In this paper, we give a brief review of the contemporary theory of nonlinear waves in the solar atmosphere. The choice of topics reflects personal interests of the author. Historically the theory of nonlinear waves was first applied to the solar atmosphere to explain the chromospheric and coronal heating. It was assumed that the turbulent motion in the solar convective zone excites sound waves that propagate upwards. Due to nonlinearity these waves steepen and form shocks. The wave energy dissipates in these shocks thus heating the corona. We give a brief description of propagation and damping of nonlinear sound waves in the stratified solar atmosphere, and point out that, at present, the acoustic heating remains the most popular theory of heating the lower chromosphere. Then we extend the analysis to nonlinear slow magnetosonic waves in coronal plumes and loops, and discuss its implications for interpretation of observational results. The next topic of interest is the propagation of nonlinear waves in a magnetically structured atmosphere. Here, we restrict our analysis to slow sausage waves in magnetic tubes and discuss properties of solitary waves described by the Leibovich–Roberts equation. We conclude with the discussion of nonlinear theory of slow resonant layers, and its possible application to helioseismology.
Historically the theory of nonlinear waves was first applied to the solar atmosphere in an attempt to explain the chromospheric and coronal heating (Biermann 1946, 1948; Schwarzschild 1948). At that time there were no direct observational evidences of the existence of waves in the solar atmosphere. For many years studying waves in the solar atmosphere was almost purely theoretical. At the end of the last century the situation changed dramatically. After the launch of Solar and Heliospheric Observatory (SOHO) and Transition Region and Coronal Explorer (TRACE) spacecrafts the direct observations of wave activity in the solar atmosphere became possible. As a result, EIT (or coronal Moreton) waves (Wills-Davey & Thompson 1999), compressible waves in polar plumes (Ofman et al. 1997, 1999; DeForest & Gurman 1998) and in coronal loops (Berghmans & Clette 1999; Nakariakov et al. 2000b; Robbrecht et al. 2001; De Moortel et al. 2002a–,c), global kink oscillations of coronal loops (Aschwanden et al. 1999, 2002; Nakariakov et al. 1999; Schrijver et al. 1999, 2002; Schrijver & Brown 2000), and longitudinal standing oscillations within coronal loops (Kliem et al. 2002; Wang et al. 2002, 2003) have been discovered. These observations boosted interest in the theory of waves and oscillations in the solar atmosphere and, in particular, in the theory of nonlinear waves.
In this paper, we give a brief review of the contemporary theory of nonlinear waves in the solar atmosphere. The choice of topics reflects personal interests of the author. For example, we do not discuss kinetic effects, which are definitely important in many applications of the wave theory to the solar physics. Our analysis is based on the use of the standard set of magnetohydrodynamic (MHD) equations, which can be found, e.g. in Kulikovsky & Lyubimov (1965), Landau et al. (1984) or Priest (2000).
Strictly speaking, the MHD equations can be used only for waves with periods larger than the collisional time of ions. This time is much smaller than a second in the photosphere, but it increases to a few seconds in coronal loops and to a few tens of seconds in coronal holes.
The paper is organized as follows. In §2 we consider the propagation of nonlinear sound waves in an inhomogeneous plasma with application to chromospheric heating. In §3 we present the theory of nonlinear slow magnetosonic waves in coronal plumes and loops. In §4 we discuss nonlinear waves in magnetically structured plasmas concentrating on slow sausage solitary waves in straight magnetic flux tubes. In §5 we describe the nonlinear theory of slow resonant waves and its possible application to solar physics. We give a brief summary and conclusions in §6.
2. Chromospheric heating by nonlinear sound waves
As we have already mentioned in §1, nonlinear waves in the solar atmosphere were first considered in relation to the solar corona heating. Soon after the discovery by Edlén (1941) that the solar corona is much hotter than the photosphere, the model of coronal heating due to dissipation of nonlinear acoustic waves was suggested by Biermann (1946, 1948) and Schwarzschild (1948). Their idea was the following. The turbulent motion in the solar convective zone excites sound waves that propagate upwards. Due to nonlinearity these waves steepen and form shocks. The wave energy dissipates in these shocks thus heating the corona.
