Dynamics of coronal loops oscillations

Transcription

1 Katholieke Universiteit Leuven Faculteit Wetenschappen Centrum voor Plasma-Astrofysica Dynamics of coronal loops oscillations De zon en nog iets verder... Tom Van Doorsselaere Promotor: Prof. dr. S. Poedts Co-promotor: Prof. dr. M. Goossens Juryleden: Prof. dr. R. Keppens (K.U.Leuven) Dr. R. Van der Linden (Director of the Royal Observatory Belgium) Dr. R. Oliver (Universitat de les Illes Balears, Spain) Dr. V. Nakariakov (University of Warwick, UK) Proefschrift voorgelegd tot het bekomen van de graad van Doctor in de Wetenschappen: wiskunde Leuven, 16 juni 006

2 ISBN Wettelijk depot D/006/10.705/3

3 Zwartkijken, Franquin, uitgeverij Arboris B.V.

4 Thanks I would like to thank my supervisor Stefaan Poedts. I want to make sure he knows that I realize that without him this work would not exist. When I was too far astray, he put me back on the right tracks with valuable comments. I also want to thank Marcel Goossens. Without him asking whether I was interested in applying for a PhD position at the Centrum voor Plasma- Astrofysica, I would not be writing this. Also, during my undergraduate course, he taught me my first plasma physics. Although, at that time, it was not my main interest, in retrospect, I m very grateful for these first steps. For a fruitful collaboration, I would like to thank Jesse Andries and Iñigo Arregui. I hope we continue to work together in the next few years. I want to thank my office mates Olena Sirenko and Paula Copil, and the rest of the present and former members of the Centrum voor Plasma-Astrofysica, for all scientific discussions and non-scientific breaks. For the emotional support, I d like to thank my Tuesday night board gaming friends, my Magic playing friends and my Friday night friends. Also, I want to send my special thanks to my brothers, sisters, parents and family-in-law. I have a magnificent wife Inge, who listens to what I have to say and who supports me in my scientific career. Thanks for that and much more!

5 Contents 1 Introduction The Sun Corona MHD equations MHD waves Ideal MHD Resistive MHD Quasi-modes and the Alfvén continuum Loops with a discontinuous density profile Loops with a quasi-discontinuous density profile Coronal loop oscillations & seismology Numerical tools: LEDA and POLLUX 31.1 Introduction LEDA The Finite Element Method (FEM) Essential boundary conditions POLLUX From 1D to D Quasi-modes in fully inhomogeneous loops Introduction & motivation Results Convergence in η The l R-dependency The k z -dependency Combined l R - and k z-dependency The ζ-dependency Summary & Conclusions

6 6 CONTENTS 4 Changes in the resistive spectrum Introduction Equilibrium & Equations Modifications to the spectrum Evolution of the resistive Alfvén modes Evolution of the quasi-mode Consequences Conclusions D loops: the effect of curvature Introduction Toroidal coordinates, equations & equilibrium Solutions in regions with a constant Alfvén frequency External region Internal region Matching the solutions First-order approximation Modifications to eigenfrequencies Modifications to eigenmodes Solutions including an inhomogeneous layer Realistic coronal loop models Conclusions D loops: analytical results Introduction Model and linearised equations Derivation of the dispersion relation Eigenmodes of the Alfvén operator Fast eigenmodes of radially uniform loops The thin boundary approximation for the fast quasi-modes Linear expansion in α n Numerical results for loop parameters Summary and Conclusions D loops: parameter study Introduction Equilibrium and linear resistive MHD waves Numerical Method Numerical Results Stratification and length of the loop Stratification and inhomogeneity length-scale Stratification and density contrast Fully non-uniform coronal loops Summary and Conclusions

7 CONTENTS 7 8 D loops: application Introduction Physical model, linear MHD waves and numerical solutions Analysis Seismology using the damping rate of loop oscillations Seismology using the period and damping rate of loop oscillations Summary and Conclusions Summary Nederlandstalige samenvatting 131 A Complexity of parameter studies using LEDA or POLLUX 133 B Expansions for d log F e,m du 135 Bibliography 140

8 8 CONTENTS

9 Chapter 1 Introduction 1.1 The Sun The Sun provides the Earth with the necessary energy to sustain life. If the Sun would cease to exist, the Earth would cool down to temperatures slightly above the absolute zero. As such, the Sun is a mother to us all and without it, nobody can survive. At first sight, the Sun may seem a rather dull astrophysical body: in visual light a few sunspots appear and disappear, apparently without any consequences. However, because it is so close, the Sun is the best observable star and images with a high spatial resolution are available. Since our Sun is a common, main sequence star, it is an excellent space laboratory unveiling the mysteries of other stars as well. The Sun has the internal structure that consists of several concentric layers like an onion (see Fig. 1.1). In the core, energy is produced through nuclear fusion. Surrounding the core, a radiative zone is found. In this layer, the energy is transmitted radiatively. When the opacity of the gas is too high, radiative energy transport is not efficient enough. As a result, convection is initiated. In the so-called convection zone, the energy is transported in rising hot gas bubbles, complemented with cool dropping material. The layer from which the visual light originates is called the photosphere, which is the actual observed surface of the Sun. Throughout all the above-mentioned layers, the temperature decreases from 10 7 K in the core to 6000 K at photospheric levels (the further away from the heat source, the cooler the layer). The photosphere is surrounded by the chromosphere. In the chromosphere, a slight temperature increase is observed (to 10 4 K). Outside the chromosphere, a narrow transition region is found where the temperature steeply increases from its chromospheric value to an average of 1 MK in the corona (Priest 1984), and temperatures as high as 10 MK may be reached.

