Güneş koronasındaki manyetik alan yapılarında magnetohidrodinamik denge ve kararsızlıklar

Abstract

Bu çalışmada güneşin korona bölgesindeki plazma içeren kapalı manyetik alan yapılan ele alınmaktadır ve güneş atmosferindeki iki boyutlu yapıların magnetostatik dengesi ideal stabilite problemi olarak incelenmektedir. Bu denge problemi özel olarak halka benzeri yapılarda örneklenmeye çalışılmaktadır. Güneşin yüzeyindeki ve atmosferindeki yapı oldukça karmaşıktır. Manyetik alanlar, güneş olaylarının ve özellikle de koronada meydana gelen güneş alevleri gibi ani ve hızlı bir biçimde ortaya çıkan patlamaların en önemli nedenidir. Bu alanların etkisiyle plazma hareketinde karmaşık etkileşimlere yol açan ve enerjinin bir kaynaktan diğerine taşınmasına neden olan olaylar meydana gelir. Güneş koronasında magnetize olmuş plazmayı kararsız yapan koşullan belirlemek önemli bir problemdir. Güneş atmosferindeki manyetik yapılar günlerce veya haftalarca denge konumunda kalabilirler, ve böylece çok miktarda manyetik enerji depolarlar. Sonradan bir tetikleme ile bu enerji bir güneş alevi veya bir prominans olarak açığa çıkar. Koronadaki büyük miktarlardaki enerjinin açığa çıkmasına, içinde plazmayı hapsetmiş olan manyetik alan konfigürasyonlarındaki bazı kararsızlıklar neden olmaktadır. MHD kararsızlıkları plazma içindeki büyük ölçekli kararsızlıkları yaratır. Çalışmada iki boyutlu kartezyen koordinat takımında izotermal atmosfer yaklaşımı içinde "aynksız" manyetik alan yapılan, gravitasyonun da etkisi gözönüne alınarak magnetostatik denge durumu için hesap edilmektedir. Burada fotosfer aşağı sınır şartlan oluştururan katman olarak ele alınmaktadır. Daha sonra magnetohidrodinamik denklemleri aracılığı ile elde edilen denge durumunun, belirli sınır şartlan altında verilen pertürbasyonlara karşı kararlılık analizi yapılmaktadır. Manyetize olmuş plazma içeren bu iki boyutlu denge sistemleri için kararlılık koşullan, MHD enerji ilkesi aracılığı ile araştırılmaktadır. Sonuç olarak güneşin atmosfer koşullarındaki bir sistemde, manyetik alan, basınç ve gravitasyonel alan etkileşimleri bulunup, sistemin kararlılığı incelenmektedir. Sonuçta bu varsayımlar altında korona halkaları için bulununan magnetostatik dengenin yerel modlara karşı kararsız olduğu görülmektedir. Kararsızlık analizi sonucunda potansiyel alan durumu hariç tüm magnetostatik çözüm ailesi, yatay deplasmanlara karşı kararsız kalmaktadır. Basıncın arttırılması ile birlikte korona halkalarında ilk manyetik adaların ortaya çıkması ile sistem marjinal kararsız duruma düşmektedir; daha büyük basınçlarda manyetik adalar çoğalmakta ve sistem kararsız olmaktadır. Korona halkalarındaki bu kararsızlıklar büyük güneş alevlerini oluşturacak etkiye sahip olmasalar da global olarak taşınım sabitleri üzerinde etkilidirler.

