THE EFFECT OF A TWISTED MAGNETIC FIELD ON THE PERIOD RATIO P1/P2 OF NONAXISYMMETRIC MAGNETOHYDRODYNAMIC WAVES

K. Karami; K. Bahari

doi:10.1088/0004-637X/757/2/186

1. INTRODUCTION

Transverse coronal loop oscillations triggered by explosive events, such as flares or filament eruptions, were first identified by Aschwanden et al. (1999) and Nakariakov et al. (1999) using the observations of TRACE (Transition Region And Coronal Explorer). These oscillations have been interpreted as the kink magnetohydrodynamic (MHD) modes of a cylindrical coronal flux tube by Nakariakov et al. (1999).

One of the most important functions of coronal seismology is determining the period ratio P₁/P₂ between the period P₁ of the fundamental mode and the period P₂ of its first harmonic. The deviation of the period ratio from its canonical harmonic value of 2 has been observed in coronal loop oscillations. Verwichte et al. (2004), using the observations of TRACE, have identified the fundamental and its first harmonic of the transverse kink mode in two coronal loops. The period ratios observed by Verwichte et al. (2004) are 1.81 ± 0.25 and 1.64 ± 0.23. However, these values were corrected with the improvement of the observational error bars to 1.82 ± 0.08 and 1.58 ± 0.06, respectively, by Van Doorsselaere et al. (2007). Also, Verth et al. (2008) added some further corrections by considering the effects of loop expansion and estimating a period ratio of 1.54. All these values are clearly lower than 2. This may be caused by different factors such as the effects of density stratification (see, e.g., Andries et al. 2005; Erdélyi & Verth 2007; Karami & Asvar 2007; Safari et al. 2007; Karami et al. 2009) and magnetic twist (see Erdélyi & Carter 2006; Erdélyi & Fedun 2006; Karami & Barin 2009; Karami & Bahari 2010) in the loops. Note that in some cases the period ratio is shifted to higher values than 2. For instance, in Table 1 of Andries et al. (2009) there are two observational examples with P₁/P₂ > 2. This may be caused by the effect of magnetic field expansion (see, e.g., Verth & Erdélyi 2008; Ruderman et al. 2008; Verth et al. 2008; Karami & Bahari 2011). Also, there are some observational cases in which the period ratios do not show any significant departures from their canonical harmonic values. For instance, in Table 1 of Andries et al. (2009) there is an example with P₁/P₂ = 2. Also Van Doorsselaere et al. (2009) found P₁/P₂ ≈ 2 and P₁/P₃ ≈ 3 in a highly twisted loop structure that is certainly not homogeneous and could be classified as a sigmoid.

The twisted magnetic tubes have been investigated in ample detail by Bennett et al. (1999) and Carter & Erdélyi (2007, 2008). For a good review see Karami & Barin (2009).

Ruderman (2007) studied the nonaxisymmetric oscillations of a compressible zero-beta thin twisted magnetic tube surrounded by a straight and homogeneous magnetic field, taking the density stratification into account. Using asymptotic analysis, he showed that the eigenmodes and the eigenfrequencies of the kink and the fluting oscillations are described by a classical Sturm–Liouville problem. The main result of Ruderman (2007), which has already been obtained by Goossens et al. (1992), was that the twist does not affect the kink mode. Note that in these analytical works, and in the present work, the azimuthal component of the equilibrium magnetic field B_ϕ is taken to be proportional to r. This is essentially because it makes the governing equations easier to solve analytically. However, this makes the equilibrium magnetic field a very particular and artificial case study that may be far from reality. It may be that more general B_ϕ equilibria do affect the kink mode.

Karami & Barin (2009) investigated both the oscillations and damping of the MHD surface and hybrid waves in coronal loops in the presence of a twisted magnetic field. They considered a straight cylindrical incompressible flux tube with a magnetic twist just in the annulus and a straight magnetic field in both the internal and external regions. They showed that the frequencies and the damping rates of both the kink and fluting modes increase when the twist parameter increases. They found that the period ratio P₁/P₂ of the fundamental and first overtone for both the kink and fluting surface modes are lower than 2 (for an untwisted loop) in the presence of a twisted magnetic field.

Karami & Bahari (2010) examined the effect of a twisted magnetic field on the resonant absorption of MHD waves in coronal loops. They concluded that by increasing the twist, the ratio of the oscillation frequency to the damping rate of the kink modes changes from 39.3 to 43.5, which is approximately one order of magnitude greater than the ratio reported by Nakariakov et al. (1999), Verwichte et al. (2004), and Wang & Solanki (2004) deduced from the TRACE data. Note that the twisted cylinder model proposed by Karami & Bahari (2010) was found to produce too weak a damping rate to explain the observed strong kink wave damping.

In the present work, our main aim is to investigate the effect of a twisted magnetic field on the frequencies of nonaxisymmetric MHD waves in coronal loops to justify the deviation of the period ratio P₁/P₂ from 2 observed by TRACE. This paper is organized as follows. In Section 2 we use the asymptotic analysis obtained by Ruderman (2007) to derive the equations of motion. In Section 3 and its subsections we present the dispersion relation, two reductions of known cases, and an analytical solution. In Section 4, we give numerical results. Section 5 is devoted to conclusions.

2. EQUATIONS OF MOTION

The linearized MHD equations for a compressible zero-beta plasma are

$\begin{equation} \frac{\partial \delta \mathbf {v}}{\partial t}=\frac{1}{4\pi \rho }[(\nabla \times \delta \mathbf {B})\times \mathbf {B} +(\nabla \times \mathbf {B})\times \delta \mathbf {B}], \end{equation} \tag{ 1 }$

$\begin{equation} \frac{\partial \delta \mathbf {B}}{\partial t}=\nabla \times (\delta \mathbf {v}\times \mathbf {B}), \end{equation} \tag{ 2 }$

where $\delta \bf {v}$ and $\delta \bf {B}$ are the Eulerian perturbations in the velocity and magnetic fields; ρ is the mass density.

The simplifying assumptions are as follows.