10 10 Introduction Figure 1.1: An overview of the different layers of the sun. (source: 1. Corona The Sun is surrounded by a high temperature, low density corona. This corona is a dynamic, structured plasma cloud around the Sun governed by the ubiquitously present magnetic field. In visible light, the radiation from the solar disc is much stronger than the light we receive from the corona. Only during a total solar eclipse the corona is visible with the naked eye (see Fig. 1.). At other times, the corona has to be observed by a coronagraph (i.e. a device which covers the solar disc) or in other wave lengths. Since the corona is so hot, it is often observed in extreme ultraviolet (EUV) and X-rays. In these high energy, short wavelength pass-bands, a wealth of structures are observable. In the radio pass-band, a high time resolution is obtained by measuring the gyrosynchrotron emission of electrons (i.e. radiation generated by the acceleration of electrons). The solar corona provides us with several enigmatic problems, such as the physics of solar flares. Solar flares are disruptive and explosive events and are the most energetic and most powerful events in our solar system. They are often associated with a violent coronal mass ejection (CME). When these CMEs interact with the Earth s magnetosphere, a geomagnetic storm is caused. Such geomagnetic storms are known to affect and sometimes disrupt satellites and

11 1. Corona 11 Figure 1.: A solar eclipse, with a prominently displayed corona. power grids on earth. Also, they influence pipe lines, telecommunication and navigation systems. On a positive note, when a CME impacts with the Earth, beautiful aurorae are observed. Another unexplained problem is the high temperature of the corona which amounts to a few million Kelvin. To sustain a corona of the observed temperature a constant supply of energy must be available. It is clear that the chaotic motions in the convective region provide that energy. Nonetheless, it is still unclear how this energy is converted to heat. Several mechanisms have been proposed, the majority of them can be classified as either wave heating or impulsive heating. The impulsive heating mechanisms all claim that the reorganisation of the magnetic field by the turbulent photosphere is slow enough. By the turbulent motions of the photosphere, the coronal magnetic field lines are braided and twisted, which results in a storage of energy in the magnetic field. This magnetic energy is subsequently released in impulsive large scale violent events: solar flares.

12 1 Introduction Figure 1.3: A picture of coronal loops, as observed with the TRACE satellite. (source: ID=191) The wave heating mechanisms, on the other hand, suggest that the reorganisation of the magnetic field is too fast for the magnetic field to adjust. The turbulence in the corona thus generates waves along the magnetic field lines. These waves are eventually dissipated and generate heat and energy deposition in the solar corona. Of particular relevance for this thesis are solar coronal loops (see Fig. 1.3). Coronal loops are over-dense coronal structures aligned with the magnetic field. They are omnipresent and seem to be a basic building block of the solar corona. Often these loops are observed to oscillate after an explosive event. These oscillations are rapidly damped, and thus possibly deposit a substantial amount of energy in coronal loops. The mechanism behind the rapid damping of the oscillations is still under debate, but resonant absorption is a viable candidate. In this thesis, basic models for coronal loop oscillations involving resonant absorption are investigated and improved.

13 1.3 MHD equations MHD equations The mentioned coronal loop oscillations will be studied within the framework of magnetohydrodynamics (MHD). The MHD equations form a mathematical system to describe physical plasmas. A plasma is an ionised gas, which behaves collectively under the influence of the magnetic field. The plasma state can be looked upon as a fourth state of matter. On earth, most matter is either a solid, fluid or gas; a plasma occurs only in fusion experiments, lightnings, neon lamps, plasma TVs,.... In the universe, on the other hand, this fourth state of matter actually constitutes the vast majority of the visible matter. Because they are scale-independent, the MHD equations describe plasma phenomena ranging from extra-galactic jets (up to 10 3 m) to laser fusion experiments (10 7 m). They are only valid to describe phenomena with length-scales larger than the Debye length and oscillations with a frequency lower than the plasma electron frequency (see Goossens 003, for a detailed derivation). The MHD equations are given by: ρ t ρ d V dt dp dt B t ( = ρv ) = p + ρ g + 1 µ (1.1a) ( B ) B (1.1b) = γp V (1.1c) ( ) ( = V B η B ) B = 0. (1.1d) (1.1e) In these equations ρ denotes the plasma density, V the plasma velocity, p the gas pressure, g the gravity and B the magnetic field. The parameter µ denotes the magnetic permeability, γ is the ratio of specific heats and η is the magnetic diffusivity of the plasma. The above equations actually describe the interaction of a plasma with a magnetic field B. To study waves in the corona, often a static model suffices. On this static equilibrium model, small amplitude motions may be superimposed. The evolutions of these small amplitude motions is adequately described by the linearised MHD equations below. In these equations, the back reaction of the perturbations on the equilibrium is ignored. The linearised MHD equations for a static gravitationless equilibrium may be

14 14 Introduction given as: ρ t ρ V t p t B t = (ρv ), (1.a) = p + 1 µ ( B ) B + 1 ( B µ ) B, (1.b) = V p γp V, (1.c) ( = V B ) (η B ). (1.d) Additionally, the solenoidal constraint (Eq. 1.1e) also has to be fulfilled for B. ξ = r r is the Lagrangian displacement and V = ξ t. In a static equilibrium (which is assumed throughout the entire thesis), Eulerian and Lagrangian perturbations are equal (in the linear approximation). All primed quantities Q denote Eulerian perturbations of the equilibrium quantity Q: Q ( r, t) = Q( r, t) Q( r). When the resistivity η is constant in space, the last term in Eqs. (1.1d) and (1.d) reduces to η B, η B, respectively. Throughout the rest of this thesis, the linear MHD equations will be employed. Although the displacements of the loops seen in the TRACE movies (see also Fig. 1.3) are large, linear perturbation theory may still be valid. To investigate this, the radial velocity should be compared to a typical speed in the plasma, and not to the equilibrium value, which is zero in our case. As will be discussed in Sec. 1.4, the Alfvén velocity is such a typical velocity characterising a plasma. Taking numbers from Aschwanden et al. (00), we obtain Alfvén Mach numbers all smaller than 6%. This is a good indication that linear perturbation theory can still be used. 1.4 MHD waves In this section, the normal modes of the linearised MHD equations will be determined in a homogeneous plasma. We will assume that both the spatial and time derivatives of equilibrium quantities are zero. We roughly follow Goedbloed & Poedts (004) and Goossens (003) Ideal MHD First, we will consider ideal MHD, i.e. we will assume η = 0. Differentiating Eq. (1.b) with respect to time and using Eqs. (1.c) and (1.d) to eliminate

