Issue 
A&A
Volume 553, May 2013



Article Number  A23  
Number of page(s)  6  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201321175  
Published online  25 April 2013 
Nonlinear damped standing slow waves in hot coronal magnetic loops
^{1}
Solar Physics and Space Plasma Research Centre
(SP 2RC), University of Sheffield,
Hicks Building, Hounsfield Road,
Sheffield
S3 7RH,
UK
^{2}
Space Research Institute (IKI), Russian Academy of Sciences,
117997
Moscow,
Russia
email: m.s.ruderman@sheffield.ac.uk
Received:
27
January
2013
Accepted:
19
March
2013
We study the nonlinear standing slow waves in coronal magnetic loops. We assume that the wave amplitude is sufficiently small and use it as a small parameter in the asymptotic expansions. In addition, we assume that the wave damping is sufficiently slow, which enables us to introduce the “slow” time related to the evolution of the wave amplitude and shape. We show that, in the leadingorder approximation, the nonlinear standing wave is a superposition of two identical nonlinear waves propagating in the opposite directions. The two waves are governed by the Burgers equation. Using the ColeHopfe substitution, we obtain an analytical solution describing a standing wave that has the form of the linear fundamental eigenmode at the initial moment of time. This solution is used for the parametric study of nonlinear standing waves. In particular, the effect of nonlinearity on the oscillation damping time is investigated.
Key words: magnetohydrodynamics (MHD) / plasmas / waves / methods: analytical
© ESO, 2013
1. Introduction
Standing waves in hot (T ≳ 6 MK) coronal loops were observed by the Solar Ultraviolet Measurements of Emitted Radiation (SUMER) spectrometer onboard the SOHO spacecraft (Kliem et al. 2002; Wang et al. 2002, 2003a,b). Observations of 17 events revealed that the oscillation periods ranged from 11 to 31 min, and the decay time from 5.5 to 29 min. Ofman & Wang (2002) interpreted these oscillations as slow standing waves. The velocity amplitude in these oscillations was up to one quarter of the phase speed, so at least some of them were moderately nonlinear. Ofman & Wang (2002) modelled these oscillations numerically and found that the main damping mechanism causing their damping is thermal conduction. They also found that the damping prevents strong nonlinear steepening and shock formation. For a recent review of standing slow waves in coronal loops see Wang (2011).
In their modelling Ofman & Wang (2002) used the direct numerical solution of the onedimensional nonstationary nonlinear equations describing the motion of compressible fluid with the account of viscosity and thermal conduction. They carried out the modelling only for a restricted range of parameters. More recently, a similar numerical modelling has been carried out by MendozaBriceño et al. (2004), Sigalotti et al. (2007), and Verwichte et al. (2008) for a wider range of coronal loop parameters and initial perturbation amplitude.
In this paper we aim to present a semianalytical study of slow standing waves in hot coronal loops. Our analysis is based on the fact that the oscillation amplitude is quite moderate, so the dimensionless amplitude can be used as a small parameter in the asymptotic expansion. As a result we derive a simple differential equation describing the perturbation evolution with time. This enables us to carry out the parametric study of oscillations for a wide range of parameters.
The paper is organized as follows. In the next section we formulate the problem, discuss the main assumptions, and present the governing equations. In Sect. 3 we derive the nonlinear equation describing the timeevolution of the velocity in the slow standing wave. In Sect. 4 we obtain the analytic solution describing the evolution with time of the perturbation corresponding to the fundamental mode of slow standing waves. We then use this solution for the parametric study of the nonlinear damping of slow standing waves. Section 5 contains the summary of the obtained results and our conclusions.
