Magnetohydrodynamic Processes in Solar Plasmas

Magnetohydrodynamic Processes in The Solar Plasma provides comprehensive and up-to-date theory and practice of the fundamentals of heliospheric research and the Sun’s basic plasma processes, covering the dynamics of the solar interior to its exterior in the framework of magnetohydrodynamics. The book covers novel aspects of solar and heliospheric physics, astrophysics and space science, and fundamentals of the fluids and plasmas. Topics covered include key phenomena in the solar interior such as magnetism, dynamo physics, and helioseismology; dynamics and plasma processes in its exterior including fluid processes such as waves, shocks, instabilities, reconnection, and dynamics in the partially ionized plasma; and physics and science related to coronal heating, solar wind, and eruptive phenomena. The content has been developed to specifically cover fundamental physics-related descriptions and up-to-date developments of the scientific research related to these significant topics. The book therefore provides the entire fundamental and front-line research aspects of solar and heliospheric plasma processes, mainly in the context of solar plasma, however, the content also has larger implications for the astrophysical plasma, and laboratory plasma, fluid dynamics, and associated basic theories. It also includes additional supplementary content such as key instruments and experimental techniques in the form of appendices, boxed-off key information highlighting the most fundamental and key aspects, and worked examples with additional question sets.Magnetohydrodynamic Processes in The Solar Plasma covers both the fundamentals of the topics included as well as up-to-date and future developments in this research field, forming an essential, foundational reference for researchers, academics, and advanced students, in the field of solar physics and astrophysics, as well as neighboring disciplines.

• Applies fundamental solar science and research in magnetohydrodynamic processes to practice, and uses in teaching and research
• Covers the latest developments in solar plasma processes in terms of both theoretical and fundamental aspects.
• Includes the large cohort of plasma processes (e.g., waves, shocks, instabilities, reconnection, heating, magnetism, seismology) significant for the diverse scales of the plasmas and fluids.
• Provides detailed physical and mathematical descriptions of the theories in each chapter, along with scientific details, which will enhance understanding of basic phenomena and aid in applying the practical content to current research

• Language: English
• Publisher: Elsevier Science
• Release date: May 10, 2024
• ISBN: 9780323956659

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Preface

Abhishek Kumar Srivastava; Marcel Goossens; Iñigo Arregui     

The solid, liquid, and gaseous states of matter, which occur at the surface of our Earth, are not typical states of matter in the universe. Most of the visible matter in the universe exists as plasma. Lightning and the aurora are the only natural manifestations of the plasma state on the Earth. Alfvén has suggested that 99,99% of the visible matter is in the plasma state. Plasma is a collection of charged particles. The name was introduced by Langmuir in 1928 during his experiments with discharge tubes. Plasmas are quasineutral and exhibit collective behavior. Understanding the physics of astrophysical fluids is considered an important aspect of natural science. With the advent of the space-borne experiments, and the development in theoretical astrophysics, the understanding of plasma became richer in scope and in context of its range of applicability.

The Sun, like most stars, is composed of plasma. It is the only object in the universe that is spatially resolved with details recorded through multiwavelength observations from ground and space. The Sun is the best cosmic laboratory where we can test fundamental aspects of plasmas, their various configurations, and different dynamical processes that occur in them. The electrically conductive solar plasma has a wide range of physical processes occurring at diverse mass density and temperature. The observations and theory of these physical processes in the solar interior and exterior have progressed with great detail at disparate scales from several megameters down to several tens of kilometers. We also witness significant developments in our understanding of the magnetism of the Sun in terms of its origin, evolution, and progress of its cycle. Moreover, Sun's internal oscillations and their various modes provide the basic diagnostics tool of the solar interior which cannot be observed directly.

The physical processes in the atmosphere of the Sun that occur at much larger spatial and temporal scales than the Debye length scale and ion gyro-period, respectively, can be understood by the model of magnetohydrodynamics (MHD). Therefore, a wide range of dynamical solar plasma processes, e.g., waves and oscillations, shocks, instabilities, turbulence, reconnection, heating, etc., can be studied using the basic principles of MHD. When the processes at MHD scale compared with kinetic and two fluid regimes, the former is basically found to be a simplified formalism. However, MHD is a potential tool to understand many dynamical processes and physical conditions of the solar plasma. MHD is diversely applicable to other astrophysical systems like the Sun, interplanetary space, planetary magnetospheres, interstellar medium, astrophysical jets, high energy transients, etc. Apart from natural plasma systems, it is also used to understand the physics of different laboratory plasmas and high energy experiments. It is also utilized in engineering studies, mechanics of nano plasma fluids, plasma devices, medical science, and energy studies.

This book contains a collection of 10 chapters describing fundamental magnetohydrodynamic processes in solar plasmas. Chapter 1 is a brief introduction about these processes in the interior and exterior of the Sun, which is basically setting their preliminary descriptions. The following chapters describe fundamental concepts of the various plasma processes (e.g., waves and oscillations, shocks, instabilities, turbulence, reconnection, heating, etc.) and also survey some details of ongoing research and front-line problems. The chapters are written by experts in the corresponding topics to provide comprehensive knowledge about the basics of these MHD processes. Chapters 2 and 3 deal with the details of the helioseismology and magnetism of the Sun, respectively. In Chapters 4 to 10, the physics of waves, shocks, instabilities, turbulence, magnetic reconnection, and heating of solar plasmas are depicted in a detailed manner in the framework of magnetohydrodynamics. Each chapter makes use of its own notation/symbols to define physical parameters, however, these symbols have their usual meaning already defined there. Chapter 6 is specifically devoted to the comparison between two-fluid/multi-fluid aspects of wave processes in the solar plasma. This comparison is necessary as there is a fundamental linkage between kinetic theory, two- or multi-fluid, and MHD scales over which the multitude of physical processes are evident in the solar plasma. The book provides a guide to the UG, PG, PhD students, as well as researchers, working in a wide range of areas including Solar and Heliospheric Physics; Planetary Science, Astronomy and Astrophysics; Laboratory and Fusion Plasma Research; Energy Studies, Engineering Studies, etc.

