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Issue
A&A
Volume 526, February 2011
Article Number A75
Number of page(s) 7
Section The Sun
DOI https://doi.org/10.1051/0004-6361/201015460
Published online 23 December 2010

© ESO, 2010

1. Introduction

Observations of various coronal wave phenomena have become increasingly common in the literature within the last twenty years. From the launch of the Solar and Heliospheric Observatory (SoHO) and the Transition Region and Corona Explorer (TRACE) in the late 1990’s to the present day, with new missions such as the Solar Dynamics Observatory (SDO) and Hinode, and instruments such as the Coronal Multi-Channel Polarimeter (CoMP) at Sacramento Peak Observatory, the evidence of waves in the solar corona is ever growing. Magnetoacoustic waves both slow and fast have been reported. Slow waves have been observed both as standing modes (Ofman et al. 1997; DeForest & Gurman 1998; Ofman et al. 1999; De Moortel et al. 2000, 2002a,b; Robbrecht et al. 2001; Ofman & Wang 2002; McEwan & De Moortel 2006; Marsh et al. 2009) and propagating modes (Wang et al. 2002, 2003, 2009; Srivastava & Dwivedi 2010). Standing fast waves are recorded in the form of transverse or vertical kink modes (Aschwanden et al. 1999, 2002; Nakariakov et al. 1999; Wang & Solanki 2004; Verwichte et al. 2004; Van Doorsselaere et al. 2007) or sausage modes (Nakariakov et al. 2003; Melnikov et al. 2005). Alfvénic or propagating kink waves have been reported by Tomczyk et al. (2007) and Tomczyk & McIntosh (2009), and interpreted by Van Doorsselaere et al. (2007) as propagating kink waves. Torsional Alfvén waves have been identified by Jess et al. (2009).

An increase in the observations of waves has led to an increased growth in the field of coronal seismology of magnetic loops, suggested over twenty five years ago by Roberts et al. (1984) who exploited the dispersion diagrams derived in Edwin & Roberts (1982, 1983). Coronal seismology helps to unveil the nature of the solar atmosphere; studying observed waves and drawing on their specific properties, it is possible to diagnose aspects of the coronal structure which might otherwise remain unknown. The period ratio between the fundamental mode and its first overtone has been noted as an effective tool for coronal seismology (Andries et al. 2005a,b; Goossens et al. 2006; McEwan et al. 2006, 2008; Donnelly et al. 2006; Dymova & Ruderman 2007; Díaz et al. 2007; Roberts 2008; Verth & Erdélyi 2008; Ruderman et al. 2008; Andries et al. 2009; Erdélyi & Morton 2009; Morton & Erdélyi 2009). This topic has recently been reviewed in Andries et al. (2009).

Observations of multi-periodicities (typically the fundamental mode and its first overtone) were first reported in standing fast waves (Verwichte et al. 2004; Van Doorsselaere et al. 2007; De Moortel & Brady 2007; O’Shea et al. 2007; Srivastava et al. 2008) and have very recently been found in slow modes (Srivastava & Dwivedi 2010). Using TRACE observations, Van Doorsselaere et al. (2007) found period ratios P1/P2 = 1.81 (P1/2P2 = 0.91), P1/P2 = 1.58 (P1/2P2 = 0.79) and P1/P2 = 1.795 (P1/2P2 = 0.90) for the fast kink mode. Using observations from the Solar Tower Telescope at Aryabhatta Research Institute of Observational Sciences, Srivastava et al. (2008) report a period ratio P1/P2 = 1.68 (P1/2P2 = 0.84) which has been attributed to the sausage mode. The observed tendency for the period ratio P1/2P2 between the fundamental mode of period P1 and twice the period P2 of its first overtone to be less than unity (the value for a simple wave on a string) has led to an interest in this ratio.

A number of physical effects have been assessed for their influence on the period ratio: wave dispersion, gravitational stratification, longitudinal and transverse density structuring, loop cross-sectional ellipticity, the overall geometry of a loop and magnetic field expansion (Andries et al. 2005a,b; McEwan et al. 2006, 2008; Díaz et al. 2007; Ruderman et al. 2008; Verth & Erdélyi 2008; Erdélyi & Morton 2009; Morton & Erdélyi 2009; Inglis et al. 2009). The overall conclusion seems to be that longitudinal structuring plays the most marked role (Andries et al. 2009). Longitudinal structuring may take the form of density stratification (e.g. Andries et al. 2005a,b; McEwan et al. 2006, 2008) or magnetic structuring (Verth & Erdélyi 2008).

In this work we consider analytically the behaviour of the period ratio for fast kink and sausage modes with transverse density structuring in the form of an Epstein profile. Previously transverse density structuring has been considered analytically in the form of a step function (Edwin & Roberts 1982, 1983), or as a smooth Epstein profile (Edwin & Roberts 1988) or its generalisation to a profile that combines both the step function and the Epstein profile through use of a free parameter (Nakariakov & Roberts 1995); the generalised case has to be treated numerically. Here we consider the Epstein profile as representative of a smoothly changing plasma density, providing a physically more realistic representation of density change than is given by the step function. The Epstein profile has been considered numerically in Pascoe et al. (2007, 2009), and the period ratio for the sausage mode has been determined numerically by Inglis et al. (2009). Our emphasis is on an analytical treatment of period ratios for the kink and sausage modes. We compare our results for the Epstein profile with the case of a simple magnetic slab with a step function density profile. The dispersion curves of a magnetic slab were given in Edwin & Roberts (1982, 1983); period ratios for such a slab have not hitherto been discussed. Our treatment is entirely in a cartesian geometry; we do not consider the case of a cylindrical tube (which requires a separate, probably numerical, treatment).

