Data-Driven Simulations of AR8210 W.P. Abbett Space Sciences Laboratory, UC Berkeley SHINE Workshop 2004.

Data-Driven Data-Driven Simulations of AR8210Simulations of AR8210

WP AbbettWP AbbettSpace Sciences Laboratory UC BerkeleySpace Sciences Laboratory UC Berkeley

SHINE Workshop 2004SHINE Workshop 2004

Incorporating Observations of Magnetic Incorporating Observations of Magnetic Fields into MHD Models of the Corona --- the Fields into MHD Models of the Corona --- the

ChallengesChallenges Determining electric fields and flows consistent with Determining electric fields and flows consistent with

the observed evolution of the magnetic field at the the observed evolution of the magnetic field at the photosphere (Data-driven modeling)photosphere (Data-driven modeling)

Generating initial atmospheres consistent with X-Generating initial atmospheres consistent with X-ray observations of the corona (Relevant to both ray observations of the corona (Relevant to both data-driven and ldquodata-inspiredrdquo modeling) data-driven and ldquodata-inspiredrdquo modeling)

Developing a physically-consistent means of Developing a physically-consistent means of incorporating newly emerging flux from a separate incorporating newly emerging flux from a separate system into fully magnetized atmospheres (Critical system into fully magnetized atmospheres (Critical to both data-driven and coupled models)to both data-driven and coupled models)

Developing standard techniques of testing and Developing standard techniques of testing and validating the new methodsvalidating the new methods

Data-driven Modeling Magnetic Fields and Data-driven Modeling Magnetic Fields and Flows at the PhotosphereFlows at the Photosphere

MHD models require boundary flows (eg to update electric MHD models require boundary flows (eg to update electric fields along the edges of control volumes within the boundary fields along the edges of control volumes within the boundary layers) that are consistent with the observed evolution of the layers) that are consistent with the observed evolution of the magnetic field in the photosphere magnetic field in the photosphere

Note that the above system of equations is Note that the above system of equations is under-determinedunder-determined

In the absence of simultaneous chromospheric and photospheric In the absence of simultaneous chromospheric and photospheric vector magnetograms we cannot use data to directly update the vector magnetograms we cannot use data to directly update the transverse components of the magnetic field since there is no transverse components of the magnetic field since there is no means to specify the needed vertical gradients means to specify the needed vertical gradients

New Inversion TechniquesNew Inversion Techniques

MEF (ldquoMinimum Energy Fittingrdquo) MEF (ldquoMinimum Energy Fittingrdquo) Constrains the under-determined system by requiring that Constrains the under-determined system by requiring that

the spatially integrated square of the velocity field at the the spatially integrated square of the velocity field at the photosphere be minimized (Longcope amp Regnier ApJ 2004 in photosphere be minimized (Longcope amp Regnier ApJ 2004 in press)press)

ILCT ldquoInductive Local Correlation Trackingrdquo ILCT ldquoInductive Local Correlation Trackingrdquo Uses velocities determined via local correlation tracking Uses velocities determined via local correlation tracking

(applied to magnetic elements) along with the Demoulin amp (applied to magnetic elements) along with the Demoulin amp Berger (2003) hypothesis to generate all three components Berger (2003) hypothesis to generate all three components of a flow field that is consistent with both the observed of a flow field that is consistent with both the observed evolution of the magnetic field and the vertical component of evolution of the magnetic field and the vertical component of the ideal induction equation (Welsch Fisher Abbett amp the ideal induction equation (Welsch Fisher Abbett amp Regnier ApJ 2004 in press)Regnier ApJ 2004 in press)

ldquoldquoMinimum Structure Reconstructionrdquo Poster 51 Minimum Structure Reconstructionrdquo Poster 51 SHINE 2004 (Georgoulis M amp LeBonte B J)SHINE 2004 (Georgoulis M amp LeBonte B J)

ILCTILCT

Consider the ideal induction equationConsider the ideal induction equation part partBBpartt = partt = x ( x (vv x x BB) )

Re-cast the z-component of the induction equation Re-cast the z-component of the induction equation asas

part partBBzzpartt + partt + middot (middot (vvBBz z minus minus vvzz BB) = 0) = 0