We acknowledge the time and effort the authors devoted to the endeavor of assembling this knowledge resource and intellectual property belonging to the scientific community. We also acknowledge collective use of various sources and figures as cited and adapted in various chapters of this book by the authors that immensely helped in developing the sequential and reasonable description of the topics here. The context discussed in the present book is the set of crucial physical processes on which significant research is ongoing in the community, thereby this book and its holistic description will serve as a ready reference for a larger community working in astrophysics and plasma physics. The current version is a timely description of these various topics.

Chapter 1: Dynamical processes in the solar plasma

Abhishek K. Srivastava    Department of Physics, Indian Institute of Technology (BHU), Varanasi, India

Abstract

In this chapter, we provide some description about the key plasma processes in the Sun's interior and exterior. We also delineate basic information about the solar magnetism and its cycle. The various chapters in the present book describe basic physics and scientific advancements of different magnetohydrodynamic (MHD) processes in the solar plasma, therefore, we provide a brief introduction on those processes to further link with their detailed scientific descriptions as outlined in various chapters.

Keywords

Sun; Plasma; Magnetic fields; Magnetohydrodynamics (MHD)

1.1 Introduction

The Sun is a magnetically active star that constitutes our solar system and planetary magnetospheres. The variable and dynamic Sun is an extraordinary energy source for our solar system, planets, and interplanetary space. It is a G2V type star, which has 4.8 stellar magnitude with some of its typical basic properties as listed in Table 1.1. The Sun is considered as a spherically symmetric structure, which is maintained by a balance between gravity and plasma pressure related forces (e.g., Golub and Pasachoff, 1997). The different plasma processes occurring in the Sun's interior and exterior have a direct influence on space weather and our life (e.g., Pulkkinen, 2007). The Sun is a grand cosmic laboratory in our best proximity, which provides an opportunity to study its physics in greater details. Moreover, its substantial understanding further generates a pathway to learn the physical nature of other similar stars where we can discern their surface characteristics and various dynamical processes (e.g., spots and magnetism, flares, coronal mass ejections, supersonic wind, etc.) to widely explore the analogous physical phenomena in stellar atmospheres (e.g., Brun et al., 2015). The continuous observations of the solar atmosphere and its various layers using the ground- and space-based instruments put forward the developments of our understanding about radiative transfer in the solar plasma and its numerous dynamical processes. The solar atmosphere is also treated as a magnetohydrodynamic (MHD) laboratory, exhibiting different dynamical plasma and wave processes at diverse spatio-temporal scales. To understand the basic physical phenomena and eruptive processes at the Sun, we must understand the behavior of the magnetic field and coupled plasma system, radiative processes, and energetic particles. The various chapters in the present book focus on describing the basic physics and some key scientific advancements in the MHD processes of solar plasma. In this chapter, we delineate a brief introduction on various dynamical processes in the solar plasma. Their substantial details are further discussed and outlined in various chapters of this book.

Table 1.1

1.2 The solar interior and its plasma processes

The Sun has well-defined regions (Fig. 1.1), a much opaque interior and its exterior magnetic atmosphere. It contains different elements with different proportions, e.g., H (92%), He (7.8%), and C, N, O as well as higher elements which contribute ∼0.2% to the solar elemental abundance (e.g., Asplund et al., 2009; Lodders, 2019). The interior of the Sun has the highest temperature ( K) and the largest density ( kg m−3) at its core where the energy is produced due to the thermo-nuclear fusion reactions (e.g., Chitre, 2003). This physical possess in its inner core (⩽0.2 R⊙) generates 99% of the Sun's energy. In this process, the ionized hydrogen nuclei (protons) fuse together to form helium (He) nuclei, and a large amount of energy (i.e., 26.73 MeV). This fusion process can take place by two different cycles namely proton–proton (PP) cycle and carbon–nitrogen–oxygen (CNO) cycle (e.g., Priest, 2014). The proton–proton (PP) cycle generates ≈98% of the solar energy, while the CNO cycle liberates ≈2% of the solar energy in the core (e.g., Chitre, 2003). The energetic gamma-ray photons further propel out, first by means of radiation and thereafter by getting absorbed in the gas lying below the solar surface to initiate the convection process. Apart from huge energy release, the nuclear fusion reaction also yields solar neutrinos, which do not interact with the matter, thus directly escaping from the solar interior. A major part of the neutrino flux is given by the proton–proton chain reaction, while the CNO cycle contributes to its smaller fraction (Orebi Gann et al., 2021). Smaller neutrino fluxes were detected compared to those predicted and estimated by the Standard Solar Model (SSM) and nuclear fusion theory. This problem is known as the Solar Neutrino Paradox (Bahcall et al., 2003). The neutrino oscillations, i.e., the conversion of electronic neutrinos in other forms of the neutrinos may be a viable explanation to the solar neutrino paradox (e.g., Ahmad et al., 2002; Bahcall and Peña-Garay, 2004; BOREXINO Collaboration et al., 2014; Orebi Gann et al., 2021, and references cited therein).

Figure 1.1 The structure of Sun's interior and its exterior atmosphere is displayed in the unlabeled articulate image. The yellowish central region is the core (⩽0.2 R ⊙ ) of the Sun where the nuclear fusion process takes place to generate the energy. The yellow rays are representing the transport of energy through radiative processes in the radiation zone lying in the range of 0.2⩽ R ⊙  ⩽ 0.7. The thin layer beneath the Sun's surface, within spatial extent of 0.7⩽ R ⊙  < 1.0, is known as convection zone where the energy is transported by the convection process. Here and elsewhere the symbol R ⊙ represents one solar radius. The different magnetic structures (e.g., loops, prominences, coronal mass ejection, coronal rays, etc.) are also illustrated in various parts of the Sun's atmosphere between the photosphere and the outer corona. (Courtesy: NASA-GSFC outreach; Mary Pat Hrybyk-Keith; Image is taken from Sun Today ( https://www.thesuntoday.org/sun/solar-structure/ ) maintained by C. Alex Young.)

