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Alfvénic field line resonances in arbitrary magnetic field topology
Advances in Space Research 38 (2006) 1720–1729 www.elsevier.com/locate/asr
Alfve´nic field line resonances in arbitrary magnetic field topology R. Rankin, K. Kabin *, R. Marchand Department of Physics, University of Alberta, Edmonton, Canada T6G 2J1 Received 28 September 2004; received in revised form 22 September 2005; accepted 23 September 2005
Abstract In this paper we present a general covariant–contravariant formalism suitable for describing standing shear Alfve´n waves in a general magnetic field topology. We define a non-orthogonal field-aligned coordinate system based on Euler potentials and compute the metric tensor coefficients for these coordinates numerically. An eigenvalue problem for the system of four ordinary equations is then solved to compute the eigenfrequency and polarization of the standing Alfve´n wave for a given field line. As examples, dipole and Tsyganenko 96 magnetic field models are used. The results are compared to the predictions of older models based on either dipole or axisymmetric magnetic fields. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Shear Alfve´n waves; Alfve´n continuum; Non-orthogonal coordinate systems; Euler potentials
1. Introduction Geomagnetic pulsations of Pc-5 type are commonly observed in ground-based magnetometers and are usually associated with standing shear Alfve´n waves along the closed geomagnetic field lines (e.g., Samson et al., 1971; Samson, 1991). The spectrum of these waves in dipole or straight magnetic fields has been modeled very extensively in the past decades (Walker, 1980; Singer et al., 1981; Southwood and Kivelson, 1986; Chen and Cowley, 1989). Extension of such modeling to non-dipole fields is, however, fairly complicated. Some fundamental difficulties of such an extension are the choice of a suitable coordinate system and the loss of a clear distinction between the poloidal and toroidal modes. One particular non-dipole geometry for which standing Alfve´n waves can be investigated fairly straightforwardly is that of a stretched (or compressed) locally axisymmetric magnetic field. This situation was considered by Rankin et al. (2000) who derived and studied the corresponding equations. The analysis under these conditions is greatly facilitated by the existence of a field-aligned orthogonal *
Corresponding author. Tel.: +1 780 492 8179. E-mail address: [email protected] (K. Kabin).
coordinate system, as also pointed out by Salat and Tataronis (2000). Furthermore, it is shown by Rankin et al. (2000) that the metric coefficients of such a coordinate system may be easily evaluated without any numerical differentiation. A simple model of Singer et al. (1981) is often applied for general geometry of magnetic field lines. This model relies on an estimation of metric coefficients which are assumed to be proportional to the distance between closely spaced magnetic field lines. However, this formulation implicitly assumes the existence of an orthogonal field aligned coordinate system, as discussed, for example by Salat and Tataronis (2000). Unfortunately, such a coordinate system typically does not exist, as discussed in Section 2.1, and therefore a more general analysis is required. More complicated models, including plasma pressure effects, which are in principle suitable for application in nondipole magnetic fields have been developed (Lui and Cheng, 2001; Leonovich, 2001; Proehl et al., 2002; Cheng, 2003; Cheng and Zaharia, 2003). However, to date these models were mostly used in dipole or axisymmetric configurations. Such models also require significant computational resources. Usually, they have to be externally driven to excite a standing Alfve´n wave somewhere in the
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.09.034
R. Rankin et al. / Advances in Space Research 38 (2006) 1720–1729
simulation domain. Some of these models, for example that of Lui and Cheng (2001), also require solution of plasma equilibrium (Grad–Shafranov) equations for a prescribed magnetic field topology to initialize the plasma pressure distribution. The main purpose of this paper is to present the details of an ordinary equation (ODE)-based model for standing shear Alfve´n waves which incorporates the effects of both curvature and torsion of magnetic field lines. We focus our attention on the development of a relatively simple cold plasma model suitable for ‘‘magneto-seismology’’ applications (Waters et al., 1996; Dent et al., 2003; Rankin et al., 2005). Thus, primarily we seek to improve the wellknown model of Singer et al. (1981) by performing a more rigorous analysis of the Alfve´n waves in an arbitrary magnetic field. To this end, we define a non-orthogonal field aligned coordinate system which is similar to that used, for example, in (Cheng, 2003; Salat and Tataronis, 2001), and discuss practical aspects of numerically evaluating the components of the metric tensor for this coordinate system. We then formulate the modified eigenvalue problem which has to be solved in order to compute the frequency and polarization of standing shear Alfve´n waves. This eigenproblem is solved for several specific magnetic field models, such as dipole and Tsyganenko 96 and the results are compared to a very simple but not self-consistent in general approximation which is described in Section 2.3.
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used in space physics. Therefore, below we provide a very brief overview of the material related to this study. A formal and detailed description of these techniques can be found, for example in (Borisenko and Tarapov, 1968; Landau and Lifshitz, 1975; Dhaeseleer et al., 1991). First, we simply assume that we have new coordinates defined as functions of the old Cartesian coordinates: u1(x,y,z), u2(x,y,z), u3(x,y,z). Later (Section 3) we will return to the question of computing these mappings for a given magnetic field. There are two sets of basis vectors which can be computed from the coordinate functions u1(x,y,z), u2(x,y,z), u3(x,y,z). These are the tangent basis vectors (using definition R = xex + yey + zez): ei ¼
oR oui
and cotangent (sometimes also called reciprocal, dual, or gradient) basis vectors: ei ¼ rui . In an orthogonal coordinate system, these two bases coincide and there is no need to distinguish between the two. In general, however, all these vectors point in distinct directions, although they still satisfy the following conditions: e1 e1 ¼ e2 e2 ¼ e3 e3 ¼ 1 and e1 e2 ¼ e1 e3 ¼ e2 e1 ¼ e2 e3 ¼ e3 e1 ¼ e3 e2 ¼ 0.
2. Derivation of the equations 2.1. Covariant–contravariant formalism Given a prescribed magnetic field B(x,y,z) where x,y,z are usual Cartesian coordinates (e.g., GSM), we want to define a new field-aligned coordinate system (u1,u2,u3) in which B has only one component. We choose u3 to be the direction of the magnetic field. For simple magnetic field configurations, such as dipole field, there are wellknown orthogonal field-aligned coordinates. In general, however, such coordinates do not exist, which can be easily proved by contradiction. Indeed, if we assume that such a coordinate system exists and compute $ · B we will immediately see that $ · B does not have a component in the direction of B. Therefore, if the magnetic field configuration is such that there are field aligned currents, there can be no orthogonal field-aligned coordinate system. Thus, zero field-aligned currents provide a necessary condition for the existence of an orthogonal field-aligned coordinate system. A detailed discussion of the sufficient conditions for the existence of orthogonal field-aligned coordinate system is given by Salat and Tataronis (2000). It is obvious, however, that for typical magnetospheric configurations we have to deal with general non-orthogonal coordinate systems in which the covariant–contravariant formalism is required. The corresponding mathematical apparatus is well-known, for example, in general relativity (e.g., Landau and Lifshitz, 1975), but is only occasionally
Any physical vector can be represented using either its covariant components A = A1e1 + A2e2 + A3e3 or contravariant components A = A1e1 + A2e2 + A3e3. If one set of components is known and the other is desired, they can be converted using metric tensor: Ai = gijAj and Ai = gijAj (using summation notation). The metric tensor is usually defined as (Dhaeseleer et al., 1991) gij = ei Æ ej and gij = ei Æ ej, however we find an alternative definition in terms of the Jacobian more convenient for our purposes. Matrixes gij and gij are symmetric and inverse of each other. If J is the Jacobian of the transformation from (x,y,z) to (u1,u2,u3): J¼
oðu1 ; u2 ; u3 Þ oðx; y; zÞ
then gij ¼ JJ T ;
T
gij ¼ ðJ 1 Þ J 1 .
Given the mapping ui(x,y,z) from the Cartesian coordinates to the new system, the derivatives appearing in the definition of this Jacobian are easy to compute numerically. Once J (which is a 3 · 3 matrix) is known it can be easily inverted numerically to obtain J1. In contrast, a direct numerical evaluation of the elements of the J1 matrix may be quite difficult, if mappings x(u1,u2,u3), etc. are not known, as in the case we will consider. A practical way to define the coordinate system and its metric is discussed in Section 3.
