Coupled Models for the Emergence of Magnetic Flux into the Solar Corona W. P. Abbett UC Berkeley SSL G. H. Fisher, Y. Fan, S. A. Ledvina, Y. Li, and D. J. Bercik

Overview: Numerical Modeling of Active Region Magnetic Fields  The Solar Interior: From 1D thin flux tube and 2D axisymmetric MHD models to fully compressible 3D MHD, and large-scale 3D MHD in the anelastic regime  The Surface Layers: 3D ideal MHD at active region spatial scales to radiative- MHD at the scale of surface granulation  The Coronal Field: Potential and force- free field extrapolations vs. dynamic models

Modeling the Interior The Flux Tube Picture: Toroidal flux layer near the tachocline succumbs to an instability, and creates a buoyant flux rope that ascends through the CZ as an Omega-loop. The loop emerges through the photosphere, and is observed as a magnetic bipole. (Cauzzi et al. 1996)

Modeling the Interior  The Thin Flux Tube Approximation: Assumptions: Assumptions: Active region fields behave as distinct, tube-like entities Active region fields behave as distinct, tube-like entities embedded in a field-free plasma. The flux tube diameter embedded in a field-free plasma. The flux tube diameter is small compared with all other relevant length scales, is small compared with all other relevant length scales, and pressure balance exists across the tube at all times. and pressure balance exists across the tube at all times. Advantages: Advantages: One can derive a simplified equation of motion for a 1D One can derive a simplified equation of motion for a 1D tube moving within a 3D model of the solar interior. tube moving within a 3D model of the solar interior.

Modeling the Interior  3D local MHD in the anelastic approximation: Assumptions: Approximation results from a scaled variable expansion of the 3D MHD equations about a zero-th order, stratified reference state. This approximation is valid in the high beta, gravitationally stratified plasma of the solar convection zone below the photosphere. Advantages: Fast-moving acoustic waves are effectively filtered out of the simulations. Time steps are less restrictive, and a large amount of parameter space can be explored.

Modeling Flux Ropes in the Interior 3D vs 2D axisymmetric (Abbett et al. 2000,2001)

ANMHD Examples: LHS --- magneto-convection and the local solar dynamo; RHS --- emerging magnetic flux (Abbett, Fan & Fisher 2002 in prep).

The Surface Layers  A fully compressible treatment is required.  Two approaches for modeling magnetic fields at or near the solar surface: 1. Realistic radiative-magnetoconvection over 1. Realistic radiative-magnetoconvection over small spatial scales (Stein & Nordlund 2001, small spatial scales (Stein & Nordlund 2001, Bercik 2002, Gudiksen et al. 2002) Bercik 2002, Gudiksen et al. 2002) 2. 3D MHD simulation of the local photosphere / 2. 3D MHD simulation of the local photosphere / transition region / low corona employing transition region / low corona employing an approximate treatment of the energy equation an approximate treatment of the energy equation (Fan 2001, Magara & Longcope 2001) (Fan 2001, Magara & Longcope 2001)

Granular-scale surface magneto-convection (Bercik 2002) Computationally expensive calculation; Computationally expensive calculation; thus, the domain size is restricted. thus, the domain size is restricted. Surface Layers

Zeus3D fully-compressible 3D ideal MHD (Fan 2001) Calculations of this Calculations of this type are important to type are important to test theoretical models test theoretical models of CME initiation. of CME initiation. Do flux ropes exist in Do flux ropes exist in the corona, and can they the corona, and can they be formed self-consistently be formed self-consistently through emergence of a through emergence of a twisted magnetic structure twisted magnetic structure from below? from below? Are multipolar magnetic Are multipolar magnetic configurations necessary configurations necessary prerequisites for an prerequisites for an eruptive event? eruptive event? Surface Layers: Modeling Large-scale Flux Emergence into the Corona

Fully-compressible 3D ideal MHD (Magara & Longcope 2001)

