Calculation of magnetic helicity of cylindrical flux rope



[1] We examine a method of calculating the magnetic helicity of cylindrical magnetic flux rope of arbitrary cross section from in-situ spacecraft data. First we examine the gauge invariance of the definition of relative helicity by Finn and Antonsen (1985). Two approaches based on Grad-Shafranov equation are then described. A case study is presented to illustrate the method. The results show significant discrepancies from an earlier study utilizing axially symmetric models. We briefly discuss the potential applications for Sun-Earth connection.

1. Introduction

[2] Magnetic helicity, Hm = ∫Vequation image · BdV, where magnetic field B = ∇ × equation image, is generally not unique due to the arbitrary gauge transformation of the vector potential, equation imageequation image + ∇ϕ. It is only uniquely defined for a closed volume, V, with Bn = 0, i.e., the normal component of magnetic field vanishes on the volume surface, S, of normal equation image. For an open-ended magnetic flux rope, a gauge-invariant relative helicity has to be defined in order to calculate a unique value of Hm [see, e.g., Berger and Field, 1984; Berger, 1999].

[3] Following the definition by Finn and Antonsen [1985], a gauge-invariant relative helicity can be written as

display math

[4] Here a reference magnetic field, B′, is chosen such that Bn = Bn, and B′ = ∇ × A′. Equation (1) can be proven to be invariant under the transformations, equation imageequation image + ∇ϕ and A′ → A′ + ∇ϕ′. The change in Kr due to the above transformations is ΔK = ∫V[(BB′) · ∇ϕ + (BB′) · ∇ϕ′]dV. From the vector identity, ∇ · [(BB′)ϕ] = ϕ∇ · [(BB′)] + (BB′) · ∇ϕ = (BB′) · ∇ϕ, and Bn = Bn, it follows that ΔK = 0.

[5] For a cylindrical magnetic flux rope, i.e., a flux rope of helical field lines lying on nested cylindrical surfaces with axis, z, and ∂/∂z ≈ 0, a convenient choice of B′ is B′ = Bzequation image. The volume, V, has to be bounded by a cylindrical flux surface of length, L, with two ends perpendicular to equation image. Dasso et al. [2003] used this reference field for deriving the relative helicity for axially symmetric flux ropes, such as the well-known force-free, constant α model. In our present study, we extend the calculation to a flux rope of an arbitrary cross section. In a Cartesian coordinate system, (x, y, z), the cross-sectional structure of a cylindrical flux rope in the (x, y) transverse plane is governed by the Grad-Shafranov (GS) equation [e.g., Sturrock, 1994; Hu and Sonnerup, 2002]

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Here the variable A(x, y) is the z component of the vector potential, as B = ∇ × Aequation image + Bz(A)equation image. The right-hand side is a total derivative of the single-variable function, Pt(A), which is the sum of plasma pressure, p(A), and the axial magnetic pressure. By choosing the reference field B′ = Bzequation image with ∇ × A′ = B′, equation (1) is reduced to

display math

as equation image = Aequation image + A′. For axisymmetric cases, equation (3) is equivalent to the definition obtained by Dasso et al. [2003].

[6] The GS reconstruction technique has been developed and utilized to derive the cross section of a cylindrical flux rope from spacecraft measurements [Hu and Sonnerup, 2001, 2002; Hu et al., 2003, 2004]. The reconstruction begins from obtaining an explicit function Pt(A) through the direct curve fitting of spacecraft data, Pt(x, 0) vs. A(x, 0), as well as Bz(A) from Bz(x, 0) vs. A(x, 0). Then the GS equation is utilized to solve for A(x, y) in a rectangular domain from known values along y = 0 (for details, see Hu and Sonnerup [2002]). Thus the transverse magnetic field, Bt = (Bx, By) = (∂A/∂y, −∂A/∂x), is derived. So is Bz(A) over a 2D grid. The approaches to determining A′ will be presented in the next section. In Section 3, we present a case study and compare our results with previous ones. Lastly, we make conclusions and briefly discuss the relevance to Sun-Earth connection.