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ð1Þ
ant component: B = B3e3. Then, B Æ e1 = B Æ e2 = 0 because e1 and e2 are perpendicular to e3. In general, however, all three covariant components of B will be non-zero. B3 is relatedpffiffiffiffiffiffiffiffiffiffi to the magnitude of B by the equation pffiffiffiffiffiffi jBj ¼ B3 B3 ¼ g33 B3 . We introduce the electric field as dE = dV · B. Because of this definition dE has only two covariant components: dE3 = 0. Eq. (2) can be written in components as
ð2Þ
g11
2.2. Equations for shear Alfve´n waves The simplest physical approximation which can be used to describe Alfve´n waves is that of a stationary cold plasma. In this approximation we need to consider the following linearized equations: odV ¼ ðr BÞ dB þ ðr dBÞ B; ot odB ¼ r ðdV BÞ. ot
q
Here, B is the prescribed background magnetic field, dB is the magnetic field perturbation, associated with the wave, and dV is the corresponding perturbation of the plasma velocity (it is assumed that the background plasma is stationary). These equations describe shear Alfve´n waves as well as compressional Alfve´n waves and, therefore, are suitable for modeling field line resonances (Samson, 1991). Note, that these equations do not describe the slow mode which is included in some of the more complicated models of ultra-low frequency waves in the magnetosphere (e.g., Lui and Cheng, 2001; Proehl et al., 2002). The expression for the $· operator in general curvilinear coordinates is (Borisenko and Tarapov, 1968; Dhaeseleer et al., 1991): 1 oAk oAj ðr AÞi ¼ pffiffiffi . g oxj oxk Here, g is the determinant of the metric tensor gij. Note, that many textbooks give an expression for the $· operator which involves covariant, rather than partial, derivatives. The two expressions, however, are the same because of the symmetry of Chistoffel symbols (e.g., Dhaeseleer et al., 1991). It is also important to note that $· in general curvilinear coordinates is defined in such a way that covariant components of the vector A are used to produce the contravariant components of the vector $ · A. There is no similar simple formula which takes contravariant components of A and returns covariant components of $ · A. If any such conversion is needed, it should be performed separately using the appropriate metric tensor. We will also need an expression for a vector product. Two formulas can be used (Dhaeseleer et al., 1991): pffiffiffi ðA CÞk ¼ eijk gAi C j and eijk k ðA CÞ ¼ pffiffiffi Ai C j . g Here, eijk is the usual antisymmetric permutation tensor. In the first case we use contravariant components of A and C to get covariant components of A · C and in the second case we use covariant components of A and C to get contravariant components of A · C. If a coordinate system is field aligned then there is no component of B perpendicular to the tangent vector e3. This means, that the background field has only one contravari-
odB1 odB2 odB3 1 odE2 þ g12 þ g13 ¼ pffiffiffi ; g ou3 ot ot ot odB1 odB2 odB3 1 odE1 g21 þ g22 þ g23 ¼ pffiffiffi ; ot ot ot g ou3 odB1 odB2 odB3 1 odE2 odE1 þ g32 þ g33 ¼ pffiffiffi g31 . ot ot ot ou2 g ou1
Now, we introduce a notation J = $ · B and multiply Eq. (1) by ·B on the right. Then, we have o g odB2 2 odB3 q dE ¼ ðJ dBÞ B e1 p33ffiffiffi ðB3 Þ ot ou2 ou3 g g33 odB3 1 2 odB1 2 e2 pffiffiffi ðB3 Þ 1 þ e3 pffiffiffi ðB3 Þ ou3 ou g g odB1 odB3 odB3 odB2 g23 þ g . ð3Þ 13 ou3 ou1 ou2 ou3 In this expression, we can introduce Alfve´n speed as 2 v2A ¼ g33 ðB3 Þ =q. Finally, assuming that the shear Alfve´n wave is localized on a particular field line, from Eqs. (2) and (3) we can write the following system of coupled ODEs: 1 odB2 c 11 ½ðJ dBÞ B1 ¼ 2 ðg xdE1 þ g12 xdE2 Þ ; pffiffiffi 2 3 g ou vA jBj
ð4Þ
1 odB1 c ½ðJ dBÞ B2 ¼ 2 ðg21 xdE1 þ g22 xdE2 Þ þ ; ð5Þ pffiffiffi 3 vA g ou jBj2 1 odE1 1 ¼ ðg12 xdB1 þ g22 xdB2 Þ; pffiffiffi c g ou3 1 odE2 1 11 ¼ ðg xdB1 þ g12 xdB2 Þ. pffiffiffi c g ou3
ð6Þ ð7Þ
Detailed expressions for (J·dB) · B term in coordinate notations are quite lengthy and are omitted here for the sake of brevity. Eqs. (4)–(7) are the main equations of our model, which describes shear Alfve´n waves in a general magnetic field topology. We have an eigenvalue problem for this system: we have to find a non-trivial solution satisfying dE1,dE2 = 0 at both ends of the field line. This formulation is somewhat different from the classical Strum–Liouville problem because we have four first-order equations and two parameters (frequency x and the polarization of the wave at the northern ionosphere) instead of a single second-order equation with a single parameter. The most straightforward way to solve this eigenvalue problem is by using the shooting method (Press et al., 1992, Chapter 17.1). As a part of the shooting method we have to find a root of a system of two equations
R. Rankin et al. / Advances in Space Research 38 (2006) 1720–1729
for the frequency and polarization which is easiest to accomplish by using a multidimensional Newtons method (Press et al., 1992, Chapter 9.7). Unfortunately, Newtons method may sometimes converge to a random non-fundamental mode (or even fail to converge to a meaningful root at all). Thus, the robustness of our model is still somewhat of an issue. As an initial guess for Newtons method we use the eigenvalue predicted by the WKB approximation. 2.3. Equations in orthogonal coordinates In the case of orthogonal coordinate systems, which exist for situations such as axisymmetric fields, the metric tensor becomes diagonal (g12 = g21 = 0) and the equations are conventionally written using scale factors h1, h2, h3. These 22 are related to the metric tensor by g11 ¼ h2 ¼ h2 1 ; g 2 ; pffiffiffi g ¼ h1 h2 h3 . Then, the coupled equations split into two sets of independent equations which can be written as h1 o h2 odE1 x2 ¼ 2 dE1 ; 3 3 h2 h3 ou h1 h3 ou vA h2 o h1 odE2 x2 ¼ 2 dE2 . 3 3 h1 h3 ou h2 h3 ou vA Recognizing that 1 o o ¼ ; h3 ou3 os where s is the distance along the field line, these equations are cast into a form in which they are commonly solved (e.g., Rankin et al., 2000). The relation between the components of electric and magnetic fields is given by Faradays law (e.g., Walker, 1980) as ih2 odE1 dB2 ¼ ; xh1 os ih1 odE2 dB1 ¼ ; xh2 os which allows the equivalent eigenmode equations to be written for the components of magnetic field, rather than electric field. The expressions for the scale factors in a dipole field are well-known (e.g., Walker, 1980) and for reference are given in Appendix A. If the background magnetic field is locally axisymmetric, then there exists an orthogonal coordinate system which can be used for the study of Alfve´n waves (Salat and Tataronis, 2000). In this coordinate system the scale factor associated with the azimuthal coordinate is h1 r where r is the distance from the symmetry axis (Rankin et al., 2000). Then, using the relations between components of the metric tensor given in Appendix B we can write the equations for the two modes as d2 dE1 ddE1 d x2 lnðBðsÞr2 ðsÞÞ þ 2 dE1 ¼ 0; þ 2 ds ds ds vA
ð8Þ
d2 dE2 ddE2 d x2 2 lnðBðsÞr ðsÞÞ þ dE2 ¼ 0. ds2 ds ds v2A
ð9Þ
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Here, Eq. (8) describes what we refer in this paper as mode 1 (poloidal electric field and toroidal perturbation magnetic field); while Eq. (9) describes mode 2 (toroidal electric field and poloidal perturbation magnetic field). These equations are fundamentally the same as those of Singer et al. (1981), however they have an advantage of using the information only along a single magnetic field line. This eliminates numerical uncertainties associated with computing the distance between two close magnetic field lines required by Singer et al. (1981) approach. By derivation, Eqs. (8) and (9) hold only in (locally) axisymmetric magnetic fields, however, at least formally, they may be used for arbitrary magnetic field models. They have an advantage of being computationally very simple while capturing the two most important aspects determining the eigenfrequency of a field line: the distribution of magnetic field intensity and plasma density along the field line. As shown in Section 5 this simple approximation often provides a very good estimate of the eigenfrequency of the field line. It should be noted, however, that Eqs. (8) and (9) assume that there are two distinct polarizations of the standing Alfve´n waves along a field line. Thus, while providing information about the frequency spectrum of the Alfve´n waves, Eqs. (8) and (9) cannot provide any information about the polarizations of the two modes. These equations also completely miss all the effects associated with the coupling between the two modes. 3. Defining a coordinate system based on Euler potentials All metric coefficients in Eqs. (4)–(7) have to be evaluated numerically. A sensible way to introduce a fieldaligned coordinate system is to use Euler potentials a and b (e.g., Stern, 1970; Dhaeseleer et al., 1991) together with a coordinate along the magnetic field line. Note, that for historic reasons Euler potentials are sometimes referred to by various authors as flux coordinates or Clebsch coordinates. The magnetic field is calculated from the Euler potentials by B ¼ B0 ra rb. Here, B0 is in principle an arbitrary normalization constant. In our work we use the dipole strength at the magnetic equator at 1 Earth radius as a convenient choice for B0. Note that, unfortunately, there is no superposition for Euler potentials in general. If a1,b1 describe magnetic field B1 and a2,b2 magnetic field B2, then there is no simple way to compute the Euler potentials for the magnetic field B1 + B2. By definition, a and b are constant along the field line. Therefore, given a magnetic field model we can trace a field line from any given point to the northern ionosphere, where the Euler potentials are evaluated. Then, these would be the values of a and b for the whole field line, including the point in question. For a dipole field, the Euler potentials can be chosen to be
