A Novel Numerical Implementation for Solar Wind Modeling by the Modified Conservation Element/Solution Element Method

Abstract

In this paper, the spacetime conservation element and solution element (CESE ) method is applied to threedimensional magnetohydrodynamics ( MHD) equations in Cartesian coordinates for solar wind plasma, with the purpose of modeling the steady state solar atmospheric study. To illustrate this newly developed scheme we have studied two examples: (1) two-dimensional coronal dynamical structure with multipole magnetic fields and (2) threedimensional 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 quasiYsteady state. From these examples we have shown that the newly developed modified spacetime CESE scheme possesses the ability to model the Sun-Earth environment and other astrophysical flows.

THREE-DIMENSIONAL SOLARWINDMODELING FROM THE SUN TO EARTH BY A SIP-CESE MHD MODEL WITH A SIX-COMPONENT GRID

“…Since the creative work by Pneuman & Kopp (1971) empirically describing the source terms for the deposition of energy and/or momentum into the solar wind, many works have been devoted to the numerical study of producing solar steady-state conditions (e.g., Usmanov 1993;McKenzie et al 1997;Mikić et al 1999;Suess et al 1996Suess et al , 1999Wu et al 1999;Groth et al 2000;Feng et al 2005Feng et al , 2007Kleimann et al 2009;Nakamizo et al 2009). …”

“…Here, the SIP-CESE MHD model will be carried out on our newly introduced six-component grid system using governing Equations (1). Following the SIP-CESE MHD method given by Feng et al (2007), solving the governing Equations (1) leads us to the discretized integral equation Φ(U)(≡ U − Δt 2 η(U)) = U H and its iteration procedure for the corresponding system of linear equations…”

“…∂U (i) like Equation (17) of Feng et al (2007), one can directly solve Equation (4) by "intel MKL" with the Fortran library function 4 according to William et al (1997).…”

“…The same procedure has been stated (Balsara & Spicer 1999;Cunningham et al 2009) for their flux-CT algorithm to preserve thermal energy at the expense of energy conservation. In order to assess the ∇·B = 0 constraint numerically, as done in Equation (18) by Feng et al (2007), we define the following error measurement as:…”

“…This is a feature common to the grid mesh in the spherical shell. Even if the numerical schemes are implemented in Cartesian or cylindrical coordinates (Tanaka 1994(Tanaka , 1995(Tanaka , 1999(Tanaka , 2000Groth et al 2000;Cohen et al 2008;Feng et al 2007; Hu et al 2008; Kleimann et al 2009; Nakamizo et al 2009), the computational domain used in solar wind modeling has to approximate the spherical shell geometry.…”