Let us give a brief mathematical description of this phenomenon. We assume that there is a driver at the base of the solar atmosphere that excites a periodic planar sound wave propagating upwards. If the distances at which we study the wave are much smaller than the solar radius, then we can neglect the curvature of the solar surface and consider the solar atmosphere as planar. In what follows we study waves with the wavelengths much smaller than the scale height, so that we can use the WKB approximation. In particular, we can neglect the wave reflection. In addition, we assume that the wave amplitude is small. Then, applying the reductive perturbation method (e.g. Kakutani et al. 1968; Taniuty and Wei 1968; Engelbrecht et al. 1988) to ideal hydrodynamic equations we can derive a simple equation describing the dependence of the vertical velocity, w, in the wave field on the time t and the vertical distance from the base of the solar atmosphere z. Omitting all the details of the derivation we write down only the final result (the derivation of a similar equation, however, with the account of the wave spherical expansion can be found, e.g. in the electronic Appendix to the paper by Izmodenov et al. 2005)(2.1)where the square of the sound speed is given by , ρ and p are the density and pressure, respectively, γ is the adiabatic exponent, and the subscript ‘0’ indicates a base quantity. Note that both cS and ρ0 are, in general, functions of z. The function w has to satisfy the boundary condition at z=0, , where ω is the wave frequency.
Using the variable substitution and considering z and as independent variables, and t as a dependent variable, we can easily solve equation (2.1). We do not give either the details of the solution or the final result. Instead we describe the qualitative properties of the solution. The solution determines as an implicit function of t and z. The evolution of the dependence of on t at a fixed z when z increases is shown in figure 1. For convenience we use the variable τ=t−φ(z) instead of t, where . In accordance with the boundary condition at z=0, is sinusoidal at z=0. Then, when z increases, the wave profile steepens due to nonlinearity, and eventually it has an infinite gradient at τ=0 when z=zc.
If we formally use the implicit formula determining for z>zc, then we obtain as a multi-valued function of τ as shown in figure 1c, which is physically meaningless. To obtain a physically meaningful solution we have to allow a discontinuity at a particular position of the wave profile. This discontinuity corresponds to a shock. It follows from the Rankine–Hugoniot relations that, for small-amplitude shocks, the velocity perturbation at the two sides of the shock have the same magnitude and opposite signs (e.g. Whitham 1974; Rudenko & Soluyan 1977).
Let us denote the shock intensity, which is equal to the jump of w across the shock, as 2ws(z). The wave behaviour is especially simple in an isothermal atmosphere, where cS=const. In such an atmosphere ws tends to a limiting value as z increases, so that for ez/2H≫1, where is the scale height. A typical snapshot of the wave profile is shown in figure 2. We see that at large z the wave takes a characteristic tooth-like shape.
Using this formula we can calculate the rate of atmospheric heating due to the wave damping.
After the theory of the solar atmospheric heating was suggested by Biermann (1946, 1948) and Schwarzschild (1948), it was further developed by many authors (e.g. Osterbrock 1961; Bird 1964; Kuperus 1969; Ulmschneider 1970, 1976, 1989; Stein & Schwartz 1972; Stein & Leibacher 1974; Gonczi et al. 1977; Carlsson & Stein 1992, 1997; Kalcofen et al. 1999). The discussion of this development is far beyond the scope of this paper, so that we refer the readers to books on solar physics (e.g. Priest 2000), and review papers on the solar atmospheric heating (e.g. Narain & Ulmschneider 1990, 1996) and waves in the chromosphere (e.g. Rutten 2003). Here, we only note that, at present, the widely accepted point of view in the solar community is that the acoustic shock damping is the most probable mechanism of the lower chromosphere heating, however, it is not appropriate for heating of the upper chromosphere and corona.
3. Slow magnetosonic waves in coronal loops and plumes
As it is well known, there are three MHD wave modes that can propagate in a homogeneous non-dissipative plasma. These modes are Alfvén waves and fast and slow magnetosonic waves (e.g. Jeffrey & Taniuti 1964; Kulikovsky & Lyubimov 1965; Goedbloed 1983; Landau et al. 1984; Priest 2000). The stratification modifies these modes. However, when the wavelength is sufficiently smaller than the characteristic scale of stratification, the local properties of the three MHD modes remain practically the same as in a homogeneous plasma.
Because the solar coronal plasma is very rarefied and the coronal magnetic fields are sufficiently strong, the transport coefficients in the coronal plasma are strongly anisotropic. The viscous force can be written as Fvis=∇.vis, where vis is the viscosity tensor. This tensor is equal to the sum of five terms, each term being proportional to its own viscosity coefficient (e.g. Braginskii 1965). The first term in this sum is often called the compressional viscosity. For typical coronal conditions, the coefficient of compressional viscosity, η0, is a few orders of magnitude larger than the other viscosity coefficients (e.g. Ofman et al. 1994; Porter et al. 1994; Erdélyi & Goossens 1995; Ruderman et al. 2000). This implies that we can neglect four other terms in the expression for vis in comparison with the compressional viscosity unless we consider the Alfvén waves that do not perturb the density.