15 1.4 MHD waves 15 p and B yields: ρ V t = ( γp V ) + 1 ( ( ( V B µ ))) B. (1.3) We now adopt a Cartesian coordinate system with the z-axis coinciding with the homogeneous magnetic field so that B = B e z. Because all the equilibrium quantities are homogeneous, we can Fourier analyse V and define the x-axis to be in the plane of the wave vector k and B. Because we are searching normal modes, all perturbations can be put proportional to: V = V exp (i(k x x + k z z ωt)). With the use of this coordinate system, Eq. (1.3) can be written in terms of V : ( ρω V x = γp k ) xv x + k x k zv z + 1 µ k B V x, ρω V y = 1 µ k z B V ( y, ρω V z = γp k x k zv x + kz V ) (1.4) z, where k = kx + k z. ω V = AV, with A = k x This system can be written as an eigenvalue problem ( V s ) + VA + k z VA 0 k x k z Vs 0 kz V A 0 k x k z Vs 0 kz V s where we have defined the square of the sound speed as Vs of the Alfvén speed as VA = B µρ, = γp ρ and the square. As characteristic equation (i.e. the equation that has to be fulfilled to make the determinant of A zero), we find ( ω kz V A )) ( ω 4 k ( Vs + V A ) ω + kz k Vs V A) = 0, with roots ω A = k zv A, ω s,f = k ( V s + V A ± (V s + V A ) 4 k z k V s V A where we have defined the Alfvén frequency ω A and the frequencies of the fast (ω f with the +-sign) and the slow wave (ω s with the -sign). Alfvén waves From Eqs. (1.4) it is clear that for waves oscillating with the Alfvén frequency only the V y component is non-zero. Eq. (1.d) consequently yields that B y is the only non-zero component of the magnetic field perturbation. The other components of the corresponding perturbations vanish, only the components ),

16 16 Introduction perpendicular to both the wave vector and magnetic field survive in this case. Such waves are called Alfvén waves. Their group speed can be calculated as V gr,a = k ω A = ±V A e z. Because their group speed is directed along the magnetic field, it can be concluded that Alfvén waves only transmit energy along the magnetic field. Magnetosonic waves For the magnetosonic waves, all the components of the perturbations in the plane of the wave vector and the magnetic field are not identically zero. Using the factor of the characteristic equation containing the slow and fast frequency, it is possible to show that k x V s ω s,f k z V s From Eq. (1.4), we then find that V x = ω s,f k z V s kx V s = ω s,f k VA ωs,f. k x k z V z = ω s,f ω s,f k V A k x k z V z. Because the factor ω s,f ω s,f k V A is positive (negative) for fast (slow) magnetosonic waves, we see that the x-component of the velocity has the same (a different) sign than the z-component. The calculation of the group speed is more involved for magnetosonic waves (when compared to Alfvén waves), and the reader is referred to Goossens (003) for a detailed derivation. Summarising Goossens results, it can be concluded that the fast magnetosonic waves are fairly isotropic (the difference in group speed for perpendicular or parallel wave vectors never exceeds 41%) and that the slow magnetosonic waves are extremely anisotropic (no perpendicular propagation possible) and, analogous to the Alfvén wave, mainly guided along the magnetic field. When the plasma is not homogeneous anymore, the natural frequencies of the plasma become spatially dependent. The respective frequencies are not a point spectrum anymore, but form a slow, Alfvén continuum and fast point spectrum. As a consequence of the inhomogeneity, it is not possible anymore to strictly separate the three different waves. They will be mutually coupled and resonances will take place. Moreover, as explained in Sec. 1.5, quasi-modes can occur in this case.

17 1.4 MHD waves Resistive MHD Now, let us add a finite resistivity to the problem and examine the influence of this non-ideal effect on the spectrum. It is to be expected that the eigenmodes will be damped due to the finite dissipation. From now on, we will assume that the plasma pressure p vanishes. As a consequence, the sound speed V s = 0, and the slow waves are removed from the system. This assumption is justifiable by the fact that in the solar corona the plasma-β (the ratio of the gas pressure versus the magnetic pressure) is very small ( 0.01). In a first step, the eigenfrequency spectrum of a homogeneous resistive plasma will be determined. In a second part, the effect of an inhomogeneity of the plasma will be considered. Homogeneous resistive plasma The linearised MHD equations governing a pressureless and gravitationless plasma with a static equilibrium are ρ V t B t = 1 ( B µ ) B, ( = V B ) + η B. Employing the same coordinate system and Fourier analysis as in the previous section, these equations reduce to ( ( ω ω + iηωk ) kz V A) V y = 0, ( ( ω ω + iηωk ) k VA ) V x = 0. We again find two wave modes: the first equation determines the Alfvén waves, the second equation the fast magnetosonic waves. Focusing on the Alfvén waves, we find that ω ( ω + iηωk ) = ωa. Splitting the frequency in a real and imaginary part (ω = ω R + iω I ), we find There are two subspectra: ω R ω I(ω I + ηk ) = ω A, ω R (ω I + ηk ) = 0. ω R = 0 & ω I = ηk ± η k 4 4ωA, ω I = ηk & ω R + ω I = ω A. Because the real part of frequency is zero and the imaginary part of the frequency is negative, the first subspectrum represents non-oscillatory purely