In the radiative zone (0.2⩽ R ), the liberated energy in the form of gamma-ray photons is transported by means of radiative processes. In the convection zone (0.7⩽ R ) lying above the radiative zone, the radiative temperature gradient becomes greater due to an increase in the opacity, while the gradient in the adiabatic temperature drops. In this way, the Schwarzschild instability criterion is satisfied, thereby the heat energy flux is predominantly transferred upward, up to the solar surface due to the convection process (see Schwarzschild, 1906; Choudhuri, 1998). The convective process is modeled with the inference of the local mixing length theory as given by Böhm-Vitense (1958). Apart from energy generation and transport, the solar interior also consists of the millions of resonant p-modes trapped within it. These modes appear on the solar photosphere accompanied by periodicities lying in the range of a few minutes to hours (see Fig. 1.2). Several decades ago, such global phenomenon was observed in the form of a distribution of velocity oscillations at the surface photosphere of the Sun. These oscillations were a manifestation of the resonant sound waves, termed as pressure modes or p-modes propagating inside the Sun (e.g., Leighton et al., 1962; Ulrich, 1970; Leibacher and Stein, 1971; Deubner, 1981; Scherrer et al., 1982). The pressure gradient serves as a restoring force for these p-mode waves. On the other hand, the buoyancy force takes action as a restoring force for the evolution of g-mode oscillations, which are critically damped in the solar convection zone. Therefore, the g-mode amplitudes at the top of the convection zone are considerably smaller than in the Sun's radiative zone and core. This makes the g-mode indistinguishable and unobservable at the surface of the Sun. In contrast to the p-modes having observable amplitude at the solar photosphere, the g-modes evince their maximum amplitudes in the Sun's core, so if observable they would be a valuable tool to probe the solar core (e.g., Appourchaux et al., 2010). The helioseismology has emerged as an efficient method to know the properties of p-mode oscillations, employing inversion techniques, and determining the crucial physical properties of the solar interior, e.g., solar tachocline, depth of the convection zone, superadiabatic transition layer, convection zone helium abundance, solar rotation, solar structure and neutrino problem, solar abundance issues, effect of departure from spherical symmetry, solar cycle related issues, effect of magnetic field, etc. (e.g., Demarque and Guenther, 1999; Basu, 2016, and the references cited therein). Chapter 2 of this book discusses in detail about the seismic Sun and helioseismology.

Figure 1.2 (From left to right) An illustration of the spherical harmonics of global p-mode solar oscillations for l = 6, m = 0, m = 3, and m = 6, respectively, which are appearing at the surface of the Sun (Courtesy: J.W. Leibacher; Demarque and Guenther (1999)); PINAS.

Another significant property of the solar interior is its differential rotation (see Fig. 1.3), i.e., the equator is rotating with a period of 25 days while the regions at higher latitudes take relatively prolonged time in their rotatory motion. The rotation in the deepest parts of the Sun is not investigated in a detailed manner as there is a lack of the observations of p-modes as well as precise inversion (Fig. 1.3). The helioseismology is also used to estimate the rate of the rotation of solar interior (see Chapter 2). The rotation causes the splitting in the oscillation frequencies. It provides us with knowledge of the solar rotation (Howe, 2009) by estimating variation in the rotation rate with radius, latitudinal positions, and time. Fig. 1.3 (bottom-right panel) manifests the rotation rate of the solar interior over a few selected latitudes as a function of the radius. The shearing layer near the solar surface is clearly seen, as well as the rotation rate at all the latitudes, which increases with the depth up to about 0.95 R⊙. The differential rotation is crucial in determining the properties of the solar magnetism (see Section 1.3).

Figure 1.3 (Top) A transverse cross-section of the spherical Sun through its interior, shown with the contours, is representing a constant rotation and the prime features of the solar rotation profile. This map is made up using a temporal average of about 12 years of Michelson Doppler Imager (MDI) observational data recorded from the Solar and Heliospheric Observatory (SoHO). The cross-hatched regions specify the areas in which it is hard to get reasonable inversion of parameters with the given observational data. (Bottom) The mean rotation profile derived from the Global Oscillations Network Group (GONG) observations and contours of constant rotation are depicted in bottom-left panel along with lines at 25 ∘ to the rotation axis drawn with dashed lines ( Howe et al., 2005). The variation of the branch cuts at constant latitude with respect to the solar radius is demonstrated in the bottom-right panel ( Howe et al., 2000) . (Courtesy: Rachel Howe; the figure is adapted from Howe (2009), LRSP/Springer under the Creative Common CC BY License.)

Owing to the processes mentioned above, the Standard Solar Model (SSM) depicts the evolution of the Sun from the point of initiation of its thermonuclear fusion reactions to its current age ( Gyr). The SSM takes into account one solar mass (see Table 1.1), a number of assumptions as given below, and determines the evolution of solar properties up to the current epoch of time. These physicswise assumptions describe: (i) hydrostatic equilibrium; (ii) spherical symmetry; (iii) energy production by nuclear fusion; (iv) radiative and convective transport of energy; and (v) homogeneous initial physical state along with corresponding mass fraction of hydrogen, helium, and other massive elements (e.g., Haxton and Serenelli, 2008; Ianni, 2014, and the references therein). According to the postulates of SSM as stated above, the Sun is considered as a spherically symmetric system in which the effects of magnetic field, rotation, tidal forces, and mass loss are insignificant in understanding its global structure (e.g., Guenther et al., 1992; Christensen-Dalsgaard et al., 1996; Bahcall et al., 2001; Chitre, 2003, and the references therein). Some descriptions of such aspects can also be found in various literature (e.g., Christensen-Dalsgaard, 2002; Basu and Antia, 2008; Howe, 2009; Aerts et al., 2010; Basu, 2016; Gough, 2013; Christensen-Dalsgaard, 2021, and the references therein), and also in Chapter 2.

1.3 The solar magnetism and its cycle

Another aspect of the dynamical plasma inside the Sun is the genesis of its magnetic field, which is originated inside the Sun and further transported to the solar surface. Thereafter, it permeates into the solar atmosphere. The magnetic field and plasma couple in various layers of the atmosphere to generate the dynamics and heating at diverse spatio-temporal scales (e.g., Solanki et al., 2006). A dynamo process, operating at the interface between the radiative and convection zones, is considered to be the source of the solar magnetism (e.g., Charbonneau, 2005; Choudhuri, 2007). The surface magnetism of the Sun can be basically categorized into two major classes: (i) the large-scale active regions and (ii) the small-scale quiet-Sun magnetic fields (e.g., Solanki et al., 2006; Bellot Rubio and Orozco Suárez, 2019). The dark sunspots in the active regions are the region of intense magnetic field of about 1000 gauss, i.e., in the range of ≈1000–3000 G (Fan, 2021). The strength of the magnetic field at the surface in the quiet-Sun is found to be of the order of a few hundred gauss (e.g., Solanki et al., 2006; Bellot Rubio and Orozco Suárez, 2019). The source of magnetic fields appears at the photosphere (see Fig. 1.4), however, it is deeply rooted at the base of the convection zone (Fan, 2021).