2. Method of Helicity Calculations

[7] The reference magnetic vector potential, A′, is related to Bz by

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One intuitive and simple approach is to approximate Bz by an analytic function,

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where, bi, i ∈ [1.6] are generally non-zero constant coefficients. After algebraic manipulation, we obtain

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and Az = 0 that satisfy equations (4) and (5). Therefore, for any given Bz distribution over a grid, we choose a number of grid points (greater than or equal to six in the present case with one at the center of maximum Bz and the rest scattering around arbitrarily), then solve the linear system resulted from (5) to obtain the coefficients. In principle, one can always use higher order polynomials to achieve better approximation of any given Bz field, and work out the explicit expressions for Ax and Ay. By visual inspection, the equi-value contours of expression (5) represent ellipses of arbitrary aspect ratio, shift and rotation, which may resemble the transverse magnetic field lines of a flux rope (note that, in turn, these lines are also equi-value contours of A, i.e., the transverse field lines, Bt).

[8] A mathematically more vigorous approach, applied by Chae [2001] to calculate magnetic helicity transport by using solar magnetograms, is to solve for A′ directly by imposing one more condition, ∇ · A′ = 0, i.e.,

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By performing 2D Fourier Transform (FT) on both sides of equations (4) and (6), one obtains two coupled linear equations on FT(Ax) and FT(Ay), using the FT property, FT(f′) = ikFT(f), where the prime denotes the first-order derivative, k is the wave number, and i = equation image. Then it follows that

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This is easy to implement by employing Fast Fourier Transform (FFT) algorithm which is available in many software packages.

[9] Once Ax and Ay are known, integral (3) is calculated numerically by summing 2(AxBx + AyByxΔy over the transverse domain to obtain relative helicity per unit length, Kr/L, and the relative helicity per volume, Kr/V, when further divided by the area, within certain boundary.

3. An Example: 24–25 October 1995

[10] We examine the same Wind event on 24–25 October 1995, as discussed by Dasso et al. [2003] to facilitate intercomparison. The time series are shown in Figure 1. Magnetic field exhibits an elevated magnitude and smooth rotation, but the temperature is fairly high, resulting in a moderate plasma β value, ∼1.3 on average, within the analyzed interval.

Figure 1.

Time series of Wind spacecraft measurements on 24–25 October 1995. Panels from top to bottom are, magnetic field components, and plasma bulk flow velocity in GSE coordinate (see legend), proton number density and total temperature (Tp + Te), and plasma β. The interval being analyzed, as indicated below the bottom panel, is enclosed by two vertical lines.

[11] Specifically we carried out computations for both the force-free (p = 0) and non-force free (p = Nk(Tp + Te)) conditions, whereas in the work by Dasso et al. [2003], only the magnetic field data were fitted to models. The resulting Pt(A) curves with and without pressure contributions are shown in Figure 2. Addition of plasma pressure increases the Pt value significantly, and modifies the slope of Pt(A) curve, which effectively changes the right-hand side of the GS equation (2), with respect to the force-free case. A boundary, A = Ab, corresponding to a flux surface, is identified such that the requirement of Bz(A) being a single-variable function of A is better satisfied for A > Ab [Hu et al., 2004] in this case. Therefore, nested cylindrical surfaces of A > Ab compose a flux rope configuration, carrying helical magnetic field lines.

Figure 2.

Plot of Pt(x, 0) versus A(x, 0) (symbols) and the fitted curve Pt(A) (thick lines) for both force-free (lower curves) and non-force free (upper curves) calculations. Circles and stars are measured data points along positive x axis (y = 0) on inbound and outbound paths with respect to the center of the flux rope, respectively. Thick lines are fitted function Pt(A). The quantity, Rf, is a measure of the quality of the fit, and the value, Ab, represents the identification of boundary, A = Ab (for details, see Hu et al. [2004]).

[12] The reconstructed cross section of the flux rope is shown in top panels of Figure 3. The yellow circles are data points employed in polynomial fitting of Bz(x, y). The outermost thick white contour represents boundary, A = Ab. The recovered Bz distributions from equation (4) with A′ derived via FFT approach are shown in corresponding lower panels. These solutions agree very well with the original data, indicating an accurate determination of A′, except for a few boundary points, probably due to spurious oscillation near boundaries during FT operations. The results are summarized in Table 1. The first four rows show the global properties of the flux rope, i.e., the average β value, the maximum axial magnetic field, Bz0, the axial current Iz, and the axial flux Φz. The helicity calculation results are shown in the next two rows from both approaches by polynomial fitting and FFT for the force-free and non-force free conditions, respectively. The value, equation image, which evaluates the deviation between the original Bz from GS reconstruction and that recovered by two different approaches in least-squares sense, is given in the last row. Although the global properties agree with Dasso et al. [2003], the relative helicity results differ substantially. The results from polynomial fitting and FFT approaches also show discrepancies, which must be due to the errors in the 2nd order polynomial fit of Bz, since their equation image values are almost one order of magnitude larger than those of FFT approach. In addition, a test case on an analytic Bz profile of exact 2nd order polynomial form yields excellent agreement among helicity calculations by different forms of A′ (divergence free or not), and the FFT approach. Therefore, we are certain that the helicity calculation via FFT approach is more accurate, provided a much smaller equation image value is obtained. Since the helicity is an integrated quantity, the effect of the shape of flux-rope cross section is not clear. But plasma pressure (gradient) definitely plays a role (see Figure 2 and associated discussions). For instance, the different cross-sectional structures for force-free and non-force free conditions as given in Figure 3 (especially the different Bz distribution) must have contributed to the discrepancies shown in Table 1.