2. Model

We consider a magnetic slab aligned with the z-axis of a cartesian coordinate system Oxyz. The equilibrium magnetic field is B0ez and is taken to be uniform. Across the magnetic field, the equilibrium plasma density is ρ0(x); this produces a non-uniform Alfvén speed cA(x)=(B02/μρ0(x))1/2\hbox{$c_{\rm A}(x) = (B_0^2/ \mu \rho_0(x))^{1/2}$} which varies across the magnetic field. The slab is of length 2L with its ends at z =  ±L; we consider the slab to be anchored at its ends (modelling line tying of a coronal loop by the photosphere). Both gravity and acoustic effects are not considered.

We take the following set of ideal zero-β MHD equations: ρ(∂v∂t+(v·∇)v)=1μ(∇×B)×B,∂B∂t=∇×(v×B),∇·B=0,∂ρ∂t+∇·(ρv)=0.\begin{eqnarray} \label{eq:mhd1} \displaystyle &&\rho \left(\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \vec{\nabla})\vec{v}\right) = \frac{1}{\mu}(\nabla \times \vec{B})\times \vec{B}, \\ \label{eq:mhd2} &&\frac{\partial \vec{B}}{\partial t} = \nabla \times (\vec{v} \times \vec{B}), \hspace{0.5cm} \nabla \cdot \vec{B}= 0, \\ \label{eq:mhd3}&&\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0. \end{eqnarray}The set of ideal MHD Eqs. (1)–(3) are linearised about the non-uniform equilibrium density ρ0(x) embedded within the uniform field B0ez. The linearised form of Eq. (3) gives the resulting density perturbations. We take the perturbed motions to be v = (vx,vy,0), there being no force in the direction of the applied magnetic field in a zero-β plasma (vz = 0). Then the linearised form of Eqs. (1) and (2) may be manipulated to yield the coupled partial differential equations 1cA2(x)∂2vx∂t2−∂2vx∂z2=∂∂x(∂vx∂x+∂vy∂y)1cA2(x)∂2vy∂t2−∂2vy∂z2=∂∂y(∂vx∂x+∂vy∂y)·\begin{eqnarray} \label{eq:diffeq1} \frac{1}{c_{\rm A}^2(x)}\frac{\partial^2 v_x}{\partial t^2} - \frac{\partial^2 v_x}{\partial z^2} &=& \frac{\partial}{\partial x}\left(\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y}\right) \\ \label{eq:diffeq2}\frac{1}{c_{\rm A}^2(x)}\frac{\partial^2 v_y}{\partial t^2} - \frac{\partial^2 v_y}{\partial z^2} &=& \frac{\partial}{\partial y}\left(\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y}\right)\cdot \end{eqnarray}In a uniform medium Eqs. (4) and (5) describe the Alfvén and fast magnetoacoustic waves of a zero-β plasma. Here we are interested in the fast waves in a non-uniform medium; accordingly, we suppose that vy = 0 and ∂/∂y = 0 so that the perturbed motions v = (vx,0,0) are perpendicular to the applied magnetic field and lie entirely in the plane of the density non-uniformity. These motions are compressible. Equations (4) and (5) then give the wave equation ∂2vx∂t2=cA2(x)(∂2vx∂x2+∂2vx∂z2)·\begin{equation} \frac{\partial^2 v_x}{\partial t^2}= c_{\rm A}^2(x) \left(\frac{\partial^2 v_x}{\partial x^2}+\frac{\partial^2 v_x}{\partial z^2}\right)\cdot \label{eq:diffwaveeq} \end{equation}(6)Finally, setting vx(x,z,t)=vx(x)ei(ωt−kzz)\begin{equation} v_x(x,z,t) = v_x(x) {\rm e}^{{\rm i}(\omega t -k_z z)} \label{eq:vx} \end{equation}(7)for frequency ω and wavenumber kz, we obtain the ordinary differential equation d2vxdx2+(ω2cA2(x)−kz2)vx=0.\begin{equation} \frac{{\rm d}^2 v_x}{{\rm d}x^2} + \left(\frac{\omega^2}{c_{\rm A}^2(x)}-k_z^2\right)v_x = 0. \label{eq:waveeq} \end{equation}(8)The generalisation of (8) to include acoustic effects (non-zero β) and also three dimensional motions is given in Roberts (1981).

We are particularly interested in the Epstein profile for the equilibrium plasma density ρ0(x)=ρe+(ρ0−ρe)sech2(xa),\begin{equation} \rho_0(x)=\rho_{\rm e} + (\rho_0-\rho_{\rm e})\textnormal{sech}^2\left(\frac{x}{a}\right), \label{eq:epstein} \end{equation}(9)where ρ0 and ρe are the internal and external densities respectively and a is the spatial scale over which the density varies. This profile provides a smooth representation of a physically realistic situation. The density varies smoothly from ρ0 at x = 0 to ρe as x →  ±∞. The corresponding Alfvén speeds are cA0=(B02/μρ0)1/2\hbox{$\co=(B_0^2/ \mu \rho_0)^{1/2}$} at x = 0 and cAe=(B02/μρe)1/2\hbox{$\ce=(B_0^2/ \mu \rho_{\rm e})^{1/2}$} as x →  ±∞; thus ρ0cA02=ρecAe2\hbox{$\rho_0 \co^2 = \rho_{\rm e} \ce^2$}. The Epstein profile is a convenient model for both analytical (Edwin & Roberts 1988; Nakariakov & Roberts 1995) and numerical (Cooper et al. 2003; Pascoe et al. 2007; Inglis et al. 2009) studies. The Epstein profile may also be compared with the simple step function profile of a magnetic slab investigated in Edwin & Roberts (1982, 1983).