For typical coronal conditions, the thermal conductivity in the directions perpendicular to the magnetic field is strongly suppressed. This enables us to assume that the heat flux is directed along the magnetic field.
When studying the wave propagation in the solar photosphere and lower chromosphere we can neglect dissipation everywhere except in shocks that we can consider as discontinuities because the Reynolds and Peclet numbers characterizing the importance of viscosity and thermal conductivity are very large. However, this is, in general, not true when we study waves in the solar corona, so that we have to take dissipation into account. Similar to stratification, dissipation modifies MHD waves. However, the local properties of MHD waves remain practically the same as in non-dissipative plasmas if these waves are only weakly damped, i.e. if their wavelengths are sufficiently smaller than the characteristic damping length.
In what follows we concentrate on slow magnetosonic waves in coronal loops and plumes. Coronal loops are thin closed structures in the solar corona elongated in the magnetic field direction. They are characterized by the enhanced plasma density and temperature. Polar plumes are cool, dense, linear, magnetically open structures in the solar polar coronal holes. Observations on board of the SOHO spacecraft showed signatures of quasiperiodic compressional waves with periods of 10–15 min (DeForest & Gurman 1998). Ofman et al. (1999) interpreted these waves as slow magnetosonic waves. Later propagating slow magnetosonic waves have been detected in coronal loops (Berghmans & Clette 1999; Robbrecht et al. 2001; De Moortel et al. 2002a–,c).
The wavelengths of observed slow magnetosonic waves are usually larger than the typical transverse size of magnetic loops and plumes. However, in low-beta plasmas like the solar coronal plasma, slow magnetosonic waves practically do not disturb the magnetic field. This observation enables us to consider slow magnetosonic waves propagating in coronal magnetic loops and plumes as waves propagating in a thin tube with rigid boundaries and, in general, with the variable cross-section. If we neglect the variation of unperturbed plasma quantities across the tube and assume that the wave fronts are planar and perpendicular to the magnetic field, then we arrive at a one-dimensional problem. Slow magnetosonic waves propagating along the magnetic field are, in fact, ordinary sound waves. So, in what follows we study sound waves propagating along a tube with the rigid boundaries, variable cross-section and background quantities varying along the tube. While plumes are almost straight, magnetic loops are curved. To simplify the analysis we neglect the tube curvature.
The propagation of slow magnetosonic waves in plumes and coronal magnetic loops was studied by Ofman et al. (2000) and Nakariakov et al. (2000b), respectively. In both papers, the authors used the assumptions described above. In addition they assumed that the equilibrium plasma is isothermal, so that the sound speed is constant. They considered nonlinear waves with small amplitudes and derived the Burgers-type equations. Ofman et al. (2000) studied sound waves in a radially divergent tube with the cross-section proportional to the distance from the centre of the Sun squared, which is a good approximation for coronal plumes. They also took γ=1, which means that they considered isothermal perturbations. In that case thermal conductivity does not contribute to the wave damping, so that the only dissipative process causing the wave damping is viscosity.
Using the reductive perturbation method (e.g. Kakutani et al. 1968; Taniuty & Wei 1968) we derived a Burgers-type equation for acoustic waves propagating in a rigid tube with variable cross-section and arbitrary variation of the equilibrium quantities along the tube,(3.1)where(3.2)
S is the tube cross-section, gas constant, mean atomic weight and κ thermal conductivity along the magnetic field. Using the approximate expressions for η0 and κ valid for fully ionized plasmas (Spitzer 1962; Priest 2000), and taking and γ=5/3, we obtain that the ratio of the second term in equation (3.2) to the first one is approximately equal to 10. This implies that, for adiabatic perturbations, the wave damping is mainly due to thermal conductivity. However, it is necessary to emphasize that this conclusion is only valid for weakly damped waves.
We obtain the equation derived by Nakariakov et al. (2000b) if we assume that the temperature and the tube cross-section are constant, and introduce the running variable ξ=z−cSt. To obtain the equation derived by Ofman et al. (2000) we once again have to assume that the temperature is constant, introduce ξ, take γ=1, substitute w=cSρ′/ρ0 with ρ′=ρ−ρ0, and put with being the solar radius.
