A&A 506, 885-893 (2009)
On the vertical and horizontal transverse oscillations of curved coronal loops
M. S. Ruderman
Department of Applied Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK
Received 8 June 2009/ Accepted 10 August 2009
Kink oscillations of curved coronal loops with the density varying along the loop are studied in the thin tube approximation. The equilibrium magnetic field is assumed to be potential, and the field potential and flux function are used as curvilinear coordinates. It is also assumed that the loop expansion is weak, and the solution to the problem is looked for in the form of power series with respect to the tube expansion parameter . The main result of the study is that the eigenfrequencies of the vertical and horizontal tube oscillations are, in general, different, their difference being proportional to . As an example a simple equilibrium with the magnetic field magnitude exponentially decaying with the height is considered. The implication of the obtained results for the interpretation of observational data is discussed.
Key words: magnetohydrodynamics (MHD) - waves - Sun: oscillations - Sun: corona
Since transverse oscillations of coronal loops were first observed by TRACE (Aschwanden et al. 1999; Nakariakov et al. 1999), they received ample attention from both observers and theorists. The majority of observed transverse oscillations of coronal loops were polarised in the horizontal direction. However, Wang & Solanki (2004) and Wang et al. (2008) reported the observations by TRACE of the vertically polarised transverse oscillations of coronal loops. These observations raise a question: what is the difference between properties of the horizontally and vertically polarised oscillations? In particular, do they have the same frequencies, or their frequencies are different?
In first theoretical studies of transverse oscillations of coronal loops the simplest model of a coronal loop was used. In this model a coronal loop is considered as a straight homogeneous magnetic tube with the footpoints frozen in the dense photospheric plasma. Since this magnetic plasma configuration is axisymmetric, the loop oscillations can be polarised in any direction, and the oscillation frequency is independent of the polarisation direction. Then more complicated models taking such effects as the density variation along and across a loop, the loop expansion and curvature were developed (see review paper by Ruderman & Erdélyi 2009, and references therein). In particular, Van Doorsselaere et al. (2004) considered transverse oscillations of curved coronal loops. They assumed that the loop has the form of a half-circle and neglected the coronal density stratification. Using the toroidal coordinates they studied the transverse loop oscillations in the thin tube approximation. Their conclusion was that the eigenmode frequencies of the transverse loop oscillations remain unchanged up to first order in curvature. The curvature has more pronounced effect on the damping rate caused by resonant absorption in a thin boundary layer near the tube boundary. The correction to the damping rate due to the curvature is of the first order in curvature.
Later Terradas et al. (2006) studied the same problem as Van Doorsselaere et al. (2004), however taking the density stratification into account. Since Terradas et al. (2006) did not use the thin tube approximation but rather solved the exact linearised MHD equations, they managed to calculate corrections to the oscillation frequencies due to curvature. As a result, they found that the curvature removes the degeneration of eigenmodes of the transverse loop oscillations in a sense that now the eigenmodes can be polarised either in the horizontal or in the vertical direction only. However for thin coronal loops the difference between the eigenfrequencies of the two modes is very small.
Terradas et al. (2006) also studied the damping of transverse oscillations. In contrast to Van Doorsselaere et al. (2004) they did not assume that the inhomogeneous layer where the density varies in the radial direction is thin. Rather they allowed the density to vary in a layer with an arbitrary thickness. Terradas et al. (2006) confirmed the conclusion by Van Doorsselaere et al. (2004) that curvature has mode pronounced effect on the damping rate than on the eigenfrequencies. In addition, they found that, due to the density stratification, the oscillations become leaky and can be in resonance with the local Alfvén oscillations in the external plasmas. However the damping due to resonant absorption in the boundary layer near the tube boundary strongly dominates the damping due to leakage and resonance in the external plasma.