Figure 1.4 A full disk magnetogram from the Kitt Peak Solar Observatory with estimates of the magnetic fields at the solar surface photosphere on May 11, 2000 during the maximum of solar cycle 23. It measures the line-of-sight density of the magnetic flux on the photosphere. The white (black) color indicates the estimated magnetic fields of positive (negative) polarities. A closer inspection of this magnetogram demonstrates that two larger sunspots emerge side by side as a pair, and possess opposite magnetic polarities satisfying Hale's law ( Hale et al., 1919). The spot ahead in the direction of solar rotation is termed as a leading one, while the other in the pair is known as a trailing sunspot. These pairs possess opposite polarities in both hemispheres. The line joining the leading and trailing spots shows a tilt with respect to the equatorial line, and it becomes greater with the increment in the latitude. This rule is known as the famous Joy's law ( Hale et al., 1919). The polarities of any sunspot pair are reversed in an 11-year solar cycle, and regain their original configuration in the next 11 years, completing a full time-span of one magnetic cycle in 22 years. These sunspot groups form active solar regions and contribute to the large-scale magnetism of the Sun, while the rest of the tiny magnetic fluxes constitute the quiet-Sun region at the solar surface. (Courtesy: Y. Fan; Kitt Peak Solar Observatory; the figure is adapted from Fan (2021), LRSP/Springer under the Creative Common CC BY License.)

It was noticed more than a century ago that the sunspot number at the surface of the Sun increases and thereafter decreases periodically with the period of 11 years, therefore, constitutes the 11-year solar cycle (Schwabe, 1844). The solar activity is defined in terms of the number of sunspots present at the photosphere. If there are fewer sunspots at the photosphere, it is defined as a solar minimum, whereas if the number of sunspots is large the phase is termed as a solar maximum (Martin and Harvey, 1979). The large-scale magnetism related to these sunspots changes its direction every 11 years, and it takes a total of 22 years to restore the original configuration of the distribution of magnetic polarities appearing at the photosphere. In the initial phase of any solar magnetic cycle, most of the sunspots appear at the higher latitudes around 30∘–40∘, while they move closer to the equator at lower latitudes when the magnetic cycle progresses (Carrington, 1858).

The butterfly diagrams of the previous four solar cycles 20–24 are displayed in Fig. 1.5. These diagrams illustrate various features of the solar cycle, such as Hale's polarity law, Joy's law of active region tilt, equatorial movement of sunspot groups, behavior of overlapping solar cycles at their minimum, poleward migrations, etc. (Maunder, 1903, 1904; Hale et al., 1919). The magnetic polarities of the leading sunspots in the two hemispheres are shown in each half of the wings of the butterfly diagram following Hale's law (Hale et al., 1919). The opposite polarity poleward stream migration begins from each wing of the butterfly diagram. This physical process further debilitates the polar fields by canceling them around the solar maximum, and causes the changes in the magnetic polarities of the polar regions. The leading spots and their magnetic field polarities move towards the equator, while the trailing polarities drift towards the poles (Fig. 1.5) where they cancel the existing fields of opposite polarities. This process is repetitive and ongoing until all the polarities get reversed. This physical phenomenon is known as the polarity reversal in a solar cycle. The strength of the polar fields becomes stronger (i.e., a few tens of gauss) at the termination of each magnetic cycle while the strongest magnetic fields of a few kilogauss are found close to the solar equator. The polar field distributions since 2007 onward are shown in Fig. 1.5, which could be an important component in the form of a diffused field in representing the dynamo action, along with the strongest sunspot fields present equatorward (see Choudhuri, 2003). The main features of solar cycle variations and origin of the surface magnetic fields, linked with the dynamo action in the solar interior, are described in Chapter 3 of the present book. Magnetic fields, which are permeating into the solar atmosphere, are responsible for various dynamical and heating processes in the solar atmosphere.

Figure 1.5 The butterfly diagrams of the previous four solar cycles numbered from 20 to 24 are displayed in this figure. The butterfly diagram is produced making use of the radial components of magnetic fields of the synoptic magnetograms averaged over the longitude in the course of each solar rotation. It should be noted that each half of the butterfly wing exhibits the polarity of the leading sunspots in the corresponding hemisphere at the Sun ( Hale et al., 1919). The strongest magnetic fields are appearing close to the equator during the advancement of the solar cycle. The opposite-polarity poleward migrations also arise from each butterfly wing. This process enfeebles the polar fields in due course during the time of a solar maximum, and it alters the magnetic polarity of the Sun's poles. (Courtesy: D. Hathaway, private communication.)