1.
The background magnetic field is assumed to be
$\begin{equation} {\mathbf {B}}= \left\lbrace \begin{array}{@{}ll}\mathbf {B}_{\rm i}= (0,A_{\rm i}r,B_{z{\rm i}} (r)),& r < a, \\ {\mathbf {B}}_0= (0,A_0 r,B_{z0} (r)),& a < r < R, \\ {\mathbf {B}}_{\rm e}= (0,0,B_{z{\rm e}}),&r>R, \\ \end{array} \right. \end{equation} \tag{ 3 }$
where A_i, A₀, B_ze are constant and a, R are radii of the core and the tube, respectively. From both the equilibrium equation, i.e., dB²/dr = −2B²_ϕ/r, and the continuity condition of the magnetic pressure across the boundaries of the tube, i.e., B²_i(a) = B₀²(a), B²₀(R) = B_e²(R), the z-component of the equilibrium magnetic field can be obtained as
$\begin{eqnarray} B_{z{\rm i}}^2 (r) &=& B_0^2 + A_{\rm i}^2 (a^2 - 2r^2), \nonumber\\ B_{z0}^2 (r) &=& B_0^2 + A_0^2 (a^2 - 2r^2), \\ B_{z{\rm e}}^2 &=& B_0^2 + A_0^2 (a^2 - R^2),\nonumber \end{eqnarray} \tag{ 4 }$
where B₀ is an integration constant. The above magnetic field configuration in the absence of the annulus is the same as the background magnetic field considered by Ruderman (2007).
2.
ρ is constant along the loop but different in the interior, annulus, and exterior regions and denoted by ρ_i, ρ₀, and ρ_e, respectively.
3.
We consider the flux tube to be a cylinder and therefore implement the cylindrical coordinates, (r, ϕ, z).
4.
The plasma equilibrium is fixed in a steady state, i.e., without flow.
5.
t-, ϕ-, and z-dependence for any of the components δv and δB is exp {i(mϕ + k_zz − ωt)}. Here k_z = lπ/L and L is the tube length. Also l and m are the longitudinal and azimuthal mode numbers, respectively.

Here, like Ruderman (2007), we consider epsilon := Aa/B₀ ∼ k_za ≪ 1, which is in good agreement with the observations, and we also look for the low-frequency eigenmodes. Following the second-order perturbation method in terms of epsilon given by Ruderman (2007), solutions for Equations (1) and (2) in terms of $\delta P={\mathbf {B}\cdot \delta \mathbf {B}}/{4\pi }$ , the Eulerian perturbation in the magnetic pressure, and ξ_r = −δv_r/iω, the Lagrangian perturbation in the radial displacement, for the interior and annulus regions yield

$\begin{equation} \delta P(r) = \frac{r}{{m^2 }}\left (\rho \omega ^2 - \frac{{B_0^2 }}{{4\pi }}F^2 \right)\frac{{{\rm d}(r\xi _r)}}{{{\rm d}r}} + \left (\frac{{B_0 AF}}{{2\pi m}}\right)r\xi _r, \end{equation} \tag{ 5 }$

$\begin{equation} \frac{{\rm d}}{{{\rm d}r}} \left(r\frac{{{\rm d}(r\xi _r)}}{{{\rm d}r}}\right) - m^2 \xi _r = 0, \end{equation} \tag{ 6 }$

where F = k_z + m(A/B₀). Equations (5) and (6) are the same as Equations (19) and (21), respectively, in Ruderman (2007).

In the interior and annulus regions, solutions for Equation (6) are

$\begin{equation} \xi _r (r) = \left\lbrace \begin{array}{@{}lll}\alpha r^{m - 1},&r < a, \\ \beta r^{m - 1} + \gamma r^{ - m - 1},& a < r < R,\\ \end{array} \right. \end{equation} \tag{ 7 }$

and solutions for δP(r) are obtained by substituting Equation (7) into Equation (5) as

$\begin{equation} \delta P(r) = \left (\rho _{\rm i} \omega ^2 - \frac{{B_0^2F_{\rm i}^2 }}{{4\pi }} + \frac{{B_0 A_{\rm i} F_{\rm i} }}{{2\pi }}\right)\frac{{\alpha r^m }}{m}, \quad r < a, \end{equation} \tag{ 8 }$

$\begin{eqnarray} \delta P(r) &=& \left (\rho _0 \omega ^2 - \frac{{B_0^2F_0^2 }}{{4\pi }} + \frac{{B_0 A_0 F_0 }}{{2\pi }}\right)\frac{{\beta r^m }}{m} \nonumber \\ && - \left (\rho _0 \omega ^2 - \frac{{B_0^2F_0^2 }}{{4\pi }} - \frac{{B_0 A_0 F_0 }}{{2\pi }}\right)\frac{{\gamma r^{ - m} }}{m},\,\,\,\,\, a < r < R. \nonumber\\ \end{eqnarray} \tag{ 9 }$

For the exterior region, r > R, we obtain

$\begin{equation} \frac{{{\rm d}^2 \delta P}}{{{\rm d}r^2 }} + \frac{1}{r}\frac{{{\rm d}\delta P}}{{{\rm d}r}} - \left (k^{\prime 2} + \frac{{m^2 }}{{r^2 }}\right)\delta P = 0, \end{equation} \tag{ 10 }$

$\begin{equation} \xi _r (r) =-\frac{4\pi }{k^{\prime 2}B_0^2}\frac{{\rm d}\delta P}{{\rm d}r}, \end{equation} \tag{ 11 }$

where

$\begin{equation} k^{\prime 2} = k_z^2 - \frac{{4\pi \rho _e \omega ^2 }}{{B_0^2 }}. \end{equation} \tag{ 12 }$

Equations (10) and (11) are the same as Equations (26) and (25a), respectively, in Ruderman (2007). In the exterior region, r > R, the waves should be evanescent. Solutions are

$\begin{equation} \delta P(r) = \varepsilon K_m (k^{\prime }r),\quad k^{\prime 2}>0, \end{equation} \tag{ 13 }$

$\begin{equation} \xi _r (r) = - \varepsilon \frac{{4\pi }}{{k^{\prime }B_0^2 }}K_m ^\prime (k^{\prime }r), \end{equation} \tag{ 14 }$

where K_m is the modified Bessel function of the second kind and a prime on K_m indicates a derivative with respect to its appropriate argument. The coefficients α, β, γ, and ε in Equations (7), (8), (9), (13), and (14) are determined by the appropriate boundary conditions.

