1 Magnetic waves in sheared field regions E.J. Zita, The Evergreen State College, Olympia, WA Abstract How do magnetohydrodynamic (MHD) waves change as they propagate along a twisted magnetic field? Observations show that sheared field regions such as "sigmoids" can give rise to solar coronal mass ejections. Understanding MHD processes in sheared fields can provide insight into mechanisms by which magnetic energy is transformed. To that end, we find exact analytic solutions for displacements in a sheared, force- free magnetic field with a current sheet. With appropriate transformations, we can write the MHD wave equation in terms of ordinary differential equations with constant coefficients and regular singularities. The solutions can describe changes to incoming waves as they travel through the sheared field region. We aim to describe reflections and transmissions near critical points, and transformations between sonic, Alfvénic, and magnetosonic waves in a twisted magnetic field.

2 Zita, The Evergreen State College, SHINE meeting, Banff BC, 2002 Aug. Motivation: sheared fields  coronal mass ejections Observations show that CMEs often follow sigmoidal twists in solar magnetic fields How do magnetic waves propagate and transform in sheared fields? Image courtesy of Montana State University:

3 Zita, The Evergreen State College, SHINE meeting, Banff BC, 2002 Aug. Method: 1. Model sheared field region mathematically. 2. Transform the wave equation to a frame of reference which rotates in space with the sheared magnetic field. 3. Analytically solve coupled differential equations. 4. Displacements reveal how waves propagate in magnetic shear region.

4 Zita, The Evergreen State College, SHINE meeting, Banff BC, 2002 Aug. 1: Mathematically model sheared field region ObservationsSchematicMathematical model This sheared magnetic field is force-free and stable: [B.C. Low, 1988] B x =0, B y = B 0 sech(ax), B z = B 0 tanh(ax) x

5 Zita, The Evergreen State College, SHINE meeting, Banff BC, 2002 Aug. A strong current sheet sustains the magnetic shear layer: J =  B Current sheets can ohmically heat solar plasma. x - y z

6 Zita, The Evergreen State College, SHINE meeting, Banff BC, 2002 Aug. 2: The wave equation describes how forces displace the plasma:  = frequency,  = displacement, c s = sound speed, v A = Alfvén speed B = total magnetic field, B 0 = mean field, b 1 = magnetic oscillation  B k || B k || 

7 Zita, The Evergreen State College, SHINE meeting, Banff BC, 2002 Aug. Mathematical trick: look at the problem from a twisted perspective, literally. For example, the displacement vector becomes: (2) Transform everything ( , k,  ) to a reference frame which rotates in space with the sheared magnetic field. Cartesian frame: where Rotating frame: where  x

8 Zita, The Evergreen State College, SHINE meeting, Banff BC, 2002 Aug. 3: Wave equation in rotating frame yields  coupled differential equations for  x (x),  || (x),   (x)  simple ODE in  x (x) with constant coefficients  Analytically solve for the displacements:

9 Zita, The Evergreen State College, SHINE meeting, Banff BC, 2002 Aug. 4: Displacements show how waves propagate through sheared magnetic region Waves oscillate along x when k x = real (p 0 > 0 and p 2 > 0), for frequencies  2 >  2 2 and  2 >  0 2 (high frequencies). Waves damp along x when k x = imaginary: LF case: (p 0 0)  2 <  0 2 MF case: (p 0 > 0 and p 2 < 0)  0 2 <  2 <  2 2 Critical frequencies: p 2 = 0 when  2 = and p 0 = 0 when  2 =

10 Zita, The Evergreen State College, SHINE meeting, Banff BC, 2002 Aug. References and Acknowledgements B.C. Low, High Altitude Observatory (HAO), NCAR, Astrophysical Journal 330, 992 E.J. Zita, TESC, Jan 2002 poster at ITP, UCSB: Dick Canfield et al., Montana St. Univ., IEEE Transactions on Plasma Physics, special issue on Space Plasmas, 2000 Thanks to Dr. B.C.Low (HAO) for suggesting this sheared field for analysis, and to Dr. Tom Bogdan (HAO and NSF) for patient discussions and clever strategies. This work is supported by NASA under the Sun-Earth Connection Guest Investigator Program, NRA 00--OSS--01 SEC

11 Zita, The Evergreen State College, SHINE meeting, Banff BC, 2002 Aug. Preprints