1.4 Dynamical processes in the solar plasma

The plasma is confined to the magnetic fields formed at diverse spatial scales. The magnetic fields fan out from the surface photosphere of the Sun to various its overlying atmospheric layers, i.e., chromosphere, transition region, and corona to couple them. By ignoring the cross-field structuring, magnetic field topology, and underlying complexities in the solar atmosphere, plasma is found to be structured with the height and constitutes a plane parallel atmosphere above the solar surface. These atmospheric layers are termed as the solar photosphere, chromosphere, transition region (TR), and corona, as we move upward radially. When we move from the solar photosphere to the chromosphere, the temperature minimum exists at the top of the photosphere, while the temperature reaches 20,000–30,000 K in the chromosphere. The temperature abruptly rises in the transition region (TR) and attains 1.0 MK value on average in the inner corona. The density is largest at the solar photosphere, 3 × 10−7 gm cm−3, while it drops sharply in the TR and inner corona (Fig. 1.6). The details of the various atmospheric layers, their magnetic fields, emissions, and dynamics are illustrated in many textbooks and literature sources (e.g., Aschwanden, 2004; Priest, 2014). All these layers are coupled magnetically and are inevitable to various plasma and wave processes (e.g., waves, shocks, instabilities, reconnection, etc.), thus leading to the transport of mass and energy in the solar atmosphere. The structuring of the plasma density and temperature with height vis-á-vis variation of the magnetic field sets in the magnetic structures and fluxtubes expanding out from the photosphere into the corona both in the quiet-Sun and active regions (e.g., Jess et al., 2023). These structures act as viable and conduit guides of the waves and shocks. Moreover, the radial or cross-field structuring (a departure from homogeneous plasma configuration), as well as vertical magnetic field structuring, generates a more complex physical scenarios in the form of the mode conversion/coupling of waves, their trapping/reflection, resonances, as well as dissipation both in the quiet-Sun and active regions (e.g., Jess et al., 2023). The inhomogeneous magnetized plasma in the localized solar atmosphere is also subject to a variety of instabilities and turbulence. Moreover, the nonideal physical conditions of resistivity/magnetic diffusion in the specific magnetic field configurations (e.g., current sheets) generate the dissipation of current, causing the heating and plasma dynamics due to the magnetic reconnection process (e.g., Pontin and Priest, 2022). These waves and reconnection processes are considered as two important mechanisms by which the solar atmosphere can be heated to its extremely high temperatures (e.g., Van Doorsselaere et al., 2020; Pontin and Priest, 2022, and the references therein).

Figure 1.6 (Top) The full-disk image of the Sun at visible, H- α 6563 Å, He II 256 Å, and Fe XII 195 Å emissions (left-to-right) representing the photosphere, chromosphere, transition region, and corona, respectively, at the same time epoch. (Bottom) The variation of mass density and temperature as functions of height while moving outward from the photosphere to the corona as per the VAL (Vernazza–Avrett–Loeset) model. (Courtesy: BBSO, GONG, SDO/NASA, SoHO/ESA; Eugene Avrett, Smithsonian Astrophysical Observatory; E.R. Priest; IOP Publishing; Oxford University Press.)

1.4.1 Wave processes in the solar atmosphere

In the light of the large cohort of space-borne and ground-based observations, it is well established that different kinds of MHD waves are ubiquitously detected in different magnetic structures of the solar atmosphere. Understanding their generation, driving mechanisms, physical properties, and damping processes in the localized solar atmosphere are important to unveil their possible contributions to the chromospheric and coronal heating, diagnostics, and plasma dynamics, etc.

1.4.1.1 Waves and oscillations in the stratified lower solar atmosphere

The p-mode is an inherent global acoustic oscillation in the subphotospheric layers. These modes also appear on the photosphere where 5.0-min oscillations are the dominant power. These modes may leak and propagate in the magnetized solar atmosphere upward if their natural oscillation frequency is found to be greater than the local acoustic cut-off frequency (e.g., Ulmschneider, 1971). Therefore, in principle these acoustic waves may not be capable of propagating into the overlying solar atmosphere whose frequencies lie below the cut-off frequency. Carlsson and Stein (1995) have reported that chromosphere is magnetic in nature, and waves may propagate there in the form of magnetoacoustic waves. The strong magnetic field in the chromosphere may change the local acoustic cut-off frequency (Bel and Leroy, 1977). Magnetic portals can be created in the chromosphere permitting the propagation of waves along the inclined magnetic field lines (Jefferies et al., 2006). The magnetic field inclination channels long-period wave propagation (see, e.g., De Pontieu et al., 2004; de Pontieu and Erdélyi, 2006; Jess et al., 2013; Kontogiannis et al., 2014; Kayshap et al., 2018, and the references therein). The inclined magnetic fields yield the long-period (or short frequency) wave propagation in the chromosphere as the cut-off period increases (or cut-off frequency decreases) (Jefferies et al., 2006). It also exhibits the leakage of photospheric oscillations into the solar chromosphere (e.g., De Pontieu et al., 2004). These magnetic portals may permit the magnetoacoustic wave propagation well below 5 mHz into the overlying solar atmosphere carrying energy along with them both in the quiet Sun as well as active regions (see, e.g., Wikstøl et al., 2000; Centeno et al., 2009; Bostancı et al., 2014; Felipe, 2021, and the references therein). The large magnetic fields can also alter the radiative relaxation time locally in the chromosphere, thus increasing the value of cut-off period (e.g., McAteer et al., 2002, 2003; Centeno et al., 2006; Khomenko et al., 2008; Centeno et al., 2009; Srivastava and Dwivedi, 2010a, and the references cited therein). In conclusion, even though naturally the acoustic oscillations near the highest power are not admissible into the upper atmosphere, the local physical conditions may favor their propagation upwards. See Fig. 1.7.

Figure 1.7 A patch of the quiet atmosphere is demonstrated between the photosphere and upper chromosphere (left). The power maps are displayed in the right panel at different solar atmospheric heights. The green, red, and yellow colors are depicting respectively the dominant-period around 3, 5, and 7 min. The bottom-most panels show the line-of-sight magnetic fields estimated using Stokes inversions of Fe I 6302 Å. (Courtesy: D.B. Jess; T. Samanta; IOP; the figure is adapted from Jess et al. (2023), LRSP/Springer under the Creative Common CC BY License.)