3. BOUNDARY CONDITIONS AND DISPERSION RELATION

The necessary boundary conditions for the perturbed tube boundary are that the plasma displacement in the radial direction and the magnetic pressure should be continuous as

$\begin{equation} {\xi _{r{\rm i}} } \big |_{r = a} = {\xi _{r0} } \big |_{r = a},\quad {\xi _{r0} } \big |_{r = R} = {\xi _{r{\rm e}} } \big |_{r = R}, \end{equation} \tag{ 15 }$

$\begin{equation} \begin{array}{l}\delta P_{\rm i} - \dsty\frac{{B_{\phi {\rm i} }^2}}{{4\pi a}}\xi _{r{\rm i}} \Big |_{r = a} = {\delta P_0 - \dsty\frac{{B_{\phi 0}^2 }}{{4\pi a}}\xi _{r0} } \Big |_{r = a},\, \\ \ms\ms {\delta P_0 - \dsty\frac{{B_{\phi 0}^2 }}{{4\pi R}}\xi _{r0} } \Big |_{r = R} = {\delta P_{\rm e} } \Big |_{r = R}. \\ \end{array} \end{equation} \tag{ 16 }$

Using the above boundary conditions and the solutions given by Equations (7)–(9) for the internal and annulus regions and Equations (13) and (14) for the exterior region, the dispersion relation is derived as

$\begin{eqnarray} && \big (\Xi _m^{0}\Pi _m^0-\Xi _m^{\rm i}\Xi _m^{\rm e} \big) [1-(a/R)^{2m} ] -\Xi _m^{\rm i}\big [\Xi _m^{0}-(a/R)^{2m}\Pi _m^0\big ] \nonumber \\ && \quad +\Xi _m^{\rm e}\big [\Pi _m^0-(a/R)^{2m}\Xi _m^{0}\big ] = 0, \end{eqnarray} \tag{ 17 }$

with

$\begin{equation} \Xi _m^{\rm j} = \frac{1}{m}\left (\rho _{\rm j} \omega ^2 - \frac{{B_0^2 k_z^2 }}{{4\pi }}\right) + \frac{{A_{\rm j} }}{{4\pi m}}(2B_0 k_z + mA_{\rm j})(1 - m), \end{equation} \tag{ 18 }$

$\begin{equation} \Pi _m^0=- \frac{1}{m}\left (\rho _0 \omega ^2 - \frac{{B_0^2 k_z^2 }}{{4\pi }}\right) + \frac{{A_0 }}{{4\pi m}}(2B_0 k_z + mA_0)(1 + m), \end{equation} \tag{ 19 }$

$\begin{eqnarray} &&\Xi _m^{\rm e}=\frac{B_0^2}{4\pi }\frac{k^{\prime 2}}{k^{\prime }R}\frac{K_m (k^{\prime }R)}{K_m ^\prime (k^{\prime }R)}, \end{eqnarray} \tag{ 20 }$

where the superscript j in Ξ^j_m stands for i and 0 corresponding to the interior and annulus regions, respectively.

Note that if we remove the annulus region by setting a = R in dispersion relation (17) and using the thin flux tube approximation for K_m(x)∝x^−m at small x, the result yields

$\begin{eqnarray} &&\omega ^2=C_k^2\left \lbrace k_z^2+\frac{A_{\rm i}(m-1)}{2B_0^2}(2B_0k_z+A_{\rm i}m)\right \rbrace, \end{eqnarray} \tag{ 21 }$

where C²_k = B₀²/2π(ρ_i + ρ_e). Equation (21) is the same as Equation (40) in Ruderman (2007). The main result of Ruderman (2007) is that the twist does not affect the kink modes in the particular case where B_ϕ∝r, and Equation (21) shows that we get the same frequencies as in the case where A_i = 0. This result has already been obtained by Goossens et al. (1992). Note that Equation (17) shows that even in the presence of an annulus, the internal twist does not affect the kink (m = 1) modes. Because the internal twist, A_i, only appears in Equation (18) and when m = 1, it therefore has no contribution.

In Sections 3.1 and 3.2, we show that the dispersion relation (17) for the cases of A_i = 0 and A₀ = 0, respectively, reduces to two known cases. Also in Section 3.3, we give an analytical solution for the dispersion relation (17).

3.1. Case A_i = 0

Here we show that the dispersion relation (17) in the absence of internal twist, i.e., A_i = 0, can be obtained from the dispersion relation, Equation (6b), given by Carter & Erdélyi (2008) under the thin tube (TT) approximation (or long-wavelength limit), i.e., k_za ≪ 1. The dispersion relation Equation (6b) in Carter & Erdélyi (2008) for body waves is

$\begin{eqnarray} && \frac{\Xi _{aY}-\Xi _{\rm i}+\Xi _{aY}\Xi _{\rm i}\frac{A_0^2}{4\pi }}{\Xi _{aJ}-\Xi _{\rm i}+\Xi _{aJ}\Xi _{\rm i}\frac{A_0^2}{4\pi }}\frac{Y_m(n_0a)}{J_m(n_0a)} = \frac{Y_m(n_0R)}{J_m(n_0R)} \nonumber\\ && \qquad \times \frac{\Xi _{RY}-\Xi _{\rm e}+\Xi _{RY}\Xi _{\rm e} \frac{A_0^2}{4\pi }}{\Xi _{RJ}-\Xi _{\rm e}+\Xi _{RJ}\Xi _{\rm e}\frac{A_0^2}{4\pi }}, \end{eqnarray} \tag{ 22 }$

which is valid for thick magnetic tubes with twisted annulus in the incompressible limit. Note that according to Edwin & Roberts (1983) there are no surface waves in a zero-beta approximation, which is compatible with coronal conditions. Although the plasma in our model is compressible, the results concerning the kink (m = 1) modes for incompressible plasmas given by Carter & Erdélyi (2008) can be applied to coronal loops despite the fact that the coronal plasma is a low-beta plasma (see Carter & Erdélyi 2007; Erdélyi & Fedun 2007). Recently, Goossens et al. (2009) showed that in the TT approximation, neglecting contributions proportional to (k_za)² of the kink wave frequencies are the same in the three cases, including a compressible pressureless plasma, an incompressible plasma, and a compressible plasma that allows for MHD radiation.

