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George H. Fisher Space Sciences Laboratory University of California, Berkeley

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## George H. Fisher Space Sciences Laboratory University of California, Berkeley

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**The Use of Vector Magnetogram Data in MHD Models of the**Solar Atmosphere and Prospects for an Assimilative Model George H. Fisher Space Sciences Laboratory University of California, Berkeley (Errors and lack of clarity introduced by Brian T. Welsch)**What would an ``Assimilative Model’’ of the solar**atmosphere consist of? A time-evolving physical model of the Sun’s atmosphere, or a portion of the Sun’s atmosphere, which can be corrected by time-dependent measurements that can be related in some manner to properties of the solar atmosphere. In particular, this means a 3D-MHD model of the Sun’s atmosphere, from photosphere to corona, that is updated by means of vector magnetograms.**What are the most important elements of a physics-based**model of the Sun? • Nearly all transient phenomena, such as solar-initiated “space weather” events, are driven by, or strongly affected by, magnetic fields. • A fluid treatment (MHD) is reasonable most of the time (except, probably, during solar flares). • Magnetic fields thread all layers of the Sun’s convection zone and atmosphere. • Maps of the estimated solar magnetic field (line-of-sight component) can be performed regularly in the photosphere. • In the near future, maps of all 3 components of the estimated magnetic field (vector magnetograms) will be taken regularly. • Vector magnetograms are essential for determining the free energy available in the solar atmosphere to drive violent phenomena. Without vector magnetograms, solar models are not meaningfully constrained.**Schematic diagram of an assimilative model employing the**Kalman filter approach: A is the physical model time-advance operator H operator relates state variable x to observable z K is the Kalman filter Q is estimated process or model error R is measurement error P is estimate of state variable error This Diagram taken from Welch and Bishop (2006) “An introduction to the Kalman Filter”**Needed ingredients for an assimilative (e.g. Kalman filter)**model of the solar atmosphere: • A reasonably good physical model • Measurements with a good enough time cadence and accuracy to be useful • A well-understood connection between physical and measured variables • A good understanding of the data and model errors Where do we stand with respect to these requirements?**1. What are the minimum requirements for a “reasonably**good” physical model? • The model must accommodate the range of conditions from the photosphere, where magnetic fields can be routinely measured, into the corona, where ``space weather’’ events occur • The model must include the dominant terms in the energy equation that apply in the photosphere-corona system. The dominant terms are drastically different in the different parts of the domain • The model must be able to accommodate the wide range of physical and temporal scales from the photosphere to the corona. • The model must be able to accommodate vector magnetic field maps, as a time-dependent boundary condition. This is required whether or not the model is truly assimilative! Until recently, no existing models satisfied these requirements. Here is a brief summary of the challenges:**Numerical challenges:**• A dynamic numerical model extending from below the photosphere out into the corona must: • span a ~ 10 - 15 order of magnitude change in gas density and a thermodynamic transition from the 1 MK corona to the optically thick, cooler layers of the low atmosphere, visible surface, and below; • resolve a ~ 100 km photospheric pressure scale height while simultaneously following large-scale evolution (we use the Mikic et al. 2005 technique to mitigate the need to resolve the 1 km transition region scale height characteristic of a Spitzer-type conductivity); • remain highly accurate in the turbulent sub-surface layers, while still employing an effective shock capture scheme to follow and resolve shock fronts in the upper atmosphere • address the extreme temporal disparity of the combined system**The Solar Photosphere**The solar photosphere is an extremely thin, corrugated, and complex layer, in which the plasma β in strong field regions is of order unity. This is the layer in which magnetic fields can be measured most routinely. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway)**The solar corona:**• The corona is a low-density, low-β, optically-thin, hot plasma • Plasma entrained within coronal loops evolves rapidly • compared to sub-surface structures • The magnetically-dominated corona can store energy over long • periods of time, but will often undergo sudden, rapid, and dramatic • topological changes as magnetic energy is released. • The size scale of coronal structures is generally much larger than the depth of the photosphere Movies courtesy of LMSAL, TRACE & LASCO consortia**RADMHD (Abbett, 2007, ApJ, in press): Numerical techniques**• We use a semi-implicit, operator-split method. • Explicit sub-step: We use a 3D extension of the semi-discrete method of Kurganov & Levy (2000) with the third order-accurate central weighted essentially non-oscillatory (CWENO) polynomial reconstruction of Levy et al. (2000). • CWENOinterpolation provides an efficient, accurate, simple shock capture scheme that allows us to resolve shocks in the transition region and corona without refining the mesh. The solenoidal constraint on B is enforced implicitly.**RADMHD: Numerical techniques**• We use a semi-implicit, operator-split method • Implicit sub-step: We use a “Jacobian-free” Newton-Krylov (JFNK) solver (see Knoll & Keyes 2003). The Krylov sub-step employs the generalized minimum residual (GMRES) technique. • JFNK provides a memory-efficient means of implicitly solving a non-linear system, and frees us from the restrictive CFL stability conditions imposed by e.g., the electron thermal conductivity and radiative cooling.**Characteristics of the Quiet Sun model atmosphere:**Note: Above movie is not a timeseries!**We drive RADMHD with photospheric velocities determined from**magnetogram sequences. • Velocities match tBz determined by induction equation. • We have spent a lot of effort deriving and evaluating techniques to do this, with our leading techniques being MEF (Longcope 2004) and ILCT (Welsch et al., 2004). • In addition to its importance for driving the code, these techniques are useful on their own to derive Poynting and helicity fluxes directly from magnetogram observations. • By itself, this approach is NOT assimilative: • the model is driven to match the observed photospheric field; • it does not predict the photospheric field.