Nakariakov et al. (2000b) assumed that the loop has the shape of a half-circle with the radius RL and obtained the expression for the equilibrium density taking into account the variation of projection of the gravity acceleration on the loop axis. Using this expression Nakariakov et al. solved the Burgers-type equation numerically and provided a comprehensive analysis of nonlinear slow wave propagation in coronal loops. Their main conclusions are as follows. For typical values of equilibrium plasma quantities in coronal loops the amplitudes of descending waves are much smaller than the amplitudes of ascending waves. The growth of the ascending wave amplitude due to the equilibrium density decrease is efficiently suppressed by dissipation. These results explain why the descending waves are not registered in observations.
Ofman et al. (2000) also solved the Burgers-type equation that they derived numerically. As a result they studied the dependence of properties of nonlinear slow magnetosonic waves on their initial amplitudes. They also compared their results with numerical solutions of full nonlinear one- and two-dimensional MHD equations and found that the agreement is very good.
As we have already mentioned, Ofman et al. (2000) took γ=1 in their analysis, which means that they considered isothermal perturbations. They did not give any particular reason for this assumption; however, the results of their numerical modelling are in good agreement with the observations. At present the physical mechanisms beyond this good agreement are not clear. It is quite possible that taking γ=1 is equivalent to the account of some unknown sources of energy in the lower part of plumes. Further studies of slow wave propagation in coronal plumes are needed to explain why modelling with γ=1 gives so good an agreement with observations as well as to find out the mechanisms of excitation of these waves.
4. Nonlinear waves in magnetically structured solar atmosphere
The solar atmosphere is strongly magnetically structured with the photosphere structured by magnetic flux tubes (e.g. Harvey 1977), and the corona structured by magnetic loops and coronal holes (e.g. Priest 1978; Vaiana & Rosner 1978). Magnetic structuring dramatically changes the properties of MHD waves. For example, surface MHD waves can exist (e.g. Roberts 1981a).
The linear theory of MHD waves in magnetically structured plasmas has been developed by many authors (e.g. Cram & Wilson 1975; Defouw 1976; Ryutov & Ryutova 1976; Roberts & Webb 1978; Wilson 1979; Wentzel 1979a,b; Roberts 1981a,b; Edwin & Roberts 1982, 1983; Miles & Roberts 1989; Ruderman 1991; Ballai et al. 2002; see also reviews by Roberts 1991a; Roberts & Ulmschneider 1997; Roberts & Nakariakov 2003).
The nonlinear theory of MHD waves in magnetically structured plasmas is more complicated than the linear theory. However, substantial progress has been made in this field as well. Early reviews on nonlinear waves in magnetically structured plasmas were given by, e.g. Ryutova (1990) and Roberts (1991b). A review of the recent progress in the theory of nonlinear MHD waves in magnetically structured plasmas was given by Ruderman (2003).
In this paper, we consider only one type of nonlinear MHD waves that can propagate in magnetically structured plasmas: slow sausage waves in magnetic flux tubes. The magnetic tube configuration consists of a cylindrical surface with homogeneous but different plasmas inside and outside.
In what follows we consider only the case where the magnetic field is straight and in the direction of the cylinder axis, and assume that the plasma is ideal. In addition, we assume that the plasma motion is axisymmetric, i.e. all quantities are independent of θ in cylindrical coordinates r, θ, z with the z-axis coincident with the cylinder axis, and the θ-components of the velocity and magnetic field are zero. And finally, we restrict our analysis to long small-amplitude waves, i.e. to waves with wavelengths much larger than r0 that cause displacements of the tube surface much smaller than r0, where r0 is the tube radius.
In what follows we use the sound speed, cS, introduced earlier, and also the Alfvén speed νA=B0(μρ0)−1/2 and tube speed . We use the same notation for the similar quantities in the external plasma, however, with the subscript ‘e’. The slow sausage waves are evanescent far from the tube only if the characteristic speeds inside and outside the tube satisfy(4.1)
An important quantity in theory of waves in magnetic tubes is given by(4.2)where ω and k are the wave frequency and number. The wave mode is called surface when , and body when .
There are many different wave modes that can propagate in a magnetic tube. In particular, there is an infinite sequence of slow sausage wave modes characterized by the property that ω/k→cT when k→0. Among these modes there is exactly one mode satisfying the condition as k→0. For all other slow sausage modes as k→0, which in particular, implies that these modes are body waves. In what follows we only concentrate on the slow sausage wave mode satisfying the condition as k→0. It can be shown that this mode is a surface wave when vAe<cT<cSe, while it is a body wave when either cSe<cT<vAe or cT<cTe. However, the properties of this body wave are quite similar to the properties of the surface wave, while they are completely different from the properties of the other slow sausage body waves. To distinguish this slow sausage body wave from all other slow sausage body waves Ruderman (2005) suggested calling this wave the pseudobody wave. We will use this name in what follows.