A very important assumption made by both Van Doorsselaere et al. (2004) and Terradas et al. (2006) was that the loop cross-section is circular and its radius does not very along the loop. These conditions are hardly satisfied in real coronal loops. The loop expansion leads to the variation of the loop cross-section area along the loop. In addition, the loop curvature can cause the variation of the cross-section shape. In this paper we study the effect of the variation of the loop cross-section on the transverse coronal loop oscillations using a simple model where the loop is embedded in a two-dimensional potential magnetic field. The paper is organised as follows. In the next section we formulate the problem, describe the unperturbed state, and present the governing equations in curvilinear coordinates. In Sect. 3 we simplify the governing equations using the thin tube approximation. In Sect. 4 we study the eigenmodes of the vertical and horizontal loop oscillations in the approximation of weak tube expansion. In Sect. 5 we apply general results obtained in the previous sections to a particular equilibrium with exponentially decaying magnetic field. Section 6 presents the summary of the obtained results and our conclusions.
Let us introduce Cartesian coordinates x, y, z with the z-axis in the
vertical direction, and consider an
equilibrium magnetic field
with the magnetic field lines in the
xz-plane that is independent of y. Since the magnetic field is
solenoidal, it follows that
can be expressed in terms of magnetic flux
We also assume that
is potential, so that it can
be expressed in terms of potential .
As a result we have
where B0 is the characteristic value of the equilibrium magnetic field. We assume that the magnetic field is closed with the footpoints of the magnetic field lines frozen in the dense photospheric plasma. The sketch of a typical equilibrium state is shown in Fig. 1.
The sketch of the equilibrium state. The ends of the magnetic loop are assumed to be frozen in a dense photospheric plasma. The axis of the magnetic loop is shown by the thick line. A few cross-sections of the loop are shown by ellipses.
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A magnetic field line is determined by the equations
The equation of the loop axis is
is a constant. We assume that this axis is in the xz-plane. The
equation of the loop boundary can be written in a parametric form as
interior of the loop is defined by the equation
while its exterior is defined by the equation
We assume that the plasma density, , is equal to inside the loop, and outside, where and are functions of only, and .
The plasma motion is described by linearised ideal MHD equations for cold
Here is the plasma displacement, the magnetic field perturbation, and the magnetic permeability of free space.
Now we introduce the curvilinear coordinates
vectors in this coordinate system are
It follows from Eq. (1) that
and the Cartesian
are given by
It is straightforward to see that the introduced curvilinear coordinate system is orthogonal and right-oriented ( ). Since the coordinate system is orthogonal, the metric tensor is diagonal, and its diagonal elements are given by
Let us introduce the components of vectors and ,
Then, using Eq. (5), the expression for the operator in orthogonal curvilinear coordinates (e.g. Korn & Korn 1961), and taking into account that, in accordance with (2), , we rewrite Eqs. (2) and (3) in the coordinate form,
where is the perturbation of the magnetic pressure. Eliminating and by we rewrite Eqs. (6), (7) and (9) as
where and Q = P/B2. Equations (10)-(12) constitute a closed system of equations for u, and Q. This system has to be supplemented by the boundary conditions at the tube boundary and footpoints. At the tube boundary the normal displacement and perturbation of the magnetic pressure have to be continuous. Since B is continuous at the tube boundary, these conditions can be written as
where is the component of displacement normal to the tube boundary, and the square brackets indicate the jump of a quantity across the boundary. The tube footpoints are determined by the equations and . The magnetic field lines at the footpoints are frozen in the dense photospheric plasma, so that the displacement is equal to zero at the footpoints. This gives the boundary conditions
The system of Eqs. (10)-(12) with the boundary conditions (13) and (14) is used in what follows to study the fast kink oscillations of curved coronal loops.