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sin2 h ; b ¼ /. R Here, h is the co-latitude (h = 0 at the northern pole, h = p/ 2 at the equator, and h = p at the southern pole) and R is the distance to the center of the Earth. The magnetic field is, of course, not exactly dipolar at the Earths surface. There is, however, a simple way to compute Euler potentials numerically on a given surface using only the magnetic field intensity on this surface, as described by Stern (1970). For a sphere of radius R0 we can choose the azimuthal angle / as one of the potentials (b) and another one as Z h BR a ¼ R20 sin h0 dh0 ; 0 B0 a¼
where BR is the component of the magnetic field normal to the sphere. If we select the polar axis of the spherical coordinates (R,h,/) to coincide with the dipole axis, then for pure dipole field we recover the standard dipole coordinates. Although this is obviously a convenient choice, it is not the only one, as there are infinitely many perfectly acceptable choices for Euler potentials for a prescribed magnetic field. It is tempting to chose the third coordinate as just the distance along the field line to the northern ionosphere. This is not a very good choice, though, because such a coordinate system would not be orthogonal even in a dipole case. Instead, we should choose Z l ¼ cos h0 jB=B0 j ds; which is, in fact, a Grad–Boozer coordinate (Dhaeseleer et al., 1991). Here, h0 is the co-latitude of the point where the field line intersects the northern ionosphere and the integral is taken from the point in question to the northern ionosphere along the field line. Defining l in this way for a dipolar field coincides with the usual dipolar coordinate ðl ¼ cos h=R2 Þ along the field line. For a pure dipole field, the coordinate system (a,/,l) is orthogonal. In practice, we evaluate the metric coefficients at a few hundred points uniformly distributed in the arc length along the field line and use cubic spline interpolation to obtain their values anywhere along the field line, as needed by the numerical ODE solver. 4. Density distribution In order to proceed with the solution of the eigenvalue problem we have to prescribe the density distribution along the field line. Eventually, the density distribution may be provided by an MHD model together with the self-consistent magnetic fields. At the present stage, however, our main goal is to develop a solver suitable for usage with an empirical Tsyganenko model. In the present work, following, for example, Waters et al. (1996) and Dent et al. (2003), we assume a power-law density profile 4
q ¼ qeq ðRE =RÞ ;
where qeq is an adjustable equatorial density, set to 7 amu/ cm3 in most of our present calculations. The actual value of the equatorial density is not important for the model validation purposes since in our model the eigenfrequency sim1=2 ply scales as qeq , same as in Singer et al. (1981) approach. 5. Examples Now we apply the derived formalism to some practical situations. We use the Tsyganenko 96 (Tsyganenko and Stern, 1996; Tsyganenko and Peredo, 1994; Tsyganenko, 1995) magnetic field model as the background field. For test purposes, we set the date to the 70th day (March 10) of year 2000, UT = 18:00:00. Note, that we use this date purely for demonstration purposes, and not as an example of a specific event. For this date the dipole tilt in the GSM coordinates is 6.28 and the equatorial dipole strength is B0 = 30111.7 nT. First, for comparison with classical results, we use a simple tilted dipole field instead of the full Tsyganenko 96 model. We trace a field line from geographical coordinates 61.11N, 265.95E, which is the location of Eskimo Point magnetometer station. Fig. 1 shows the three projections of the field line traced from the above locations. The field line length is 25.33RE. Next we compute components of the metric tensor for this field line using the techniques presented above. Fig. 2 shows the components of the metric tensor as a function of the coordinate l, along the field line. The results of numerical calculation are displayed by circles, while the solid lines represent the analytical expressions, given in Appendix A. There is no solid line in the panel showing g21 = g12 because analytically this coefficient should be zero. Numerically, it is not zero exactly, but is several orders of magnitude smaller than either g11 or g22. Once the metric coefficients along the field line are computed the eigenvalue problem can be solved to give the frequency of the standing Alfve´n wave as well as the distributions of the electric and magnetic fields along the field line. Fig. 3 shows the eigenfunctions for the two modes in this case. The axisymmetric model described in Section 2.3 in this situation becomes exact. The periods of mode 1 and mode 2 computed using the simple code, designed specifically for a dipole magnetic field, in this case are 868.75 and 1200.64 s, respectively. For comparison purposes, we quote the computed periods to second decimal point, although this level of accuracy is generally not warranted in any practical situation. The period prediction from the WKB approximation (which is the same for either mode) is 680.94 s. Calculation with our new model gives for the first mode a period of 868.72 s and a polarization angle n = 0.257. Here, we define the polarization angle as tan n = dB2/dB1 at the northern ionosphere. For the second mode, we get a period of 1200.73 s and polarization of 89.996. Obviously, we have a very impressive agreement (the relative error is about 105) between the two methods of calculation in this case. The purpose of this comparison is the validation of the
R. Rankin et al. / Advances in Space Research 38 (2006) 1720–1729 4
2
2
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–2
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–4
–4
Z
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–6 –3
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X
0
Y 0
–0.5
Y
–1 –1.5 –2 –2.5 –3
0
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4
6
8
10
X
g11
Fig. 1. Projections of the dipole field line on the three GSM coordinate planes.
2
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10 –0.8
–0.6
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g21
5
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–0.8
G
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–0.8
Fig. 2. Components of the metric tensor along the field line. Continuous lines are analytical expressions, circles are the numerical results.
fairly involved numerics associated with defining the coordinate system and its metric coefficients in our new model. This test firmly establishes that our model handles the tilted dipole case correctly, which is the most non-trivial
case when a comparison with rigorously established results is possible. Next, we consider a non-dipolar case. For this example, we used the full Tsyganenko 96 routines for the same date
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R. Rankin et al. / Advances in Space Research 38 (2006) 1720–1729 E and B for mode 1 (poloidal E and toroidal B)
1
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E
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E and B for mode 2 (toroidal E and poloidal B)
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1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 normalized distance along the field line
1
0.5
1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 normalized distance along the field line
1 0
1
Fig. 3. Left panel. Electric and magnetic fields for mode 1. Right panel. Electric and magnetic fields for mode 2.
The distribution of the metric coefficients in this case are shown in Fig. 5. In this figure we have also shown the values given by the formulas of Appendix A, although this field line is not dipole any more and so these formulas, strictly speaking, do not apply. Coefficient g12 changes sign, so we have plotted its absolute value on the logarithmic scale. The WKB period for this field line is 858 s; the period for mode 1 given by the approximate code described in Section 2.3 is 1070 s and for mode 2 the period is 1240 s. Our full model gives two modes with periods 1060 and 1304 s
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2
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–2
–2
–4
–4
–6
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–6
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–3
–2
–1 Y
0 –0.5 –1 –1.5 Y
Z
as before with the following parameters: Pdyn = 3 nPa, IMF By = 3 nT, Bz = 5 nT, and DST = 20. Note that these parameters were selected for demonstration purposes only and do not specifically correspond to any particular measurements. However, generally these values of the parameters are fairly representative. The resulting field line for these conditions traced from the same location as before is shown in Fig. 4. The length of this field line is 28.37RE. It happens, that this particular field line passes very close to the magnetopause.
–2 –2.5 –3 –3.5
0
2
4
6
8
10
X Fig. 4. Projections of the Tsyganenko field line on the three GSM coordinate planes.