Figure 3.

Reconstruction results and recovered Bz profiles via FFT approach. (a) Upper panel: cross section under force-free condition, bottom panel: the corresponding Bz profile. (b) The same as Figure 3a for the non-force free condition. In each upper panel, the black contour lines of A(x, y) show the transverse magnetic field lines, and the colors show the Bz(x, y) distribution as indicated by the color bar. The yellow arrows along y = 0 denote measured transverse magnetic field vectors, scales given by the horizontal yellow arrow of magnitude 10 nT. Yellow circles mark the points utilized in polynomial fitting of Bz. The surfaces they are lying on are highlighted by thick white contours. In each lower panel, the 3D gray mesh shows the result derived from A′ by the FFT approach, while the red dots/lines are the original distribution as shown in the upper panel. The green curve marks the location of boundary, Bz = Bz(Ab) (A = Ab).

Table 1. Results of 24–25 October 1995 Event
 Force-FreeNon-Force FreeDasso et al. [2003]a
Poly. FitFFTPoly. FitFFT
  • a

    Median values taken from Table 1 therein.

  • b

    For reference only; not defined the same way as the others.

Φz(nT AU2)0.00930.0110.013
(Kr/L) × 10−3 (nT2AU3)0.480.670.420.681.35
(Kr/V) × 10−1 (nT2AU)
equation image(nT)

[13] Results from one example presented here showed discrepancies from previous study [Dasso et al., 2003]. Besides the basic difference in the model geometry and assumptions, employed in two studies, another important issue is the boundary definition. It directly affects the integral of relative helicity. In the present case, for the non-force free calculation, we were unable to recover a closed boundary within the computational domain (see top panel, Figure 3b). To examine how the values change with different chosen boundaries, we calculate the integral (3) using a series of surfaces highlighted in top panel of Figure 3b as boundaries. The results are given in Figure 4. As the enclosed area surrounding the center of the flux rope increases, so do the values of Kr/L and Kr/V. Therefore, our results may yield a lower bound (for single flux rope) when the flux rope boundary is less certain. Otherwise, definitive results can be provided.

Figure 4.

Calculated magnetic helicity enclosed by different cylindrical surfaces from the innermost white contour (“1”) to the outermost one, A = Ab (“5”), in Figure 3b upper panel.

4. Conclusion and Discussion

[14] In conclusion, we examined an approach for calculating the magnetic helicity of a cylindrical flux rope of arbitrary cross section, based on the Grad-Shafranov equation. This study augments the capability of the GS reconstruction technique, and enables us to derive one more important physical quantity without any constraint on the geometry of the cross section and on the force-free condition. We expect to extend our study and to make connections with helicity origination and transport from the Sun [e.g., Ruzmaikin et al., 2003; Rust, 1997, 1999]. A recent work by Leamon et al. [2004] related the flux rope properties, i.e., the current, the magnetic flux, the twist, and the sign of magnetic helicity of magnetic clouds observed at 1 AU with those of their solar origins (all active regions), by using linear force-free models. We have the ability to derive all these quantities for flux ropes with in-situ measurements, plus relative magnetic helicity as presented here by the non-linear non-force free Grad-Shafranov technique. Efforts in connecting Coronal Mass Ejections (CMEs) with their Interplanetary counterparts, ICMEs/magnetic clouds, are underway. The applicability of our method to solar observations, such as the vector magnetograms, is currently being sought.


[15] We thank B. Welsch at UC Berkeley for directing us to Chae's work. HQ acknowledges the NASA grant NNG04GF47G for support. We are grateful to R. P. Lepping, K. Ogilvie, and CDAWeb for Wind data.