18 18 Introduction ω I PSfrag replacements ω R Figure 1.4: The resistive Alfvén spectrum for a plasma with a finite extent. damped modes. The second subspectrum contains oscillatory damped modes of which the frequencies are located on a half circle in the complex plane with radius ω A. Since no spatial inhomogeneities are present, the wave numbers can take any arbitrary value. When the domain is limited by boundaries (for example a perfectly conducting wall at x = 0 and x = L), a quantisation will occur. In that situation, k x has to be an integer multiple of π/l. An example the resistive Alfvén spectrum of such a configuration is plotted in Fig Resistive plasma with a non-uniform density In this subsection, we will follow the analysis by Riedel (1985), Dewar & Davies (1984), and Hermans (1987). To gain a qualitative insight in the resistive MHD spectrum, we will study a pressureless, gravitationless plasma with a homogeneous magnetic field.

19 1.4 MHD waves 19 We adopt a Cartesian geometry, similar to the geometry studied in the previous section, again with the z-axis coinciding with the direction of the homogeneous magnetic field. In contrast to the previous section, however, we now consider a one-dimensional plasma density (i.e. depending on one spatial coordinate only) and we take the x-axis in the direction of the density inhomogeneity. We Fourier-analyse all quantities in the y- and z-direction (with wave numbers k y and k z ) and time. The governing equations for small amplitude disturbances in this coordinate system can be derived from Eqs. (1.): x (ρ V x x ωb x = Bk z V x + iη B x, (1.5a) ) ρk V x = k zb µω B x. (1.5b) We assume that the resistivity is small, and we introduce a WKB expansion, i.e. we assume that all perturbed quantities Q(x) have a dependency: Q(x) = Q(x) exp (Φ(x)/ η), where Q(x) and Φ(x) are assumed to be of the order 1 (in an expansion with respect to η). Substituting this formula in the governing equations (Eqs. 1.5), dropping the tildes and collecting the highest order terms (in η) we get: V x = kzb µρω B x, from Eq. (1.5b); ωb x = Bk z V x + i ( ) dφ dx B x, from Eq. (1.5a). Using the first equation to eliminate V x, yields: which is only solvable if: ( ) dφ ω B x = k z V A B x + iω B x dx, ( ) dφ = ω kz V A. (1.6) dx iω Zeros of the right-hand side are called turning points x A. For a monotonous density profile and a given (complex) frequency, at most one turning point can be found. Out of this turning point, anti-stokes lines emanate. Anti-Stokes lines are defined as the lines where 0 = Re(Φ(x)) = x x A ( ω kz V A (z) ) (1/) dz. iω

20 0 Introduction For example, in the simple case that both ω and x A are real and that V A (x) has a linear behaviour, Φ can be calculated to be: ( ) Φ(x) = ω kz V A (x x A) dv A (3/) dx, 3 iω dv A dx where all quantities are evaluated at x = x A. The real part of Φ(x) can only be zero in this case if arg(ω k zv A (x x A ) dv A dx ) = π + k π 3, (k Z). It is thus clear that three anti-stokes lines emanate from x A in this case. Since the solution specified by Eq. (1.6) is only valid in regions where Φ(x) is monotonically increasing, only one of the sectors between the anti-stokes lines yields a resistive solution. In the other sectors, solutions of the ideal MHD equations have to be used. Consequently, two situations can occur in a plasma of finite extent. Either the plasma domain is completely embedded in the sector of monotonically increasing Φ(x), or one of the anti-stokes lines crosses the plasma domain. In the first case (indicated as III in Fig. 1.5), the entire solution is completely determined by Eq. (1.6). An eigenmode then only occurs if this solutions fulfils the boundary conditions. These eigenmodes will exhibit a spatially oscillatory behaviour in the entire domain. As an example, the eigenmode belonging to the squared eigenvalue in branch III in Fig. 1.5 is plotted in the bottom panel of Fig In the second case (indicated as I or II in Fig. 1.5), the resistive solution has to be fitted with an ideal solution (outside the domain of validity of the resistive solution). These oscillations exhibit a damped behaviour at the endpoint not in the domain of validity and an oscillatory behaviour near the other ( resistive ) endpoint. Examples of this kind of eigenmodes are plotted in the top two panels of Fig. 1.6 and correspond to the squared eigenfrequencies in branches I and II in Fig The two kinds of solutions meet each other when the plasma domain is exactly contained within the domain of validity. It is thus clear that a discrete, three-branched spectrum is obtained. One branch contains the eigenmodes oscillatory at one endpoint, the second branch the solutions oscillating at the other endpoint and the third branch solutions oscillating in the entire domain. The first and second branch both meet the real axis at the Alfvén frequency belonging to the respective endpoints (the endpoints of the Alfvén continuum).