Roberts (1985) considered long small-amplitude nonlinear slow sausage surface waves in a magnetic flux tube embedded in a magnetic-free (Be0=0) plasma. Using the MHD equations averaged over the tube cross-section he derived the equation describing nonlinear slow sausage surface waves. With the accuracy up to the notation this equation can be written as(4.3)
Here, w is the z-component of the velocity. The coefficients β and Χ are given by(4.4)with vAe=0. The quantity α and the operator are determined by(4.5)where once again vAe=0. Since a similar equation was derived by Leibovich (1970) for waves in a vortex tube in an incompressible fluid, equation (4.3) is called the Leibovich–Roberts equation.
Molotovshchikov & Ruderman (1987) derived the Leibovich–Roberts equation for slow sausage waves propagating in a magnetic tube embedded in a magnetic plasma (Be0≠0). They did not use the MHD equations averaged over the tube cross-section. Instead they used the reductive perturbation method with double expansions in series with respect to two small parameters, the ratio of r0 to the characteristic wavelength, ϵ, and |ln ϵ|−1. They derived the Leibovich–Roberts equation (4.3) with the integral operator given by(4.6)where K0 and K1 are the modified Hankel functions (also called the modified Bessel functions of the second type or McDonald functions). We obtain the expression (4.5) for from equation (4.6) if we take K1(k)≈1/k valid for k≪1 (Abramowitz & Stegun 1964).
Note that the two forms of the Leibovich–Roberts equation, one with given by equation (4.5) and the other with given by equation (4.6) are asymptotically equivalent. This means that the difference between the left-hand sides of the two equations is of higher order of magnitude with respect to ϵ than any term in these equations.
Molotovshchikov & Ruderman (1987) derived the Leibovich–Roberts equation under assumption , so that they considered only surface waves. However, a straightforward inspection of their derivation shows that it remains valid also when and (Ruderman 2003). Hence, the equation derived by Molotovshchikov & Ruderman is the governing equation for long nonlinear slow sausage as well as pseudobody waves. Note that Ruderman (2003) made a terminological mistake and called the wave mode governed by the Leibovich–Roberts equation ‘surface wave’ in the case when either or . This mistake was corrected by Zhugzhda (2004) who pointed out that this wave is a body wave.
Molotovshchikov & Ruderman (1987) solved (4.3) with given by (4.6) numerically to find solutions in the form of solitary waves. They also determined the dependences of the nonlinear correction V to the solitary wave phase speed and the solitary wave width D on the solitary wave amplitude a. They found that V is almost a linear function of a, and the dependence of D on a is intermediate between characteristic for the KdV solitons, and characteristic for the Benjamen–Oho solitons. It is worth noting that Molotovshchikov & Ruderman found that a solitary wave exists for any value of a. Another interesting result is that a surface solitary wave causes the tube inflation and a pseudobody solitary wave causes the tube contraction.
Molotovshchikov (1989) studied the interaction of two solitary waves numerically and showed that they interact elastically, i.e. they preserve their shapes after the interaction.
Weisshaar (1989) solved the Leibovich–Roberts equation with given by (4.5) numerically to obtain solutions in the form of solitary waves. In particular, he found that these solutions exist only when the solitary wave amplitude is smaller than a critical value. On the other hand, similar to Molotovshchikov (1989), he also found that solitary waves interact elastically. We see that, although the two alternative forms of the Leibovich–Roberts equation are asymptotically equivalent, the properties of their solutions are quite different. This is a common situation in the asymptotic theory of nonlinear waves.
When the conditions (4.1) are not satisfied, the slow sausage waves become leaky waves. They decay due to radiation of MHD waves in the external plasma. Zhugzhda (2000) and Zhugzhda & Goossens (2001) derived the so-called extended Leibovich–Roberts equation with the modified integral operator that describes both the wave dispersion and damping due to wave leakage in the plasma surrounding the tube. Ballai & Zhugzhda (2002) generalized this derivation for anisotropic plasmas described by Chew, Goldberger and Low equations.