In the strongly magnetized structures, the power associated with global acoustic oscillations is augmented in the photosphere and lower chromosphere making the power halos (e.g., Kontogiannis et al., 2010; Rajaguru et al., 2013, and the references cited therein). On the other hand, these power halos are substantially suppressed in the higher chromosphere producing magnetic shadows (e.g., McIntosh and Judge, 2001). The power increment in the form of halos at the lower heights of the solar atmosphere may be caused by the reflection of the fast magnetoacoustic waves in the inclined magnetic fields forming the magnetic canopies. This physical phenomenon occurs due to the onset of a large Alfvén speed gradient in the localized solar atmosphere (Khomenko and Cally, 2012). The observations demonstrate that the strength of the magnetic fields and their inclination both are significant for the production of larger power in the intense and comparatively horizontal magnetic fields (e.g., Rajaguru et al., 2019). The suppression of power in the magnetoacoustic waves at the higher heights in the solar chromosphere may be caused by the mode conversion in the plasma with its beta (β) equal to 1.0 (e.g., Nutto et al., 2012). It can also occur due to the physical process of lesser wave propagation in the magnetic canopy or wave dissipation below of this region (e.g., Ulmschneider, 1971; Srivastava et al., 2021). Fig. 1.16 demonstrates a recent observations from the 1-m Swedish Solar Telescope (SST)/CRISP by Samanta et al. (2016). Researchers observed the suppression of the wave power related to 3-min oscillations in the upper part of the solar chromosphere where no oscillatory power related to this period was present in the temporal image data of H-α line-core intensity. On the contrary, they noted the existence of power halos in the lower solar atmosphere possibly due to the energization by the small-scale transient energy release. The possible complexities of the localized solar photosphere and chromosphere may trigger the mode conversion, and this may cause the disappearance of the 3.0-min wave propagation. A detailed description of basic magnetohydrodynamics (MHD) and physics of the wave propagation in the solar atmosphere is outlined in Chapter 4 of this book.

Finally, we conjecture here that the wave mode conversion is a significant physical process in the lower solar atmosphere. It is the manifestation that one of three wave modes, namely slow, Alfvén, and fast, may convert into another form where the phase velocities of these waves approximately equalize in the solar atmosphere. The two different kinds of mode conversion are noteworthy in the lower solar atmosphere. The first is the fast-to-slow mode wave conversion, while the second is the fast-to-Alfvén wave mode conversion process (e.g., Felipe et al., 2010; Cally and Hansen, 2011; Hansen and Cally, 2012; Pennicott and Cally, 2019, and the references therein). The observational manifestation of the wave mode conversion process is not directly detectable in the current regime of the observations.

The strongly magnetized sunspots are the locations above which the evolution, trapping, and mode conversion of the magnetoacoustic and Alfvén waves also efficiently take place (e.g., Khomenko and Collados, 2015; Jess et al., 2023, and references cited therein). The large and essentially vertical magnetic field of the central umbra of the sunspot makes the plasma environment low beta throughout the entire solar atmosphere above it. This makes it a conduit of the magnetohydrodynamic waves. The radial, as well as longitudinal, variations of the plasma beta, gas, and magnetic field properties in the sunspots make them an interesting waveguide where a range of physical processes influencing the intrinsic wave modes (e.g., mode conversion, trapping, resonant oscillations, etc.) occur. Such processes are utilized in diagnosing sunspot physical conditions as well as heating aspects.

The sunspot's central umbra shows the tendency of oscillations but with the reduced amplitude by the margin of 40%–60%, where the dominant period comes from 5.0 min oscillations at the photosphere, while at the top of it and in the upper chromosphere 3.0-min oscillations dominate (e.g., Braun, 1995; Bellot Rubio et al., 2000). In the upper photospheric or chromospheric lines, it was found that 3.0 min oscillations do present as solitary peaks in the power spectra of the deduced velocity time series in contrast to a natural 5.0 min peak. This put forward the physical scenario of the evolution of resonant magnetoacoustic modes of the sunspots with a cavity that is likely to be formed either in the convection zone below the umbra or in the chromosphere (e.g., Zhugzhda and Dzhalilov, 1982). The photospheric velocity oscillations in the outer part of the sunspot penumbra possess a similar power spectrum, peaking at 5.0 min as for the umbral oscillations at the photosphere (Nagashima et al., 2007). In the chromosphere above the sunspot, the running penumbral waves are also observed with the phase speed of 10–50 km s−1 and periodicity of 5.0 min (e.g., Zirin and Stein, 1972; Christopoulou et al., 2001; Rouppe van der Voort et al., 2003; Nagashima et al., 2007, and the references therein). A three-dimensional scenario of intensities and its peak power associated with dominant periods is shown in Fig. 1.8. The observed sunspot is associated with AR NOAA11823 and studied in various spectral lines formed at different heights (Löhner-Böttcher and Bello González, 2015). The intensity maps (left) depict the sunspot appeared on August 21, 2013, at 15:00:06 UTC. It is clear that the dominant power is associated with 3.0 min in the sunspot umbra at the top of the photosphere and upper chromosphere, whereas the 5.0-min power is present in the penumbra at the photospheric and lower chromospheric heights. Apart from the velocity and intensity oscillations at photospheric levels in the sunspot umbra, there were several clues regarding the oscillations of magnetic fields also present predominantly at the periods between 3 and 5 min (e.g., Bellot Rubio et al., 2000; de la Cruz Rodríguez et al., 2013). There are long-period oscillations also observed in the sunspots at the scales of several tens of minutes to hours, and at a scale of days, however, their measurements and physical interpretation are full of uncertainties (e.g., Sych et al., 2012; Smirnova et al., 2013; Griñón-Marín et al., 2020).

Figure 1.8 Various emissions at different wavelengths (left) formed at different heights above the sunspot in NOAA11823 are shown at 15:00:06 UT on August 21, 2013. Their peak powers associated with dominant periods are estimated and displayed in the right panel. The images along the z -axis correspond to different spectral line cores, as well as their wings in Fe I 6301.5 Å, Na I 5896 Å, and Ca II 8542 Å. The black contours represent the position of the inner sunspot umbra and its outer penumbra in the bottom panel. The atmosphere above the sunspot up to the chromospheric height of 1.5 Mm is displayed. (Courtesy: J. Lohner-Bottcher and N. Bello González ( 2015); A&A; EDP Sciences.)

The most plausible drivers of the magnetoacoustic waves in the sunspot are the in situ p-mode absorption and magnetoconvection, which allow explaining the reduced power of waves within these spots (e.g., Parchevsky and Kosovichev, 2007; Jacoutot et al., 2008). Sometimes the episodic flare events can generate in situ waves and oscillations in the chromosphere above the sunpots (e.g., Kosovichev and Sekii, 2007). It was found that the 3.0-min oscillations in sunspots show power enhancement prior to the trigger of solar flares, causing the view that oscillatory motions may help trigger the flares in the active regions (e.g., Sych et al., 2009). A sunspot, under specific physical conditions when fast magnetoacoustic waves convert into Alfvén oscillations, may host the propagation of Alfvén waves with a huge amount of mechanical energy (see Grant et al., 2018). The understanding of the drivers of various MHD wave modes, their unambiguous detection at multiple heights, their energetics and damping, and direct observations of the mode conversion in the sunspot still remain some of the open questions (e.g., see Jess et al., 2023).