Following Landau & Lifshitz (1958), we solve Eq. (8) in terms of hypergeometric functions and find the solution vx=(1−s2)λ/2F(A,B,C,(1−s)/2)\begin{equation} v_x=(1-s^2)^{\lambda/2}F(A,B,C,(1-s)/2) \label{eq:vxsol} \end{equation}(10)where s = tanh(x/a), A = λ − ν, B = 1 + λ + ν, C = 1 + λ for λ and ν given by λ=kza(1−c2cAe2)1/2\begin{equation} \displaystyle \lambda=k_z a \left(1-\frac{c^2}{\ce^2}\right)^{1/2} \label{eq:lambda} \end{equation}(11)and ν(ν+1)=kz2a2(c2cA02−c2cAe2)·\begin{equation} \nu(\nu+1) = k_z^2 a^2 \left(\frac{c^2}{\co^2} - \frac{c^2}{\ce^2}\right)\cdot \label{eq:nu} \end{equation}(12)The hypergeometric function F is defined by (Abramowitz & Stegun 1965) F(A,B,C,(1−s)/2)=∑m=0n(A)m(B)m(C)m(1−s2)m\begin{equation} F(A,B,C,(1-s)/2) = \displaystyle \sum_{m=0}^n \frac{(A)_m (B)_m}{(C)_m}\left(\frac{1-s}{2}\right)^{m} \label{eq:F} \end{equation}(13)where (α)m=Γ(α+m)Γ(α)\begin{equation} \displaystyle (\alpha)_m = \frac{\Gamma(\alpha + m)}{\Gamma(\alpha)} \label{eq:Freq} \end{equation}(14)and Γ(α) denotes the gamma function of argument α. The dispersion relation is given by ν=λ+n,forn=0,1,2,...\begin{equation} \nu=\lambda+n, \hspace{0.5cm} \mbox{for} \hspace{0.2cm} n=0,1,2, \ldots \label{eq:dr} \end{equation}(15)

3. The kink mode

In the case of the fast kink mode (n = 0), which disturbs the central axis of the loop (the z-axis), so vx ≠ 0 at x = 0, Eq. (8) has the solution vx=v0[sech(xa)]λ,\begin{equation} v_x = v_0\left[\textnormal{sech}\left(\frac{x}{a}\right)\right]^{\lambda}, \label{eq:solkink} \end{equation}(16)for arbitrary amplitude v0. The power λ is determined transcendentally through the dispersion relation λ = ν. The dispersion relation λ = ν implies that kza=(cAe2−c2)(c2−cA02)1/2cA02cAe·\begin{equation} k_z a = \displaystyle \frac{\left(\ce^2 - c^2\right)}{\left(c^2-\co^2\right)}^{1/2}\frac{\co^2}{\ce}\cdot \label{eq:kink} \end{equation}(17)This relation has been given by Cooper et al. (2003).

Equation (17) determines the wave speed c (= ω/kz) as a function of kza. In order that kza is real and positive we require cA0 < c < cAe. We note that for kza → 0 Eq. (17) leads to c = cAe, and for kza → ∞, c = cA0. Further, for cAe ≫ cA0 Eq. (17) yields c2=cA02(1+1kza)·\begin{equation} c^2 = \displaystyle \co^2\left(1+\frac{1}{k_za}\right)\cdot \label{eq:cspeed2} \end{equation}(18)Equation (18) provides an upper bound on the behaviour of c2. The limit cAe → ∞ described by (18) applies if ρe → 0, i.e. the environment is a vacuum.

The general solution of the kink dispersion relation (17) for the square of the wave speed is given by c2=cA022cAe2kz2a2(2cAe2kz2a2−cA02+cA04+4cAe4kz2a2−4cA02cAe2kz2a2).\begin{eqnarray} \displaystyle c^2&=&\displaystyle \frac{\co^2 }{2\ce^2k_z^2a^2}\left(2\ce^2k_z^2a^2-\co^2 \right. \nonumber \\ \label{eq:csol}&& \left. +\sqrt{\co^4+4\ce^4k_z^2a^2-4\co^2\ce^2k_z^2a^2}\right). \end{eqnarray}(19)We plot this solution in Fig. 1. The behaviour of the solution is such that as kza increases c decreases from cAe to cA0 and so we expect the period ratio to be less than unity.

thumbnail Fig. 1

A plot of the wave speed c (in units of cA0) with respect to kza for cAe/cA0 = (ρ0/ρe)1/2 = 2,5, and 10 for the solid, dashed, and dotted curves respectively. The speed c is bounded below by cA0 and above by cAe.