Let L be the loop length and a be the characteristic size of loop
cross-sections. For typical coronal loops ,
which inspires us to
To have the same spatial scales of
perturbation variation with respect to all three variables, ,
y and ,
we introduce the new scaled variable
the characteristic period of kink oscillations is equal to the transverse
Alfvén time times L/a, we also introduce the scaled time
In new variables Eqs. (10)-(12) are rewritten as
Now we look for solution to this system of equations in the form of expansions with respect to . It follows from (15) and (16) that, if the expansions for u and start from terms of the order of unity, then the expansion for Q has to start from the term of the order of . Hence, we write these expansions as
where the dots indicate terms of higher order with respect to . While the characteristic scale of variation of perturbations in the transverse directions is a, the characteristic scale of variation of equilibrium magnetic field in the transverse directions is L. This implies that, inside the loop and in its immediate vicinity we can write B as
Substituting Eqs. (18) and (19) in Eqs. (15)-(17), and collecting terms of the lowest order with respect to , we obtain
where B is calculated at . It follows from Eq. (22) that u1 and can be expressed in terms of the flux function ,
Substituting these expressions in Eqs. (20) and (21), and using cross-differentiation to eliminate Q1, we obtain the following equation for ,
Let us introduce the polar coordinates in the -plane,
(recall that at the loop axis). The loop cross-section by the surface is, in general, a curved surface. Using the Taylor expansion we obtain
where is the point of intersection of the surface with the loop axis, and the partial derivatives on the right-hand side of this expression are calculated at and . The characteristic spatial scale of variation of function is L. This implies that the ratio of the last term on the right-hand side of this expression to the first two terms is of the order of . Then it follows that, in the main order approximation with respect to , the loop cross-section by the surface is a part of plane perpendicular to the loop axis. Cartesian coordinates in this plane are y and . Since the -axis is orthogonal to y-axis and , we obtain that is the unit vector of the -axis.
In what follows we assume that the loop boundary is determined by the equation
r = a. In the first order approximation with respect to
are related by
Hence, once again in the first order approximation with respect to , the equation of the loops boundary intersection with the -plane is
This is the equation of an ellipse with the half-axes equal to a and a(B0/B), where B is calculated at the loop axis. We see that, while the axis parallel to the y-direction is constant, the perpendicular axis, in general, varies along the loop.
In what follows we are looking for eigenmodes of the loop oscillations and take
perturbations of all quantities proportional to
Then, in the new variables, Eq. (24) takes the form
The expression for is obtained from the expression for by substituting for .
Let us now rewrite the boundary conditions (13) in terms of .
To transform the first boundary condition we need to calculate the normal vector
to the loop boundary. Since r = a at the boundary, and the Cartesian
coordinates of a point are functions of ,
y and ,
equation of the boundary in Cartesian coordinates can be written as
Here we have taken into account that x and z are functions of and , while they are independent of y. Equation (28) is the equation of the boundary written in a parametric form, and being the parameters. The vectors
are tangential to the boundary. To evaluate these vectors we use the relations
We obtain with the aid of Eq. (1) that
Using this result, Eq. (30), and the relation between and , yields
The unit normal vector to the boundary is then given by
When deriving this expression we have used Eq. (4). The normal component of the displacement is given by
This expression is only valid in the lowest order approximation with respect to because, when deriving it, we substituted u1 and for u and . Using the expression for , and taking into account that B and, consequently, N is continuous at the loop boundary, we reduce the first boundary condition in (13) to
To rewrite the second boundary condition in (13) in terms of we need to express Q1 in terms of . To do this we first take u1, and Q1 proportional to . Then we transform Eqs. (20) and (21) to the variables r and , multiply the first equation by , the second by , and add the results. As a result, with the aid of (23), we obtain the expression for in terms of . Substituting this expression in the second boundary condition in (13), that has been preliminarily differentiated with respect to , we arrive at
and the expression for is obtained from the expression for by substituting for .