2. Problem formulation and main assumptions
We study slow standing waves in hot coronal loops with the temperature T ≳ 6 MK. The atmospheric scale height in these hot loops is larger than 300 Mm. This implies that, in these loops, we can safely neglect the density variation if the height of the loop apex point is smaller than or of the order of 100 Mm. The plasma number density in hot loops usually does not exceed 10^{15} m^{3}. Then, for T ≤ 10 MK, the plasma pressure does not exceed 0.3 Nm^{2}. The magnetic pressure is equal to B^{2}/2μ_{0}, where B is the magnetic field magnitude and μ_{0} ≈ 1.26 × 10^{6} H m^{1} is the magnetic permeability of free space. Then we easily find that the plasma beta is smaller than 0.1 when B ≳ 10^{3} Tesla = 10 G. Since this condition is usually satisfied in the coronal loops, we can use the lowbeta plasma approximation when studying the slow waves in hot coronal loops. This implies that we can neglect the magnetic field perturbation. If, in addition, we neglect the variation of the loop crosssection and the equilibrium quantities in the directions perpendicular to the loop axis, and the loop curvature, then we can describe the slow waves by onedimensional hydrodynamic equations (Priest 1982; Goedbloed & Poedts 2004) Here u is the velocity, ρ the density, p the pressure, T the temperature, γ the ratio of specific heats, k_{B} the Boltzmann constant, and m the mean mass for a particle (m ≈ 0.6 m_{p} in the solar corona, where m_{p} is the proton mass). The coefficients ν and κ are defined by (5)where k_{B} is the Boltzmann constant, η_{0} the first viscosity coefficient in the Braginskii’s expression for the viscosity tensor, and k_{∥} the thermal conductivity parallel to the magnetic field. These latter two quantities are given by (Braginskii 1965) (6)and (Spitzer 1962; Priest 1982) (7)where n = ρ/m_{p} is the number density and τ_{p} the proton collision time. It is given by the approximate expression (8)where Λ is the Coulomb logarithm and the number density n is measured in m^{3}. Fot typical conditions in hot loops lnΛ ≈ 20, and τ_{p} is approximately between 10 and 25 s.
The system of Eqs. (1)–(4) is essentially the same as the one used by Ofman & Wang (2002). The only difference is that we neglected the term describing the viscous heating in the energy Eq. (4). The reason for this is that this term is quadratic with respect to u, while, in what follows, we linearise the dissipative terms. The system of Eqs. (1)–(4) has to be supplemented with the boundary conditions. We assume that the magnetic loop is frozen in the dense photosphere, so that (9)where L is the loop length.
The system of Eqs. (1)–(4) with the boundary conditions (9) is used in the next section to derive the equation governing the evolution of the velocity with time in a slow standing wave.
3. Derivation of the governing equation for the oscillation velocity
In this section we derive the equation governing the evolution of the oscillation velocity with time. If we neglect the nonlinear and dissipative terms in Eqs. (1)–(4), this system of equations admits solutions describing periodic standing slow waves in the loop. We assume that the nonlinear and dissipative terms are small and only cause a slow evolution of perturbations. This assumption enables us to use the method of two scales with the “fast time” related to the quasiperiodic oscillations and “slow” time related to slow evolution of oscillations (e.g. Bender & Orszag 1978). Before starting the derivation we comment on the criterion of applicability of the twoscale method. When studying damped oscillations it is usually assumed that these oscillations can be considered as slowly evolving and the twoscale method can be applied if the characteristic damping time is much longer than the oscillation period. However, this is not completely correct. In the linear theory a typical function describing the damped oscillations is e^{−γt}sinωt. The characteristic damping time is 1/γ. It is also clear that the characteristic time related to the oscillation is 1/ω rather than the oscillation period 2π/ω. This implies that the criterion for the applicability of the twoscale method is γ/ω ≪ 1. We see that even when the characteristic damping time is equal to the oscillation period, we have γ/ω < 0.2 and the use of the twoscale method is perfectly correct. Since, in accordance with observations, the characteristic damping time of the slow standing waves in hot coronal loops is longer than or of the order of the oscillation period, we can use the twoscale method to study these oscillations. Then we use the asymptotic method based on the expansions with respect to the small amplitude of oscillations. To complete this introductory discussion we mention that a similar method of asymptotic expansions was previously successfully applied by Gusev (1984b,a) to study the steady state of oscillation of driven standing nonlinear waves in a finitelength wave guide, and by Ruderman & Nocera (1998) to study the steady state of driven Alfvén oscillations in a magnetic cavity.
