• Motivation: the interface region • Introduction: magnetostatics, plasma-beta • The principle way: nonlinear equations • The easy way: linear equations • Application: Sunrise/IMaX

• Motivation: • Introduction: • The principle way: • The easy way: • Application:

Selfconsistent magnetostatic modelling of the mixed plasma-beta

solar atmosphere Thomas Wiegelmann

Source: IRIS-proposal, Fig. E1

Aim: Towards a better under- standing of interface region between photos- phere and corona.

The interface region

Static models cannot explain all these dynamic phenomena, but knowledge of the magnetic field structure might help to guide measurements

Magneto-Hydro-Statics (MHS)

• Heritage of methods are force-free magnetic field extrapolation methods [Lorentz-force vanishes]

• We aim to solve the force-balance [Lorentz-force compensated by Pressure gradient and gravity], not the coronal heating problem.

Plasma Beta • Without gravity and curvature forces the

force balance reduces to gradients of magnetic and plasma pressure

• Plasma Beta defines relative importance of plasma and magnetic forces:

Plasma Beta Lorentz force vanishes, force-free fields

In the generic case Lorentz force is compensated by plasma forces.

On Sun: 1D barometric solution

But: Force-free fields are still possible:

• Solve nonlinear MHS by minimizing a Functional • Use potential magnetic field and 1D barometric

formular for pressure and density as initial state. • Minimize functional with measured photospheric

magnetic vector as boundary conditions • Need suitable boundary conditions for pressure and

density (How? measure or reasonable assumptions?)

Principle way: nonlinear MHS equations

Well tested with semi-analytic equilibria. As accurate as nonlinear force-free code. Very, very slow convergence. Not applied to data yet.

Easier way: linearized MHS-equations • Here we use a Cartesian system with (x,y) parallel

and z perpendicular to the Sun‘s surface. • Assumption: Currents flow in the x,y

plane[perpendicular to gravity] + optional a linear current parallel to the field lines (Low 1991):

Same decomposition is possible in spherical geometry (Bogdan&Low 86, Neukirch 95)

Linear force-free part this part contains currents perpendicular to z =>nonmagnetic forces

8 8

( Gary, 2001)

Plasma Beta in Solar atmosphere

Magnetostatic Model

Force-free Model

Linear MHS, Low 1991 solutions

• Remember: The solar atmosphere becomes (almost) force-free above photosphere and chromosphere [say thickness ~ 1/k], and the perpendicular part of the current should vanish in the corona:

• With the measured Bz(x,y,z=0) in the photosphere as boundary condition, the equation above is solved with a Fast Fourier Transformation. α0 and ɑ are free parameters.

• Plasma pressure and density are computed self-consistently from the force balance equation, here for a constant gravity g:

Linear MHS, Low 1991 solutions

Compensating the Lorentz force

Hydrostatic 1D atmosphere

MHS-equilibria from observed magnetograms • Typical pixel sizes of magnetograms are about

1400 km (SOHO/MDI) or 350 km (SDO/HMI) => Magnetostatic modelling makes hardly sense, non force-free layer (about 2 Mm) will be resolved only by 1-6 points

• Sunrise/IMaX has a pixel size of 40 km and we can resolve the layer vertically by about 50 points.

• Here we use a quiet Sun area (from first Sunrise flight in June 2009) and a linear MHS solutions with 1/k=2Mm, ɑ =0.5 and α0 =3

Quiet Sun‘s B-Field, Sunrise/IMaX

Linear MHS: Fieldlines from Sunrise/IMaX

Linear MHS: Pressure disturbance at z= 1Mm Full IMaX-FOV

Pressure must be positive, but pressure disturbance is negative. Background pressure has to fulfill:

Quiet Sun‘s B-Field, Sunrise/IMaX

P0 must be high enough to compensate largest Pressure disturbance => increased plasma-beta in the entire region! [or ɑ must be very small]

Now we concentrate on a small local FOV.

Plasma Beta in Solar atmosphere

Linear MHS: Pressure disturbance at z= 1Mm Local-FOV

Potential Field

Linear Force-Free-Field

Linear MHS-Field

Local FOV in IMaX

MHS, Local FOV, Averaged quanti- ties as function of height

MHS, local FOV, Equicontours of Pressure

MHS, local FOV, Equicontours of Beta

Linear MHS • Fast, equations can be solved

by Fast-Fourier Transformation (slower than linear-force-free [Bessel instead Exp in z] faster than nonlinear FF)

• B.C.: Line-of-sight magnetogram • Specific (unrealistic ??)

solutions. Assumption based on mathematical simplicity, not physical reasoning. Solution has (2 free parameters: α0 and ɑ )

• Reduces to linear force-free solutions for zero plasma beta. (1 free parameter : α0 )

Nonlinear MHS • Much slower (by factor ~100)

than nonlinear force-free in mixed beta plasma, if beta changes by orders of magnitudes.

• Requires vector magnetogram (not available in quiet Sun)

• General generic solution. If a magnetostatic solution exists for given boundary condition, code is likely to find it (but no proof).

• Reduces to nonlinear force-free solutions for zero plasma beta.

Preliminary conclusions