To make analytical progress we consider weakly expanding loops and assume that
the variation of the magnetic field magnitude along the loop is small. In
accordance with this assumption we take
and assume that , while the function is of the order of unity. Then we use the regular perturbation method and look for the solution in the form
If we denote the expression in the second brackets in Eq. (38) as f, then it follows from Eq. (40) that f = 0 at . As a result we obtain the Sturm-Liouville problem for the function f,
Since is the square of the Alfvén speed, this Sturm-Liouville problem describes Alfvénic oscillations (inside the loop when , and outside when ). In what follows we assume that the frequencies of the loop kink oscillations do not coincide with the Alfvénic frequencies either inside or outside the loop. Then is not an eigenvalue of the Sturm-Liouville problem (41), and the only solution of this problem is f = 0. This implies that Eq. (38) reduces to
In what follows we only study the kink oscillations, so that we take
Substituting this expression in Eq. (42) we obtain
In addition, the functions and have to satisfy the boundary conditions (39). The solution to Eq. (44) regular at r = 0, vanishing as , and satisfying the first boundary condition in (39), is
Substituting this solution in the second boundary condition in (39) yields
This equation coincides with the equation derived by Dymova & Ruderman (2005) for kink oscillations of a straight loop with the constant circular cross-section. This is not surprising at all because, in the first order approximation with respect to , we neglected the effect of variation of the cross-section along the loop. It follows from (40) that the functions and have to satisfy the boundary conditions
Equation (46) and the boundary condition (47) constitute the Sturm-Liouville problem. This problem has a non-trivial solution only when is an eigenvalue. In that case the solution is unique with the accuracy up to multiplication by an arbitrary constant. Let us denote this solution as . Then
where and are (in general complex) constants.
Equation (49) for inside the loop coincides with Eq. (42) for . It can be easily verified that , where and are arbitrary functions, is an exact solution of Eq. (26). This property is related to the fact that, in the thin tube approximation, the loop moves as rigid, i.e. the displacement of all points of a particular cross-section are the same. The solution to Eq. (49) is
where and are arbitrary functions. It follows from (53) that .
We look for the solution to Eq. (50) in the form
Substituting this expression in Eq. (50) we obtain
The general solution to this equation is given by
where is an arbitrary function, and satisfies the equation
Substituting (55) in (51) and using (57) we obtain , , and , so that eventually
It follows from (53) that .
Now we substitute (54) and (59) in the boundary condition (52), and collect terms proportional to
As a result we obtain
and A3. The equation for A3coincides with (58), and the equations for A2c and A2s are
The homogeneous counterparts to Eqs. (60) and (61) coincide with Eq. (49), so that they have a non-trivial solution . Then it follows from the general theory of linear operators that Eqs. (60) and (61) can be solved only when their right-hand sides are orthogonal to . These solvability conditions can be also obtained directly. To do this we multiply either of Eqs. (60) and (61) by A1 and integrate with respect to from to . Using integration by parts and the boundary conditions we obtain that the integrals from the left-hand sides of Eqs. (60) and (61) are equal to zero. This implies that the integrals from the right-hand sides of Eqs. (60) and (61) are also equal to zero. Once again using integration by parts we can write these conditions as
In general, . Then it follows from (60) and (61) that either and , or and . Using (23), (25), (43), (45), (48), and the relation , we easily obtain
This result implies that the case where corresponds to vertical oscillations, and the case where to horizontal oscillations. We see that the second order approximation removes the system degeneration. Now eigenmodes cannot be polarised in any direction. Rather they are polarised either in the vertical or in horizontal direction.
In the first order approximation with respect to
and eigenmodes of the loop kink oscillations are determined by Eq. (46) with the boundary conditions (47). The second order
corrections to the eigenfrequencies are given by Eqs. (62) and
(63). All these equations are written in variables that are
inconvenient for comparison with the results obtained in previous studies and
with observations. We now rewrite these equations in physical variables. We
start from introducing the length s along the loop axis. Since
We also introduce , , and . In these new variables Eq. (46) with the boundary conditions (47) takes the form
where L is the length of the loop given by
and either , or . Now we recall that and look for the solution to the boundary value problem (66) in the form and . Then we obtain in the first order approximation with respect to
where either , or . The boundary value problem (68) coincides with one obtained by Dymova & Ruderman (2005) for kink oscillations of a straight non-expanding loop with the circular cross-section and with the density varying along the loop.