Characterizing the Coronal Fields  Global or local potential field extrapolation  Constant or non-constant alpha force-free fields  3D ideal MHD (resistivity due to truncation error) or resistive MHD (ohmic heating self- consistently included in the equation of internal energy) with enhanced thermal conduction along field lines and optically thin losses  A simplified combined approach: treat the energetics in 1D along a thin loop defined by a 3D non-ideal numerical calculation

Potential Field vs. MHD models of the Global Corona  LHS: The potential field model of Li & Luhmann using an artificial photospheric boundary generated by ANMHD (Li et al. 2001)  RHS: The MHD dynamic model corona of Linker & Mikic being driven by an artificial, evolving active region generated by ANMHD (Li et al. 2001)

ANMHD Interior model drives the lower boundary of a Zeus3D model corona (Abbett & Fisher 2002). Code coupling: Does Code coupling: Does the corona significantly the corona significantly affect the sub-surface affect the sub-surface calculation (Welsch & calculation (Welsch & Longcope 2000) Longcope 2000) How important are How important are treatments of the treatments of the energy equation in energy equation in the transition layers the transition layers and corona (Mikic, and corona (Mikic, Linker, Lionello, Mok Linker, Lionello, Mok 2002)? 2002)? Toward Coupled Models of Flux Emergence:

Coupled Models: ANMHD Interior, Zeus3D Model Corona Left Column: The response of the Zeus3D model corona to three different ANMHD driving boundaries --- sub-surface Omega-loops of increasing twist. Right Column: Local potential field extrapolation calculated from the vertical component of the magnetic field using the the same photospheric lower boundary. (Abbett & Fisher 2002)

DynamicEmergence: How force-free is the coronal field during the early stages of the emergence process?

Dynamic Emergence: A measure of how force-free the coronal field becomes later in the simulation; at a time when less flux is being introduced into the corona from below.

“Sigmoid” Structures Fieldlines generated from arched flux ropes that emerge with non- zero helicity form sigmoid-shaped structures when viewed from above. However, the “direction” of a sigmoid (and other details of its structure depend on projection effects, viewing angle, and location within a given loop of emitting plasma.

What these Simple Models Tell us about the Emergence Process:  The presence and distribution of boundary flows resulting from the sub-surface evolution of a magnetic structure is important to coronal dynamics and morphology. The component of the flow perpendicular to the boundary is particularly important since (in an ideal calculation) such a flow is necessary to transport magnetic field into the corona while conserving flux.  During the initial stages of the flux emergence process, the emerging coronal structure differs substantially from a force-free configuration. As the velocity and magnetic fields at the boundary evolve, less flux is transported into the corona, and with the exception of structures close to the photospheric boundary, the overlying field relaxes to a more force-free configuration.

 PARAMESH: A domain decomposition, adaptive mesh refinement (AMR) framework developed by MacNeice et al. and distributed by GSFC  Zeus3D: A staggered mesh finite-difference (non- relativistic) MHD code originally developed by Stone, Norman, and Clarke and publicly distributed by NCSA  ZeusAMR: A fully compressible 3D MHD code with AMR which resulted from a merge of PARAMESH with a modified version of Zeus3D Toward Truly Coupled Models, and the Ability To Model the Effect of Active Region Magnetic Fields on the Global Corona

Example of driving a ZeusAMR coronal simulation with an ANMHD generated lower boundary. True “code coupling” can be achieved using the PARAMESH framework achieved using the PARAMESH framework.

Work in Progress: AMR in Global Coronal Models LHS: ZeusAMR in spherical coordinates with 3 levels of grid refinement RHS: Using ANMHD subsurface runs to drive SAIC code (Li, Linker, Mikic)

Toward Coupled Models of Flux Emergence: Summary  Existing code coupling frameworks have the potential to provide a straightforward way to self-consistently connect existing numerical treatments of local flux emergence into large- scale models of global phenomena.  Though, the devil is in the details: -- Different numerical algorithms, boundary treatments, and physical conditions between individual models of different regimes make the task of transferring information back and forth between codes in a suitably efficient, yet physically consistent manner, a non-trivial task. -- Different numerical algorithms, boundary treatments, and physical conditions between individual models of different regimes make the task of transferring information back and forth between codes in a suitably efficient, yet physically consistent manner, a non-trivial task.