3.1. The period ratio for the kink mode

For a loop of length 2L, the wavenumber kz is taken as kz = π/2L for the fundamental mode and kz = π/L for the first overtone. Thus the speed c1 of the fundamental mode is given by Eq. (19) and the frequency is given by ω1 = kzc1 where kz = π/2L. The speed c2 of the first overtone is given by Eq. (19) and the frequency is given by ω2 = kzc2 where kz = π/L. We use the subscript 1 to denote any value pertaining to the fundamental mode and the subscript 2 to denote the first overtone. The fundamental period is given by P1 = 2π/ω1 = 4L/c1 and the period of the first overtone is P2 = 2π/ω2 = 2L/c2. Thus the period ratio determined by the dispersion relation (17) is P1/2P2 = c2/c1 where c1 and c2 are determined from Eq. (19): (P12P2)2=14(2π2a2L2−cA02cAe2+cA04cAe4+4π2a2L2−4cA02cAe2π2a2L2π2a22L2−cA02cAe2+cA04cAe4+π2a2L2−cA02cAe2π2a2L2)·\begin{eqnarray} \displaystyle \left(\frac{P_1}{2P_2}\right)^{2} = \frac{1}{4}\left(\frac{\displaystyle2\frac{\pi^2 a^2}{L^2} - \frac{\co^2}{\ce^2} + \sqrt{\frac{\co^4}{\ce^4} + 4\frac{\pi^2 a^2}{L^2} - 4\frac{\co^2}{\ce^2}\frac{\pi^2 a^2}{L^2}}}{\displaystyle\frac{\pi^2 a^2}{2L^2} - \frac{\co^2}{\ce^2} + \sqrt{\frac{\co^4}{\ce^4} + \frac{\pi^2 a^2}{L^2} - \frac{\co^2}{\ce^2}\frac{\pi^2 a^2}{L^2}}}\right)\cdot\nonumber \\ \label{eq:kinkpr} \end{eqnarray}(20)Regardless of the values of the internal and external Alfvén speeds (cA0 and cAe), as a/L → ∞ the period ratio is unity. Thus for long thin loops the period ratio is unaffected by transverse density structuring in the form of the Epstein profile.

thumbnail Fig. 2

A plot of the period ratio P1/2P2 with respect to a/L for the fast kink mode with an Epstein density profile and cAe/cA0 = 2,5,10, and 50 for the solid, dotted, dashed and dot-dashed curves respectively.

In Fig. 2 we plot the period ratio for the kink mode under an Epstein profile for selected values of cAe/cA0. For all values of cAe/cA0, in the limits a/L → 0 and a/L → ∞ the period ratio is unity. Thus for long and thin or short and fat loops the period ratio is close to unity. For small a/L there is a rapid departure from unity, possibly falling to as low a value as P1/2P2=1/2\hbox{$P_1/2P_2=1/\sqrt{2}$} in the limit of cAe ≫ cA0. This tendency towards a least value is shown in Fig. 3 where we plot the minimum value of the period ratio as a function of cAe/cA0.

thumbnail Fig. 3

A plot of the minimum value of the period ratio P1/2P2 for the kink mode with respect to cAe/cA0 = (ρ0/ρe)1/2. In the extreme cAe/cA0 → ∞, the period ratio minimum approaches 1/2\hbox{$\sqrt{1/2}$}.

3.2. Approximations for small and large a/L

Consider Eq. (19) in the extreme 2cAe2kz2a2≫cA02\hbox{$2\ce^2k_z^2a^2 \gg \co^2$}, which applies for all kza as cAe/cA0 → ∞ and for all cAe/cA0 as kza → ∞. We consider this approximation for kza → ∞ (corresponding to a/L → ∞). With 2cAe2kz2a2≫cA02\hbox{$2\ce^2k_z^2a^2 \gg \co^2$}, Eq. (19) yields c2=cA02(1+μ1kza)\begin{equation} c^2 = \co^2 \left(1+ \displaystyle \frac{\mu_1}{k_z a}\right) \label{eq:ckzalarge} \end{equation}(21)where μ1=(cAe2−cA02cAe2)1/2,0<μ1<1.\begin{equation} \displaystyle \mu_1 = \left(\frac{\ce^2-\co^2}{\ce^2}\right)^{1/2}, \hspace{0.25cm} 0<\mu_1<1. \label{eq:mu1} \end{equation}(22)The period ratio is then given by P12P2=(1+μ1Lπa1+2μ1Lπa)1/2·\begin{equation} \displaystyle \frac{P_1}{2P_2} = \left(\frac{\displaystyle1+\frac{\mu_1L}{\pi a}}{\displaystyle1+\frac{2\mu_1L}{\pi a}}\right)^{1/2}\cdot \label{eq:kzalarge} \end{equation}(23)For cAe ≫ cA0, μ1 = 1 and Eq. (23) gives P1/2P2 = 1 for a/L → ∞ and P1/2P2=1/2\hbox{$P_1/2P_2=1/\sqrt{2}$} for a/L → 0, in agreement with Figs. 2 and 3. Expanding Eq. (23) gives P12P2=1−μ1L2πa+7μ12L28π2a2+...,πa2L≫1.\begin{equation} \displaystyle \frac{P_1}{2P_2} = 1-\frac{\mu_1 L}{2 \pi a} + \frac{7\mu_1^2 L^2}{8\pi^2 a^2} + \ldots, \hspace{0.5cm} \frac{\pi a}{2L} \gg 1. \label{eq:kzalarge2} \end{equation}(24)In the opposite extreme of πa/2L ≪ 1, the period ratio is given by P12P2=1−3π2a2μ28L2−21π4a4μ22128L4+...,πa2L≪1,\begin{equation} \displaystyle \frac{P_1}{2P_2} = 1 -\frac{3\pi^2 a^2 \mu_2}{8L^2} - \frac{21\pi^4 a^4 \mu_2^2}{128L^4}+ \ldots, \hspace{0.5cm} \frac{\pi a}{2L} \ll 1, \label{eq:kzasmall} \end{equation}(25)where μ2=(cAe2−cA02cA02)2.\begin{equation} \displaystyle \mu_2 = \left(\frac{\ce^2-\co^2}{\co^2}\right)^{2}. \label{eq:mu2} \end{equation}(26)We note that as a/L → 0 the period ratio is unity.