Under the TT approximation, we have

$\begin{equation} \frac{{xK_m ^{\prime }(x)}}{{K_m (x)}} = \frac{{xY_m ^{\prime }(x)}}{{Y_m (x)}} =-m+O(x^2), \end{equation} \tag{ 23 }$

$\begin{equation} \frac{{xI_m ^{\prime }(x)}}{{I_m (x)}} = \frac{{xJ_m ^{\prime }(x)}}{{J_m (x)}} = m+O (x^2),\ \end{equation} \tag{ 24 }$

where (J_m, Y_m) and (I_m, K_m) are the Bessel and modified Bessel functions of the first and second kind, respectively. Using Equations (23) and (24), the different terms appearing in Equation (22) given by Carter & Erdélyi (2008) are reduced to

$\begin{equation} \ \ \ \Xi _{aY} = \Xi _{RY}= - \frac{m}{{\rho _0 }}\frac{1}{{\big (\omega ^2 - \omega _{A_0}^2 \big) - \frac{{2A_0 \omega _{A_0} }}{{\sqrt{4\pi \rho _0 } }}}}, \end{equation} \tag{ 25 }$

$\begin{equation} \Xi _{aJ}=\Xi _{RJ}=\frac{m}{\rho _0}\frac{1}{{\big (\omega ^2 - \omega _{A_0}^2 \big) + \frac{{2A_0 \omega _{A_0} }}{{\sqrt{4\pi \rho _0 } }}}}, \end{equation} \tag{ 26 }$

$\begin{equation} \Xi _{\rm i} = \frac{m}{{\rho _{\rm i} \big (\omega ^2 - \omega _{A_{\rm i}}^2 \big)}},\ \, \end{equation} \tag{ 27 }$

$\begin{equation} \Xi _{\rm e} = \frac{{ - m}}{{\rho _{\rm e} \big (\omega ^2 - \omega _{A_{\rm e}}^2\big)}},\ \, \end{equation} \tag{ 28 }$

where, following Carter & Erdélyi (2008), $\omega _{A_0}$ , $\omega _{A_{\rm i}}$ , and $\omega _{A_{\rm e}}$ are the Alfvén frequencies in the annulus, internal, and external regions, respectively, given by

$\begin{equation} \qquad \omega _{A_0}=\frac{1}{\sqrt{4\pi \rho _0}}(mA_0+k_zB_0), \end{equation} \tag{ 29 }$

$\begin{equation} \omega _{A_{\rm i}}=\frac{k_zB_{\rm i}}{\sqrt{4\pi \rho _{\rm i}}},\qquad\ \ \end{equation} \tag{ 30 }$

$\begin{equation} \omega _{A_{\rm e}}=\frac{k_zB_{\rm e}}{\sqrt{4\pi \rho _{\rm e}}}.\qquad\ \ \, \end{equation} \tag{ 31 }$

Using Equations (18)–(20) and Equations (23)–(31), one can rewrite the terms Ξ_Y := Ξ_aY = Ξ_RY, Ξ_J := Ξ_aJ = Ξ_RJ, Ξ_i, and Ξ_e in the dispersion relation (22) as follows:

$\begin{equation} \Xi _{\rm i} = \frac{1}{{\Xi _m^{\rm i} }},\quad\ \ \ \end{equation} \tag{ 32 }$

$\begin{equation} \Xi _J = \frac{1}{{\Xi _m^0+\frac{{A_0^2 }}{{4\pi }}}}, \end{equation} \tag{ 33 }$

$\begin{equation} \Xi _Y = \frac{1}{{\Pi _m^0+\frac{{A_0^2 }}{{4\pi }}}}, \end{equation} \tag{ 34 }$

$\begin{equation} \Xi _{\rm e}=\frac{4\pi m}{B_0^2k^{\prime 2}}=-\frac{1}{\Xi _m^{\rm e}}. \end{equation} \tag{ 35 }$

Also under the TT approximation we have

$\begin{equation} \frac{Y_m(n_0R)}{Y_m(n_0a)}\frac{J_m(n_0a)}{J_m(n_0R)}=\left (\frac{a}{R}\right)^{2m}. \end{equation} \tag{ 36 }$

One can easily show that substituting Equations (32)–(36) into Equation (22) yields

$\begin{equation} R^{2m}\frac{\Xi _m^{0}+\Xi _m^{\rm e}}{\Pi _m^0+\Xi _m^{\rm e}}=a^{2m}\frac{\Xi _m^{0}-\Xi _m^{\rm i}}{\Pi _m^0-\Xi _m^{\rm i}}, \end{equation} \tag{ 37 }$

which is nothing but Equation (17) in which the terms a^2m and R^2m have been grouped.

3.2. Case A₀ = 0

Here we show that the dispersion relation (37) in the absence of twist in the annulus region, i.e., A₀ = 0, is the same as the dispersion relation Equation (11) in Carter & Erdélyi (2007) for the kink (m = 1) modes in the TT approximation. By setting A₀ = 0, Equations (18)–(20) reduce to

$\begin{equation} \Xi _1^{\rm i}=\rho _{\rm i} \big(\omega ^2-\omega _{A_{\rm i}}^2\big), \end{equation} \tag{ 38 }$

$\begin{equation} \ \ \Xi _1^{0}=\rho _{0} \big(\omega ^2-\omega _{A_{0z}}^2\big), \end{equation} \tag{ 39 }$

$\begin{equation} \ \ \ \ \, \Pi _1^0=-\rho _{0} \big(\omega ^2-\omega _{A_{0z}}^2\big), \end{equation} \tag{ 40 }$

$\begin{equation} \Xi _1^{\rm e}=\rho _{\rm e} \big(\omega ^2-\omega _{A_{\rm e}}^2\big), \end{equation} \tag{ 41 }$

where

$\begin{equation} \omega _{A_{0z}}=\frac{k_zB_0}{\sqrt{4\pi \rho _0}}. \end{equation} \tag{ 42 }$

With the help of the above expressions, Equation (37) yields

$\begin{eqnarray} &&\frac{{Q_0^{\rm i} + 1}}{{Q_0^{\rm i} - 1}} - \Big (\frac{a}{R}\Big)^{2} \frac{{Q_0^{\rm e} - 1}}{{Q_0^{\rm e} + 1}}= 0, \end{eqnarray} \tag{ 43 }$

where

$\begin{equation} Q_0^{\rm i} = \frac{{\rho _{\rm i} }}{{\rho _0 }}\left(\frac{\omega ^2 - \omega _{A_{\rm i}}^2}{\omega ^2 - \omega _{A_{0z}}^2}\right), \end{equation} \tag{ 44 }$

$\begin{equation} Q_0^{\rm e} = \frac{\rho _{\rm e}}{\rho _0}\left(\frac{\omega ^2 - \omega _{A_{\rm e}}^2}{\omega ^2 - \omega _{A_{0z}}^2}\right). \end{equation} \tag{ 45 }$

Equation (43) is the same as the dispersion relation (11) in Carter & Erdélyi (2007) for the kink (m = 1) modes in the TT approximation.