In the next order approximation we obtain
with at s = 0,L. Multiplying this equation by q, integrating the result with respect to s, and using (68) and integration by parts, we obtain the solvability condition for this boundary value problem,
Now we introduce and write . Then we have
so that . Using this result, Eq. (70), and Eqs. (62) and (63) rewritten in terms of s and , we eventually obtain expressions for the frequency of vertical, , and horizontal, , oscillations of the loop,
To verify the correctness of the obtained results we consider kink oscillations of a straight homogeneous magnetic tube with an elliptic cross-section. To do this we take q to be a non-zero constant. Then the half-axes of the elliptic cross-section are a and . The solution to the boundary value problem (68) corresponding to the fundamental mode is , . Calculation of and is straightforward, and eventually we obtain
Kink oscillations of a straight homogeneous magnetic tube with an elliptic cross-section were studied by Ruderman (2003). In particular, he obtained the expressions for the frequencies of two kink modes in the thin tube approximation (see his Eq. (60)). If we substitute and in these expressions, and make expansions with respect to up to terms proportional to , then we obtain exactly the same expressions for and .
To give an example of the obtained general results we consider the following
equilibrium state. The potential and flux function of the equilibrium magnetic
field are give by
The picture of the field lines for this magnetic field is qualitatively the same as one shown in Fig. 1. The ratio of densities inside and outside the loop is constant, . The loop is in the isothermal atmosphere, so that
where H is the atmospheric scale height. The loop footpoints are at z = 0, , so that . Then the equation of the loop axis is
Using (74) and (76) we can express z in terms of ,
It follows from (1) and (74) that
This equation implies that
The function varies from at the footpoints to zero at the loop apex, so that varies from (lk)2 - 1 to . The condition implies that
It follows from the second equation in (81) that . Since is an arbitrary small parameter, we will take
in what follows. To satisfy the first Eq. in (81) we take
where is a constant of the order of unity. Then it follows from (80) that
Now we can rewrite the equation of the loop axis (76) in an approximate form,
We see that the loop shape is approximately parabolic. The ratio of the loop height to the distance between the loop footpoints is , so that it is small.
The loop cross-section is elliptic with the horizontal half-axis equal to a. The vertical half-axis is equal to . Hence, in accordance with (74) and (82)-(84), it monotonically increases with the height from at the loop footpoints to at the loop apex.
Substituting in (85) we obtain the expression for the loop length, . Using (85) we rewrite the expression for q as
Now we assume that H and L are of the same order of magnitude. Since the loop height is of the order of , this implies that the relative density variation along the loop axis is of the order of . The account of this density variation would result in a correction to of the order of . This correction is the same for the vertical and horizontal oscillations. Since we are mainly interested in the difference of frequencies of the vertical and horizontal oscillations, we will neglect the density variation in what follows. Then and the solution to the boundary value problem (68) corresponding to the fundamental mode is
where is a constant. The calculation of integrals in (72) and (73) with q(s) given by (86) and given by (87) is straightforward, and we obtain
When deriving these expressions we have taken . The difference between the frequencies of vertical and horizontal oscillations is
We see that, in general, . They are only equal when
It is especially instructive to take . In that case the loop cross-sections at the footpoints are circles of radius a. The vertical half-axis b of the elliptic loop cross-section monotonically increases with the height from a at the loop footpoints to at the loop apex. Substituting in (90) we obtain
so that . This result is in good agreement with the results obtained by Ruderman (2003). Ruderman (2003) has shown that the frequency of kink oscillation of a magnetic tube with an elliptic cross section in the direction of the larger axis is smaller than that in the direction of the smaller axis. Since the vertical axis of the variable elliptic loop cross-section is everywhere larger than the horizontal axis, we should expect that the frequency of the vertical oscillation is smaller than that of the horizontal oscillation.
In this paper we have studied kink oscillations of curved coronal magnetic loops with the density varying along the loop. We have assumed that the loop is in a vertical plane, so that the loop torsion is zero. We have also assumed that the equilibrium magnetic field is potential. Using the magnetic field potential and flux function as curvilinear coordinates in the loop plane we derived the equation governing the loop motion in the thin tube approximation. Then we have considered a loop with the elliptic cross-section, and rewrote the governing equations and the boundary conditions at the loop boundary in terms of polar coordinates in the loop cross-section. An important property of this model is that the loop curvature causes the loop expansion. As a result the ratio of axis of the loop cross-section varies along the loop.
