In constant volume processes, the heat change in the system is equal to the change in specific internal energy. Thus:
In constant pressure processes, the heat change in the system is equal to the change in enthalpyh=²+p/ρ. Thus, we can writecp in the following way:
2Note that likeJ,Kis a Jacobian matrix, but for the inverse transformation. Since the EOS is smooth and well-behaved, we do not need to worry thatJmaybe singular.
A.4 Adiabatic temperature gradient
A.4 Adiabatic temperature gradient
To derive an expression for the adiabatic temperature gradient ∇ad, let us consider Eq.
(A.2). Since²(T, p)andρ(T, p)are functions of temperature and pressure, we can express d²anddρin terms ofdT anddp, which yields:
where we have made use of Eq. (A.8). Settingds= 0, we obtain the following expression for∇ad:
Chandrasekhar’s adiabatic exponents (Chandrasekhar 1957) are defined as γ1 := calculate∇ad, calculatingγ2 is trivial. The first adiabatic exponent gives the response of pressure to change in density in isentropic expansion/compression. It is a useful quantity because sound waves consisting of adiabatic perturbations propagate through the medium with a speed
cs = rγ1p
ρ . (A.17)
To calculate this quantity, we follow the strategy of § A.2 and make use of inversions of the Jacobian matrix. Here, we useρand²as the fundamental thermodynamic variables.
The differentials of specific entropy and pressure are given by µ ds
M =
One can use Eqn. (A.16) to calculateγ3. Alternatively, one can express the differentials dsand dT in terms ofd²and dρand the relevant partial derivatives, and then invert the resulting Jacobian matrix to findγ3.
B Diffusion of a magnetic structure with a Gaussian profile
In the absence of motion and for constant magnetic diffusivityη, the Induction Equation (3.1) reduces to
∂B
∂t = −η∇ × ∇ ×B, (B.1)
= η∇2B, (B.2)
where∇2 is the Laplacian operator. This equation establishes the fact that the diffusion of each of the cartesian components ofBis decoupled from the other two components.
In other words, we have a scalar diffusion equation for each cartesian component.
Consider a magnetic flux tube with the longitudinal component of the magnetic field described by the Gaussian profile
Bl(r, t= 0) = B0e−
r2 R2
0. (B.3)
The diffusion of the longitudinal component is not influenced by the transverse compo-nents of the field. We seek a self-similar solution of the form:
Bl(r, t) = Φ0 Substi-tution of (B.4) into Eq. (B.2) yields the simple ordinary differential equation forR(t)
RdR
The field strength at core of the flux tube (r= 0) follows Bl(r= 0, t)
This equation describes the weakening field strength due to diffusion. In order for dif-fusive effects not to weaken the tube significantly within a specified time intervalτ, we
B0−Bl(r= 0, t=τ)
B0 ¿ 1 (B.8)
→η ¿ R20
4τ. (B.9)
C Magnetic field extrapolation
In the study of solar magnetic fields, one is often restricted to measurements of the vertical component of the field on the solar surface. However, one would like to know how the magnetic field is structured in the chromosphere or corona. By assuming that the magnetic field above the surface is potential or force-free, one can extrapolate the field to the higher layers of the atmosphere.
C.1 Potential field
Consider the vertical component of the magnetic field distributed over the planez = 0.
We assume that the field distribution is periodic in thexandydirections with periodsLx andLy, respectively. If we assume that the magnetic field at and above the surface (z ≥0) is potential, we can write
B(x, y, z) =−∇Φ (C.1) whereΦis a scalar potential. To findΦ, we take the divergence of the previous equation:
∇2Φ = 0, (C.2)
and solve it subject to the boundary conditions
−∂Φ
∂z =Bz(x, y, z = 0). (C.3) andB → 0asz → ∞. Since the magnetic field is periodic in the horizontal directions, we use Fourier transforms to solve this problem. Let
F(kx, ky) = FT {f(x, y)} (C.4)
= Z Z
f(x, y)e−i(kxx+kyy)dxdy (C.5) denote the two-dimensional Fourier transform of the spatial function f(x, y). Inverse Fourier transforms are denoted by the symbolFT−1.
Let us make the following Ansatz for the scalar potentialΦ:
Φ(x, y, z) =FT−1{A(kx, ky)
|k| e−|k|z}, (C.6)
whereA(kx, ky)is a function over Fourier-space(kx, ky)and|k|=p
k2x+kz2the wavenum-ber. Applying boundary condition (C.3), we obtain
A(kx, ky) =FT {Bz(x, y, z= 0)}, (C.7)
Φ(x, y, z) = FT−1{FT {Bz(x, y, z = 0)}
|k| e−|k|z}. (C.8)
We don’t actually need to evaluateΦto calculate the magnetic field components. Substi-tution of (C.8) into (C.1) gives The factor e−|k|z in these equations tells us that small-scale features in the surface field are smoothed out with increasing height.