In this study, magnetic field structures confining the plasma in the solar corona and magnetostatic equilibrium of the two dimensional solar atmospheric structures are being investigated as an ideal stability problem. Loop-like coronal structures are given as samples for these equilibrium problems. Sun is the only star that may be observed in detail by ground based instruments or via satellites. Solar physics may be used in understanding the other stars and helps developing the basic physical concepts. The solar structure is rather complicated over the solar surface and in the solar atmosphere. The magnetic field is the most important cause for the solar events and especially for rapid eruptions as the solar flares. As a result of these effects, events, causing complicated interactions in plasma movements and causing transportation of the energy from one source to the other, arise. It is an important problem to obtain the conditions that cause instability in the magnetized solar corona. The magnetic structures of the solar atmosphere may keep their stability for days or weeks. Thus, they store a large amount of magnetic energy. After a trigger, this energy may stem out as a solar flare or prominence (Sakurai,1989). Gravitational force stratifies the photospheric and the chromospheric gases. Therefore, the gas density and pressure decrease with height. The balloon shaped magnetic field applies a mechanical pressure. Thus, it responds the pressure variations at the boundary. This results in an increase of the heights of the isolated dense flux tubes and solar spots that fill the whole space and affect the adjacent tubes and spots. These events occur above the minimum temperature. Thus a large volume in the chromosphere and the whole corona are filled with the magnetic field. The chromosphere and the corona are under the effect of magnetic forces. Magnetic fields cause the observed structural shapes and the plasma movements. These fields also cause the nonhomogeniety in the solar atmosphere. There are two important regions in the solar plasma due to the magnetic fields structures: The first one is "the closed field regions" which connect the photospheric- chromospheric region by both ends of the field and the second one is "the open field region" where one end is at the lower atmosphere, and the other end extends to the space through the upper atmosphere. The density and temperature show non- homogeneous structures in the closed field regions. The magnetic fields emerging from the solar surface, make a loop and return back to the solar surface. These closed magnetic field lines are "the coronal loops". The coronal plasma is trapped in the coronal loops. The coronal loops observed over the solar spots are called "active regions". The open field regions are the coronal holes that are the source of the solar wind. The magnetic field in the coronal loops is stronger than in the coronal holes. The field strength in the loops is about 10 G, and 10 G in the holes. The temperature is about 2-3x10 K in coronal loops, and -1.5 x 10 K in the coronal holes. As for the density in corona, it is 109-1010 cm"3 in the loops, and this quantity is less than that for loops in the open field region with 108 cm"3. The essential properties of sun, especially the characteristic dynamical behavior of corona and sun spots has been obtained with high sensitive, wide range observations by the Soft X-ray Telescope carrying with satellite Yohkoh. For the first time the interesting dynamical behaviors and interactions in not only the "active corona region", but also in "quiet background corona" have been obtained by the Yohkoh- STX with the use of video-movie representations. It can be said that "quiet background corona" is not quiet and isolated at all, but rather, forms a dynamical system with the "active corona region" (Linsky, Serio,1993). One of the most important activation movements in corona is solar flares. This event is an energy liberation coming after a slow phase energy build-up above the active 32 8 2 regions. A large flare can release an energy about 10 erg over the surface of 10 cm in some minutes. Sometimes a large amount of material (1015 gr) is also released with a speed of 100 km s"1. This process is called "coronal mass ejection" (CME). The heating of the corona from its quiescent 2 MK state to a 20-30 MK flare state indicates that the density of energy that produces flares must greatly exceed the thermal energy density of the quiescent coronal plasma. The most dominant source of energy in corona is the magnetic field. The energy that is necessary for the flare is stored in the magnetic field structure under the tension. So the flare activation is thought to be a liberation of energy stored as magnetic energy in corona. The other possible mechanisms likely to be responsible for coronal heating are quiet weak to explain the observed energy. For example over the active region, thermal energy in the corona is defined as Eth=(3nkT)V where n, T and V are respectively density, temperature and volume. When the values are taken as nV~\Q and 7=10, this yields Etk~\Q2S- This resulting value is about one of third or one of forth of the energy released during a large flare. The other source of energy is gravitational energy (Egrav=nVgh) resulting the value of 10 erg. However, the energy storing at the rate of (IT/&tt)V for 5=10 G gives 10 erg. The magnetic energy resulting from these rough calculations is adequate for a flare (Schmelz, Brown, 1994). The energy in magnetic field structure is not released totally. The ground state is potential state in which there is no current. Energy is loaded over this state to be converted to the other energy forms. Therefore, the flares must emerge, in the non- potential coronal regions where there are currents and magnetic shears (Lothian, Hood, 