4. The sausage mode

In the case of the sausage mode (n = 1) Eq. (8) has solution vx(x)=v0tanh(xa)[sech(xa)]λ,\begin{equation} \displaystyle v_x(x) = v_0 \tanh \left(\frac{x}{a}\right)\left[\textnormal{sech}\left(\frac{x}{a}\right)\right]^{\lambda}, \label{eq:solsaus} \end{equation}(27)for arbitrary amplitude v0, giving vx = 0 at x = 0. From Eqs. (11) and (12), the dispersion relation ν = 1 + λ gives kz2a2(c2cA02−1)−3kza(1−c2cAe2)1/2−2=0.\begin{equation} k_z^2 a^2 \left(\frac{c^2}{\co^2}-1\right)- 3k_z a \left(1-\frac{c^2}{\ce^2}\right)^{1/2} - 2 = 0.\label{eq:saus} \end{equation}(28)Relation (28) has been given in Cooper et al. (2003); see also Pascoe et al. (2007) and Inglis et al. (2009). The presence of the square root in (28) makes it clear that we require c2≤cAe2\hbox{$c^2 \le \ce^2$}. Moreover, since 1−c2/cAe2≤1\hbox{$1-c^2/\ce^2 \le 1$}, it follows from (28) that the wave speed squared is bounded above by the expression c2=cA02(1+3kza+2kz2a2)·\begin{equation} c^2 = \displaystyle \co^2\left(1+\frac{3}{k_za}+\frac{2}{k_z^2a^2}\right)\cdot \label{eq:cspeeds} \end{equation}(29)In fact, this bound is achieved in the limit cAe → ∞, which attains whenever ρe → 0, i.e. if the environment is a vacuum. The bound (29) shows that c2→cA02\hbox{$c^2 \rightarrow \co^2$} as kza → ∞, so altogether the solutions of (28) satisfy cA02≤c2≤cAe2\hbox{$\co^2 \le c^2 \le \ce^2$}.

The upper bound (29) is not a strong constraint for small kza. In fact it is evident from the dispersion relation (28) that when c2=cAe2\hbox{$c^2 = \ce^2$} we require kza=(2cA02cAe2−cA02)1/2,cutoff;\begin{equation} k_z a=\left(\frac{2\co^2}{\ce^2-\co^2}\right)^{1/2}, \hspace{0.5cm} \mbox{cutoff}; \label{eq:sauscutoff} \end{equation}(30)kza must exceed this cutoff value.

The dispersion relation (28) may be viewed as a quadratic equation determining kza as a function of c/cA0 for given cAe/cA0 (= (ρ0/ρe)1/2). Specifically, kza=3(1−c2cAe2)1/2+(1−9c2cAe2+8c2cA02)1/22(c2cA02−1),\begin{equation} k_z a = \displaystyle \frac{3\left(\displaystyle1- \frac{c^2}{\ce^2}\right)^{1/2}+\left(\displaystyle1-9\frac{c^2}{\ce^2}+8\frac{c^2}{\co^2}\right)^{1/2}}{2\left(\displaystyle\frac{c^2}{\co^2}-1\right)}, \label{eq:sauskza} \end{equation}(31)the second root of the quadratic being rejected (it gives kza < 0).

We may also solve (28) for the square of the wave speed, viz. c2=cA022cAe2kz2a2(2cAe2kz2a2+4cAe2−9cA02+39cA04−4cA02cAe2kz2a2−8cA02cAe2+4cAe4kz2a2).\begin{eqnarray} \displaystyle c^2&=&\frac{\co^2}{2\ce^2k_z^2a^2}\left(2\ce^2k_z^2a^2+4\ce^2-9\co^2 \right. \nonumber \\ \label{eq:csol2} && \left. + 3\sqrt{9\co^4-4\co^2\ce^2k_z^2a^2-8\co^2\ce^2+4\ce^4k_z^2a^2}\right). \end{eqnarray}(32)We plot this solution in Fig. 4 as a function of kza. The behaviour of the solution is such that the speed c exists for values of kza above the cutoff (30) after which as kza increases c decreases and so we expect the period ratio to be less than unity. Increasing the ratio cAe/cA0 between the internal and external Alfvén speeds acts to increase the speed c for a particular kza.

thumbnail Fig. 4

A plot of the wave speed c (in units of cA0) with respect to kza for cAe/cA0 = 2,5, and 10 shown as solid, dashed, and dotted curves respectively. The cutoff value for each curve is given by Eq. (30). For example, the curve with cAe/cA0 = 2 has a cutoff of kza=2/3\hbox{$k_z a = \sqrt{2/3}$}.