Define a new quantity Define a new quantity UU asas

UU vv minus (minus (BBBBzz))vvzz

(equivalent to the Demoulin (equivalent to the Demoulin amp Berger 2003 hypothesis)amp Berger 2003 hypothesis)

Then we have Then we have

part partBBzzpartt + partt + middot (middot (BBzz UU) = 0) = 0

ILCTILCT

Note that only flows perpendicular to the magnetic Note that only flows perpendicular to the magnetic field affect the evolution of field affect the evolution of BB thus we have the thus we have the freedom to set freedom to set vvmiddotmiddotBB=0=0

Then if we can somehow determine Then if we can somehow determine UU we can obtain we can obtain vv and v and vzz via a simple algebraic decomposition via a simple algebraic decomposition

vvzz = ( = (UU BB))BBzzB B 22

vv = = UU minus (minus (UU BB) ) BBB B 22

But LCT techniques applied to magnetic elements But LCT techniques applied to magnetic elements return a quantity ldquoreturn a quantity ldquouu

(LCT)(LCT)rdquo that in practice differs from rdquo that in practice differs from the true the true UU

ILCTILCT Then letrsquos define scalar quantities Then letrsquos define scalar quantities φφ and and ψψ in the in the

following wayfollowing way

BBzzuu minus minusφφ + + x ( x (ψψ zz))

Taking the curl of both sides of the equation gives a Taking the curl of both sides of the equation gives a Poisson equation for Poisson equation for ψψ

x (x (BBzzuu) = minus) = minus2 2 ψψ

If we now assume that If we now assume that uu can be approximated by can be approximated by

uu(LCT) (LCT) in the above equation we can determine in the above equation we can determine ψψ If If

we now require that we now require that uu also satisfy the induction also satisfy the induction

equation we can writeequation we can write

BBzzuu = = vvBBz z minus minus vvzz BB = = minusminusφφ + + x ( x (ψψ zz))

and the induction equation thus constrains and the induction equation thus constrains φφ

part partBBzzpartt = minus partt = minus 2 2 φφ

ILCTILCT

Then all that remains is to solve two Poisson Then all that remains is to solve two Poisson equations to obtain equations to obtain φφ and and ψψ (problem solved) (problem solved)

Note that only the vertical component of the Note that only the vertical component of the magnetic field is required to find a solution magnetic field is required to find a solution consistent with the z-component of the induction consistent with the z-component of the induction equationequation

Given the transverse magnetic field from a vector Given the transverse magnetic field from a vector magnetogram we can obtain a physically self-magnetogram we can obtain a physically self-consistent flow field suitable for incorporation into consistent flow field suitable for incorporation into the lower boundary of MHD models of the coronathe lower boundary of MHD models of the corona

ILCT Applied to NOAA AR8210 (see poster 94 Fisher et al)ILCT Applied to NOAA AR8210 (see poster 94 Fisher et al)

Testing Inversion Techniques (see Welsch et al Testing Inversion Techniques (see Welsch et al poster 50)poster 50)

Apply the inversion Apply the inversion techniques to magnetic techniques to magnetic fields and flows obtained fields and flows obtained from simulations of surface from simulations of surface and sub-surface active-and sub-surface active-region magnetic fieldsregion magnetic fields

Radiative MHD simulations Radiative MHD simulations of the surface layers can of the surface layers can also provide a test of LCT also provide a test of LCT techniques applied to techniques applied to intensity featuresintensity features

Generating an Initial StateGenerating an Initial State

We need more than just a physically consistent We need more than just a physically consistent scheme to update the photospheric boundary --- we scheme to update the photospheric boundary --- we also need an initial specification of all components of also need an initial specification of all components of the magnetic field throughout the domain that the magnetic field throughout the domain that compares favorably with eg soft X-ray images of the compares favorably with eg soft X-ray images of the corona corona (see poster 49 Barnes et al for a discussion of ldquocompares (see poster 49 Barnes et al for a discussion of ldquocompares favorablyrdquo)favorablyrdquo)

ChallengesChallenges The magnetic configuration of a complex active region is The magnetic configuration of a complex active region is

highly non-potentialhighly non-potential The atmosphere below the chromosphere is not force-free The atmosphere below the chromosphere is not force-free