1991),. The small flares may occur as a result of photospheric movements. For these flares, there is no need to store much energy. Some minor non-potential situations can be enough for this flaring. Whereas, the condition for a large flare to occure is free xi energy storing in non-potential region in time. It is observed that large flaring is also seen to be associated with strong magnetic gradients. Electric currents are the source of magnetic energy in the corona. When no electric currents exist in corona, the magnetic field in the corona is totally due to the currents flowing in the lower layers (e.g., in the photosphere and below) of the solar atmosphere. These magnetic fields are not distorted in shape and they do not contribute to any of the events releasing high energy in corona. When these currents occur in corona, the coronal magnetic fields are distorted and this results in a flare of high energy. The magnetic field in corona is anchored in the dense photosphere. The mechanisms that create fluid motions such as the flow of gas in the photosphere and convection affect the motions of the foot points to which the coronal magnetic fields connect. These mechanisms result in evolution of the field structure. Such a situation may be modeled by taking the photosphere as the lower boundary on which the boundary conditions can be specified. Therefore, the slow build-up of magnetic energy in the corona reflects the slow change in the boundary conditions near the photosphere. During the evolution time the magnetic energy is preserved in the structure. When the field structure is unstable, this energy is released as heat or motion. If the energy release occurs gradually, this results in coronal heating. But if it occurs rapidly and suddenly then, this results in flaring. "Two ribbon flare" occurs in the arcade shape structure as an example of violent flare events. However, in the coronal loops less energetic compact flares occur (Priest, Kirk, Melrose, 1994). Plasma instabilities are generally based on perturbation theory. If the given initial perturbation grows, plasma is called "unstable". A growing perturbation changes the plasma properties. These can be temperature, density, isotropy and uniformity. There are two types of instability: Configuration-space and velocity-scape instabilities. Configuration-space instability is a macro instability and this is due to the plasma structure in the configuration space. For instance, if the plasma is in a finite volume and expands or moves from one place to an other, unstable situation can occur. Coronal field structure is investigated in terms of this large scale instability. In coronal loop-like structures, the type of instabilities such as Rayleigh-Jeans, kink and interchange can occur (Pekünlü,1993). Astrophysical applications of MHD (magnetohydrodynamic) theory is quite different from the applications in laboratory. For example, in the solar atmosphere, coronal loops, filaments, prominences may have long life-time and they can be regarded as stable structures. The essential problem of solar MHD theory and MHD applications to other astrophysical situation is to establish the models of these structures such as coronal loop and solar wind. Instabilities in these systems can be observed, only if system is destroyed or broken out to space. Even if some system parameters change infinitesimally in time, the parameter reaches eventually at some critical value and this results in massive destruction of solar structures (Pekünlü,1993). To study the stability and instabilities in magnetic field configuration, first of all, these structures must be constituted theoretically. Then, unstable behavior of the structures to some critical parameters can be investigated. xii In this study, the problem of magnetostatic equilibrium of two dimensional structures in an isothermal atmosphere with gravitational field is examined as a theoretical example. This is an ideal magnetostatic equilibrium that does not include resistive and viscous effects (Velli, Hood, 1987). The formulation of this equilibrium is applied to loop-like structures in the solar atmosphere. A family of analytic solutions in Cartesian coordinates is used to illustrate the effects of an inhomogeneous distribution of plasma within magnetic loops. The structures given by these solutions provide understanding to the interaction between gravitational, pressure and magnetic forces in solar atmosphere. Also, the stability of the structures in equilibrium under these influences can be investigated. The field lines lie in the plane perpendicular to the z axis and the components of B are the functions of jc and y only; B = xBx (x,y)+yBy(x,y), and the vector potential is in the form of -A = z A{x,y). For this magnetostatic equilibrium problem, magnetic field structure satisfies the non linear differential equation V2A= -f (A)e'(y/H) where H is the scale-height in an isothermal atmosphere, and y is the height from the photosphere that is the lower boundary. As a simplifying assumption, the function, f(A)=a2A is considered. Here a2 is a constant. A family of the analytic solution is derived by solving the differential equation. The vector potentials considered by Zweibel and Hundhausen (1982) depend on three parameters: a horizontal wavenumber k, a constant a2 that is a measure of the current density, and the amplitude A 0. The potentials are A(xy)=A0J2kH(2aHe(y/2H)) cos kx, a2>0 (1) A(xy)=A0 I2kH(2\a\He(y/2H)) cos kx, a2<0 where J2m and hm are the ordinary and modified Bessel functions of the first kind, respectively. The atmosphere extends from y=0 to infinity. As a result, the magnetic field components and pressure (or plasma density) distribution are Bx = - = -\a\e-{y/2H)A0Z;kH(2\a\He-(y/2H))coskx, dy v / By - - = kA0Z'im (2|a| He~iy/2H) )sin kx, (2) dx ( a2B2Z22kH(2\a\He