21 PSfrag replacements 1.5 Quasi-modes and the Alfvén continuum II I ωi III ω R Figure 1.5: A part of the resistive spectrum. The eigenmodes are plotted with +s, whereas the ideal Alfvén continuum is indicated with s. 1.5 Quasi-modes and the Alfvén continuum To construct resonantly damped quasi-mode oscillations, we now assume a cylindrical geometry (with a cylindrical coordinate system (r, φ, z)). As in the previous section, we neglect the gravitation and gas pressure. Moreover, we again consider a static equilibrium and we assume that the equilibrium magnetic field has only a z-component: B = B ez. Using these assumptions, the density profile can be chosen freely ρ = ρ(r). Since the equilibrium density profile depends on r only, the perturbed quantities can be Fourier-analysed with respect to the ignorable coordinates φ and z and put proportional to exp[i(mφ + k z z)]. Here m (an integer) and k z = nπ/l are the azimuthal and axial wave numbers. Then, the n = 1 case represents the observed fundamental mode with no nodes in the z- direction. Again, we can propose the following time dependence: exp[ iωt], i.e. we consider normal modes. The real part of the frequency is related to the oscillation period, P = π/ω R, while the damping time is related to the negative imaginary part of the frequency, τ d = 1/ω I.

22 Introduction vx PSfrag replacements v x vx 0 PSfrag replacements x v x vx PSfrag replacements 1e x e-04 5e-05 v x 0-5e Figure 1.6: The eigenmodes belonging to the squared eigenvalues, respectively in branch I, II, or III in Fig. 1.5 x Following Appert et al. (1974), Eqs. (1.) can be reduced to = ((ω A m ω ) r V A ρva (ω ωa )d(rξ r ) dr dp dr ) rp, (1.7a) = ρ(ω ω A )ξ r. (1.7b)

23 1.5 Quasi-modes and the Alfvén continuum 3 is the total perturbed pressure. These equations are comple- Here, P = BB z mented with ρ(ω ω A)ξ φ = i m r P, (1.7c) for the azimuthal component of the perturbation ξ φ Loops with a discontinuous density profile Eqs. (1.7) admit analytical solutions in terms of (modified) Bessel functions for uniform equilibrium models. When used for coronal loops these uniform cylindrical equilibrium models are characterised by a true discontinuity surface where the equilibrium quantities vary from their constant internal to their constant external values, respectively ρ i and ρ e in our case (uniform magnetic field). The true discontinuity is situated at r = R, where R is the loop radius. By combining the two first order Eqs. (1.7), one second order equation for P can be obtained, which, for constant equilibrium quantities, reduces to the modified Bessel s equation: with: d P dr + 1 r dp ( ) m dr r + κ P = 0, (1.8) κ = ω ωa VA. (1.9) For the internal region the solutions obeying the regularity condition at the axis are: P i = A I m (κ i r), ξ ri κ i = A ρ i (ω ωai )DI m(κ i r). (1.10) For the external region the solutions have to vanish at infinity (we neglect leaky waves) so that P e = B K m (κ e r), ξ re κ e = B ρ e (ω ωae )DK m(κ e r). (1.11) I m and K m are the modified Bessel functions of order m of the first and second kind, respectively. Df(r) denotes the derivative of f(r) with respect to the argument. κ is defined as the square root with positive real part of κ. The dispersion relation is obtained by matching the internal and external solutions. At the discontinuous boundary both ξ r and P have to be continuous. This leads to the dispersion relation: D(ω, m, k z ) ξ re (r = R) P e (r = R) ξ ri (r = R) P = 0, (r = R) i

24 4 Introduction which (with the above analytical expressions) becomes (Edwin & Roberts 1983): κ i DI m (κ i R) κ e DK m (κ e R) ρ i (ω ωai ) = I m (κ i R) ρ e (ω ωae ) K m (κ e R). (1.1) From the Bessel functions in Eq. (1.10), we can see that oscillations that displace the axis of the tube necessarily have an azimuthal wavenumber m = 1. For m = 0, radially expanding and shrinking cylinders are obtained. These modes are called sausage modes. For m, fluting modes occur, i.e. oscillations perturbing the interface between the interior and exterior without actually moving the center of the cylinder. Since we are interested in the fundamental mode, the length of a coronal loop has to correspond exactly to half the longitudinal wave-length. Therefore, we can appropriately express the longitudinal wavenumber as k z = π/l, with L the length of the coronal loop. As κ roughly scales with k z (cfr. Eq. 1.9), the arguments of the Bessel functions become small when the tube is long compared to its radius L R. For small arguments we can approximate DI m (z)/i m (z) m/z and DK m (z)/k m (z) m/z (except for m = 0). Thus in the thin tube or long tube approximation the dispersion relation for all m 0 modes becomes: D = m R ( 1 ρ e (ω ω Ae ) + 1 ρ i (ω ω Ai ) ) = 0, (1.13) which can readily be solved to yield the well known result: ρ i ωai ω = + ρ eωae = k z B. (1.14) ρ i + ρ e ρ i + ρ e In the thin tube (TT) approximation the frequencies of the eigenoscillations are independent of the azimuthal wave number m Loops with a quasi-discontinuous density profile Analytical solutions to Eqs. (1.7) do not exist in general for non-uniform equilibrium models. The dispersion relation cannot be written down in closed analytical form, except possibly for special choices of the equilibrium profiles. Analytic progress is still possible however when the true discontinuity is replaced with a thin non-uniform boundary layer [R l/, R + l/] of thickness l in which the equilibrium quantities vary continuously from their constant internal to their constant external values. The currently used equilibrium configuration is plotted in Fig. (1.7). A thin non-uniform layer means that l/r 1. The presence of a non-uniform layer alters the physics significantly. Instead of having separate, isolated internal and external Alfvén frequencies, all intermediate frequencies are now also attained. As indicated on Fig. 1.8, the Alfvén