3.3. Analytical Solution of the Dispersion Relation

Here we try to solve the dispersion relation (17), analytically. To do this we apply the TT approximation, Equation (23), to the expression Ξ^e_m, Equation (20). This reduces the dispersion relation (17) to a second-order equation in terms of ω², which can now be solved analytically. To write the solutions in a simple form, we define the expressions X^j_m and Z⁰_m as follows:

$\begin{equation} X_m^{\rm j}=\frac{A_{\rm j}}{B_0k_z}\left(2+\frac{mA_{\rm j}}{B_0k_z}\right)(1-m), \end{equation} \tag{ 46 }$

$\begin{equation} Z_m^0=\frac{A_0}{B_0k_z}\left(2+\frac{mA_0}{B_0k_z}\right)(1+m), \end{equation} \tag{ 47 }$

which are proportional to the second terms that appear in Equations (18) and (19), respectively. These expressions contain all twist parameters in the dispersion relation.

Using Equations (46) and (47), one can rewrite Equation (17) as

$\begin{equation} c_m\left(\frac{\sqrt{4\pi \rho _{\rm i}}}{B_0k_z}\right)^4\omega ^4-c_{ml}\left(\frac{\sqrt{4\pi \rho _{\rm i}}}{B_0k_z}\right)^2\omega ^2+\tilde{c}_{ml}=0, \end{equation} \tag{ 48 }$

where c_m, c_ml, and $\tilde{c}_{ml}$ are constants defined as

$\begin{eqnarray} c_m &=& \left[\frac{\rho _{\rm e}}{\rho _{\rm i}}+\left(\frac{\rho _0}{\rho _{\rm i}}\right)^2\right](1-(a/R)^{2m})\nonumber\\ && +\frac{\rho _0}{\rho _{\rm i}}\left(1+\frac{\rho _{\rm e}}{\rho _{\rm i}}\right)(1+(a/R)^{2m}), \end{eqnarray} \tag{ 49 }$

$\begin{eqnarray} c_{ml}&=&2\left(1+\frac{\rho _{\rm e}}{\rho _{\rm i}}+2\frac{\rho _0}{\rho _{\rm i}}\right) +\left[\!\frac{\rho _0}{\rho _{\rm i}}+\frac{\rho _{\rm e}}{\rho _{\rm i}}+\left(1-\frac{\rho _0}{\rho _{\rm i}}\right)(a/R)^{2m} \!\right]Z_m^0 \nonumber \\ && -\left[1+\frac{\rho _0}{\rho _{\rm i}}+\left(\frac{\rho _{\rm e}}{\rho _{\rm i}}-\frac{\rho _0}{\rho _{\rm i}}\right)(a/R)^{2m}\right]X_m^0 \nonumber\\ && -\left[\frac{\rho _0}{\rho _{\rm i}}+\frac{\rho _{\rm e}}{\rho _{\rm i}}+\left(\frac{\rho _0}{\rho _{\rm i}}-\frac{\rho _{\rm e}}{\rho _{\rm i}}\right)(a/R)^{2m}\right]X_m^{\rm i}, \end{eqnarray} \tag{ 50 }$

$\begin{eqnarray} \tilde{c}_{ml} &=& 4-2\big (X_m^{\rm i}+X_m^0-Z_m^0\big)-\big (Z_m^0-X_m^{\rm i}\big)X_m^0 \nonumber\\ && -(a/R)^{2m}\big (X_m^{\rm i}-X_m^0\big)Z_m^0. \end{eqnarray} \tag{ 51 }$

Here c_m depends only on the azimuthal mode number m, but c_ml and $\tilde{c}_{ml}$ contain both the azimuthal m and longitudinal l mode numbers. Note that l appears in k_z = lπ/L.

Now solving Equation (48) gives the eigenfrequencies

$\begin{equation} \omega _{{\rm nml}}=\frac{k_zB_0}{\sqrt{4\pi \rho _i}}\left(\frac{c_{ml}\pm \sqrt{c_{ml}^2-4\tilde{c}_{ml}c_{m}}}{2c_m} \right)^{1/2}, \end{equation} \tag{ 52 }$

where the subscript n (or ± signs) denotes the radial mode number corresponding to the lower (n = 1) and upper (n = 2) frequencies of the two nonaxisymmetric modes. Note that for kink (m = 1) modes from Equation (46) we get Xⁱ_m = X_m⁰ = 0. Also, from Equations (50) and (51) for m = 1, both c_ml and $\tilde{c}_{ml}$ are positive. On the other hand, c_m is always positive. Therefore, for Equation (52) the − and + signs correspond to the radial mode numbers n = 1 and n = 2, respectively. Since our model is based on the TT approximation, the numerical solution of the dispersion relation (17) in Section 4 is approximately the same as the analytical solution (52).

Note that in addition to solution (52), there is a set of broadband body modes that has already been predicted by Carter & Erdélyi (2008). However, these modes are lost from the solutions of the dispersion relation (22) under the TT approximation. Since our model is based on the TT approximation, the infinite set of body modes is absent in our calculations. In accordance with the classification introduced by Roberts (1981), the modes both described and not described by Equation (17) are body waves in the zero-beta plasma approximation. In order to distinguish the kink mode described by Equation (17) from those not described by Equation (17), Ruderman & Roberts (2002) suggested calling the solutions (52) "global kink modes," retaining the name "body kink modes" for all other kink modes. We generalize this convention for fluting modes and call those described by Equation (17) "global fluting modes," retaining the name "body fluting modes" for all others.