4.1. The period ratio for the sausage mode

The period ratio for the sausage mode is given by P1/2P2 = c2/c1 where c1 and c2 are determined by (32). Thus the period ratio is determined by (P12P2)2=14(4+2π2a2L2−9cA02cAe2+39cA04cAe4−4cA02cAe2π2a2L2−8cA02cAe2+4π2a2L24+π2a22L2−9cA02cAe2+39cA04cAe4−cA02cAe2π2a2L2−8cA02cAe2+π2a2L2)·\begin{eqnarray} \displaystyle \left(\frac{P_1}{2P_2}\right)^2 = \hspace{7cm}\nonumber \\ \frac{1}{4}\left(\!\frac{\displaystyle4\!+\!2\frac{\pi^2 a^2}{L^2}\!-\!9\frac{\co^2}{\ce^2}\! +\! 3\!\sqrt{9\frac{\co^4}{\ce^4}\!-\!4\frac{\co^2}{\ce^2}\frac{\pi^2 a^2}{L^2}\!-\!8\frac{\co^2}{\ce^2}\!+\!4\frac{\pi^2 a^2}{L^2}}}{\displaystyle 4\! +\!\frac{\pi^2 a^2}{2L^2}\!-\!{9}\frac{\co^2}{\ce^2}\! +\! {3}\!\sqrt{{9}\frac{\co^4}{\ce^4}\!-\!\frac{\co^2}{\ce^2}\frac{\pi^2 a^2}{L^2}\!-\!8\frac{\co^2}{\ce^2}\!+\!\frac{\pi^2 a^2}{L^2}}}\!\right)\cdot \nonumber \\ \label{eq:sauspr} \end{eqnarray}(33)In the limit a/L → ∞ the period ratio P1/2P2 → 1. In Fig. 5 we plot the period ratio for the sausage mode under the Epstein profile. A similar graph is given in Inglis et al. (2009).

thumbnail Fig. 5

A plot of the period ratio P1/2P2 with respect to a/L for the fast sausage mode with an Epstein density profile for cAe/cA0 = 2,5,10 and 50 for the solid, dotted, dashed and dot-dashed curves respectively.

Just as in the case of the kink mode we note that the period ratio appears to have a limit of 1/2 as cAe/cA0 → ∞ and a/L → 0. We plot in Fig. 6 the minimum value of the period ratio with respect to cAe/cA0 and note that as cAe/cA0 → ∞ the minimum value approaches 1/2.

thumbnail Fig. 6

A plot of the minimum value of the period ratio P1/2P2 for the sausage mode with respect to cAe/cA0. In the extreme cAe/cA0 → ∞, the period ratio minimum approaches 1/2.

4.2. Approximation for large a/L

As for the kink mode we consider the approximation for 2cAe2kz2a2≫cA02\hbox{$2\ce^2k_z^2a^2 \gg \co^2$}, corresponding to the ratio cAe/cA0 being much greater than L/a. This gives c2=cA02(1+3μ1kza+2kz2a2),\begin{equation} c^2 = \displaystyle \co^2\left(1+\frac{3\mu_1}{k_za}+\frac{2}{k_z^2a^2}\right), \label{eq:capprox2s} \end{equation}(34)where μ1 is given by Eq. (22) as for the kink mode case. The period ratio is then given by P12P2=(1+3μ1Lπa+2L2π2a21+6μ1Lπa+8L2π2a2)1/2·\begin{equation} \displaystyle \frac{P_1}{2P_2} = \left(\frac{\displaystyle1+\frac{3\mu_1 L}{\pi a}+\frac{2L^2}{\pi^2a^2}}{\displaystyle1+\frac{6\mu_1 L}{\pi a}+\frac{8L^2}{\pi^2a^2}}\right)^{1/2}\cdot \label{eq:sauspr2} \end{equation}(35)For cAe ≫ cA0, μ1 = 1 and Eq. (35) gives P1/2P2 → 1/2 for a/L → 0, although this may not be attained due to the cutoff, and P1/2P2 → 1 for a/L → ∞. Expanding Eq. (35) we obtain P12P2=1−3μ1L2πa−3L2π2a2+63μ12L28π2a2+...,πa2L≫1.\begin{equation} \displaystyle \frac{P_1}{2P_2} = 1 - \frac{3\mu_1 L}{2\pi a} - \frac{3L^2}{\pi^2 a^2} + \frac{63\mu_1^2L^2}{8\pi^2a^2} + \ldots, \hspace{0.25cm} \frac{\pi a}{2L} \gg 1. \label{eq:kzalarges} \end{equation}(36)

5. Comparison between the Epstein profile and a step function slab

The results for the Epstein profile show how period ratios vary with the density ratio ρ0/ρe (or equivalently (cAe/cA0)1/2) and the length L of the magnetic field lines. It is natural to compare these results with the simpler magnetic slab model of a step function change in the plasma density. Dispersion curves for the step function slab are well known (see Edwin & Roberts 1982, 1983) but hitherto period ratios have not been determined. We give a brief discussion here.