0
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R. Rankin et al. / Advances in Space Research 38 (2006) 1720–1729
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0.4 0.5 0.6 0.7 normalized distance along the field line
Fig. 5. Components of the metric tensor along the field line. Continuous lines are analytical expressions, circles are the numerical results.
and polarizations of 95 and 79, respectively. We can loosely identify the first of these as corresponding to mode 1 and the second to mode 2. Fig. 6 is similar to Fig. 3 for this case. Note that in this case, the eigenfunctions are no longer symmetric. However, even in this case the eigenfunctions computed with either method look very similar to each other. Fig. 7 shows the change in the polarization angle tan n(s) = dB2(s)/dB1(s) as a function of the distance along the field line, s. From this figure, it is clear that a clean
1
E and B for mode 1 (poloidal E and toroidal B)
1 0.8
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E and B for mode 2 (toroidal E and poloidal B)
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E
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B
B
classification of the solution as one of the two modes is not possible in general. The distinction between the two modes with different polarizations is only possibly near the ends of the field line, where the magnetic field is more or less dipolar and the deviation of our coordinate system from an orthogonal one is not too large. However, near the center of the field line the polarization of the modes change very significantly. To some extent this is related to our definition of the polarization, because dB1 has a node somewhere in the middle of the field line. At this
–0.5 –1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 normalized distance along the field line
1
–0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 normalized distance along the field line
1
–1 0
Fig. 6. Left panel. Electric and magnetic fields for mode 1. Right panel. Electric and magnetic fields for mode 2.
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ξ
50
0
–50
–100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
90 80 70 ξ
60 50 40 30 20
normalized distance along the field line
Fig. 7. Polarization of the two modes as a function of the distance along the field line.
point tan n becomes infinite, because dB2, in general has a node at a somewhat different location.
partial differential equations as well as with the data for specific events.
6. Conclusions
Acknowledgments
In this paper we have presented a model describing standing shear Alfve´n waves suitable for application in general three-dimensional magnetic fields. Our model is based on an eigenvalue problem for four ordinary differential equations and, therefore, it is considerably less computationally intensive than some other models which require solution of partial differential equations. We also described in detail the procedure used for defining a field-aligned, in general non-orthogonal, coordinate system required for this study. This coordinate system is based on Euler potentials, and a numerical procedure for computing these potentials is laid out. We also describe a stable numerical method for evaluating the metric tensor for this coordinate system. We test our new model for the case of dipole magnetic field and find excellent agreement with the well-known results for this situation. Then we proceed to a more complicated Tsyganenko 96 magnetic field, which in general does not possess any symmetries. For the particular field lines that we considered, we found that the frequencies evaluated with our new model were in good agreement with those computed with an approximate model derived for axisymmetric fields. Future work will include more detailed comparisons of our model with fully three-dimensional models based on
This work is supported by the Canadian Space Agency and by Natural Sciences and Engineering Research Council of Canada. We also acknowledge the use of WestGrid computational resources. Appendix A. Metric coefficients for dipole field For a pure dipole field the metric coefficients can be computed analytically as follows (e.g., Walker, 1980) 1 sin2 hð1 þ 3cos2 hÞ ¼ ; R4 h21 1 1 g22 ¼ 2 ¼ 2 2 ; h2 R sin h 1 1 þ 3cos2 h g33 ¼ 2 ¼ ; R6 h3
g11 ¼
g12 ¼ g21 ¼ g13 ¼ g31 ¼ g23 ¼ g32 ¼ 0; pffiffiffi g ¼ h1 h2 h3 ¼
R6 . 1 þ 3cos2 h
Here, h is the polar angle (co-latitude) determined with respect to the dipole axis and R is the distance from the center of the Earth. For a dipole field line R = L sin2h.
R. Rankin et al. / Advances in Space Research 38 (2006) 1720–1729
Appendix B. Some relations for the coefficients of the metric tensor There are several useful relations between the components of the metric tensor. Using the definition of cotangent basis vectors, it can be easily seen that B ¼ B0 e 1 e 2 . From this equation using the definition gij = ei Æ ej, it can be shown that (Salat and Tataronis, 2001): 2 B ¼ g11 g22 ðg12 Þ2 . ðB:1Þ B 0 Another constraint may be obtained from the fact that $ Æ B = 0. Then, o pffiffiffi 3 ð gB Þ ¼ 0 ou3 or pffiffiffi gjBj pffiffiffiffiffiffi ¼ const g33
ðB:2Þ
along the field line. For orthogonal coordinates, relations (B.1) and (B.2) become the same and coincide with the one used in (Rankin et al., 2000). References Borisenko, A.I., Tarapov, I.E. Vector and Tensor Analysis with Applications. Dover Publication, New York, 1968. Chen, L., Cowley, S.C. On field line resonances of hydromagnetic Alfve´n waves in dipole magnetic field. Geophys. Res. Lett. 16, 895–897, 1989. Cheng, C.Z. MHD field line resonances and global modes in threedimensional magnetic fields. J. Geophys. Res. 108, doi:10.1029/ 2002JA009470, 2003. Cheng, C.Z., Zaharia, S. Field line resonances in quiet and disturbed time three-dimensional magnetospheres. J. Geophys. Res. 108, doi:10.1029/ 2002JA009471, 2003. Dent, Z.C., Mann, I.R., Menk, F.W., Goldstein, J., Wilford, C.R., Clilverd, M.A., Ozeke, L.G. A coordinated ground-based and image satellite study of quiet-time plasmaspheric density profiles. Geophys. Res. Lett. 30, doi:10.1029/2003GL016946, 2003. Dhaeseleer, W.D., Hitchon, W.N.G., Callen, J.D., Shohet, J.L. Flux Coordinates and Magnetic Field Structure. Springer-Verlag, Berlin, Germany, 1991.
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Landau, L.D., Lifshitz, E.M. The Classical Theory of Fields. Pergamon Press, Oxford, UK, 1975 (Chapter 10). Leonovich, A.S. A theory of field line resonance in a dipole-like axisymmetric magnetosphere. J. Geophys. Res. 106, 25803–25812, 2001. Lui, A.T.Y., Cheng, C.Z. Resonance frequency of stretched magnetic field lines based on a self-consistent equilibrium magnetosphere model. J. Geophys. Res. 106, 25793–25802, 2001. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T. Numerical Recipes in Fortran. Cambridge University Press, Cambridge, UK, 1992. Proehl, J.A., Lotko, W., Kouznetsov, I., Geimer, S.D. Ultralow-frequency magnetohydrodynamics in boundary-constrained geomagnetic flux coordinates. J. Geophys. Res. 107, doi:10.1029/2001JA000135, 2002. Rankin, R., Fenrich, F., Tikhonchuk, V.T. Shear Alfve´n waves on stretched magnetic field lines near midnight in Earths magnetosphere. Geophys. Res. Lett. 27, 3265–3268, 2000. Rankin, R., Kabin, K., Lu, J.Y., Mann, I.R., Marchand, R., Rae, I.J., Tikhonchuk, V.T., Donovan, E.F. Magnetospheric field-line resonances: Ground-based observations and modeling. J. Geophys. Res. 110, doi:10.1029/2004JA010919, 2005. Salat, A., Tataronis, J.A. Conditions for existence of orthogonal coordinate systems oriented by magnetic field lines. J. Geophys. Res. 105, 13055–13062, 2000. Salat, A., Tataronis, J.A. Shear Alfve´n mode resonances in nonaxisymmetric toroidal low-pressure plasmas. I. Mode equations in arbitrary geometry. Phys. Plasmas 8, 1200–1206, 2001. Samson, J. Geomagnetic pulsations and plasma waves in the Earths magnetosphere. in: Jacobs, J.A. (Ed.), Geomagnetism, vol. 4. Academic Press, New York, pp. 481–592, 1991. Samson, J., Jacobs, J.A., Rostoker, G. Latitude-dependent characteristics of long-period geomagnetic micropulsations. J. Geophys. Res. 76, 3675–3686, 1971. Singer, H.J., Southwood, D.J., Walker, R.J., Kivelson, M.G. Alfve´n-wave resonances in a realistic magnetospheric magnetic field geometry. J. Geophys. Res. 86, 4589–4596, 1981. Southwood, D.J., Kivelson, M.G. The effect of parallel inhomogeneity on magnetospheric wave coupling. J. Geophys. Res. 91, 6871–6876, 1986. Stern, D.P. Euler potentials. Am. J. Phys. 35, 494–501, 1970. Tsyganenko, N.A. Modeling the Earths magnetospheric field confined within a realistic magnetopause. J. Geophys. Res. 100, 5599–5612, 1995. Tsyganenko, N.A., Peredo, M. Analytical models of the magnetic field of disk-shaped current sheets. J. Geophys. Res. 99, 199–205, 1994. Tsyganenko, N.A., Stern, D.P. Modeling the global magnetic field of the large-scale Birkeland current systems. J. Geophys. Res. 101, 27187– 27198, 1996. Walker, A.D.M. Modelling of Pc5 pulsation structure in the magnetosphere. Planet. Space Sci., 213–223, 1980. Waters, C.L., Samson, J.C., Donovan, E.F. Variation of plasmatrough density derived from magnetospheric field line resonances. J. Geophys. Res. 101, 24737–24745, 1996.