1.4.1.2 The waves in the homogeneous coronal plasma

Once the wave-like perturbations occur in low-β and high Reynolds number solar coronal plasma, three specific modes profoundly evolve, namely the slow magnetoacoustic, fast magnetoacoustic, and Alfvén waves. If we consider the typical scales of the inhomogeneity in the plasma and magnetic fields in localized corona, thus defining the concept of fluxtubes, then the wave modes will have distinct appearance and physical properties as described in Section 1.4.1.3.

An Alfvén wave is an incompressible and transverse wave, which propagates along the solar coronal magnetic field lines. The tension related force on the magnetic field produces these transverse waves, which propagate with the phase speed . Here, and are respectively the background magnetic field and density of the large-scale, homogeneous corona. These Alfvén waves do not exhibit the coherent behavior while propagating along different field lines in the corona, therefore, they are always prone to the phase mixing process and subsequent dissipation (e.g., Heyvaerts and Priest, 1983). The phase mixing is termed as the process when, due to the variation of plasma density and magnetic field across, the Alfvén waves travel with different speeds and get out of phase quickly, thus producing greater evolution of the gradients and significantly increasing the viscous and ohmic dissipation (e.g., Heyvaerts and Priest, 1983). If we consider radial and fluxtube structure in the solar atmosphere (e.g., coronal loops, active surge, etc.), the Alfvén waves appear as torsional Alfvén waves (see Section 1.4.1.3; also Fig. 1.8(C)). The torsional waves (Fig. 1.8(C)) or long-wavelength planar Alfvén waves (Fig. 1.8(B)) both cannot be observed by any imaging instrument alone as they neither displace the axis of the magnetic structure as a whole nor cause the variations in the density, and thus in the intensity. The Alfvén (and torsional waves) are claimed to be observed by estimating spectral line-width variations (e.g., Banerjee et al., 1998; Harrison et al., 2002; O'Shea et al., 2005; Banerjee et al., 2009) and spectroscopic/spectro-polarimetric observations (e.g., Jess et al., 2009; Srivastava et al., 2017; Kohutova et al., 2020). See Fig. 1.9.

Figure 1.9 (A) The images in IRIS FUV SJI data demonstrating for the first time the torsional Alfvén waves present at the coronal heights in active surge plasma ( Kohutova et al., 2020). (B) A simple schematic adapted from Dwivedi and Srivastava (2010) demonstrating the long-wavelength, planar Alfvén wave motions in the solar corona. (C) Consideration of fluxtube geometry in the solar corona and demonstration of torsional Alfvén waves as illustrated in Van Doorsselaere et al. (2008). This figure is demonstrating the difference between Alfvén waves in a homogeneous plasma (C) and torsional Alfvén waves in structured coronal fluxtubes (A)–(B). (Courtesy: A&A/EDP Sciences, ApJ/IOP; Current Science/IAS.)

The slow magnetoacoustic waves are well observed in the solar corona, and they exhibit significant damping. The slow waves, while propagating through the corona, perturb the physical properties (e.g., density, temperature, pressure) of the plasma gas. These waves are observed in the coronal open (e.g., coronal hole), closed (e.g., bipolar loops), or along the fan-like active region loop arches (see, e.g., Ofman et al., 1997; DeForest and Gurman, 1998; Robbrecht et al., 2001; Wang et al., 2003, and the references therein; see also Fig. 1.10 as adapted from Krishna Prasad et al., 2019). In an ideal plasma medium, a slow wave has a sonic phase speed, equivalent to the local sound speed in the medium. It is a longitudinal and compressive wave, which has a propagation vector aligned in the direction of the magnetic field. The compression and rarefaction (thus the velocity perturbations) in plasma occur in the direction of the wave propagation with a phase speed . The propagating slow waves can also be generated in the photosphere and chromosphere to further propagate in the coronal structures (e.g., De Pontieu et al., 2005). These waves can also be generated in situ both in the solar transition region and corona (e.g., Ofman et al., 1997; DeForest and Gurman, 1998; Marsh et al., 2011; Jess et al., 2012; Krishna Prasad et al., 2012). The standing slow waves are also observed in flaring and typical warm coronal loops (e.g., Wang et al., 2002; Srivastava and Dwivedi, 2010b; Kumar et al., 2013, and the references therein). The slow waves are highly dispersive, and various literature sources suggest that thermal conduction is a dominant mechanism in the solar corona which dissipates wave energy (see, e.g., Ofman and Wang, 2002; Pandey and Dwivedi, 2006; Prasad et al., 2021, and the references therein).

Figure 1.10 (Left) SDO/AIA 171 Å image depicts a fan-loop system in NOAA AR 12553. The solid lines in blue demonstrate the boundary of the selected loop. (Right) The distance–time maps (top and bottom) display the slanted bright ridges periodically demonstrating the propagating slow magnetoacoustic waves. (Courtesy: Krishna Prasad; the figure is adapted from Krishna Prasad et al. (2019), FrSS under the Creative Common CC BY License.)

The fast magnetoacoustic waves are the compressive magnetohydrodynamic waves, which can propagate at an arbitrary angle to the background magnetic field in the homogeneous solar coronal plasma. They propagate in the coronal plasma with a phase speed , where and are respectively the Alfvén and sound speeds. These waves are the fastest waves in the solar plasma where the restoring force incorporates both the gas and magnetic pressure related forces. These waves originate due to the coupling between the magnetic field and gas pressure perturbations. The slow mode wave perturbations propagate in a narrow region aligning with the background magnetic field lines, while the fast wave propagates isotropically across the field (see Fig. 1.11). The fast magnetoacoustic waves are seen in the corona in the form of quasiperiodic fronts propagating across the magnetic fields (Liu et al., 2011). In the next subsection, we briefly discuss the waves in the inhomogeneous magnetic structures.