25 1.5 Quasi-modes and the Alfvén continuum 5 z l ρi ρ e L R B B PSfrag replacements ϕ r Figure 1.7: Sketch of the equilibrium configuration representing a straightened coronal flux tube of length L and radius R modelled as a density enhancement. The magnetic field is uniform and parallel to the z-axis and the whole configuration is invariant in the φ-direction. The density varies in a non-uniform transitional layer of width l from a constant internal value, ρ i to a constant external value in the coronal environment, ρ e. frequency now becomes spatially dependent. As a result, when the plasma oscillates with a frequency ω, a resonance will occur at r = r 0. As a result of the resonance, a singularity will occur. This singularity is also prominently present in Eqs. (1.7): when ω = ωa, the factor in front of the highest derivative vanishes. Eqs. (1.7) can be expanded around the singularity. We take s = r r 0 as a small parameter and expand ω ωa = A s, where A stands for the derivative of ωa at r = r 0. Following Goossens et al. (1995), the highest order terms in s of Eqs. (1.7) are s A dξ r ds dp ds = m ρr0 P (1.15a) = ρs A ξ r (1.15b) s A ξ φ = i m r P. (1.15c) In Eq. (1.15b) it can be seen that, when s = 0, P is a constant. We can thus conclude that the perturbed total pressure remains constant across the singularity. On the other hand, Eq. (1.15a) shows that dξ r ds 1 s and, thus, that

26 6 Introduction ω A (r) ω Ae ω ω Ai r R l/ 0 R + l/ Figure 1.8: A spatially dependent Alfvén frequency for a linear density profile in the inhomogeneous layer. r ξ r exhibits a ln s -singularity across the resonance point. Moreover, from the last equation, we learn that ξ φ 1 s. This last solution is not square integrable anymore and cannot be accepted as a physical solution. When dissipation is taken into account, however, the singularity is removed from the equations and the variations around the resonance can be obtained analytically by a local asymptotic analysis. The variations in the resonant layer can be expressed as jump conditions over the resonant layer (see Sakurai et al. 1991): [P ] = 0, (1.16a) [ξ r] = πi m /r 0 ρ 0 A P. (1.16b) The subscript 0 denotes that the quantities are evaluated at the Alfvén resonant surface r = r 0. In this thesis, we neglect poloidal components of the equilibrium magnetic fields. When poloidal magnetic fields are considered, the above jump conditions (Eqs. 1.16) are modified and the perturbed pressure is not continuous across the resonant layer (Sakurai et al. 1991). In a plasma with constant magnetic field we can rewrite A : ρ A A = ρ 0 (kzb ) 1 dρ ρ 0 dr (r 0), = ωa0 ρ i ρ e, (1.17) l

27 1.5 Quasi-modes and the Alfvén continuum 7 where = l ρ i ρ e dρ dr (r 0), is the slope of the density profile normalised to the width of the layer and to the total density difference. It is a parameter that is only dependent on the shape of the density profile. Note that the jump conditions are independent of the value of the magnetic diffusivity. In fact they can even be obtained in ideal MHD (i.e. η = 0). Because the jump conditions (Eqs. 1.16) have a complex part, complex solutions of the dispersion relation can be found. These complex frequencies cannot be an eigenvalue of the Hermitian ideal MHD operator. Therefore they are called quasi-modes. These quasi-modes are the natural and collective oscillation modes of the system and correspond to eigenmodes in dissipative MHD in the limit of vanishing resistivity (Poedts & Kerner 1991). Taking into account the variations around the resonance (Eqs. 1.16), while ignoring all other variations in the non-uniform boundary layer, we can write down the dispersion relation as: l m /r0 D(ω) = πi ωa0 ρ i ρ e. (1.18) We then search for solutions in the neighbourhood of the solution for the discontinuous model ω = ω A0 + δω and expand D(ω) in a Taylor series to find the imaginary frequency shift as: π l m /r0 δω = ωa0 ρ i ρ e D(ω i. (1.19) A0) ω Note that D(ω) is a complex function. However since ω A0 is real and D(ω) is real for real arguments, together with the Cauchy-Riemann conditions this ensures that the derivative is real. Thus the resulting frequency shift δω is purely imaginary. The procedure to find quasi-mode frequencies in the thin boundary (TB) approximation thus goes as follows. Dispersion relation (Eq. 1.1) is solved numerically, and yields the real part of the frequency. Based on the real part of the frequency the resonant position is determined and Eq. (1.19) is used to calculate the frequency shift. An analytic expression for the damping rate can be obtained when the thin tube and thin boundary approximations are combined. Using Eqs. (1.13) and (1.14) to approximate D(ω A0), the frequency correction becomes: ω δω I = π m ( ) ( ) R l ρi ρ e ω A0. (1.0) 8 r 0 R ρ i + ρ e