C.2 Linear force-free field
A force-free field is one such that the Lorentz forcej ×B/cvanishes. A potential mag-netic field configuration is an example of a force-free field. Linear force-free fields are magnetic field configurations satisfying the condition:
∇ ×B=αB, (C.12)
whereαis constant. As such, linear force-free fields are also called constant-αfields in the literature. Given the appropriate boundary conditions on a boundary ∂R enclosing the volumeR, one can solve for the force-free field enclosed in the volume. In the same manner as section C.1, suppose we are given a vertical fieldBz(x, y, z = 0)distribution over thex-yplane. This field is periodic in both horizontal directions with periodsLxand Lz. Furthermore,B →0asz → ∞.
Now we make the following Ansatz for the solution:
Bn(x, y, z) = FT−1{An(kx, ky)e−|k|z}, (C.13) where
An(kx, kz) = FT {Bn(x, y, z = 0)}. (C.14) For the moment, we do not specify the exact form of|k|. Since Bz(x, y, z = 0)is given as a boundary condition,Az(kx, ky)is known. So our goal is to solve forAxandAysuch that the field given by Eq. (C.13) satisfies the linear force-free condition. Substitution of Eq. (C.13) into Eq. (C.12) gives:
²srnAn ∂
∂xr
£ei(kxx+kyy)−|k|z¤
=αAsei(kxx+kyy)−|k|z, (C.15) where²srnis the Levi-Civita symbol. Writing out the three equations explicitly, we have:
C.2 Linear force-free field
Eq. (C.16) is a system of three linear equations in two unknowns (since Az is an im-posed boundary condition). In order for this system of equations to be consistent, the determinant of the matrix must be zero. This is equivalent to the constraint
|k|2 =kx2+ky2−α2. (C.17) For a decaying solution (i.e.B → 0 as z → ∞), we require that |k| be real and non-negative for all possible values ofkxandky. This restricts the solution space to values of αsatisfyingα2 ≤min{L−2x , L−2y }. Another restriction is that the net vertical flux through the planez = 0be zero (i.e.Az = 0forkx =ky = 0). The solution to the problem is
Ax = −i(|k|kx−αky)
kx2+k2y Az, (C.18)
Ay = −i(|k|ky +αkx)
kx2+k2y Az. (C.19)
For the caseα = 0, we obtain the potential field solution.
Publications
Publications in refereed scientific journals
• M. C. M. Cheung, F. Moreno-Insertis & M. Schüssler, “Moving magnetic tubes:
fragmentation, vortex streets and the limit of the approximation of thin flux tubes”, Astronomy & Astrophysics, in press.
• J. L. Caswell, N. M. McClure-Griffiths, N. M. & Cheung, M. C. M., “Supernova remnant G292.2-0.5, its pulsar, and the Galactic magnetic field”, Monthly Notices of the Royal Astronomical Society, Volume 352, Issue 4, pp. 1405-1412.
Contributed papers in conference proceedings
• M. C. M. Cheung, M. Schüssler & F. Moreno-Insertis, “Flux emergence at the pho-tosphere”, in: J. Leibacher, H. Uitenbroek and B. Stein (eds.) Solar MHD: Theory and Observations - a High Spatial Resolution Perspective, ASP Conf. Ser., in press.
• M. Cheung, M. Schüssler & F. Moreno-Insertis, “3D magneto-convection and flux emergence in the photosphere”, in: D. Danesy (ed.) Chromospheric and Coronal Magnetic Fields, ESA SP-596.
Acknowledgements
I would like to thank the Max Planck Institute for Solar System Research (MPS) and the International Max Planck Research School (IMPRS) for giving me the opportunity and the financial support to carry out the research presented in this dissertation.
I am indebted to my supervisors Prof. Manfred Schüssler and Prof. Fernando Moreno-Insertis (IAC, Tenerife) for their continued support, their patience and for all the stimulat-ing discussions that nurtured my fascination of solar physics.
I wish to thank Prof. Franz Kneer for accepting me as his student at the University of Göttingen and for always being helpful. I have learned a great deal from my discussions with him.
I must thank the coordinator of the IMPRS, Dieter Schmitt, for keeping the research school in such good order. This is one fine place to do a PhD.
Special thanks goes to Alexander Vögler and Robert Cameron. Both have taken a barrage of questions from me about MHD and numerical codes. The occasional sarcastic com-ment spurred me on.
A big thank you and a big hug to all the nice people in Lindau. Lindau is bearable (even wonderful!) because of you. Tien and I will never forget the bouquet you gave us.
I want to finish this dissertation so I can only name a few: Katerina, Durgesh, Fu, Hebe and Luciano, Shibu & Co., Monica and Martin T.
I thank my parents for their support through the years. I appreciate the freedom you have given me to pursue my studies in physics.
Finally, I want to thank my wife Tien for showing me all the love and support I need.
Lebenslauf
Name: Chun Ming Mark Cheung
Geburt: Am 02.04.1982 in Hong Kong Staatsbürgerschaft: Australisch
Schulbildung: 06/1996 bis 11/1998 Norwood Morialta High School, Adelaide South Australian Certificate of Education Studium: 03/1999 bis 11/2002 Physikstudium an der University of Adelaide
Bachelor of Science (Honours) Honours Dissertation:
“Supernova remnants and the interstellar medium”
02/2003 bis 02/2006 Doktorarbeit
Während dieser Zeit Tätigkeit als
wissenschaftlicher Mitarbeiter am Max-Planck-Institut für Sonnensystemforschung
in Katlenburg-Lindau