**Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets**

Ivan Zhelyazkov *Faculty of Physics, Sofia University Bulgaria*

### **1. Introduction**

134 Topics in Magnetohydrodynamics

Yamada, M., et al., (2007), A Self-Organized Plasma with Induction, Reconnection, and Injection Techniques: the SPIRT Concept for Field Reversed Configuration

Research, *Plasma and Fusion Research*, Vol. 2, 004, pp. 1-14

One of the most enduring mysteries in solar physics is why the Sun's outer atmosphere, or corona, is millions of kelvins hotter than its surface. Among suggested theories for coronal heating is one that considers the role of spicules – narrow jets of plasma shooting up from just above the Sun's surface – in that process (Athay & Holzer, 1982; Athay, 2000). For decades, it was thought that spicules might be sending heat into the corona. However, following observational research in the 1980s, it was found that spicule plasma did not reach coronal temperatures, and so this line of study largely fell out of vogue. Kukhianidze et al. (Kukhianidze et al., 2006) were first to report the observation of kink waves in solar spicules – the wavelength was found to be ∼3500 km, and the period of waves has been estimated to be in the range of 35–70 s. The authors argue that these waves may carry photospheric energy into the corona and therefore can be of importance in coronal heating. Zaqarashvili et al. (Zaqarashvili et al., 2007) analyzed consecutive height series of H*α* spectra in solar limb spicules at the heights of 3800–8700 km above the photosphere and detected Doppler-shift oscillations with periods of 20–25 and 75–110 s. According to authors, the oscillations can be caused by waves' propagation in thin magnetic flux tubes anchored in the photosphere. Moreover, observed waves can be used as a tool for spicule seismology, and the magnetic filed induction in spicules at the height of ∼6000 km above the photosphere is estimated as 12–15 G. De Pontieu et al. (De Pontieu et al., 2007) identified a new class of spicules (see Fig. 1) that moved much faster and were shorter lived than the traditional spicules, which have speeds of between 20 and 40 km s−<sup>1</sup> and lifespans of 3 to 7 minutes. These Type II spicules, observed in Ca II 854.2 nm and H*α* lines (Sterling et al., 2010), are much more dynamic: they form rapidly (in ∼10 s), are very thin (200 km wide), have lifetimes of 10 to 150 s (at any one height), and shoot upwards at high speeds, often in excess of 100–150 km s−1, before disappearing. The rapid disappearance of these jets had suggested that the plasma they carried might get very hot, but direct observational evidence of this process was missing. Both types of spicules are observed to carry Alfvén waves with significant amplitudes of order 20 km s−1. In a recent paper, De Pontieu et al. (De Pontieu et al., 2011) used new observations from the Atmospheric Imaging Assembly on NASA's recently launched *Solar Dynamics Observatory* and its Focal Plane Package for the Solar Optical Telescope (SOT) on the Japanese *Hinode* satellite. Their observations reveal "a ubiquitous coronal mass supply in which chromospheric plasma in fountainlike jets or spicules (see Fig. 2) is accelerated upward into the corona, with much of the plasma heated to temperatures between ∼0.02 and 0.1 million kelvin (MK) and a small but sufficient fraction to temperatures above 1 MK. These observations provide constraints

coronal heating – the energy flux transported into corona was estimated to be of about 3 <sup>×</sup> <sup>10</sup><sup>5</sup> erg s−<sup>1</sup> cm−2, i.e., roughly half of the flux carried by the Alfvén waves running on Type II spicules (Moore et al., 2011). Tavabi et al. (Tavabi et al., 2011), performed a statistical analysis of the SOT/*Hinode* observations of solar spicules and their wave-like behavior, and argued that there is a possible upward propagation of Alfvén waves inside a doublet spicule with a

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 137

No less effective in coronal heating are the so called X-ray jets. We recall, however, that whilst the classical spicules were first discovered in 1870's by the Jesuit astronomer Pietro Angelo Secchi (Secchi, 1877) and named as "spicules" by Roberts (Roberts, 1945), the X-ray jets are relatively a new discovered phenomenon. They, the jets, were extensively observed with the Soft X-ray Telescope on *Yohkoh* (Shibata et al., 1992; Shimojo et al., 1996), and their structure and dynamics have been better resolved by the X-Ray Telescope (XRT) on *Hinode*, in movies having 1 arc sec pixels and ∼1-minute cadence (Cirtain et al., 2007) – see Fig. 3. According