In what follows we assume that the dissipation is weak and introduce the scaled coefficients at the dissipative terms, and , where ϵ ≪ 1. We also assume that the oscillation amplitude is sufficiently small and neglect the dependence of ν and κ on T. Hence, in what follows, ν and κ are constant. Finally we introduce the “slow” time t_{1} = ϵt. Then Eqs. (1)–(3) are transformed to Equation (4) remains unchanged. We look for the solution to the system of Eqs. (4) and (10)−(12) with the boundary conditions (9) in the form of expansions (13)where f represents any of the quantities u, ρ, p and T. The first term, f_{0}, corresponds to the unperturbed state, so f_{0} = const. Note that u_{0} = 0.
3.1. Firstorder approximation
In the firstorder approximation we substitute the expansion (13) in Eqs. (4) and (10)−(12), and the boundary condition (9) and collect the terms proportional to ϵ. As a result we obtain Eliminating perturbations of all variables in favour of u_{1}, we obtain (19)where c_{s} = (γp_{0}/ρ_{0})^{1/2} is the adiabatic sound speed. When deriving this equation we have used he relation p_{0} = k_{B}ρ_{0}T_{0}/m that follows from Eq. (4). The general solution to Eq. (19) can be written as (20)where, at present, f and g are arbitrary functions, and ω is an arbitrary constant with the dimension s^{1}. Note that f and g also depend on t_{1}, however, at present, we do not write this argument explicitly. Substituting Eq. (20) in Eq. (18), we obtain (21)It follows from these equations that g(y) = −f(y), and f(y) is a function with the period equal to 2ωL/c_{s}. It is convenient to define ω such that the period of function f(y) is equal to 2π. In this case ω = πc_{s}/L. Summarizing, we obtain that (22)where f is a 2πperiodic function. Now it follows from Eqs. (14), (15), and (17) that (23)(24)
3.2. Secondorder approximation
In the secondorder approximation we collect the terms proportional to ϵ^{2} in Eqs. (4) and (10)–(12), and the boundary condition (9). Then, using Eqs. (22)−(24), we obtain where f_{−} = f(ξ,t_{1}), f_{ + } = f(η,t_{1}), and the prime denotes the partial derivative of f(y,t_{1}) with respect to y. Eliminating ρ_{2}, p_{2}, and T_{2} from Eqs. (25)−(28), we obtain the equation for u_{2}: (30)We introduce the Fourier series where , , and the asterisk denotes the complex conjugate. Without loss of generality we can always assume that f_{0} = 0. Substituting these expressions in Eq. (30), and using the identity valid for real functions f and g, we obtain (35)It follows from Eq. (9) that U_{n} satisfies the boundary conditions (36)The homogeneous counterpart of Eq. (35) together with the boundary conditions (36) constitute the SturmLiouville problem. It has a nontrivial solution U_{n} = sin(πnx/L). This implies that Eq. (35) has a solution satisfying the boundary conditions (36) only if the righthand side of it satisfies the compatibility condition. This compatibility condition is the condition that it is orthogonal to the function sin(πnx/L), i.e. that the integral of the product of the righthand side and the function sin(πnx/L) over the interval [0,L] is equal to zero. In the last sum on the righthand side of Eq. (35) only two terms, one proportional to sin(πnx/L) and the other proportional to −sin(πnx/L) can give nonzero contributions in this integral. However, due to the assumption that f_{0} = 0, these two terms are identically zero. Hence, the last sum on the righthand side of Eq. (35) does not contribute in the integral. It is easy to show that this conclusion remains correct even if we do not assume that f_{0} = 0. Then we immediately obtain that the compatibility condition is (37)Multiplying this equation by e^{iny} and taking the sum with respect to n from −∞ to ∞ we obtain (38)where τ = t/t_{dl}, (39)Here t_{dl} is the damping time given by the linear theory. We see that the function f satisfies the Burgers Eq. (38). The Burgers equation is one of the basic equations in nonlinear acoustics (e.g. Burgers 1948; Whitham 1974; Rudenko & Solujan 1977). In the leadingorder approximation with respect to the small parameter ϵ the plasma velocity, density, pressure, and temperature are given by Eqs. (22)−(24). Hence, in the leadingorder approximation, the nonlinear standing wave is a superposition of two identical nonlinear waves propagating in the opposite directions. The evolution of each of these waves is governed by the Burgers Eq. (38).