We conclude this section with the following remark. In the photosphere magnetic flux tubes are usually embedded in a magnetic-free plasma, so that . This implies that we can expect that slow sausage surface solitary waves can propagate along magnetic tubes in the solar photosphere. Recall that these solitary waves cause the tube inflation. The coronal structures, where the slow sausage solitary waves can propagate are coronal magnetic loops. The plasma inside a coronal loop is hotter and denser than the surrounding plasma. In addition the coronal plasma is a low-beta plasma, so that the magnetic field magnitude is almost the same inside and outside the loops. This implies that , , and . Hence, and slow sausage solitary waves that can propagate in coronal loops are pseudobody solitary waves. Recall that these solitary waves cause the loop contraction.
5. Nonlinear effects in resonant layers
The approach to studying waves in magnetically structured plasma based on dividing the plasma in regions with homogeneous plasma separated by tangential discontinuities turned out to be very fruitful. It enabled us to learn a lot about the properties of MHD waves propagating in plasmas with sharp variation of equilibrium quantities. However, the representation of narrow inhomogeneous layers by discontinuities is an idealization. There are no true discontinuities in real plasmas. Instead the equilibrium quantities vary continuously in space, which leads to a continuous spectrum in linear ideal MHD. Eigenfunctions that correspond to frequencies in the continuum spectrum are improper and contain a non-integrable singularity at a spatial position called the resonant position. The presence of improper eigenfunctions results in a new phenomenon called resonant absorption. This process occurs because the global wave motions are in resonance with local plasma oscillations in the vicinity of the so-called resonant magnetic surface. The resonance causes energy to build up in the vicinity of the resonant magnetic surface at the expense of the energy in the global motion. Both the wave energy and the spatial gradients become large in the vicinity of the resonant magnetic surface. If there is even a small amount of dissipation, then the energy which has been transferred to the resonant magnetic surface can be converted into heat. The possibility of plasma heating has attracted the attention of solar physicists. Ionson (1978) proposed resonant absorption as a mechanism for heating the solar corona. Since then, resonant absorption remains a popular mechanism for explaining the solar coronal heating (e.g. Davila 1987; Goossens 1991; Hollweg 1991; Poedts 1999).
There are two types of resonances: Alfvén and slow. In the case of Alfvén resonance the global plasma motion is in resonance with local Alfvén waves at the Alfvén resonant position, and in the case of slow resonance it is in resonance with local slow waves at the slow resonant position. Since nonlinear effects are more pronounced in slow resonant layers, in this paper we concentrate on the slow resonance.
In weakly dissipative plasmas large gradients are only present in the vicinities of resonant positions. This implies that we can use a mixed description of the plasmas motion: the ideal MHD is used to describe the plasma motion far from resonant positions, while the dissipative MHD is used in thin dissipative layers embracing resonant magnetic surfaces. It turned out that the linear dissipative MHD equations in thin dissipative layers can be solved analytically. After that Sakurai et al. (1991) suggested the following concept of connection formulae (see also Goossens et al. 1995; Goossens & Ruderman 1995). Using the analytical solution of the linear dissipative MHD equations in the dissipative layer we can calculate the jumps of the perturbation of the total pressure, P, and the normal component of the velocity, u, across the dissipative layer. The expressions for these jumps are called the connection formulae. Then we can consider the dissipative layer as a surface of discontinuity and use the connection formulae to connect the solutions of linear ideal MHD to the left and the right of this surface of discontinuity. Note that this approach is similar to using the Rankine–Hugoniot relations at shocks for studying flows of ideal compressional fluids.
Linear theory of resonant MHD waves predicts not only large gradients in the vicinity of a resonant position in a weakly dissipative plasma, but also large amplitude of oscillations in this vicinity even when the amplitude of oscillations far from the resonant position is small. This implies that nonlinearity can be important in the vicinity of a resonant position. In accordance with this we have to modify our approach and use the nonlinear dissipative MHD equations to describe the motion near resonant magnetic surfaces. Since the thickness of layers where we have to use the nonlinear dissipative MHD equations is now determined not only by dissipation but also by nonlinearity, we will rename dissipative layers in resonant layers.
Ruderman et al. (1997) obtained a criterion showing when nonlinear effects are important in a slow resonant layer in plasmas with finite electric conductivity and isotropic viscosity. These authors introduced the viscous, , and magnetic, , Reynolds numbers, where is the characteristic speed of the problem (usually of the order of ), the characteristic spatial scale of plasma inhomogeneity, and ν and λ are kinematic viscosity and coefficient of magnetic diffusion, respectively. Then they showed that nonlinearity is important in the resonant layer when , where R is the total Reynolds number given by , and ϵ is the characteristic dimensionless wave amplitude far from the resonant magnetic surface. When , the linear description can be used.