Fig. 3. Three X-ray jets recorded by the *Hinode* spacecraft on January 10, 2007. Credit:

to Cirtain et al. (Cirtain et al., 2007), "coronal magnetic fields are dynamic, and field lines may misalign, reassemble, and release energy by means of magnetic reconnection. Giant releases may generate solar flares and coronal mass ejections and, on a smaller scale, produce X-ray jets. *Hinode* observations of polar coronal holes reveal that X-ray jets have two distinct velocities: one near the Alfvén speed (∼800 kilometers per second) and another near the sound speed (200 kilometers per second). The X-ray jets are from 2 <sup>×</sup> 103 to 2 <sup>×</sup> 104 kilometers wide and 1 <sup>×</sup> 105 kilometers long and last from 100 to 2500 seconds. The large number of events, coupled with the high velocities of the apparent outflows, indicates that the jets may contribute to the high-speed solar wind." The more recent observations (Madjarska, 2011; Shimojo & Shibata, 2000) yield that the temperature of X-ray jets is from 1.3 to 12 MK (i.e., the jets are hotter than the ambient corona) and the electron/ion number density is of about (0.7–4) <sup>×</sup> 109 cm−<sup>3</sup> with average of 1.7 <sup>×</sup> 109 cm−3. The X-ray jets can have velocities above 103 km s−1, reach heights of a solar radius or more, and have kinetic energies of the order of

Since both spicules and X-ray jets support Alfvén (or more generally magnetohydrodynamic) waves' propagation it is of great importance to determine their dispersion characteristics

typical wave's period of 110 s.

SAO/NASA/JAXA/NAOJ.

1029 erg.

Fig. 1. Solar spicules on the Sun recorded on August 3, 2007. Credit: NASA/*STEREO*.

Fig. 2. Solar spicules recorded by the *Solar Dynamics Observatory* on April 25, 2010. Credit: NASA/*SDO*.

on the coronal heating mechanism(s) and highlight the importance of the interface region between photosphere and corona." Nevertheless, Moore et al. (Moore et al., 2011) from *Hinode* observations of solar X-ray jets, Type II spicules, and granule-size emerging bipolar magnetic fields in quiet regions and coronal holes, advocate a scenario for powering coronal heating and the solar wind. In this scenario, Type II spicules and Alfvén waves are generated by the granule-size emerging bipoles in the manner of the generation of X-ray jets by larger magnetic bipoles. From observations and this scenario, the authors estimate that Type II spicules and their co-generated Alfvén waves carry into the corona an area-average flux of mechanical energy of <sup>∼</sup><sup>7</sup> <sup>×</sup> 105 erg s−<sup>1</sup> cm−2. This is enough to power the corona and solar wind in quiet regions and coronal holes, hence indicates that the granule-size emerging bipoles are the main engines that generate and sustain the entire heliosphere. The upward propagation of highand low-frequency Alfvén waves along spicules detected from SOT's observations on *Hinode* was also reported by He et al. (He et al, 1999) and Tavabi et al. (Tavabi et al., 2011). He et al. found in four cases that the spicules are modulated by high-frequency (0.02 Hz) transverse fluctuations. These fluctuations are suggested to be Alfvén waves that propagate upwards along the spicules with phase speed ranges from 50 to 150 km s−1. Three of the modulated spicules show clear wave-like shapes with short wavelengths less than 8 Mm. We note that at the same time, Kudoh & Shibata (Kudoh & Shibata, 1999) presented a torsional Alfvén-wave model of spicules (actually the classical Type I spicules) and discussed the possibility for wave 2 Will-be-set-by-IN-TECH

Fig. 1. Solar spicules on the Sun recorded on August 3, 2007. Credit: NASA/*STEREO*.

Fig. 2. Solar spicules recorded by the *Solar Dynamics Observatory* on April 25, 2010. Credit:

on the coronal heating mechanism(s) and highlight the importance of the interface region between photosphere and corona." Nevertheless, Moore et al. (Moore et al., 2011) from *Hinode* observations of solar X-ray jets, Type II spicules, and granule-size emerging bipolar magnetic fields in quiet regions and coronal holes, advocate a scenario for powering coronal heating and the solar wind. In this scenario, Type II spicules and Alfvén waves are generated by the granule-size emerging bipoles in the manner of the generation of X-ray jets by larger magnetic bipoles. From observations and this scenario, the authors estimate that Type II spicules and their co-generated Alfvén waves carry into the corona an area-average flux of mechanical energy of <sup>∼</sup><sup>7</sup> <sup>×</sup> 105 erg s−<sup>1</sup> cm−2. This is enough to power the corona and solar wind in quiet regions and coronal holes, hence indicates that the granule-size emerging bipoles are the main engines that generate and sustain the entire heliosphere. The upward propagation of highand low-frequency Alfvén waves along spicules detected from SOT's observations on *Hinode* was also reported by He et al. (He et al, 1999) and Tavabi et al. (Tavabi et al., 2011). He et al. found in four cases that the spicules are modulated by high-frequency (0.02 Hz) transverse fluctuations. These fluctuations are suggested to be Alfvén waves that propagate upwards along the spicules with phase speed ranges from 50 to 150 km s−1. Three of the modulated spicules show clear wave-like shapes with short wavelengths less than 8 Mm. We note that at the same time, Kudoh & Shibata (Kudoh & Shibata, 1999) presented a torsional Alfvén-wave model of spicules (actually the classical Type I spicules) and discussed the possibility for wave

NASA/*SDO*.

coronal heating – the energy flux transported into corona was estimated to be of about 3 <sup>×</sup> <sup>10</sup><sup>5</sup> erg s−<sup>1</sup> cm−2, i.e., roughly half of the flux carried by the Alfvén waves running on Type II spicules (Moore et al., 2011). Tavabi et al. (Tavabi et al., 2011), performed a statistical analysis of the SOT/*Hinode* observations of solar spicules and their wave-like behavior, and argued that there is a possible upward propagation of Alfvén waves inside a doublet spicule with a typical wave's period of 110 s.

No less effective in coronal heating are the so called X-ray jets. We recall, however, that whilst the classical spicules were first discovered in 1870's by the Jesuit astronomer Pietro Angelo Secchi (Secchi, 1877) and named as "spicules" by Roberts (Roberts, 1945), the X-ray jets are relatively a new discovered phenomenon. They, the jets, were extensively observed with the Soft X-ray Telescope on *Yohkoh* (Shibata et al., 1992; Shimojo et al., 1996), and their structure and dynamics have been better resolved by the X-Ray Telescope (XRT) on *Hinode*, in movies having 1 arc sec pixels and ∼1-minute cadence (Cirtain et al., 2007) – see Fig. 3. According

Fig. 3. Three X-ray jets recorded by the *Hinode* spacecraft on January 10, 2007. Credit: SAO/NASA/JAXA/NAOJ.

to Cirtain et al. (Cirtain et al., 2007), "coronal magnetic fields are dynamic, and field lines may misalign, reassemble, and release energy by means of magnetic reconnection. Giant releases may generate solar flares and coronal mass ejections and, on a smaller scale, produce X-ray jets. *Hinode* observations of polar coronal holes reveal that X-ray jets have two distinct velocities: one near the Alfvén speed (∼800 kilometers per second) and another near the sound speed (200 kilometers per second). The X-ray jets are from 2 <sup>×</sup> 103 to 2 <sup>×</sup> 104 kilometers wide and 1 <sup>×</sup> 105 kilometers long and last from 100 to 2500 seconds. The large number of events, coupled with the high velocities of the apparent outflows, indicates that the jets may contribute to the high-speed solar wind." The more recent observations (Madjarska, 2011; Shimojo & Shibata, 2000) yield that the temperature of X-ray jets is from 1.3 to 12 MK (i.e., the jets are hotter than the ambient corona) and the electron/ion number density is of about (0.7–4) <sup>×</sup> 109 cm−<sup>3</sup> with average of 1.7 <sup>×</sup> 109 cm−3. The X-ray jets can have velocities above 103 km s−1, reach heights of a solar radius or more, and have kinetic energies of the order of 1029 erg.

Since both spicules and X-ray jets support Alfvén (or more generally magnetohydrodynamic) waves' propagation it is of great importance to determine their dispersion characteristics