**The Ideal MHD Induction Equation**• How can we ensure that estimated velocities are physically consistent with the magnetic induction equation? • Only the normal component of the induction equation contains no unobservable vertical derivatives: Demoulin & Berger argue tracked motions, U, relate to v via: The ideal MHD induction equation simplifies to this form:**MEF & ILCT: constrain solutions of the induction equation**• Solve for φwith 2D divergence, tBz = -2φ • MEF: Minimize ∫dA (v2 + vz 2)to find ψ • ILCT: Assume u = uLCT, then solve x uBz = -2ψ. • Note that if only Bz (or an approximation to it, BLOS) is known, ILCT can still solve for φ, ψ! ^ Let uBz = v Bz - vzB = - φ + x ψ z.**Apply ILCT to IVM vector magnetogram data for AR 8210**• Vector magnetic field data enables us to find 3-D flow field from ILCT via the equations shown on the previous slide. Transverse flows are shown as arrows, up/down flows shown as blue/red contours.**2. Measurements of the magnetic field at the photosphere**Slide courtesy of Tom Metcalf, CoRA/NWRA**Slide courtesy of Tom Metcalf, CoRA/NWRA**How is the vector magnetic field determined? • Magnetic fields will be split by Zeeman effect, but using the split itself not useful in most cases. Spot in 5250 A (normal Zeeman triplet)**Zeeman Effect: Normal Zeeman Triplet**σ π • Pi component is unshifted in wavelength (1) • Sigma components shifted to either side of pi component (2). σ • If the magnetic field is directed along the line of sight, the sigma components are left and right circularly polarized and the pi component is unpolarized. • If the magnetic field is directed perpendicular to the line of sight, the sigma and pi components have mutually orthogonal linear polarizations. Slide courtesy of Tom Metcalf, CoRA/NWRA**How is Polarization Measured?**• Polarization is measured as the difference between data obtained using two different polarizers. • For example a Wollaston prism or a calcite beam splitter produces two output beams of orthogonal linear polarization: I+Q,I-Q. • U and V follow in the same way with a retarder in the path. Slide courtesy of Tom Metcalf, CoRA/NWRA**The Stokes Profiles**• A magnetograph observes the Stokes profiles. • V/I is circular polarization and gives the LOS field • U/I and Q/I are linear polarization and give the transverse field Slide courtesy of Tom Metcalf, CoRA/NWRA**Observed Stokes Profiles**Stokes I Stokes Q Stokes U Stokes V • Na-D line observations from the IVM • They look more or less as expected with a few differences: • Noise is clearly present • prefilter distorts spectrum Relative Wavelength (nm) Slide courtesy of Tom Metcalf, CoRA/NWRA**3. Inverting the Polarization Observations to get B**• The best method is to observe the Zeeman splitting directly. • Not generally possible for optical observations since the fields on the Sun are too weak. • The Zeeman splitting goes as so this works better in the IR. • Gives the magnetic field directly without worrying about the filling factor. • The next best method is to fit the Stokes profiles to the Unno profiles (Milne-Eddington atmosphere: source function linear with optical depth). • This gives the magnetic field, filling factor, thermodynamic parameters Slide courtesy of Tom Metcalf, CoRA/NWRA**4. Known sources of error in vector magnetograms:**• Photon statistics • Polarization is computed as a difference of two signals • polarization cross talk • Polarization signal “leaks” between Stokes parameters • Is corrected on an instrument-by instrument basis • calibration constant • The calibration constant in magnetographs is very approximate and is not constant at all. • atmospheric seeing • Will induce spurious polarization, sometimes strong • polarization bias • Should be correctable in most instruments by looking at the continuum or regions of very weak field. • bad 180 degree ambiguity resolution (how to quantify??) Bottom line – vector magnetogram errors can be characterized, at least statistically Slide courtesy of Tom Metcalf, CoRA/NWRA**Summary of assimilative model requirements**• Reasonably good physical model – good progress! • Measurements with good time cadence, accuracy – rapidly improving! • Well-understood connection between physical variables and measurements – reasonably good • Understanding of data errors – reasonably good; understanding of model errors - unknown**Issues that must be resolved for an assimilative solar MHD**model • Currently, the data are used directly to determine the flow-field at the photosphere. How can this be made consistent with the Kalman “corrector” step, since the data have already been used? • Can the Kalman filter approach be used in a “sub-step” process to determine the photospheric velocity field instead of using the ILCT or MEF procedures? • Noise in the vector magnetogram data will probably introduce spurious Alfven waves into the model, even with the “filtering”. How do we cope with this? • How do we estimate “model” errors? “Ensemble” runs with Monte Carlo’d magnetogram errors? (Non-linearity Chaotic response?)**Conclusions**• Difference between assimilative and models directly driven by data (as we now perform): Assimilative models have the potential to accommodate data errors more consistently • Assimilative techniques are worth detailed investigation for solar MHD models. • There are other, simpler solar models that may be more immediately amenable to assimilation techniques.**Alternative data for application of Kalman approach with**time-dependent coronal models: • Input: photospheric BOutput: n, T, B, v • Possible data to assimilate: 1. Coronal B, from radio magnetography • LOS integration, B on voxels unkown 2. Helioseismic estimates of sub-photospheric v • Spatial resolution quite coarse 3. H- fibrils (direction of chromospheric B) • Can reveal errors, but how to update model’s B? 4. w/Emissivity model, coronal EUV/SXR emission • LOS integration, unknown heating function**Alternative data for Kalman approach, cont’d:**• Input: photospheric BOutput: n, T, B, v • Possible data to assimilate: 5. Chromosperic BLOS, from, e.g., SOLIS • Precise altitude along LOS unknown 6. Tomographic density reconstruction • Stereoscopy from solar rotation too slow • During STEREO mission, this could work!

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