Best solution (at the moment) Perform a non-linear Best solution (at the moment) Perform a non-linear force-free extrapolationforce-free extrapolation Note however not all techniques produce results that can Note however not all techniques produce results that can

be used to initiate MHD models (eg mismatches in the be used to initiate MHD models (eg mismatches in the transverse field at the lower boundary are problematic)transverse field at the lower boundary are problematic)

Generating an Initial State Testing Extrapolation Generating an Initial State Testing Extrapolation Techniques Against MHD Simulations of Flux EmergenceTechniques Against MHD Simulations of Flux Emergence

A comparison of a local PFSS and the Wheatland et al A comparison of a local PFSS and the Wheatland et al 2000 non-constant-alpha force-free extrapolation 2000 non-constant-alpha force-free extrapolation technique applied to the Magara 2004 MHD simulation of technique applied to the Magara 2004 MHD simulation of flux emergence (from Abbett et al 2004)flux emergence (from Abbett et al 2004)

Synthetic ldquomagnetogramsrdquo taken at different heights in the model atmosphere from the model photosphere to the model chromosphere (bottom right) from Magara et al 2004

Generating an Initial State AR-8210Generating an Initial State AR-8210

Above Wheatland et al 2000 method applied to NOAA Above Wheatland et al 2000 method applied to NOAA AR-8210 (May 1 1998) --- from J M McTiernanAR-8210 (May 1 1998) --- from J M McTiernan

Note that to compare with observed X-ray emission Note that to compare with observed X-ray emission one must perform additional calculations eg assume one must perform additional calculations eg assume a loop heating mechanism and solve the energy a loop heating mechanism and solve the energy equation along individual loops (Lundquist Schrijver)equation along individual loops (Lundquist Schrijver)

Emerging Flux into a Fully Magnetized Model Emerging Flux into a Fully Magnetized Model CoronaCorona

Calculations like the Calculations like the one shown on the left one shown on the left represent a very simple represent a very simple case here sub-case here sub-photospheric flux photospheric flux emerges into an emerges into an initially field-free model initially field-free model atmosphereatmosphere

If we now assume that If we now assume that the model corona is the model corona is initially filled with field initially filled with field we must consider how we must consider how the pre-existing the pre-existing structure interacts with structure interacts with the introduction of new the introduction of new flux when updating the flux when updating the boundary valuesboundary values

A simulation of flux emergence into an initially field-free model corona (from Abbett amp Fisher 2003) The color table indicates the degree to which the model corona is force-free during the dynamic emergence process

To address this problem an assumption must be made in our To address this problem an assumption must be made in our case we choose to ignore the back-reaction from coronal case we choose to ignore the back-reaction from coronal forces --- that is we assume that photospheric flows dominate forces --- that is we assume that photospheric flows dominate the dynamics of the boundary layerthe dynamics of the boundary layer

Then the ideal induction equation is linear and we can express Then the ideal induction equation is linear and we can express the magnetic field in the boundary layer as a superposition of the magnetic field in the boundary layer as a superposition of two vector fields two vector fields BB = = BB11+ + BB22

Emerging Flux into a Fully Magnetized Model Emerging Flux into a Fully Magnetized Model CoronaCorona

Here Here vv11 represents the imposed boundary flow represents the imposed boundary flow BB11 represents represents new flux introduced into the system from below (assumed zero new flux introduced into the system from below (assumed zero at t=0) and at t=0) and BB22 which at t=0 represents the portion of the initial which at t=0 represents the portion of the initial coronal flux system that permeates the boundary layerscoronal flux system that permeates the boundary layers

Since the emerging flux system satisfies partSince the emerging flux system satisfies partBB11partt=partt= x ( x (vv11 x x BB11) ) BB22(t=0) is known and (t=0) is known and vv11 is specified for all t we can advance is specified for all t we can advance BB22 in time and thus specify a boundary field in time and thus specify a boundary field BB that satisfies the that satisfies the ideal MHD induction equation for all time t given a standard ideal MHD induction equation for all time t given a standard boundary condition for boundary condition for BB22

partpart((BB11++BB22)partt = )partt = x x vv11 x ( x (BB11 + + BB22))