4. NUMERICAL RESULTS

To solve the dispersion relation (17) numerically, we chose the physical parameters L = 10⁵ km, a/L = 0.01, ρ_e/ρ_i = 0.1, ρ₀/ρ_i = 0.5, ρ_i = 2 × 10⁻¹⁴ g cm⁻³, and B₀ = 100 G. For such a loop, one finds $v_{A_{\rm i}}=({B_0}/{\sqrt{4\pi \rho _{{\rm i}}}})=2000$ km s⁻¹, $\omega _{A_{\rm i}}{:} \;=({v_{A_{\rm i}}}/{{\rm L}})=0.02$ rad s⁻¹. In what follows, we illustrate our numerical studies in the two separate equilibrium cases where there is (1) twist in both the core and annulus regions and (2) twist in the core and no twist in the annulus.

4.1. Case A_i ≠ 0 and A₀ ≠ 0

The effect of a twisted magnetic field on the frequencies ω is calculated by the numerical solution of the dispersion relation, Equation (17). Figures 1 and 2 show the frequencies of the fundamental and first overtone l = 1, 2 kink (m = 1) modes with radial mode numbers n = 1, 2 versus the twist parameter of the annulus, B_ϕ/B_z: = (A₀a/B₀), and for different relative core widths a/R = (0.65, 0.9, 0.99). Note that here the parameter A_i does not need to be set explicitly, because the second term in Equation (18) containing the contribution of the parameter A_i is automatically removed for the kink (m = 1) modes.

**Figure 1.** Frequencies of the fundamental and its first overtone kink (m = 1) modes with radial mode number n = 1 vs. the twist parameter of the annulus, B_ϕ/*B_z* = A₀a/B₀, for different relative core widths a/R = 0.65 (dash-dotted), 0.9 (dashed), and 0.99 (solid). The loop parameters are L = 10⁵ km, a/L = 0.01, ρ_e/ρ_i = 0.1, ρ₀/ρ_i = 0.5, ρ_i = 2 × 10⁻¹⁴ g cm⁻³, and B₀ = 100 G. Frequencies are in units of the interior Alfvén frequency, $\omega _{\rm A_i}= 0.02\,{\rm rad \,s^{-1}}$ .
Download figure:
Standard image High-resolution image

**Figure 1.** Frequencies of the fundamental and its first overtone kink (m = 1) modes with radial mode number n = 1 vs. the twist parameter of the annulus, B_ϕ/*B_z* = A₀a/B₀, for different relative core widths a/R = 0.65 (dash-dotted), 0.9 (dashed), and 0.99 (solid). The loop parameters are L = 10⁵ km, a/L = 0.01, ρ_e/ρ_i = 0.1, ρ₀/ρ_i = 0.5, ρ_i = 2 × 10⁻¹⁴ g cm⁻³, and B₀ = 100 G. Frequencies are in units of the interior Alfvén frequency, $\omega _{\rm A_i}= 0.02\,{\rm rad \,s^{-1}}$ .
Download figure:
Standard image High-resolution image

**Figure 2.** Same as Figure 1, for kink (m = 1) modes with a radial mode number of n = 2.
Download figure:
Standard image High-resolution image

Figures 1 and 2 reveal that (1) for a given a/R, the frequencies increase when the twist parameter of the annulus increases. The result is in good agreement with that obtained by Carter & Erdélyi (2008) and Karami & Barin (2009). (2) For a given n and a/R, when the longitudinal mode number, l, increases, the frequencies increase. (3) For a given l, a/R, and B_ϕ/B_z, when the radial mode number, n, increases, the frequencies increase. (4) For n = 1, when a/R goes to unity then the frequencies become independent of B_ϕ/B_z. Therefore in the absence of the annulus, the twist does not affect the kink modes in the specific case of having B_ϕ∝r. This is in good agreement with the results obtained by Goossens et al. (1992) and Ruderman (2007). (5) For n = 2, when a/R reaches to unity, then the second mode is removed. This was expected to occur because we have only one boundary in the tube corresponding to one mode for a/R = 1.

To compare our results with those of Carter & Erdélyi (2008), we use Equation (22) and obtain the frequencies of the kink (m = 1) modes with radial mode number (n = 2) versus the twist parameter of the annulus and for different values of k_za = πa/L = (π/100, 0.1, 1). The results are displayed in Figure 3. The result for k_za = 1 has already been obtained by Carter & Erdélyi (2008) for incompressible flux tubes with twisted annulus. Figure 3 shows that for a slender tube with k_za = π/100 there is a much greater variation with the twist in the annulus than what Carter & Erdélyi (2008) found for a thicker tube with k_za = 1.

**Figure 3.** Frequencies of kink (m = 1) modes with a radial mode number of n = 2 vs. the twist parameter of the annulus, B_ϕ/*B_z* = A₀a/B₀, for different values of *k_za* = 1 (dash-dotted), 0.1 (dashed), and π/100 (solid). Here a/R = 0.8 and other auxiliary parameters are the same as in Figure 1.
Download figure:
Standard image High-resolution image

One important problem with the frequencies displayed in Figures 1 and 2 is that in all four panels of these figures, all the curves for different values of a/R intersect at a single point. For instance, in Figure 1 for the fundamental and first overtone kink modes, the frequencies of the intersection points are ω₁₁₁ = 4.23 and ω₁₁₂ = 2ω₁₁₁ = 8.46, respectively. Also the location of the intersection point for the fundamental kink mode ω₁₁₁ occurs at B_ϕ/B_z = (A₀a/B₀) = 0.0055, and for the first overtone kink mode ω₁₁₂, the intersection occurs at exactly twice the value as the fundamental mode. The frequency and twist of the intersection point can be obtained from the dispersion relation. To do this, we rewrite the dispersion relation (37) for the kink (m = 1) modes as

$\begin{equation} f-\Big (\frac{a}{R}\Big)^{2}g=0, \end{equation} \tag{ 53 }$

where

$\begin{equation} f=\frac{\Pi _1^0-\Xi _1^{\rm i}}{\Xi _1^0-\Xi _1^{\rm i}}, \end{equation} \tag{ 54 }$

$\begin{equation} g=\frac{\Pi _1^0+\Xi _1^{\rm e}}{\Xi _1^0+\Xi _1^{\rm e}}. \end{equation} \tag{ 55 }$

If we set f = g = 0, then the result of the dispersion relation (53) would be independent of the term a/R. Therefore, the set of equations Π⁰₁ − Ξ₁ⁱ = 0 and Π⁰₁ + Ξ₁^e = 0, with the help of Equations (18)–(20) in the TT approximation, yields