Alfve´nic field line resonances in arbitrary magnetic field topology R. Rankin, K. Kabin *, R. Marchand Department of Physics, University of Alberta, Edmonton, Canada T6G 2J1 Received 28 September 2004; received in revised form 22 September 2005; accepted 23 September 2005
Abstract In this paper we present a general covariant–contravariant formalism suitable for describing standing shear Alfve´n waves in a general magnetic field topology. We define a non-orthogonal field-aligned coordinate system based on Euler potentials and compute the metric tensor coefficients for these coordinates numerically. An eigenvalue problem for the system of four ordinary equations is then solved to compute the eigenfrequency and polarization of the standing Alfve´n wave for a given field line. As examples, dipole and Tsyganenko 96 magnetic field models are used. The results are compared to the predictions of older models based on either dipole or axisymmetric magnetic fields. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Shear Alfve´n waves; Alfve´n continuum; Non-orthogonal coordinate systems; Euler potentials
1. Introduction Geomagnetic pulsations of Pc-5 type are commonly observed in ground-based magnetometers and are usually associated with standing shear Alfve´n waves along the closed geomagnetic field lines (e.g., Samson et al., 1971; Samson, 1991). The spectrum of these waves in dipole or straight magnetic fields has been modeled very extensively in the past decades (Walker, 1980; Singer et al., 1981; Southwood and Kivelson, 1986; Chen and Cowley, 1989). Extension of such modeling to non-dipole fields is, however, fairly complicated. Some fundamental difficulties of such an extension are the choice of a suitable coordinate system and the loss of a clear distinction between the poloidal and toroidal modes. One particular non-dipole geometry for which standing Alfve´n waves can be investigated fairly straightforwardly is that of a stretched (or compressed) locally axisymmetric magnetic field. This situation was considered by Rankin et al. (2000) who derived and studied the corresponding equations. The analysis under these conditions is greatly facilitated by the existence of a field-aligned orthogonal *
Corresponding author. Tel.: +1 780 492 8179. E-mail address: [email protected] (K. Kabin).
coordinate system, as also pointed out by Salat and Tataronis (2000). Furthermore, it is shown by Rankin et al. (2000) that the metric coefficients of such a coordinate system may be easily evaluated without any numerical differentiation. A simple model of Singer et al. (1981) is often applied for general geometry of magnetic field lines. This model relies on an estimation of metric coefficients which are assumed to be proportional to the distance between closely spaced magnetic field lines. However, this formulation implicitly assumes the existence of an orthogonal field aligned coordinate system, as discussed, for example by Salat and Tataronis (2000). Unfortunately, such a coordinate system typically does not exist, as discussed in Section 2.1, and therefore a more general analysis is required. More complicated models, including plasma pressure effects, which are in principle suitable for application in nondipole magnetic fields have been developed (Lui and Cheng, 2001; Leonovich, 2001; Proehl et al., 2002; Cheng, 2003; Cheng and Zaharia, 2003). However, to date these models were mostly used in dipole or axisymmetric configurations. Such models also require significant computational resources. Usually, they have to be externally driven to excite a standing Alfve´n wave somewhere in the
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.09.034
R. Rankin et al. / Advances in Space Research 38 (2006) 1720–1729
simulation domain. Some of these models, for example that of Lui and Cheng (2001), also require solution of plasma equilibrium (Grad–Shafranov) equations for a prescribed magnetic field topology to initialize the plasma pressure distribution. The main purpose of this paper is to present the details of an ordinary equation (ODE)-based model for standing shear Alfve´n waves which incorporates the effects of both curvature and torsion of magnetic field lines. We focus our attention on the development of a relatively simple cold plasma model suitable for ‘‘magneto-seismology’’ applications (Waters et al., 1996; Dent et al., 2003; Rankin et al., 2005). Thus, primarily we seek to improve the wellknown model of Singer et al. (1981) by performing a more rigorous analysis of the Alfve´n waves in an arbitrary magnetic field. To this end, we define a non-orthogonal field aligned coordinate system which is similar to that used, for example, in (Cheng, 2003; Salat and Tataronis, 2001), and discuss practical aspects of numerically evaluating the components of the metric tensor for this coordinate system. We then formulate the modified eigenvalue problem which has to be solved in order to compute the frequency and polarization of standing shear Alfve´n waves. This eigenproblem is solved for several specific magnetic field models, such as dipole and Tsyganenko 96 and the results are compared to a very simple but not self-consistent in general approximation which is described in Section 2.3.
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used in space physics. Therefore, below we provide a very brief overview of the material related to this study. A formal and detailed description of these techniques can be found, for example in (Borisenko and Tarapov, 1968; Landau and Lifshitz, 1975; Dhaeseleer et al., 1991). First, we simply assume that we have new coordinates defined as functions of the old Cartesian coordinates: u1(x,y,z), u2(x,y,z), u3(x,y,z). Later (Section 3) we will return to the question of computing these mappings for a given magnetic field. There are two sets of basis vectors which can be computed from the coordinate functions u1(x,y,z), u2(x,y,z), u3(x,y,z). These are the tangent basis vectors (using definition R = xex + yey + zez): ei ¼
oR oui
and cotangent (sometimes also called reciprocal, dual, or gradient) basis vectors: ei ¼ rui . In an orthogonal coordinate system, these two bases coincide and there is no need to distinguish between the two. In general, however, all these vectors point in distinct directions, although they still satisfy the following conditions: e1 e1 ¼ e2 e2 ¼ e3 e3 ¼ 1 and e1 e2 ¼ e1 e3 ¼ e2 e1 ¼ e2 e3 ¼ e3 e1 ¼ e3 e2 ¼ 0.
2. Derivation of the equations 2.1. Covariant–contravariant formalism Given a prescribed magnetic field B(x,y,z) where x,y,z are usual Cartesian coordinates (e.g., GSM), we want to define a new field-aligned coordinate system (u1,u2,u3) in which B has only one component. We choose u3 to be the direction of the magnetic field. For simple magnetic field configurations, such as dipole field, there are wellknown orthogonal field-aligned coordinates. In general, however, such coordinates do not exist, which can be easily proved by contradiction. Indeed, if we assume that such a coordinate system exists and compute $ · B we will immediately see that $ · B does not have a component in the direction of B. Therefore, if the magnetic field configuration is such that there are field aligned currents, there can be no orthogonal field-aligned coordinate system. Thus, zero field-aligned currents provide a necessary condition for the existence of an orthogonal field-aligned coordinate system. A detailed discussion of the sufficient conditions for the existence of orthogonal field-aligned coordinate system is given by Salat and Tataronis (2000). It is obvious, however, that for typical magnetospheric configurations we have to deal with general non-orthogonal coordinate systems in which the covariant–contravariant formalism is required. The corresponding mathematical apparatus is well-known, for example, in general relativity (e.g., Landau and Lifshitz, 1975), but is only occasionally
Any physical vector can be represented using either its covariant components A = A1e1 + A2e2 + A3e3 or contravariant components A = A1e1 + A2e2 + A3e3. If one set of components is known and the other is desired, they can be converted using metric tensor: Ai = gijAj and Ai = gijAj (using summation notation). The metric tensor is usually defined as (Dhaeseleer et al., 1991) gij = ei Æ ej and gij = ei Æ ej, however we find an alternative definition in terms of the Jacobian more convenient for our purposes. Matrixes gij and gij are symmetric and inverse of each other. If J is the Jacobian of the transformation from (x,y,z) to (u1,u2,u3): J¼
oðu1 ; u2 ; u3 Þ oðx; y; zÞ
then gij ¼ JJ T ;
T
gij ¼ ðJ 1 Þ J 1 .
Given the mapping ui(x,y,z) from the Cartesian coordinates to the new system, the derivatives appearing in the definition of this Jacobian are easy to compute numerically. Once J (which is a 3 · 3 matrix) is known it can be easily inverted numerically to obtain J1. In contrast, a direct numerical evaluation of the elements of the J1 matrix may be quite difficult, if mappings x(u1,u2,u3), etc. are not known, as in the case we will consider. A practical way to define the coordinate system and its metric is discussed in Section 3.