The starting point for our discussion is again the wave Eq. (8), but now in place of the Epstein profile (9) we consider the step function ρ0(x)={\begin{eqnarray} \rho_0(x) = \left\{\begin{array}{cc} \rho_{\rm e}, & |x| > a, \\ \rho_0, & |x|<a,\end{array}\right. \end{eqnarray}(37)the internal density ρ0 changing to ρe discontinuously at the slab boundaries at x =  ± a. Following the notation of Edwin & Roberts (1982), we write n02=ω2cA02−kz2,me2=kz2−ω2cAe2\begin{eqnarray} \displaystyle n_0^2=\frac{\omega^2}{\co^2}-k_z^2, \hspace{0.5cm} \displaystyle m_{\rm e}^2=k_z^2 -\frac{\omega^2}{\ce^2} \label{eq:n0ne} \end{eqnarray}(38)where cA0 is the Alfvén speed in the slab (−a < x < a) and cAe is the Alfvén speed in the environment (|x|  > a). Equation (8) has solution vx(x)={\begin{eqnarray} v_x (x) = \left\{\begin{array}{cc} \ale {\rm e}^{-m_{\rm e}(x-a)}, & x>a \\ \alo \cos n_0 x + \bo \sin n_0 x, & |x| < a\\ \be {\rm e}^{m_{\rm e}(x+a)}, & x<-a. \end{array}\right. \end{eqnarray}(39)In order that the velocity disturbance is effectively confined to the interior of the slab, so that vx → 0 as  |x|  → ∞, we require me > 0. It is also required (see Roberts 1981; Edwin & Roberts 1982) that both vx and the total pressure perturbation pT are continuous at x =  ±a, where pT(x)=iρ0ωcA2(x)dvxdx·\begin{eqnarray} p_T (x) = \displaystyle \frac{{\rm i} \rho_0}{\omega} c_{\rm A}^2 (x) \frac{{\rm d} v_x}{\rm dx}\cdot \label{eq:pt} \end{eqnarray}(40)Thus, we have the dispersion relations (Edwin & Roberts 1982) cotn0a=n0meandtann0a=−n0me\begin{eqnarray} \displaystyle \cot n_0 a = \frac{n_0}{m_{\rm e}} \hspace{0.5cm} \mbox{and} \hspace{0.5cm} \displaystyle \tan n_0 a = -\frac{n_0}{m_{\rm e}} \label{eq:disprels} \end{eqnarray}(41)for the kink and sausage modes respectively. Since ω = ckz we have tan⎧⎪⎪⎨⎪⎪⎩kza(c2−cA02cA02)1/2⎫⎪⎪⎬⎪⎪⎭=cA0cAe(cAe2−c2c2−cA02)1/2\begin{eqnarray} \displaystyle \tan \left\{k_z a \left(\frac{c^2-\co^2}{\co^2}\right)^{1/2} \right\} = \frac{\co}{\ce}\left(\frac{\ce^2-c^2}{c^2-\co^2}\right)^{1/2} \label{eq:kinkg} \end{eqnarray}(42)for the kink mode, and tan⎧⎪⎪⎨⎪⎪⎩kza(c2−cA02cA02)1/2⎫⎪⎪⎬⎪⎪⎭=−cAecA0(c2−cA02cAe2−c2)1/2\begin{eqnarray} \displaystyle \tan \left\{k_z a \left(\frac{c^2-\co^2}{\co^2}\right)^{1/2} \right\} =- \frac{\ce}{\co}\left(\frac{c^2-\co^2}{\ce^2-c^2}\right)^{1/2} \label{eq:sausg} \end{eqnarray}(43)for the sausage mode. The principal kink mode has no cutoff; the cutoff for the principal sausage mode occurs when c2=cAe2\hbox{$c^2=\ce^2$} with kza=π2(cA02cAe2−cA02)1/2,cutoff,\begin{eqnarray} k_z a = \displaystyle \frac{\pi}{2} \left(\frac{\co^2}{\ce^2-\co^2}\right)^{1/2}, \hspace{0.5cm} \mbox{cutoff}, \end{eqnarray}(44)which may be compared with the cutoff value (30) for the Epstein profile.

Figure 7 gives the behaviour of the wave speed c as a function of kza, for both the Epstein and step function density profiles.

thumbnail Fig. 7

A plot of the wave speed c (in units of cA0) with respect to kza for the Epstein profile (solid curves) and the step function profile (dashed curves), for both kink and sausage modes. Here cAe/cA0 = 2 (ρ0 = 4ρe).

Figure 8 gives a plot of the period ratio in the kink mode for both the Epstein profile and the step function profile. We note that at a/L = 1 the curves are not converging but rather at this point they cross, exhibiting a changeover in behaviour and then converge for a/L → ∞. The period ratio for the sausage mode is shown in Fig. 9. In both cases the period ratio has the same general behaviour, although the period ratio achieves a lower minimum in the step function profile than the Epstein profile, for both kink and sausage modes.

thumbnail Fig. 8

A plot of the period ratio P1/2P2 with respect to a/L for the kink mode with an Epstein profile (solid curve) and for the step function profile (dashed curve) for cAe/cA0 = 2. Beyond a/L = 1, the two curves cross over and later converge together as a/L → ∞.

thumbnail Fig. 9

A plot of the period ratio P1/2P2 with respect to a/L for the sausage mode with an Epstein profile (solid curve) or for step function profile (dashed curve), for cAe/cA0 = 2. Wave cutoff restricts the formation of the period ratio.