$\begin{equation} \omega =C_kk_z= \frac{B_0}{\sqrt{2\pi (\rho _{\rm i} + \rho _{\rm e})}}\frac{l\pi }{L},\ \ \ \ \, \end{equation} \tag{ 56 }$

$\begin{equation} \frac{B_{\phi }}{B_z}=\frac{A_0a}{B_0}=\left(\sqrt{\frac{\rho _{\rm i}+\rho _0}{\rho _{\rm i}+\rho _{\rm e}}}-1\right)\frac{l\pi a}{L}, \end{equation} \tag{ 57 }$

which show that the values of both the frequency and the twist of the intersection point depend on the wavelength as observed. Taking L = 10⁵ km, a/L = 0.01, ρ_e/ρ_i = 0.1, ρ₀/ρ_i = 0.5, and B₀ = 100 G, then Equations (56) and (57) give $\omega /\omega _{A_{\rm i}}=4.24 l$ and B_ϕ/B_z = (A₀a/B₀) = 0.0053l, which are in good agreement with the numerical results obtained for the frequency and twist of the intersection point in Figure 1.

To investigate this problem from another point of view, the radial component of the fundamental kink eigenfunctions, ξ_r(r), is studied. To do this, using the eigenfrequencies obtained from the dispersion relation (17) and applying the boundary conditions (15) and (16) to the solutions (7), (8), (9), (13), and (14), one can obtain the coefficients α, β, γ, and ε. Then with the help of Equations (7) and (14) the radial component of the fundamental kink eigenfunctions ξ_r(r) can be obtained. The results of ξ_r(r) for different relative core widths a/R = (0.65, 0.9, 0.99) and for two different values of the twist parameter of the annulus B_ϕ/B_z = (A₀a/B₀) = 0.0055 and 0.02 are plotted in Figures 4 and 5, respectively. Note that the intersection occurs at B_ϕ/B_z = 0.0055. Comparing Figure 4 with 5, we find out that the eigenfunctions at the second boundary behave smoothly for the twist parameter of the annulus 0.0055. This means that at the intersection point, the location of the second boundary is not important, or physically, the variation of the thickness of the annulus does not affect the eigenfunctions or eigenfrequencies. Figure 5 clearly shows that, for the twist parameter of the annulus 0.02 where the intersection does not occur, the behavior of the eigenfunctions depends on the thickness of the annulus region.

**Figure 4.** Radial component of the eigenfunctions of the fundamental kink (m = 1) modes with radial mode number n = 1 against fractional radius x = r/a for B_ϕ/*B_z* = (A₀a/B₀) = 0.0055 and different relative core widths a/R = 0.65, 0.9, and 0.99.
Download figure:
Standard image High-resolution image

**Figure 5.** Same as Figure 4, for B_ϕ/*B_z* = (A₀a/B₀) = 0.02.
Download figure:
Standard image High-resolution image

To compare the kink oscillations of different radial mode numbers n, the radial component of the kink eigenfunctions with n = 2 is plotted in Figure 6 for the twist parameter of the annulus B_ϕ/B_z = (A₀a/B₀) = 0.02 and for different relative core widths a/R = (0.65, 0.9, 0.99). Figure 6 shows that, contrary to the n = 1 mode in which the core and annulus regions oscillate with the same phase (see Figure 5), for n = 2 the kink oscillations of the core and annulus region, having opposite signs of ξ_r(r), are out of phase with each other. This is in agreement with the previous results obtained by Mikhalyaev & Solov'ev (2005) and Ruderman & Erdélyi (2009).

**Figure 6.** Radial component of the eigenfunctions of the fundamental kink (m = 1) modes with radial mode number n = 2 against fractional radius x = r/a for B_ϕ/*B_z* = (A₀a/B₀) = 0.02 and different relative core widths a/R = 0.65, 0.9, and 0.99.
Download figure:
Standard image High-resolution image

The period ratio P₁/P₂ of the fundamental and first overtone, l = 1, 2 modes of the kink (m = 1) waves with n = 1, and 2 versus the twist parameter of the annulus are plotted in Figures 7 and 8. Figures 7 and 8 show that (1) the period ratio P₁/P₂, for n = 1 decreases from 2 by increasing the twist parameter of the annulus (for the untwisted loop), comes down to a minimum, and then increases whereas for n = 2, it decreases from 2 and approaches below 1.6 for a/R = 0.5, for instance. Note that when the twist is zero, the diagrams of P₁/P₂ do not start exactly from 2. This may be caused by the radial structuring (ρ₀ ≠ ρ_i, ρ_e ≠ ρ_i), but for the selected TT with a/L = 0.01, this departure is very small, O(10⁻⁴), and does not show itself in the diagrams (see McEwan et al. 2006). (2) For a given B_ϕ/B_z, the period ratio P₁/P₂ for n = 1 increases and for n = 2 decreases when the relative core width increases. Figure 7 shows that for the kink modes (m = 1, n = 1) with a/R = 0.5, for both B_ϕ/B_z = 0.0107 and 0.0153, the ratio P₁/P₂ is 1.82. This is in good agreement with the period ratio observed by Van Doorsselaere et al. (2007), 1.82 ± 0.08, deduced from the observations of TRACE. See also McEwan et al. (2008). For more observational examples of the period ratio, we estimate the twist parameter of the annulus for the kink modes (m = 1, n = 1) with different relative core widths. The results, which give the same P₁/P₂ observed by TRACE, are summarized in Table 1.