4. Nonlinear damping of slow waves
In this section we use the results obtained in the previous section to study the damping of nonlinear standing waves. We assume that, at the initial moment of time, the function f is given by (40)This implies that in the leadingorder approximation (41)We see that the initial wave amplitude is 2ϵc_{s}. Hence, 2ϵ is the perturbation Mach number at the initial moment of time. To obtain the 2πperiodic solution of Eq. (38) satisfying the initial condition (40), we use the famous ColeHopf (Cole 1951; Hopf 1950) substitution (42)Substituting this expression in Eq. (38) yields (43)The function h has to be 2πperiodic. It follows from Eqs. (40) and (42) that h has to satisfy the initial condition (44)The general 2πperiodic solution to Eq. (43) is (45)Using the formulae (e.g. Abramowitz & Stegun 1964) (46)where n = 0,1,... and I_{n} is the modified Bessel function of the first kind and order n, we obtain the Fourier expansion (47)It follows from Eq. (44), and the comparison of the Fourier expansion (45) calculated at τ = 0 and the Fourier expansion (47) that (48)Substituting these expressions in Eq. (45), we eventually arrive at (49)This equation together with Eq. (42) define the 2πperiodic solution to Eq. (38) satisfying the initial condition (40). We see that, in the dimensionless variables, the evolution of the initial perturbation depends on one dimensionless parameter λ only. Using Eqs. (4)−(8), we obtain the approximate expression (50)n_{0} is in m^{3}, L is in m, and T_{0} is in Kelvin, and we have taken the Coulomb logarithm lnΛ = 20.
The parameter λ can vary in a quite large limits. For instance, in the example considered by Ofman & Wang (2002) the loop parameters are as follows: L ≈ 400 Mm, n = 5 × 10^{14} m^{3} and T = 6.3 MK. The initial perturbation Mach number is about 0.2, so ϵ ≈ 0.1. Substituting these numbers in Eq. (50), we obtain λ ≈ 0.25. If we take higher values of ϵ and/or n, we can obtain the value of λ exceeding unity. However, for typical hot loop parameters and moderate Mach numbers of the initial amplitude we can expect that λ is sufficiently smaller than unity. In that case we can obtain the approximate solution using the expansion in the power series with respect to λ. Using the expansions of the modified Bessel functions in the power series (e.g. Abramowitz & Stegun 1964), we obtain from Eq. (49) (51)Substituting this expression in Eq. (42) yields (52)In order to study how nonlinearity affects the damping rate we calculate the average over period energy of oscillations per unit area of the loop crosssection. In the leadingorder approximation with respect to ϵ it is given by (53)We define the dimensionless damping time τ_{d} by the equation ℰ(τ_{d}) = e^{2} (note that ℰ(0) = 1). Substituting Eq. (52) in Eq. (53), we obtain (54)Using this result we easily find (55)We see that nonlinearity reduces the damping time. This is an expected result because the nonlinearity transfers a part of the wave energy from the fundamental mode to the overtones that damp faster than the fundamental mode. As we have already pointed out, in the example considered by Ofman & Wang (2002)λ ≈ 0.4. Then, using Eq. (55), we find that the nonlinear damping time is by about 1.3% shoter than that given by the linear theory. We see that even for a moderately high value of λ the correction to the damping time given by the nonlinear theory is very small.