As a final comparison we plot in Figs. 10 and 11 the period ratio for the step function profile for both the kink and sausage modes, for various values of cAe/cA0. For cAe/cA0 → ∞ the period ratio for the kink mode may be as little as 1/2\hbox{$1/\sqrt{2}$}, and for the sausage mode the period ratio may fall to 1/2. Comparing these plots with Figs. 2 and 5 we note that the limits of 1/2\hbox{$1/\sqrt{2}$} and 1/2 for the period ratio are the same regardless of whether there is an Epstein profile or a step function profile.

thumbnail Fig. 10

A plot of the period ratio P1/2P2 with respect to a/L for the fast kink mode in a step function slab with for cAe/cA0 = 2,5,10 and 50 for the solid, dotted, dashed and dot-dashed curves respectively.

thumbnail Fig. 11

A plot of the period ratio P1/2P2 with respect to a/L for the fast sausage mode with step function profile for cAe/cA0 = 2,5,10 and 50 shown as solid, dotted, dashed and dot-dashed curves respectively. Wave cutoff restricts the formation of the period ratio.

6. Discussion

There is a striking similarity between the results for a magnetic slab with Epstein density profile and one with a step function profile. This means that either profile serves as a useful and robust guide as to the expected behaviour in a slab, the Epstein profile perhaps being most useful for numerical investigations with the step function being more readily discussed analytically. Although there are differences in the period ratio from one model to another, it is perhaps unlikely that observations will be able to distinguish between the two cases given the additional complications that other factors, such as longitudinal density and magnetic field variation (see Andries et al. 2009) or non-adiabatic effects (see Macnamara & Roberts 2010) are also likely to impose. Moreover, our assumptions of a slab geometry makes application to flux tube structures problematic.

In future, it would be important to explore the cylindrical case with various density profiles, though it is likely that this would require a largely numerical approach given the expected loss of compact expressions for the wave speed c that a slab geometry provides.

Comparison of our slab results with available observations (such as by Van Doorsselaere et al. 2007, or Srivastava et al. 2008) may be inappropriate until both slab and cylinder results are available, for otherwise it may be that the geometry of a magnetic structure has a larger effect than the lateral profile of density variation.

Acknowledgments

C.K.M. acknowledges financial support from the Carnegie Trust for Scotland. We are grateful to the referee for constructive suggestions which helped improve our paper.

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All Figures

thumbnail Fig. 1

A plot of the wave speed c (in units of cA0) with respect to kza for cAe/cA0 = (ρ0/ρe)1/2 = 2,5, and 10 for the solid, dashed, and dotted curves respectively. The speed c is bounded below by cA0 and above by cAe.
In the text
thumbnail Fig. 2

A plot of the period ratio P1/2P2 with respect to a/L for the fast kink mode with an Epstein density profile and cAe/cA0 = 2,5,10, and 50 for the solid, dotted, dashed and dot-dashed curves respectively.
In the text
thumbnail Fig. 3

A plot of the minimum value of the period ratio P1/2P2 for the kink mode with respect to cAe/cA0 = (ρ0/ρe)1/2. In the extreme cAe/cA0 → ∞, the period ratio minimum approaches 1/2\hbox{$\sqrt{1/2}$}.
In the text
thumbnail Fig. 4

A plot of the wave speed c (in units of cA0) with respect to kza for cAe/cA0 = 2,5, and 10 shown as solid, dashed, and dotted curves respectively. The cutoff value for each curve is given by Eq. (30). For example, the curve with cAe/cA0 = 2 has a cutoff of kza=2/3\hbox{$k_z a = \sqrt{2/3}$}.
In the text
thumbnail Fig. 5

A plot of the period ratio P1/2P2 with respect to a/L for the fast sausage mode with an Epstein density profile for cAe/cA0 = 2,5,10 and 50 for the solid, dotted, dashed and dot-dashed curves respectively.
In the text
thumbnail Fig. 6

A plot of the minimum value of the period ratio P1/2P2 for the sausage mode with respect to cAe/cA0. In the extreme cAe/cA0 → ∞, the period ratio minimum approaches 1/2.
In the text
thumbnail Fig. 7

A plot of the wave speed c (in units of cA0) with respect to kza for the Epstein profile (solid curves) and the step function profile (dashed curves), for both kink and sausage modes. Here cAe/cA0 = 2 (ρ0 = 4ρe).

In the text
thumbnail Fig. 8

A plot of the period ratio P1/2P2 with respect to a/L for the kink mode with an Epstein profile (solid curve) and for the step function profile (dashed curve) for cAe/cA0 = 2. Beyond a/L = 1, the two curves cross over and later converge together as a/L → ∞.
In the text
thumbnail Fig. 9

A plot of the period ratio P1/2P2 with respect to a/L for the sausage mode with an Epstein profile (solid curve) or for step function profile (dashed curve), for cAe/cA0 = 2. Wave cutoff restricts the formation of the period ratio.
In the text
thumbnail Fig. 10

A plot of the period ratio P1/2P2 with respect to a/L for the fast kink mode in a step function slab with for cAe/cA0 = 2,5,10 and 50 for the solid, dotted, dashed and dot-dashed curves respectively.
In the text
thumbnail Fig. 11

A plot of the period ratio P1/2P2 with respect to a/L for the fast sausage mode with step function profile for cAe/cA0 = 2,5,10 and 50 shown as solid, dotted, dashed and dot-dashed curves respectively. Wave cutoff restricts the formation of the period ratio.
In the text

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