filled with ideal compressible plasma of density *<sup>ρ</sup>*<sup>i</sup> <sup>∼</sup> <sup>3</sup> <sup>×</sup> <sup>10</sup>−<sup>13</sup> g cm−<sup>3</sup> (Sterling, 2000) and immersed in a constant magnetic field **B**<sup>i</sup> directed along the *z* axis. Such a cylinder is usually termed *magnetic flux tube* or simply 'flux tube.' The most natural discontinuity, which occurs at the surface binding the cylinder, is the tangential one because it is the discontinuity that ensures an equilibrium total pressure balance. Moreover, it is worth noting that the jet is non-rotating and without twist – otherwise the centrifugal and the magnetic tension forces should be taken into account. Due to the specific form of the real flux tube which models a spicule, that part of the whole flux tube having a constant radius actually starts at the height of 2 Mm from the tube footpoint. The flow velocity, **U**i, like the ambient magnetic field, is directed along the *z* axis. The mass density of the environment, *ρ*e, is much, say 50–100 times, less than that of the spicule, while the magnetic field induction *B*e might be of the order or less than *B*<sup>i</sup> ∼ 10–15 G. Both the magnetic field, **B**e, and flow velocity, **U**<sup>e</sup> (if any), are also in the *z*ˆ-direction. We note that while the parameters of classical Type I spicules are well-documented (Beckers, 1968; 1972) those of Type II spicules are generally disputed; Centeno et al. (Centeno et al., 2010), for example, on using a novel inversion code for Stokes profiles caused by the joint action of atomic level polarization and the Hanle and Zeeman effects to interpret the observations, claim that magnetic fields as strong as ∼50 G were detected in a very localized area of the slit, which might represent a lower field strength

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 139

The flux tube modelling of the X-ray jets is actually the same as that for spicules, however, with different magnitudes of the mass densities, flow velocities, and background magnetic fields. When studying waves' propagation and their stability/instability status for a given solar structure (spicule or X-ray jet), the values of the basic parameters will be additionally specified. Now let us see what are the basic magnetohydrodynamic equations governing the

Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas and liquid metals. The field of MHD was initiated in 1942 by the Swedish physicist Hannes Alfvén (1908–1995), who received the Nobel Prize in Physics (1970) for "fundamental work and discoveries in magnetohydrodynamics with fruitful applications in different parts of plasma physics." The fundamental concept behind MHD is that magnetic fields can induce currents in a moving conductive fluid, which in turn creates forces on the fluid and also changes the magnetic field itself. The set of equations, which describe MHD are a combination of the equations of motion of fluid dynamics (Navier–Stokes equations) and Maxwell's equations of electromagnetism. These partial differential equations have to be solved simultaneously, either analytically or numerically. Magnetohydrodynamics is a macroscopic theory. Its equations can in principle be derived from the kinetic Boltzmann's equation assuming space and time scales to be larger than all inherent scale-lengths such as the Debye length or the gyro-radii of the charged particles (Chen, 1995). It is, however, more convenient to obtain the MHD equations in a phenomenological way as an electromagnetic extension of the hydrodynamic equations of ordinary fluids, where the main approximation is to neglect the displacement current ∝*∂***E**/*∂t*

of organized network spicules.

motions in a flowing solar plasma.

in Ampère's law.

**2.1 Basic equations of ideal magnetohydrodynamics**

and more specifically their stability/instability status. If while propagating along the jets MHD waves become unstable and the expected instability is of the Kelvin–Helmholtz type, that instability can trigger the onset of wave turbulence leading to an effective plasma jet heating and the acceleration of the charged particles. We note that the Alfvénic turbulence is considered to be the most promising source of heating in the chromosphere and extended corona (van Ballegooijen et al., 2011). In this study, we investigate these travelling wave properties for a realistic, cylindrical geometry of the spicules and X-ray jets considering appropriate values for the basic plasma jet parameters (mass density, magnetic fields, sound, Alfvén, and jet speeds), as well as those of the surrounding medium. For detailed reviews of the oscillations and waves in magnetically structured solar spicules we refer the reader to (Zaqarashvili & Erdélyi, 2009) and (Zaqarashvili, 2011). Our research concerns the dispersion curves of kink and sausage modes for the MHD waves travelling primarily along the Type II spicules and X-ray jets for various values of the jet speed. In studying wave propagation characteristics, we assume that the axial wave number *kz* (*z*ˆ is the direction of the embedded constant magnetic fields in the two media) is real, while the angular wave frequency, *ω*, is complex. The imaginary part of that complex frequency is the wave growth rate when a given mode becomes unstable. All of our analysis is based on a linearized set of equations for the adopted form of magnetohydrodynamics. We show that the stability/instability status of the travelling waves depends entirely on the magnitudes of the flow velocities and the values of two important control parameters, namely the so-called density contrast (the ratio of the mass density inside to that outside the flux tube) and the ratio of the background magnetic field of the environment to that of the spicules and X-ray jets.
