disciplines.