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ð1Þ
ant component: B = B3e3. Then, B Æ e1 = B Æ e2 = 0 because e1 and e2 are perpendicular to e3. In general, however, all three covariant components of B will be non-zero. B3 is relatedpffiffiffiffiffiffiffiffiffiffi to the magnitude of B by the equation pffiffiffiffiffiffi jBj ¼ B3 B3 ¼ g33 B3 . We introduce the electric field as dE = dV · B. Because of this definition dE has only two covariant components: dE3 = 0. Eq. (2) can be written in components as
ð2Þ
g11
2.2. Equations for shear Alfve´n waves The simplest physical approximation which can be used to describe Alfve´n waves is that of a stationary cold plasma. In this approximation we need to consider the following linearized equations: odV ¼ ðr BÞ dB þ ðr dBÞ B; ot odB ¼ r ðdV BÞ. ot
q
Here, B is the prescribed background magnetic field, dB is the magnetic field perturbation, associated with the wave, and dV is the corresponding perturbation of the plasma velocity (it is assumed that the background plasma is stationary). These equations describe shear Alfve´n waves as well as compressional Alfve´n waves and, therefore, are suitable for modeling field line resonances (Samson, 1991). Note, that these equations do not describe the slow mode which is included in some of the more complicated models of ultra-low frequency waves in the magnetosphere (e.g., Lui and Cheng, 2001; Proehl et al., 2002). The expression for the $· operator in general curvilinear coordinates is (Borisenko and Tarapov, 1968; Dhaeseleer et al., 1991): 1 oAk oAj ðr AÞi ¼ pffiffiffi . g oxj oxk Here, g is the determinant of the metric tensor gij. Note, that many textbooks give an expression for the $· operator which involves covariant, rather than partial, derivatives. The two expressions, however, are the same because of the symmetry of Chistoffel symbols (e.g., Dhaeseleer et al., 1991). It is also important to note that $· in general curvilinear coordinates is defined in such a way that covariant components of the vector A are used to produce the contravariant components of the vector $ · A. There is no similar simple formula which takes contravariant components of A and returns covariant components of $ · A. If any such conversion is needed, it should be performed separately using the appropriate metric tensor. We will also need an expression for a vector product. Two formulas can be used (Dhaeseleer et al., 1991): pffiffiffi ðA CÞk ¼ eijk gAi C j and eijk k ðA CÞ ¼ pffiffiffi Ai C j . g Here, eijk is the usual antisymmetric permutation tensor. In the first case we use contravariant components of A and C to get covariant components of A · C and in the second case we use covariant components of A and C to get contravariant components of A · C. If a coordinate system is field aligned then there is no component of B perpendicular to the tangent vector e3. This means, that the background field has only one contravari-
odB1 odB2 odB3 1 odE2 þ g12 þ g13 ¼ pffiffiffi ; g ou3 ot ot ot odB1 odB2 odB3 1 odE1 g21 þ g22 þ g23 ¼ pffiffiffi ; ot ot ot g ou3 odB1 odB2 odB3 1 odE2 odE1 þ g32 þ g33 ¼ pffiffiffi g31 . ot ot ot ou2 g ou1
Now, we introduce a notation J = $ · B and multiply Eq. (1) by ·B on the right. Then, we have o g odB2 2 odB3 q dE ¼ ðJ dBÞ B e1 p33ffiffiffi ðB3 Þ ot ou2 ou3 g g33 odB3 1 2 odB1 2 e2 pffiffiffi ðB3 Þ 1 þ e3 pffiffiffi ðB3 Þ ou3 ou g g odB1 odB3 odB3 odB2 g23 þ g . ð3Þ 13 ou3 ou1 ou2 ou3 In this expression, we can introduce Alfve´n speed as 2 v2A ¼ g33 ðB3 Þ =q. Finally, assuming that the shear Alfve´n wave is localized on a particular field line, from Eqs. (2) and (3) we can write the following system of coupled ODEs: 1 odB2 c 11 ½ðJ dBÞ B1 ¼ 2 ðg xdE1 þ g12 xdE2 Þ ; pffiffiffi 2 3 g ou vA jBj
ð4Þ
1 odB1 c ½ðJ dBÞ B2 ¼ 2 ðg21 xdE1 þ g22 xdE2 Þ þ ; ð5Þ pffiffiffi 3 vA g ou jBj2 1 odE1 1 ¼ ðg12 xdB1 þ g22 xdB2 Þ; pffiffiffi c g ou3 1 odE2 1 11 ¼ ðg xdB1 þ g12 xdB2 Þ. pffiffiffi c g ou3
ð6Þ ð7Þ
Detailed expressions for (J·dB) · B term in coordinate notations are quite lengthy and are omitted here for the sake of brevity. Eqs. (4)–(7) are the main equations of our model, which describes shear Alfve´n waves in a general magnetic field topology. We have an eigenvalue problem for this system: we have to find a non-trivial solution satisfying dE1,dE2 = 0 at both ends of the field line. This formulation is somewhat different from the classical Strum–Liouville problem because we have four first-order equations and two parameters (frequency x and the polarization of the wave at the northern ionosphere) instead of a single second-order equation with a single parameter. The most straightforward way to solve this eigenvalue problem is by using the shooting method (Press et al., 1992, Chapter 17.1). As a part of the shooting method we have to find a root of a system of two equations
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for the frequency and polarization which is easiest to accomplish by using a multidimensional Newtons method (Press et al., 1992, Chapter 9.7). Unfortunately, Newtons method may sometimes converge to a random non-fundamental mode (or even fail to converge to a meaningful root at all). Thus, the robustness of our model is still somewhat of an issue. As an initial guess for Newtons method we use the eigenvalue predicted by the WKB approximation. 2.3. Equations in orthogonal coordinates In the case of orthogonal coordinate systems, which exist for situations such as axisymmetric fields, the metric tensor becomes diagonal (g12 = g21 = 0) and the equations are conventionally written using scale factors h1, h2, h3. These 22 are related to the metric tensor by g11 ¼ h2 ¼ h2 1 ; g 2 ; pffiffiffi g ¼ h1 h2 h3 . Then, the coupled equations split into two sets of independent equations which can be written as h1 o h2 odE1 x2 ¼ 2 dE1 ; 3 3 h2 h3 ou h1 h3 ou vA h2 o h1 odE2 x2 ¼ 2 dE2 . 3 3 h1 h3 ou h2 h3 ou vA Recognizing that 1 o o ¼ ; h3 ou3 os where s is the distance along the field line, these equations are cast into a form in which they are commonly solved (e.g., Rankin et al., 2000). The relation between the components of electric and magnetic fields is given by Faradays law (e.g., Walker, 1980) as ih2 odE1 dB2 ¼ ; xh1 os ih1 odE2 dB1 ¼ ; xh2 os which allows the equivalent eigenmode equations to be written for the components of magnetic field, rather than electric field. The expressions for the scale factors in a dipole field are well-known (e.g., Walker, 1980) and for reference are given in Appendix A. If the background magnetic field is locally axisymmetric, then there exists an orthogonal coordinate system which can be used for the study of Alfve´n waves (Salat and Tataronis, 2000). In this coordinate system the scale factor associated with the azimuthal coordinate is h1 r where r is the distance from the symmetry axis (Rankin et al., 2000). Then, using the relations between components of the metric tensor given in Appendix B we can write the equations for the two modes as d2 dE1 ddE1 d x2 lnðBðsÞr2 ðsÞÞ þ 2 dE1 ¼ 0; þ 2 ds ds ds vA
ð8Þ
d2 dE2 ddE2 d x2 2 lnðBðsÞr ðsÞÞ þ dE2 ¼ 0. ds2 ds ds v2A
ð9Þ
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Here, Eq. (8) describes what we refer in this paper as mode 1 (poloidal electric field and toroidal perturbation magnetic field); while Eq. (9) describes mode 2 (toroidal electric field and poloidal perturbation magnetic field). These equations are fundamentally the same as those of Singer et al. (1981), however they have an advantage of using the information only along a single magnetic field line. This eliminates numerical uncertainties associated with computing the distance between two close magnetic field lines required by Singer et al. (1981) approach. By derivation, Eqs. (8) and (9) hold only in (locally) axisymmetric magnetic fields, however, at least formally, they may be used for arbitrary magnetic field models. They have an advantage of being computationally very simple while capturing the two most important aspects determining the eigenfrequency of a field line: the distribution of magnetic field intensity and plasma density along the field line. As shown in Section 5 this simple approximation often provides a very good estimate of the eigenfrequency of the field line. It should be noted, however, that Eqs. (8) and (9) assume that there are two distinct polarizations of the standing Alfve´n waves along a field line. Thus, while providing information about the frequency spectrum of the Alfve´n waves, Eqs. (8) and (9) cannot provide any information about the polarizations of the two modes. These equations also completely miss all the effects associated with the coupling between the two modes. 3. Defining a coordinate system based on Euler potentials All metric coefficients in Eqs. (4)–(7) have to be evaluated numerically. A sensible way to introduce a fieldaligned coordinate system is to use Euler potentials a and b (e.g., Stern, 1970; Dhaeseleer et al., 1991) together with a coordinate along the magnetic field line. Note, that for historic reasons Euler potentials are sometimes referred to by various authors as flux coordinates or Clebsch coordinates. The magnetic field is calculated from the Euler potentials by B ¼ B0 ra rb. Here, B0 is in principle an arbitrary normalization constant. In our work we use the dipole strength at the magnetic equator at 1 Earth radius as a convenient choice for B0. Note that, unfortunately, there is no superposition for Euler potentials in general. If a1,b1 describe magnetic field B1 and a2,b2 magnetic field B2, then there is no simple way to compute the Euler potentials for the magnetic field B1 + B2. By definition, a and b are constant along the field line. Therefore, given a magnetic field model we can trace a field line from any given point to the northern ionosphere, where the Euler potentials are evaluated. Then, these would be the values of a and b for the whole field line, including the point in question. For a dipole field, the Euler potentials can be chosen to be