The governing equation for the loop motion is a partial differential equation for one function that depends of the polar coordinates in the loop cross-section, r and . The coefficients of this equation depend both on r and . We managed to find the solution to the governing equation corresponding to kink oscillations inside the loop. However, it seems highly improbable that a similar solution describing the motion outside the loop can be found. To make analytical progress we assumed that the loop expansion is small and introduced the small parameter, , characterising this expansion. In this approximation the loop cross-section is almost circular. The ratio of axes of the loop elliptic cross-section differs from unity by a quantity of the order of . After that we have used the regular perturbation method and looked for the solution to the problem in the form of power series expansions with respect to .
In particular, we looked for the eigenmodes of kink oscillations in the form . In the first order approximation we found that is determined by the eigenvalue problem (68). Equation (68) is exactly the same as one obtained by Dymova & Ruderman (2005) for kink oscillations of a straight non-expanding loop with the circular cross-section and the density varying along the loop. The eigenvalue problem is degenerate in the sense that kink oscillations can be polarised in any direction.
In the next order approximation the kink eigenmodes of the loop oscillation can be polarised only either in the vertical or horizontal direction, so that the degeneration of the eigenvalue problem is removed. The corrections to the kink oscillation frequency are different for the vertical and horizontal oscillations. They are given by Eqs. (72) and (73) for the fundamental modes of the vertical and horizontal kink oscillations.
As an example we have considered a simple equilibrium with the magnetic field magnitude exponentially decaying with the height. Neglecting the density variation along the loop we reduced the eigenvalue problem (68) to one describing kink oscillations of a straight thin homogeneous magnetic tube. Then we easily calculated the corrections to the frequencies of the vertical and horizontal oscillations, and (see Eqs. (88) and (89)). These corrections depend on the parameter characterising the shape of the loop cross-section at the footpoints. In general, this cross-section is an ellipse with the ration of the vertical and horizontal axes equal to . When , which corresponds to the circular footpoint cross-section, , i.e. the frequency of the vertical loop oscillation is smaller than that of the horizontal oscillation.
The main conclusion of this work is that the frequencies of the vertical and horizontal kink oscillations are, in general, different. The effect of loop curvature on the frequencies of the vertical and horizontal kink oscillations is not direct. Rather the curvature results in variation of the shape of the loop cross-section along the loop, and it is this variation of the cross-section shape that causes the difference in the frequencies. It is instructive to compare the results of this work with those obtained by Van Doorsselaere et al. (2004) and Terradas et al. (2006). These authors came to a conclusion that the difference between the frequencies of the vertical and horizontal oscillations of a curved loop is only of the order of , where is the ration of the loop radius to its length, so that it can be neglect in the thin tube approximation. Our study clearly shows that this result in related to the model used by Van Doorsselaere et al. (2004) and Terradas et al. (2006). In this model a curved loop has a circular cross-section with the constant radius.
The difference between the frequencies of the vertical and horizontal oscillations is of the order of , i.e. it is small. This result is directly related to our assumption that the loop expansion is small. For loops with sufficiently large expansions this difference can be quite substantial.
Although, as we have already mentioned, the majority of observed kink oscillations of coronal loops are horizontally polarised, an arbitrary disturbance should cause both vertically and horizontally polarised oscillations. Since, in general, the frequencies of the vertical and horizontal oscillations are different, it would be interesting to look for signatures of two different frequencies in the observational data. If the ratio of two observed frequencies is sufficiently large (say, larger than 1.5), then they are usually attributed to the fundamental mode and first overtone of kink oscillations. However if two frequencies with a smaller ratio (say, 1.2 or 1.3) are found, then it is quite probable that they are the frequencies of the vertical and horizontal kink oscillations.Acknowledgements
The author acknowledges support by an STFC grant.
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The sketch of the equilibrium state. The ends of the magnetic loop are assumed to be frozen in a dense photospheric plasma. The axis of the magnetic loop is shown by the thick line. A few cross-sections of the loop are shown by ellipses.
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