28 8 Introduction Ruderman & Roberts write their corresponding equation in terms of l/a. Since l/r 1 and a = R + l/, the difference between the two formulae is second order and, therefore, they are equivalent in the TB approximation. Likewise R r 0. We thus obtain: δω I ω 0 = πm 8 l R ρ i ρ e ρ i + ρ e. (1.1) This relation shows that under thin tube and thin boundary conditions (TTTB) the damping rate is linearly proportional to the length scale of the inhomogeneity l/r. It was first obtained by Hollweg & Yang (1988) for nearly perpendicular propagation of a surface wave on a Cartesian interface. The result was also obtained by Goossens et al. (199) and recently retrieved by Ruderman & Roberts (00), who related it to the observed damping of coronal loop oscillations. 1.6 Coronal loop oscillations & seismology In 1999, coronal loop oscillations were observed for the first time by the TRACE spacecraft (Aschwanden et al. 1999). Since then, several oscillating loops have been reported and thoroughly studied (Aschwanden et al. 00; Schrijver et al. 00; Nakariakov et al. 1999). The observed coronal loop oscillations have been modelled as fast kink oscillations by e.g. Nakariakov et al. (1999), Ruderman & Roberts (00) and Goossens et al. (00) and as phase mixed torsional Alfvén waves (Ofman & Aschwanden 00). The rapid damping of the oscillations has been the subject of speculation. Nakariakov et al. (1999) concluded that Reynolds numbers smaller than the classical value of by 8 to 9 orders of magnitude, are needed to explain the rapid damping. A similar conclusion was drawn by Ofman & Aschwanden (00) who compared several damping mechanisms and, based on the observed periods and damping times, found phase-mixing of fast kink modes to be most likely. However, Goossens et al. (00) pointed out that damping by resonant absorption of quasi-mode kink-oscillations is also a very attractive explanation as it does not require to change the estimates of the Reynolds numbers. Like Ruderman & Roberts (00), they used the analytical formula for the damping rate (Eq. 1.1) to calculate the length scales of the inhomogeneity. They concluded that resonant absorption was ruled out by Ofman & Aschwanden, because they did not allow for the radial length scales to vary from loop to loop. Goossens et al. used the observed periods and damping rates combined with analytic results for thin loops with thin non-uniform layers to deduce the width of the non-uniform layer for 11 loops. Most of the values for the width of the non-uniform layers are too large for the TTTB formula to be an accurate

29 1.6 Coronal loop oscillations & seismology 9 approximation. Goossens et al. interpreted this as a motivation for an eigenvalue analysis for 1D non-uniform equilibrium states where the non-uniformity is not restricted to a thin layer. Eigenmodes of such highly inhomogeneous loop models have not been calculated before. As a result of Goossens et al. s urge for a numerical computation of the eigenvalues of fully inhomogeneous equilibrium models for coronal loops, the research project that led to this thesis was initiated. The results of these investigations and several numerical and analytical extensions of the standard model in Sec are reported in the subsequent chapters. An additional merit of studying and improving the models for coronal loop oscillations is coronal loop seismology. Through the remote sensing of waves in coronal loops, several properties of the equilibrium quantities can be obtained. Often those quantities (e.g. B, ρ) are difficult to measure directly. Coronal loop seismology provides an extra constraint to limit the range of the values of these quantities. In an early attempt, Nakariakov & Ofman (001) estimated the magnetic field to be 13 ± 9 G. Even though the error bars are huge, it nonetheless is an estimate and provides an order of magnitude for the strength of the magnetic field. Using improved and more accurate models in combination with better observational data the error bars will be decreased significantly. More recently, the value of the density stratification height was estimated by Andries et al. (005a). They found values of 36 Mm and 65 Mm in two different cases. These estimations are both compatible with the observational value of 50 Mm.

30 30 Introduction

31 Chapter Numerical tools: LEDA and POLLUX The core of the numerical tools used in this thesis was based on Kerner et al. (1985) for LEDA and on Van der Linden (1991) for POLLUX. The author of the present thesis implemented additional boundary conditions both in LEDA and POLLUX, and added a modern and more efficient eigenvalue solver to POLLUX..1 Introduction It is evident that the use of computers and numerical solutions has become an additional branch in modern research. Also in solar physics, numerical models have been used to overcome the difficulties in finding (semi-)analytical solutions to the MHD equations (Eqs. 1.1). Even for the simpler linearised ideal MHD equations (Eqs. 1.), it is not trivial to find closed formula solutions. The scientific results reported in this thesis have been obtained by applying three numerical codes to model coronal loops oscillations. Often these models extend previously studied analytical models into regions where they are not valid anymore, e.g. because the simplifying assumptions made to enable the analytical solution are not satisfied anymore. This thesis will employ two eigenvalue codes (LEDA (1D) and POLLUX (D)) to refine our current analytical models for resonantly damped coronal loop oscillations. The codes were originally developed by Kerner et al. (1985) and Van der Linden (1991) and have been used by several authors (see e.g. Poedts & Kerner 1991; van der Holst et al. 1999). The purpose of this chapter is to provide a general introduction to the codes, to acquaint the reader with the Finite Element Method (FEM), and to justify the boundary conditions and the surface terms used to implement these

32 3 Numerical tools: LEDA and POLLUX boundary conditions. It is not the goal of this chapter to recalculate all the matrix elements. For more detailed information about the numerical codes, the reader is referred to Van der Linden (1991).. LEDA LEDA stands for Large-scale Eigenvalue solver for the Dissipative Alfvén spectrum, and is a numerical code that solves the linear MHD equations as an eigenvalue problem. The author implemented a loop around the entire numerical code in order to efficiently execute parameter studies. Moreover, he implemented additional boundary conditions to investigate the MHD spectrum of plasma-vacuum-wall configuration...1 The Finite Element Method (FEM) As a first step to construct a 1D numerical code in a cylindrical coordinate system (r, φ, z), the linearised MHD equations (Eqs. 1.) are undimensionalised. Distances are normalised to the radius of the computational domain, magnetic field and densities to the values on the axis and times to the Alfvén travel time. The normalised equations are then rewritten in terms of other variables, viz. V 1 = rv r, V = iv, V 3 = irv ρ = rρ, T = rt, a 1 = ia r, a = ra φ, a 3 = A z, where A denotes the vector potential. It is defined so that B = A and it ensures that the solenoidal constraint (Eq. 1.1e) is automatically fulfilled. Furthermore, we consider a static cylindrical equilibrium in which the physical quantities only vary in the radial direction. It is assumed that the perturbed quantities f can be Fourier analysed in the trivial spatial directions and in time: f(r, φ, z; t) = f(r) exp (i(mφ + k z z ωt)), (.1) where ω is an unknown, to be solved for, complex frequency. As a next step, the equations are spatially discretised. For this purpose, a discrete grid is introduced as well as a set of finite elements. These functions are only different from zero on a very limited amount of grid-points. All the perturbed variables of the system are then expressed in terms of these finite elements: n u k (r) = u k j h k j (r). j=1 Here h k j (r) is the finite element on the jth grid-point for the kth variable uk in the state vector u = ( ρ, V 1, V, V 3, T ), a 1, a, a 3. For k = 1, 3, 4, 5, 6 ( ρ, V, V 3,