We introduce the oscillation period P = 2π/ω = 2L/c_{s} and the damping time (56)Then we obtain for the ratio of damping time to the wave period t_{d}/P = τ_{d}t_{dl}/P, where (57)where t_{dl} is the damping time given by linear theory, and n_{0}, L and T_{0} are measured in SI units. When deriving this formula we have used the relation m = 0.6 m_{p}. Using Eqs. (50) and (57) we obtain one more useful formula, (58)In Fig. 1 the evolution of the velocity profile during one oscillation period is shown for t_{dl} = 5P and ϵ = 0.1, which corresponds to λ ≈ 2.1. We can clearly see the wave steepening related to the generation of higher harmonics due do nonlinearity.
Fig. 1 Evolution of the velocity profile with time for t_{dl} = 5P and ϵ = 0.1. The upper, middle, and lower panel correspond to t = 0, P/2, and P respectively. 
Fig. 2 Dependence of the dimensionless damping time τ_{d} on the nonlinearity parameter λ. The solid line shows the dependence calculated numerically using the full Fourier expansion (49), while the dashed line shows the approximate dependence for small λ given by Eq. (55). 
Figure 2 shows the dependence of the dimensionless damping time, τ_{d}, on the nonlinearity parameter λ. Once again we see that nonlinearity reduces the damping time. It is interesting to note that, although Eq. (55) was derived under the assumption that λ ≪ 1, in fact it approximates τ_{d} for λ ≲ 1 fairly well.
We can see from Fig. 2 that the nonlinearity effect can be neglected for λ ≲ 1. Then it follows from Eq. (58) that the nonlinear correction to the damping time must be taken into account only when ϵt_{dl}/P ≳ 0.3. Once again using Fig. 2 we conclude that, for realistic values of λ ≲ 3, nonlinearity can reduce the damping time by no more than about one third, so t_{dl} ≲ 1.5t_{d}. Hence, eventually, the criterion for the importance of nonlinearity effect on the damping time is ϵt_{d}/P ≳ 0.2. Since, in accordance with observations, t_{d} ≲ 2P, we conclude that the nonlinearity effect is important only when ϵ ≳ 0.1, i.e. when the Much number of the initial perturbation is of the order of or larger than 0.2.
It is expedient to compare the results obtained in this paper with those presented by Verwichte et al. (2008). Unfortunately, we cannot carry out a quantitative comparison. The reason for this is that, for the loop parameters considered by Verwichte et al. (2008) Eq. (57) gives t_{dl}/P ≈ 0.43, so, for this set of loop parameters, the condition t_{dl}/P ≳ 1 is not satisfied, i.e. one of the main assumptions made in this paper is violated. It follows from Fig. 5 of Verwichte et al. (2008) that the correct value is t_{dl}/P ≈ 1.4. This discrepancy is not surprising at all. It is well known that t_{dl}/P is inversely proportional to k_{∥} only when k_{∥} is not too large. When k_{∥} increases, t_{dl}/P takes its minimum value, and then starts to grow (e.g. De Moortel & Hood 2003). However, we can still make a qualitative comparison. First of all, similar to Verwichte et al. (2008), we found that nonlinearity reduces the damping time, which is a result expected on physical grounds. If, in accordance with the results obtain by Verwichte et al. (2008), we take t_{dl}/P ≈ 1.4, then, using Eq. (58), we have λ ≈ 6ϵ = 3V_{0}/c_{s}, where V_{0} is the initial perturbation amplitude. Then it follows from Fig. 2 that the nonlinear correction to the damping time becomes important for V_{0}/c_{s} ≳ 0.3, and for this values of V_{0}/c_{s} the nonlinear correction to the damping time is an almost linear function of V_{0}/c_{s}. Taking into account that this correction is sufficiently small, we can conclude that the nonlinear correction to P/t_{d}, which is proportional to 1/τ_{d}, is also an approximately linear function of V_{0}/c_{s}. The inspection of Fig. 5 of Verwichte et al. (2008) reveals that P/t_{d} is also a linear function of V_{0}/c_{s} when V_{0}/c_{s} is approximately between 0.3 and 0.6, i.e. for values of V_{0}/c_{s} that can be considered as moderate. Our analytical results are not applicable for higher values of V_{0}/c_{s}. It is also worth noting that, for low values of V_{0}/c_{s}, the dependence of P/t_{d} on V_{0}/c_{s} given by both Fig. 2 of this paper and by Fig. 5 of Verwichte et al. (2008) is approximately quadratic.