It can be shown that the dimensionless amplitude of the wave motion in the nonlinear resonant layer is of the order of , i.e. it remains small. However, this motion still can be strongly nonlinear in the sense that the nonlinearity dominates dissipation. The fact that the wave amplitudes in the resonant layer are small enables us to use the asymptotic theory and derive the approximate governing equation for plasma motion in a slow resonant layer. From the mathematical point of view this equation is much simpler than the full system of nonlinear dissipative MHD equations.
The nonlinear governing equation for plasma motion in a slow resonant layer was first derived by Ruderman et al. (1997). These authors assumed that the equilibrium state is static one-dimensional with all equilibrium quantities dependent only on x in Cartesian coordinates x, y, z. They only considered two-dimensional plasma motions with all quantities independent of y and the y-components of the velocity and the magnetic field perturbation equal to zero. We adopt these assumptions in what follows. The plasma motion in a slow resonant layer is a slow wave propagating almost perpendicular to the equilibrium magnetic field. The wave front of such a wave propagates along the magnetic field with the velocity equal to . Ruderman et al. considered plasmas with finite resistivity and isotropic viscosity, and restricted their analysis to the motion in the resonant layer that has the form of a nonlinear wave with permanent shape, i.e. they were looking for the solutions that depend on . Ballai et al. (1998) extended their analysis and derived the nonlinear governing equation for plasma motion in a slow resonant layer in a plasma with strongly anisotropic viscosity and thermal conductivity. Ballai & Erdélyi (1998) obtained the nonlinear governing equation for plasma motion in a slow resonant layer for a steady equilibrium, i.e. for an equilibrium with a flow. Then the derivation was extended to cylindrical geometry. Ballai et al. (2000) studied slow waves in a straight magnetic cylinder assuming that the perturbations of all variables depend on r and in cylindrical coordinates r, z, θ, where h is a constant. And Ballai & Erdélyi (2002) and Erdélyi & Ballai (2002) derived the nonlinear governing equation for axisymmetric slow waves in twisted magnetic flux tubes.
Ruderman & Erdélyi (2000) relaxed the assumption that perturbations of all variables depend on . They used the reductive perturbation method combined with the method of matched asymptotic expansions to derive a non-stationary version of the nonlinear governing equation for plasma motion in a slow resonant layer in a planar geometry. They considered a steady equilibrium, i.e. an equilibrium with a flow, and assumed that the equilibrium magnetic field is constant, parallel to the xy-plane, and the angle between this field and the z-axis is arbitrary. Here, for the sake of simplicity, we consider an equilibrium without flow and assume that the equilibrium magnetic field is parallel to the z-axis. Under these simplifying assumptions the equation derived by Ruderman & Erdélyi reduces to(5.1)
Here, w and P are the z-component of the velocity and the perturbation of the total pressure, respectively, β is given by (4.4), , is the ideal slow resonant position determined by the condition that, at , coincides with the phase velocity of external perturbation in the magnetic field direction, and all coefficient-functions in (5.1) are calculated at . For what follows it is also important that the x-component of the velocity, u, is related to w by(5.2)where once again all coefficient-functions are calculated at .
The left-hand side of equation (5.1) describes slow magnetosonic waves in the resonant layer. The term on the right-hand side of (5.1) is determined by the solution of the external problem, which is the solution of linear ideal MHD equations far from the resonant layer. Hence, it has to be considered as an external driver.
Ruderman (2001) derived an equation similar to (5.1) for a plasma with strongly anisotropic viscosity and thermal conductivity. The main difference between this equation and equation (5.1) is that the last term on the left-hand side is proportional to rather than to .
Now we only consider solutions of (5.1) that depend on and are periodic with respect to θ. In general, such solutions can be found only numerically. This means that, in contrast to the linear theory, we cannot separate the external and internal problems and have to solve (5.1) together with the linear MHD equations outside the resonant layer.
However, there are two special cases when we can separate the internal and external problems. In the first case dissipation strongly dominates nonlinearity, so that the nonlinearity parameter is small (). This implies that we can consider the nonlinear term in (5.1), which is the third term on the right-hand side, as a perturbation that gives a small correction to the linear solution. This correction can be calculated with the use of the regular perturbation method.