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sin2 h ; b ¼ /. R Here, h is the co-latitude (h = 0 at the northern pole, h = p/ 2 at the equator, and h = p at the southern pole) and R is the distance to the center of the Earth. The magnetic field is, of course, not exactly dipolar at the Earths surface. There is, however, a simple way to compute Euler potentials numerically on a given surface using only the magnetic field intensity on this surface, as described by Stern (1970). For a sphere of radius R0 we can choose the azimuthal angle / as one of the potentials (b) and another one as Z h BR a ¼ R20 sin h0 dh0 ; 0 B0 a¼
where BR is the component of the magnetic field normal to the sphere. If we select the polar axis of the spherical coordinates (R,h,/) to coincide with the dipole axis, then for pure dipole field we recover the standard dipole coordinates. Although this is obviously a convenient choice, it is not the only one, as there are infinitely many perfectly acceptable choices for Euler potentials for a prescribed magnetic field. It is tempting to chose the third coordinate as just the distance along the field line to the northern ionosphere. This is not a very good choice, though, because such a coordinate system would not be orthogonal even in a dipole case. Instead, we should choose Z l ¼ cos h0 jB=B0 j ds; which is, in fact, a Grad–Boozer coordinate (Dhaeseleer et al., 1991). Here, h0 is the co-latitude of the point where the field line intersects the northern ionosphere and the integral is taken from the point in question to the northern ionosphere along the field line. Defining l in this way for a dipolar field coincides with the usual dipolar coordinate ðl ¼ cos h=R2 Þ along the field line. For a pure dipole field, the coordinate system (a,/,l) is orthogonal. In practice, we evaluate the metric coefficients at a few hundred points uniformly distributed in the arc length along the field line and use cubic spline interpolation to obtain their values anywhere along the field line, as needed by the numerical ODE solver. 4. Density distribution In order to proceed with the solution of the eigenvalue problem we have to prescribe the density distribution along the field line. Eventually, the density distribution may be provided by an MHD model together with the self-consistent magnetic fields. At the present stage, however, our main goal is to develop a solver suitable for usage with an empirical Tsyganenko model. In the present work, following, for example, Waters et al. (1996) and Dent et al. (2003), we assume a power-law density profile 4
q ¼ qeq ðRE =RÞ ;
where qeq is an adjustable equatorial density, set to 7 amu/ cm3 in most of our present calculations. The actual value of the equatorial density is not important for the model validation purposes since in our model the eigenfrequency sim1=2 ply scales as qeq , same as in Singer et al. (1981) approach. 5. Examples Now we apply the derived formalism to some practical situations. We use the Tsyganenko 96 (Tsyganenko and Stern, 1996; Tsyganenko and Peredo, 1994; Tsyganenko, 1995) magnetic field model as the background field. For test purposes, we set the date to the 70th day (March 10) of year 2000, UT = 18:00:00. Note, that we use this date purely for demonstration purposes, and not as an example of a specific event. For this date the dipole tilt in the GSM coordinates is 6.28 and the equatorial dipole strength is B0 = 30111.7 nT. First, for comparison with classical results, we use a simple tilted dipole field instead of the full Tsyganenko 96 model. We trace a field line from geographical coordinates 61.11N, 265.95E, which is the location of Eskimo Point magnetometer station. Fig. 1 shows the three projections of the field line traced from the above locations. The field line length is 25.33RE. Next we compute components of the metric tensor for this field line using the techniques presented above. Fig. 2 shows the components of the metric tensor as a function of the coordinate l, along the field line. The results of numerical calculation are displayed by circles, while the solid lines represent the analytical expressions, given in Appendix A. There is no solid line in the panel showing g21 = g12 because analytically this coefficient should be zero. Numerically, it is not zero exactly, but is several orders of magnitude smaller than either g11 or g22. Once the metric coefficients along the field line are computed the eigenvalue problem can be solved to give the frequency of the standing Alfve´n wave as well as the distributions of the electric and magnetic fields along the field line. Fig. 3 shows the eigenfunctions for the two modes in this case. The axisymmetric model described in Section 2.3 in this situation becomes exact. The periods of mode 1 and mode 2 computed using the simple code, designed specifically for a dipole magnetic field, in this case are 868.75 and 1200.64 s, respectively. For comparison purposes, we quote the computed periods to second decimal point, although this level of accuracy is generally not warranted in any practical situation. The period prediction from the WKB approximation (which is the same for either mode) is 680.94 s. Calculation with our new model gives for the first mode a period of 868.72 s and a polarization angle n = 0.257. Here, we define the polarization angle as tan n = dB2/dB1 at the northern ionosphere. For the second mode, we get a period of 1200.73 s and polarization of 89.996. Obviously, we have a very impressive agreement (the relative error is about 105) between the two methods of calculation in this case. The purpose of this comparison is the validation of the
R. Rankin et al. / Advances in Space Research 38 (2006) 1720–1729 4
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Fig. 1. Projections of the dipole field line on the three GSM coordinate planes.
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Fig. 2. Components of the metric tensor along the field line. Continuous lines are analytical expressions, circles are the numerical results.
fairly involved numerics associated with defining the coordinate system and its metric coefficients in our new model. This test firmly establishes that our model handles the tilted dipole case correctly, which is the most non-trivial
case when a comparison with rigorously established results is possible. Next, we consider a non-dipolar case. For this example, we used the full Tsyganenko 96 routines for the same date
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R. Rankin et al. / Advances in Space Research 38 (2006) 1720–1729 E and B for mode 1 (poloidal E and toroidal B)
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Fig. 3. Left panel. Electric and magnetic fields for mode 1. Right panel. Electric and magnetic fields for mode 2.
The distribution of the metric coefficients in this case are shown in Fig. 5. In this figure we have also shown the values given by the formulas of Appendix A, although this field line is not dipole any more and so these formulas, strictly speaking, do not apply. Coefficient g12 changes sign, so we have plotted its absolute value on the logarithmic scale. The WKB period for this field line is 858 s; the period for mode 1 given by the approximate code described in Section 2.3 is 1070 s and for mode 2 the period is 1240 s. Our full model gives two modes with periods 1060 and 1304 s
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as before with the following parameters: Pdyn = 3 nPa, IMF By = 3 nT, Bz = 5 nT, and DST = 20. Note that these parameters were selected for demonstration purposes only and do not specifically correspond to any particular measurements. However, generally these values of the parameters are fairly representative. The resulting field line for these conditions traced from the same location as before is shown in Fig. 4. The length of this field line is 28.37RE. It happens, that this particular field line passes very close to the magnetopause.
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X Fig. 4. Projections of the Tsyganenko field line on the three GSM coordinate planes.