Of course this treatment allows for differences between the Of course this treatment allows for differences between the magnetic field imposed in the boundary layers and the vector magnetic field imposed in the boundary layers and the vector field observed at the photospherefield observed at the photosphere

If we impose a further condition and require that the vertical If we impose a further condition and require that the vertical component of the field evolve exactly in accordance with the component of the field evolve exactly in accordance with the z-component of the field observed at the photosphere our z-component of the field observed at the photosphere our previous condition can be re-cast asprevious condition can be re-cast as

Emerging Flux into a Fully Magnetized Model Emerging Flux into a Fully Magnetized Model CoronaCorona

partpartBBpartt = partt = x x vv11 x x BB11 + + x x z z [([(vv11 x x BB22)middot)middotzz]]ˆ ˆ

In this approximation we neglect the components of In this approximation we neglect the components of part partBB22partt=partt= x ( x (vv11 x x BB22) that either alter the prescribed evolution ) that either alter the prescribed evolution

of Bof Bzz at the boundary at the boundary zzmiddot (partmiddot (partBB22partt) or involve vertical gradients partt) or involve vertical gradients of of BB22

Emerging Flux into a Fully Magnetized Model Emerging Flux into a Fully Magnetized Model CoronaCorona

We demonstrate the We demonstrate the previous technique by previous technique by driving the SAIC model driving the SAIC model corona with the vector corona with the vector magnetic field obtained magnetic field obtained from an ANMHD sub-from an ANMHD sub-surface simulation surface simulation

We emerge flux into a pre-We emerge flux into a pre-existing dipole field In one existing dipole field In one case the arcade field has case the arcade field has an opposite polarity to an opposite polarity to that of the emerging that of the emerging bipole and in another bipole and in another case the arcade field has case the arcade field has the same polaritythe same polarity

Consider this a ldquotest runrdquo Consider this a ldquotest runrdquo for a data-driven for a data-driven calculationcalculation

Image from Abbett Mikic Linker et al 2004

Putting it all TogetherPutting it all Together

Two fully-coupled codesTwo fully-coupled codes

Boundary code Flows prescribed by ILCT Boundary code Flows prescribed by ILCT the magnetic induction equation continuity the magnetic induction equation continuity equation and a simple energy equation are equation and a simple energy equation are solved implicitly in a thin boundary layersolved implicitly in a thin boundary layer

MHD corona the system of ideal MHD MHD corona the system of ideal MHD equations are solved on a non-uniform grid equations are solved on a non-uniform grid the boundary code is fully coupled to the the boundary code is fully coupled to the model coronamodel corona

Simulation of AR-8210 The Boundary Simulation of AR-8210 The Boundary LayersLayers

Vertical magnetic Vertical magnetic field from a 3D field from a 3D calculation initiated calculation initiated by an IVM vector by an IVM vector magnetogram of magnetogram of AR-8210 at 1940 AR-8210 at 1940 (Regnier) and a (Regnier) and a NLFFF extrapolation NLFFF extrapolation (McTiernan)(McTiernan)

The simulation is The simulation is driven by ILCT flows driven by ILCT flows applied to the applied to the magnetogram at magnetogram at 1940 and one 1940 and one approximately four approximately four hours laterhours later

Simulation of AR-8210 The chromosphereSimulation of AR-8210 The chromosphere

Preliminary MHD Simulation of AR-8210Preliminary MHD Simulation of AR-8210

ProgressProgress

Developed necessary inversion techniquesDeveloped necessary inversion techniquesDeveloped 3D boundary code and applied it to Developed 3D boundary code and applied it to

AR-8210 as a test of the inversion techniqueAR-8210 as a test of the inversion technique Coupled boundary code to 3D MHD coronaCoupled boundary code to 3D MHD corona

Remaining ChallengesRemaining Challenges

Incorporate global topology into the local Incorporate global topology into the local model coronamodel corona

Refine lower boundary condition Refine lower boundary condition (energetics temporal scaling flows parallel (energetics temporal scaling flows parallel to the magnetic field)to the magnetic field)