In the second case the situation is opposite: nonlinearity strongly dominates dissipation, so that the nonlinearity parameter is large (). This case is much more interesting than the first one because in practically all applications to solar physics the inequality is satisfied. However, this case is also much more complicated mathematically because we now have a small parameter at the last term on the left-hand side of (5.1) which contains the highest derivative. As a result, to solve (5.1) we have to use the singular perturbation method. Ruderman (2000) applied this method to obtain an implicit solution of (5.1) under the assumption that this solution depends on x and θ and it is periodic with respect to θ. Using this solution he derived the connection formulae similar to ones obtained in the linear theory. The first connection formula is the same as the linear one. It is , where the square brackets indicate the jump of a quantity. The second connection formula gives the jump of u across a slow resonant layer. To obtain this formula equation (5.2) was used. For the particular equilibrium considered in this paper this connection formula can be written as(5.3)
Here, L is the period with respect to θ, is the point where P takes its maximum value in the interval , and . All quantities in this expression are calculated at .
Although the solution of (5.1) can be found only in an implicit form, it is possible do describe its qualitative behaviour. In figure 3 the dependence of w on the dimensionless variables and is shown. The function w is continuous for and tends to zero when . Inside the strip the function w is discontinuous at the solid curves. These discontinuities are slow MHD shocks. The intensity of each shock is zero at . It increases to its maximum value at σ=0, and then decreases back to zero at . Note that figure 3 is strongly stretched in the σ-direction. In the real scale, and the shocks almost coincide with the -axis.
It follows from the derivation of (5.1) that the dimensionless amplitude of the wave motion in the resonant layer is of the order of no matter how large is the nonlinearity parameter (recall that ϵ is the characteristic dimensionless amplitude of the wave motion far from the resonant layer). This result is in sharp contrast to the linear theory that predicts unbounded growth of the amplitude of the wave motion in the resonant layer as R is increased.
Ruderman (2000) used the nonlinear connection formulae to calculate the coefficient of wave energy resonant absorption for incoming sound waves interacting with an inhomogeneous magnetized plasma. He then compared the results with those obtained on the basis of the linear theory. It turned out that, for a wide range of parameters, the difference between the linear and nonlinear coefficients of resonant absorption does not exceed 20%. Hence, while the linear theory fails to describe properly the wave motion in strongly nonlinear slow resonant layers, it gives fairly good approximation for the coefficient of resonant absorption.
Until recently, in all applications to solar physics, the most important part of theory of resonant MHD waves was the effect of resonant absorption on the solution of the external problem. The wave motion in resonant layers was considered as of minor importance. However, the situation changed when Burgess et al. (2003, 2004) and Semikoz et al. (2004) suggested that resonant MHD waves can be used for seismology of the deep solar interior. Their idea can be described as follows. It seems quite reasonable to assume that there are gravity modes (g-modes) trapped deep in the solar interior. If, in addition, we assume that there is magnetic field in the solar interior, then these modes become magneto-gravity modes. We can expect that these modes are in resonance with slow magnetosonic modes at some spatial positions. As a result there are slow resonant layers in the deep solar interior. Density fluctuation in the slow resonant layers can cause oscillation of the neutrino flux that can be measured at the Earth. This measurement can be used to provide information about the plasma and magnetic field parameters in the deep solar interior. For correct interpretation of measurements of the neutrino flux oscillations an accurate description of the plasma motion in slow resonant layers is of crucial importance.
In this paper, we gave a brief review of the contemporary theory of nonlinear waves in the solar atmosphere. We started our review with nonlinear sound waves that historically first appeared in theoretical solar physics. Then we proceeded to nonlinear slow magnetosonic waves in coronal plumes and loops. The next topic was nonlinear waves in a magnetically structured atmosphere. Here, we restricted our analysis to slow sausage waves in magnetic tubes and discussed properties of solitary waves described by the Leibovich–Roberts equation. The last topic was nonlinear effects in slow resonant layers, and possible application of the nonlinear theory of slow resonant layers to helioseismology.
In this brief review it was impossible to consider all applications of theory of nonlinear waves to solar physics. We did not discuss nonlinear theory of Alfvén waves in coronal holes (Nakariakov et al. 2000a), and nonlinear effects in phase mixing of Alfvén waves (Nakariakov & Roberts 1997). We did not mention a very interesting theory of solar coronal heating based on the nonlinear conversion of large-scale fast magnetosonic waves in small-scale kinetic Alfvén waves (Voitenko & Goossens 2001, 2002). And finally, a huge number of papers on the direct numerical modelling of nonlinear hydrodynamic and MHD waves in the solar atmosphere were not discussed in this review. However, we hope that even this incomplete review can convince the readers that nonlinear theory of hydrodynamic and MHD waves is of great importance for solar physics, and its development with the intended application to solar physics deserves great effort of theorists.
One contribution of 20 to a Discussion Meeting Issue ‘MHD waves and oscillations in the solar plasma’.
- © 2005 The Royal Society