33 . LEDA 33 T and a 1 ) quadratic elements are used, whereas for k =, 7, 8 (V 1, a and a 3 ) cubic elements are used. In LEDA the finite elements h j (r) are only non-zero in the interval [r j 1, r j+1 ]. Per grid-point, two elements are needed to uniquely determine a variable. Consequently, by using the above expansion, 16N g unknowns u k j are introduced (N g is the number of grid-points). As a next step, the expansion for u k (r) is substituted in the linearised MHD equations. The equations are solved in the so-called weak form. This means that the equations are multiplied by all the 16N g finite elements and integrated over the whole domain. This standard solution strategy is known as the Galerkin method. Because of the localised nature of the finite elements, only a limited number of matrix elements survive, i.e. the coefficient matrix of the obtained system is sparse. The integral of the product of two finite elements is zero, unless their grid-points are adjacent. Since every grid-point has two neighbours in 1D and there are 8 unknowns on each grid point, a tri-diagonal block system is obtained. The blocks have a size of 16 16: two elements for each of the 8 variables in every interval. For the eigenvalue problem, this system is then written in the form: A U = λb U, (.) where U is the vector formed by u k j and iλ = ω. The obtained linear eigenvalue system can be solved by the Jacobi-Davidson algorithm (Sleijpen & van der Vorst (1996), but we used LEDAFLOW (Nijboer et al. 1997). For Jacobi-Davidson method, a target eigenvalue σ has to be provided. This target eigenvalue is then used to rewrite the system as: or (A σb)u = (λ σ)bu, 1 λ σ U = (A σb) 1 BU. This system is an alternative eigenvalue system. The eigenvalues are easy to compute, since the dominant eigenvalues are in the neighbourhood of the target eigenvalue σ... Essential boundary conditions Some boundary conditions cannot be implemented by adding contributions of surface terms to the matrices A and B. They need to be enforced by manipulating the space of finite elements, i.e. by deleting those elements that do not satisfy these boundary conditions. For example, at the axis of symmetry (r = 0), V r needs to be finite. This cannot be achieved by calculating the surface terms. The boundary condition is equivalent to putting V 1 = rv r to zero. To do this

34 34 Numerical tools: LEDA and POLLUX in an efficient manner (programming-wise), the original matrix equations will be erased and replaced by the equation MV 1 = λv 1, (.3) where M is an arbitrary complex number. Using Eq. (.3), an artificial eigenvalue M is introduced in the spectrum. Taking M to be out of the range of interest in the spectrum (e.g. M 1) implies that for all eigenvalues λ other than M, V 1 has to be 0 to satisfy Eq. (.3). Additionally, all contributions of V 1 to the other equations are put to zero. As such, the other equations are solved as if V 1 is exactly 0. Practically, all the matrix elements of the 3rd and 4th row and column (corresponding to V 1 ) in matrices A and B will be put to zero. The diagonal elements of the 3rd row will be made M and 1 in respectively A and B..3 POLLUX For this code, the author only implemented the new (previously described) Jacobi-Davidson solver. The old matrix elements remained intact..3.1 From 1D to D POLLUX stands for Program On Line-tied Loops Under excitation and is a generalisation of LEDA to D. It is also an MHD spectral code but additionally it allows for variation of the equilibrium quantities in both the radial and longitudinal direction. As in LEDA, the radial direction is discretised by using finite elements. In the longitudinal direction, however, a spectral discretization is used. Since now D equilibria will be allowed, it is not possible anymore to Fourier analyse in the z-direction and to study each Fourier mode in this direction separately. The z-dependence of the equilibrium quantities, and thus of the coefficients in the PDEs, causes a coupling of Fourier modes. Thus, in contrast to Eq. (.1), only the Fourier modes in the φ-direction and time can be studied separately. The state vector u now has to be written as a superposition of different longitudinal Fourier modes: u = N n= N U n exp ( inπz L ), where U n = ( ρ n, V 1n, V n, V 3n, T n, a 1n, a n, a 3n ). Analogous to the 1D case, the MHD equations are written down for each of these 8 (N 1) variables and discretised in the radial direction by using the

35 .3 POLLUX 35 finite element method. As a result of adding additional longitudinal Fourier modes, the dimension of the matrices A and B in Eq. (.) is now raised to 16N g (N 1). The matrices still have tridiagonal block structure (due to the finite elements) but the blocks now have a dimension 16(N 1). Additional boundary conditions are supplied. To mimic a line-tied loop, rigid wall boundary conditions are assumed and both the velocity and temperature perturbation have to vanish at the end points of our domain [0, L]: V (r, 0) = V (r, L) = 0 T (r, 0) = T (r, L) = 0. These boundary conditions can be fulfilled by demanding that ( r): N n= N V 1n (r) = N n= N V n (r) = N n= N V 3n (r) = N n= N T n (r) = 0. Practically, in the code this is achieved by subtracting the equation of component N from all the other components and, analogous to the essential boundary conditions in LEDA, replacing it by: M N n= N V 1n = λ N n= N V 1n. In contrast to the application in LEDA, this has to be done for all the gridpoints now, because the longitudinal boundary conditions have to be satisfied at each radial position.