We now compare the results obtained in this paper with the dependence of the damping time on the initial perturbation amplitude obtained from the observations. This dependence is presented in Fig. 6 of Verwichte et al. (2008). If we approximate the solid curve in Fig. 2 by a linear curve, then we obtain τ_{d} ≈ 1−0.215λ, so . Using this result yields We see that the slope of the curve showing the dependence of P/t_{d} on V_{0}/c_{s} is about 0.44, which is much smaller than the slope 1.8 found by Verwichte et al. (2008) from observations. Hence, similar to Verwichte et al. (2008), we conclude that our theory cannot explain the strong dependence of the damping time on the initial perturbation amplitude.
5. Summary and conclusions
We have studied the nonlinear damping of slow standing waves in hot coronal loops. We assumed that the wave amplitude is small and used it as a small parameter in the asymptotic expansions. In addition, we assumed that the wave damping is sufficiently slow and introduced the “slow” time related to the evolution of the wave amplitude and shape. We showed that, in the leadingorder approximation, the nonlinear standing wave is a superposition of two identical nonlinear waves propagating in the opposite directions. The evolution of the two waves is governed by the Burgers equation.
When written in the dimensionless variables, the Burgers equation contains only one dimensionless parameter λ that defines the ratio of the nonlinear and dissipative term. Using the ColeHopfe substitution, we obtained an analytical solution
describing the evolution of a standing wave that has the form of the linear fundamental eigenmode at the initial moment of time. Then we used this solution for the parametric study of the nonlinear damping of standing waves. In particular, we studied the dependence of t_{d}/t_{dl} on λ, where t_{d} and t_{dl} are the damping time given by linear and nonlinear theory, respectively. We showed that, for λ ≤ 3, nonlinearity can reduce the damping time by no more than about one third. Moreover, for realistic coronal loop parameters and observed periods and damping times, the nonlinearity effect on the damping time must be taken into account only when the Mach number of the initial velocity perturbation is sufficiently large, of the order of or larger than 0.2.
We compared the results obtained in this paper with those obtained from observations. Similar to Verwichte et al. (2008), we found that out theory cannot explain strong dependence of the damping time on the initial perturbation amplitude.
Finally, we note that the analysis presented in this paper can be modified to study the effect of loop cooling on the nonlinear damping of slow standing waves in hot coronal loops. Recently, this effect has been studied in the linear approximation by AlGhafri & Erdélyi (2013).
Acknowledgments
M.S.R. acknowledges the support by the STFC grant, and thanks the unknown referee for very useful comments.
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All Figures
Fig. 1 Evolution of the velocity profile with time for t_{dl} = 5P and ϵ = 0.1. The upper, middle, and lower panel correspond to t = 0, P/2, and P respectively. 

In the text 
Fig. 2 Dependence of the dimensionless damping time τ_{d} on the nonlinearity parameter λ. The solid line shows the dependence calculated numerically using the full Fourier expansion (49), while the dashed line shows the approximate dependence for small λ given by Eq. (55). 

In the text 
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