**Figure 7.** Period ratio P₁/P₂ of the fundamental and its first overtone kink (m = 1) modes with radial mode number n = 1 vs. the twist parameter of the annulus for different relative core widths a/R = 0.5 (dotted), 0.65 (dash-dotted), 0.9 (dashed), and 0.99 (solid). Auxiliary parameters are the same as in Figure 1.
Download figure:
Standard image High-resolution image

**Figure 8.** Same as Figure 7, for the kink (m = 1) modes with a radial mode number of n = 2.
Download figure:
Standard image High-resolution image

Table 1. Coronal Seismology Using the Period Ratio P₁/P₂ of the Kink Modes (m = 1, n = 1): the Twist Parameter of the Annulus B_ϕ/B_z = (A₀a/B₀) and the Relative Core Width a/R

Reference	P₁/P₂	B_ϕ/B_z = A₀a/B₀	a/R
Van Doorsselaere et al. (2007)	1.795 ± 0.051	0.0097, 0.0204	0.35
Van Doorsselaere et al. (2009)	1.980 ± 0.002	0.0014, 0.0219	0.92
Ballai et al. (2011)	1.82 ± 0.02	0.0079, 0.0231	0.35

Download table as: ASCII Typeset image

Note that the results of Figure 7 and Table 1 show that the period ratio P₁/P₂ of the kink (m = 1) modes for n = 1 is not a monotonic function of the twist parameter of the annulus. The analytical solution (52) also confirms this behavior. Therefore, we conclude that the value of the twist parameter in coronal loops cannot be determined uniquely using the model studied here.

Finally, it is worth mentioning the coronal seismology using the period ratio P₁/P₂ of the kink (m = 1) modes. Although the kink modes with n = 1 and n = 2 can be excited in the flux tube, to compare the results with the observation we considered only the period ratio P₁/P₂ of the fundamental (l = 1) and its first overtone (l = 2) kink (m = 1) modes with n = 1, because, as we already mentioned in the case of n = 2, the core and annulus oscillate with opposite phases. This yields small transverse global displacement for the tube, which is not compatible with the observed displacements of coronal loops, roughly 20 Mm from the loop top (see Verwichte et al. 2004). The other possibility for explaining the observed P₁/P₂ is the longitudinal fundamental (l = 1) kink modes of n = 1 and n = 2. But this case also cannot justify the observations. Following Verwichte et al. (2004), the long-period oscillation is the fundamental mode, with a maximum amplitude at the loop top. The short-period oscillation is its second harmonic, and it has a node at the loop top, i.e., l = 2. One notes that besides the two global kink modes with n = 1, 2, an infinite set of body kink modes can also be excited in the tube. However, due to oscillatory displacements inside the tube the body modes cannot drive the loop to oscillate globally.

4.2. Case A_i ≠ 0 and A₀ = 0

As we mentioned before, the internal twist does not affect the kink (m = 1) modes. Hence, we extended our investigation to the fluting (m = 2) modes and studied the effect of the internal twist on their frequencies through a numerical solution of the dispersion relation, Equation (17), in the absence of twist in the annulus region. Figure 9 shows the frequencies of the fundamental and first overtone l = 1, 2 fluting (m = 2) modes with radial mode number n = 1 versus the internal twist parameter, B_ϕ/B_z: = (A_ia/B₀), when A₀ = 0 and for different relative core widths a/R = (0.65, 0.9, 0.99). Figure 9 shows that (1) for a given a/R, the frequencies increase when the internal twist parameter increases. (2) For a given a/R, when the longitudinal mode number, l, increases, the frequencies increase. (3) When a/R goes to unity then the frequencies obey Equation (21).

**Figure 9.** Frequencies of the fundamental and its first overtone fluting (m = 2) modes with a radial mode number of n = 1 vs. the internal twist parameter, B_ϕ/*B_z* = (A_ia/B₀), for different relative core widths a/R = 0.65 (dash-dotted), 0.9 (dashed), and 0.99 (solid). The twist in the annulus region is absent. Auxiliary parameters are the same as in Figure 1.
Download figure:
Standard image High-resolution image

Figure 10 shows the period ratio P₁/P₂ of the fundamental and first overtone, l = 1, 2 modes of the fluting (m = 2) waves with n = 1 versus the internal twist parameter when A₀ = 0. Figure 10 shows that (1) the period ratio P₁/P₂, with a increase in the internal twist parameter, decreases from 2 (for an untwisted loop) and approaches below 1.6 for a/R = 0.5, for instance. (2) For a given B_ϕ/B_z, the period ratio P₁/P₂ decreases when the relative core width increases.

5. CONCLUSIONS

Oscillations of nonaxisymmetric MHD waves in coronal loops in the presence of the twisted magnetic field are studied. To do this, a coronal loop is considered as a straight cylindrical compressible zero-beta thin flux tube with a magnetic twist in the internal region and the annulus and straight magnetic field in the external region. Using the perturbation method given by Ruderman (2007), the dispersion relation is obtained and solved both analytically and numerically by obtaining the frequencies of the nonaxisymmetric modes. Our dispersion relation confirms the results of other studies for different cases. For instance, (1) it shows that in the absence of an annulus region, the twist does not affect the kink (m = 1) modes, which is the same as the result obtained by Goossens et al. (1992) and Ruderman (2007). (2) In the absence of internal twist, the dispersion relation reduces to the same result derived by Carter & Erdélyi (2008) in the TT approximation. (3) In the absence of twist in the annulus region, the dispersion relation yields the same result obtained by Carter & Erdélyi (2007) for the kink (m = 1) modes in the TT approximation. Furthermore, the effect of the internal twist on the fluting (m = 2) modes is investigated. Our numerical results show the following.

1.
For a given relative core width, frequencies of the fundamental and first overtone l = 1, 2 kink (m = 1) modes with radial mode numbers n = 1, 2 increase when the twist parameter of the annulus increases. The same behavior holds for the frequencies of the fluting (m = 2) modes with n = 1 when the internal twist parameter increases.
2.
When the relative core width, a/R, goes to unity then the kink (m = 1) modes with n = 1 become independent of the twist and, in the case of a/R = 1, the second mode labeled by n = 2 is removed from the system.
3.
The period ratio P₁/P₂ for the kink (m = 1) modes with n = 1, 2 is lower than 2 (for an untwisted loop) in the presence of the twisted magnetic annulus. The results of P₁/P₂ for the kink (m = 1) modes with n = 1 are in accordance with some observations of TRACE. The period ratio for the fluting (m = 2) modes with n = 1 decreases from 2 when the internal twist parameter increases.

The authors thank the anonymous referee for a number of valuable suggestions. The work of K. Karami has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project No. 1/1551.

THE EFFECT OF A TWISTED MAGNETIC FIELD ON THE PERIOD RATIO P₁/P₂ OF NONAXISYMMETRIC MAGNETOHYDRODYNAMIC WAVES

Article metrics

Permissions

Author e-mails

Author affiliations

Dates

ABSTRACT

1. INTRODUCTION

2. EQUATIONS OF MOTION