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Fig. 5. Components of the metric tensor along the field line. Continuous lines are analytical expressions, circles are the numerical results.
and polarizations of 95 and 79, respectively. We can loosely identify the first of these as corresponding to mode 1 and the second to mode 2. Fig. 6 is similar to Fig. 3 for this case. Note that in this case, the eigenfunctions are no longer symmetric. However, even in this case the eigenfunctions computed with either method look very similar to each other. Fig. 7 shows the change in the polarization angle tan n(s) = dB2(s)/dB1(s) as a function of the distance along the field line, s. From this figure, it is clear that a clean
1
E and B for mode 1 (poloidal E and toroidal B)
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classification of the solution as one of the two modes is not possible in general. The distinction between the two modes with different polarizations is only possibly near the ends of the field line, where the magnetic field is more or less dipolar and the deviation of our coordinate system from an orthogonal one is not too large. However, near the center of the field line the polarization of the modes change very significantly. To some extent this is related to our definition of the polarization, because dB1 has a node somewhere in the middle of the field line. At this
–0.5 –1 0
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1
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Fig. 6. Left panel. Electric and magnetic fields for mode 1. Right panel. Electric and magnetic fields for mode 2.
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R. Rankin et al. / Advances in Space Research 38 (2006) 1720–1729 100
ξ
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90 80 70 ξ
60 50 40 30 20
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Fig. 7. Polarization of the two modes as a function of the distance along the field line.
point tan n becomes infinite, because dB2, in general has a node at a somewhat different location.
partial differential equations as well as with the data for specific events.
6. Conclusions
Acknowledgments
In this paper we have presented a model describing standing shear Alfve´n waves suitable for application in general three-dimensional magnetic fields. Our model is based on an eigenvalue problem for four ordinary differential equations and, therefore, it is considerably less computationally intensive than some other models which require solution of partial differential equations. We also described in detail the procedure used for defining a field-aligned, in general non-orthogonal, coordinate system required for this study. This coordinate system is based on Euler potentials, and a numerical procedure for computing these potentials is laid out. We also describe a stable numerical method for evaluating the metric tensor for this coordinate system. We test our new model for the case of dipole magnetic field and find excellent agreement with the well-known results for this situation. Then we proceed to a more complicated Tsyganenko 96 magnetic field, which in general does not possess any symmetries. For the particular field lines that we considered, we found that the frequencies evaluated with our new model were in good agreement with those computed with an approximate model derived for axisymmetric fields. Future work will include more detailed comparisons of our model with fully three-dimensional models based on
This work is supported by the Canadian Space Agency and by Natural Sciences and Engineering Research Council of Canada. We also acknowledge the use of WestGrid computational resources. Appendix A. Metric coefficients for dipole field For a pure dipole field the metric coefficients can be computed analytically as follows (e.g., Walker, 1980) 1 sin2 hð1 þ 3cos2 hÞ ¼ ; R4 h21 1 1 g22 ¼ 2 ¼ 2 2 ; h2 R sin h 1 1 þ 3cos2 h g33 ¼ 2 ¼ ; R6 h3
g11 ¼
g12 ¼ g21 ¼ g13 ¼ g31 ¼ g23 ¼ g32 ¼ 0; pffiffiffi g ¼ h1 h2 h3 ¼
R6 . 1 þ 3cos2 h
Here, h is the polar angle (co-latitude) determined with respect to the dipole axis and R is the distance from the center of the Earth. For a dipole field line R = L sin2h.
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Appendix B. Some relations for the coefficients of the metric tensor There are several useful relations between the components of the metric tensor. Using the definition of cotangent basis vectors, it can be easily seen that B ¼ B0 e 1 e 2 . From this equation using the definition gij = ei Æ ej, it can be shown that (Salat and Tataronis, 2001): 2 B ¼ g11 g22 ðg12 Þ2 . ðB:1Þ B 0 Another constraint may be obtained from the fact that $ Æ B = 0. Then, o pffiffiffi 3 ð gB Þ ¼ 0 ou3 or pffiffiffi gjBj pffiffiffiffiffiffi ¼ const g33
ðB:2Þ
along the field line. For orthogonal coordinates, relations (B.1) and (B.2) become the same and coincide with the one used in (Rankin et al., 2000). References Borisenko, A.I., Tarapov, I.E. Vector and Tensor Analysis with Applications. Dover Publication, New York, 1968. Chen, L., Cowley, S.C. On field line resonances of hydromagnetic Alfve´n waves in dipole magnetic field. Geophys. Res. Lett. 16, 895–897, 1989. Cheng, C.Z. MHD field line resonances and global modes in threedimensional magnetic fields. J. Geophys. Res. 108, doi:10.1029/ 2002JA009470, 2003. Cheng, C.Z., Zaharia, S. Field line resonances in quiet and disturbed time three-dimensional magnetospheres. J. Geophys. Res. 108, doi:10.1029/ 2002JA009471, 2003. Dent, Z.C., Mann, I.R., Menk, F.W., Goldstein, J., Wilford, C.R., Clilverd, M.A., Ozeke, L.G. A coordinated ground-based and image satellite study of quiet-time plasmaspheric density profiles. Geophys. Res. Lett. 30, doi:10.1029/2003GL016946, 2003. Dhaeseleer, W.D., Hitchon, W.N.G., Callen, J.D., Shohet, J.L. Flux Coordinates and Magnetic Field Structure. Springer-Verlag, Berlin, Germany, 1991.
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Landau, L.D., Lifshitz, E.M. The Classical Theory of Fields. Pergamon Press, Oxford, UK, 1975 (Chapter 10). Leonovich, A.S. A theory of field line resonance in a dipole-like axisymmetric magnetosphere. J. Geophys. Res. 106, 25803–25812, 2001. Lui, A.T.Y., Cheng, C.Z. Resonance frequency of stretched magnetic field lines based on a self-consistent equilibrium magnetosphere model. J. Geophys. Res. 106, 25793–25802, 2001. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T. Numerical Recipes in Fortran. Cambridge University Press, Cambridge, UK, 1992. Proehl, J.A., Lotko, W., Kouznetsov, I., Geimer, S.D. Ultralow-frequency magnetohydrodynamics in boundary-constrained geomagnetic flux coordinates. J. Geophys. Res. 107, doi:10.1029/2001JA000135, 2002. Rankin, R., Fenrich, F., Tikhonchuk, V.T. Shear Alfve´n waves on stretched magnetic field lines near midnight in Earths magnetosphere. Geophys. Res. Lett. 27, 3265–3268, 2000. Rankin, R., Kabin, K., Lu, J.Y., Mann, I.R., Marchand, R., Rae, I.J., Tikhonchuk, V.T., Donovan, E.F. Magnetospheric field-line resonances: Ground-based observations and modeling. J. Geophys. Res. 110, doi:10.1029/2004JA010919, 2005. Salat, A., Tataronis, J.A. Conditions for existence of orthogonal coordinate systems oriented by magnetic field lines. J. Geophys. Res. 105, 13055–13062, 2000. Salat, A., Tataronis, J.A. Shear Alfve´n mode resonances in nonaxisymmetric toroidal low-pressure plasmas. I. Mode equations in arbitrary geometry. Phys. Plasmas 8, 1200–1206, 2001. Samson, J. Geomagnetic pulsations and plasma waves in the Earths magnetosphere. in: Jacobs, J.A. (Ed.), Geomagnetism, vol. 4. Academic Press, New York, pp. 481–592, 1991. Samson, J., Jacobs, J.A., Rostoker, G. Latitude-dependent characteristics of long-period geomagnetic micropulsations. J. Geophys. Res. 76, 3675–3686, 1971. Singer, H.J., Southwood, D.J., Walker, R.J., Kivelson, M.G. Alfve´n-wave resonances in a realistic magnetospheric magnetic field geometry. J. Geophys. Res. 86, 4589–4596, 1981. Southwood, D.J., Kivelson, M.G. The effect of parallel inhomogeneity on magnetospheric wave coupling. J. Geophys. Res. 91, 6871–6876, 1986. Stern, D.P. Euler potentials. Am. J. Phys. 35, 494–501, 1970. Tsyganenko, N.A. Modeling the Earths magnetospheric field confined within a realistic magnetopause. J. Geophys. Res. 100, 5599–5612, 1995. Tsyganenko, N.A., Peredo, M. Analytical models of the magnetic field of disk-shaped current sheets. J. Geophys. Res. 99, 199–205, 1994. Tsyganenko, N.A., Stern, D.P. Modeling the global magnetic field of the large-scale Birkeland current systems. J. Geophys. Res. 101, 27187– 27198, 1996. Walker, A.D.M. Modelling of Pc5 pulsation structure in the magnetosphere. Planet. Space Sci., 213–223, 1980. Waters, C.L., Samson, J.C., Donovan, E.F. Variation of plasmatrough density derived from magnetospheric field line resonances. J. Geophys. Res. 101, 24737–24745, 1996.