Figure 1.11 (Top) The Solar Dynamic Observatory (SDO)/Atmospheric Imaging Assembly observed large-scale active region where a flare-CME eruption launches the quasiperiodic and large-scale perturbations propagating in the corona. (Middle) The different images of 171 Å AIA time series depict the dark and bright consecutive strips moving across the magnetic field lines with the speed of ≈2000 km s −1 . (Bottom) The Fourier filtered images at 69-s periodicity and 110-Mm wavelength demonstrate the fast magnetoacoustic waves propagating in the solar corona. (Courtesy: Liu Wei & Leon Ofman; IOP Publishing; Liu, W. et al. ( 2011).)

1.4.1.3 Waves in the inhomogeneous magnetic structures

Astrophysical plasmas are often nonuniform and also structured by the magnetic field. This structuring is perceptible in the solar atmosphere when we move from the surface photosphere to the overlying corona (see, e.g., Roberts, 2019). The physical properties of the MHD waves and oscillations in a structured and inhomogeneous medium can be very different from that found in a uniform medium. For example, it is improbable that pure Alfvén waves exist in the structured and magnetized solar plasma. The traditional theory of Alfvén waves needed the presence of homogeneous and uniform plasma of infinite extent, along with an isotropically constant unidirectional magnetic field. Since the magnetic plasma structures have a finite extent and are almost always nonuniform, it would be surprising that pure Alfvén waves are abundantly present in the magnetized and structured solar atmosphere. Moreover, the basic property of an Alfvén wave infers that the sum of plasma and magnetic pressure remains invariant during its propagation. For an inhomogeneous plasma, however, the total plasma pressure couples with the wave dynamics (e.g., Hasegawa and Uberoi, 1982). Therefore, the consideration to neglect the total pressure perturbations set-off is incapacitated. However, this does not imply that physicswise the concept of pure Alfvén waves is outworn. In customary, i.e., inhomogeneous and structured, plasma, MHD waves possess mixed physical properties that can be related to the properties of the classical slow, fast, and Alfvén waves in the homogeneous and infinite extended plasma. The extent to which the classic physical properties of pure waves are present depends on the properties of the background medium through which the MHD waves propagate. The nonuniformity across the magnetic field has a distinctly different effect on MHD waves compared to the nonuniformity along the magnetic field. The nonuniformity across the magnetic field is the cause of mixed properties of MHD waves. When an MHD wave, originating as a primarily fast magnetoacoustic wave, travels in an inhomogeneous background medium, it can convert into an MHD wave with equally strong fast and Alfvén wave properties and after some time further turn into a predominately Alfvén wave. This is the natural cause of resonant absorption. MHD waves in inhomogeneous structures are important in their own right. In addition, since the properties of the waves depend on the properties of the magnetic structures they live in, MHD waves can be used as seismic tools. MHD waves transport energy and their dissipation can heat plasmas. A possible mechanism in that context is resonant absorption, which is a fundamental mechanism that is difficult to avoid in an inhomogeneous plasma. In Chapter 5, mixed properties and resonant absorption will be discussed in detail. The effects of nonuniformity along the magnetic field will not be discussed. These effects influence the periods of MHD waves, but they are largely unimportant for the dissipation of MHD waves. Early scientific works on the effect of longitudinal stratification were performed, for example, by Andries et al. (2005b,a, 2009), Dymova and Ruderman (2005); Díaz and Roberts (2006), Arregui et al. (2005), and Verth and Erdélyi (2008). The effects of equilibrium flows are not considered here. Chapter 6 is focused on linear MHD waves superimposed on a static equilibrium state. Much of the analysis is based on the concept of a fluxtube which is modeled by a 1D axisymmetric straight cylindrical plasma that is radially stratified. Since the background is time-independent, it is a standard procedure to adopt an dependence on time with ω, the frequency of the wave. The spatial variation of the wave solution is more involved and depends on the degree of nonuniformity that is present in the background. In a 1D straight cylindrical plasma, the equilibrium quantities depend only on r and are invariant of φ and z. Hence the wave variables can be taken as an exponential factor , with m, the azimuthal and axial wave numbers, the azimuthal wave number m being an integer. When the complex functions , are transformed into real ones, the waves can be axially standing, axially propagating, or spinning. The radial dependencies of the waves are described by differential equations which might allow analytical solutions but often require numerical treatment. The MHD waves with azimuthal wave number are axisymmetric. These modes are sausage oscillations (e.g., Pascoe et al., 2007; Srivastava et al., 2008; Li et al., 2020, and the references therein). Conversely, corresponds to nonaxisymmetric MHD waves. The case is particular since MHD waves with are the only wave modes which displace the axis of the magnetic fluxtube. They are called kink waves. Waves with are called fluting waves. A demonstration of these wave modes is shown in Fig. 1.12. By using an imaging instrument (like TRACE, SDO/AIA, STEREO/EUVI), the fast magnetoacoustic kink waves are discovered as collective transversal displacements of the axis of fluxtubes (see, e.g., Nakariakov et al., 1999; Aschwanden et al., 1999; Aschwanden and Schrijver, 2011; Goddard and Nakariakov, 2016, and the references cited therein). A detailed review on the kink oscillations is recently given by Nakariakov et al. (2021). In contrast to the kink mode, the torsional Alfvén mode neither exhibits the displacement of the tube as a whole nor variations in its density due to incompressibility, thus no intensity oscillations can be detected by any imaging instruments. In spectrographs and spectro-polarimeters (such as CoMP for corona, and CRISP/SST for chromosphere), both transverse waves can be observed as the kink mode may generate a periodic Doppler shift of the spectral lines, while the torsional Alfvén waves exhibit both blue and red shifts simultaneously in various parts of the fluxtubes, further causing periodic nonthermal line broadening (e.g., Van Doorsselaere et al., 2008; Jess et al., 2009; Williams, 2004; Srivastava et al., 2017; Kohutova et al., 2020). In linear MHD there are infinitely many waves. However, these linear waves do not interact and there is no coupling. The equations might be coupled but the MHD waves are not. The properties of MHD waves change when the wave travels through a nonuniform medium. It might start as a wave that resembles what is known in a uniform plasma as a fast magneto-acoustic wave and arrive at a position where it resembles an Alfvén wave. This might be called mode conversion or a wave with mixed properties, which depend on the background. There is no coupling of waves. In the next subsection, we briefly discuss the physical manifestation of the