Magnetic field decay with Hall drift in neutron star crusts

Abstract

1 INTRODUCTION

The recent discovery of soft gamma repeaters (SGRs) with weak fields (Rea et al. 2010) has again raised problems related to magnetic field evolution in isolated neutron stars. The activity of magnetars with SGRs and anomalous X-ray pulsars (AXPs) was believed to be powered by the decay of ultrastrong magnetic fields (Thompson & Duncan 1995, 1996). Their surface dipole fields, which are observationally inferred from the spin period and its time derivative, typically exceed the electron quantum magnetic field B_Q= 4.4 × 10¹³ G. The field strength of SGR 0418+5729 is, however, relatively weak at <7.5 × 10¹² G (Rea et al. 2010). The critical boundary between magnetars and radio pulsars has thus become less clear and magnetars are not sufficiently characterized by their dipole field strength alone. Their activity may be explained by hidden magnetic fields such as poloidal components with higher-order multipoles or internal toroidal components. In either case, the field strength should be greater than B= 10¹⁴–10¹⁵ G, since other energy sources are insufficient (see, for example, the review by Mereghetti 2008).

The importance of Hall drift on the magnetic evolution of neutron stars was pointed out before evidence of the existence of magnetars was available (Jones 1988; Goldreich & Reisenegger 1992). The effect, which depends on the field strength, becomes more important in strong regimes where B > 10¹⁴ G. Furthermore, Hall drift may induce instability under certain conditions (Rheinhardt & Geppert 2002). The effect has been considered in analytic treatments (Cumming, Arras & Zweibel 2004; Reisenegger et al. 2007) and for plane-parallel slab geometry (Vainshtein, Chitre & Olinto 2000; Geppert, Rheinhardt & Gil 2003; Rheinhardt, Konenkov & Geppert 2004). These studies are useful to understand some aspects of the mechanism, but non-linear numerical analysis is also required to examine the behaviour in more realistic stars. Several simulations have also been performed, assuming that the fields are located in the spherical crust region of a neutron star (Naito & Kojima 1994; Shalybkov & Urpin 1997; Hollerbach & Rüdiger 2002, 2004; Pons & Geppert 2007). In particular, Hollerbach & Rüdiger (2002, 2004) extensively studied the effect in a crust with uniform conductivity and density and subsequently extended the discussion to stratified stellar models. Pons & Geppert (2007) also calculated the evolution in a realistic stratified model with a thermal history. These numerical calculations are based on spectral or quasi-spectral methods, for example expanding angular functions using spherical harmonics. The limitations of such an approach are discussed therein. For example, evolution of a purely toroidal field forms a steep gradient that cannot be calculated (Pons & Geppert 2007). A recent promising approach is one using a finite-difference scheme to examine the non-linear evolution of Hall instability in a two-dimensional (2D) slab (Pons & Geppert 2010). Each numerical scheme has advantages and disadvantages, so multiple complementary approaches are needed.

In this paper, we calculate magnetic field evolution using a finite-difference scheme to understand non-linear Hall drift dynamics. The model is simplified by assuming an axisymmetric magnetic field located in a crust with uniform conductivity and electron number density. This paper is organized as follows. In Section 2, the model and its assumptions are described. Magnetic field evolution is governed by an induction equation with a non-linear Hall drift term. The relevant boundary conditions and initial configurations are also discussed. Sections 3 and 4 give numerical results for two distinct initial configurations. One condition is a purely toroidal field in which the poloidal part is always zero if it is exactly zero at the initial state. This is in contrast to a purely poloidal initial case, for which the toroidal part is inevitably induced. The evolution of a purely toroidal field is furthermore of interest, due to the similarity to Burgers’ equation (e.g. Whitham 1974). Vainshtein et al. (2000) study the problem in a stratified plane-parallel slab, which corresponds to small-scale dynamics on scales much smaller than the stellar radius. The results clarify the local mechanism, but our concern is with the global aspects. How do the results change in a spherical shell like the crust? This problem is studied numerically in Section 3. Section 4 describes a second study related to the evolution of mixed fields with poloidal and toroidal components. Magnetohydrodynamics simulations (Braithwaite & Nordlund 2006; Braithwaite & Spruit 2006; Braithwaite 2009) show that dynamically stable configurations are mixed ones, in which poloidal and toroidal field strengths are of the same order. There is little known about the initial configuration of neutron stars, in particular the location, topology and field strength. If both components coexist, their strengths are likely to be similar. It is therefore important to explore energy transfer by Hall drift between the components over long-term evolution through Ohmic dissipation. Section 5 presents our conclusions.

2 MODEL AND FORMULATION

2.1 Equations

Magnetic field evolution is governed by the induction equation with the Hall term,

1

where σ is the electric conductivity, n_e the electron number density, e the charge density and c the speed of light (Goldreich & Reisenegger 1992; Naito & Kojima 1994). There are two typical time-scales associated with the first and second terms in equation (1), the Ohmic decay time-scale 4πσL²/c² and the Hall drift time-scale 4πen_eL²/(cB₀). Here, L and B₀ are typical values for the spatial length and magnetic field strength. In general, σ and n_e depend on the spatial position and the time through the thermal history. In this paper, however, for simplicity they are assumed to be constant to explore the dynamics. Thus, the behaviour of this system is specified by the ratio of two time-scales, ⁠, which is called the magnetized parameter by Pons & Geppert (2007). The parameter is given by the initial maximum value of the magnetic field. The typical value for a neutron star is 1–10× (B₀/10¹³ G) (Pons & Geppert 2007), but this depends significantly on spatial position and temperature (Cumming et al. 2004). The overall physical time-scale is scaled by ⁠, where the stellar radius r_s is used for normalization. Note that the characteristic decay time-scale is τ_d for a nodeless field filling a sphere, but it becomes smaller for a field localized in a crust. A magnetic field with is initially given, and the evolution is numerically followed until decay at ∼τ_d. In the case where ⁠, which is relevant to strong fields like those of magnetars, the second term on the right-hand side of equation (1) dominates. The advection term is non-linear and treating it becomes complicated.

Magnetic fields with axial symmetry (∂/∂φ= 0) are described by two functions, a flux function G describing the poloidal magnetic field and a stream function S describing poloidal current flow:

2

where R=r sin θ is the cylindrical radius in spherical coordinates (r, θ, φ). Ampere’s equation gives the current density as

3

where is given by

4

or in cylindrical coordinates (R, Z, φ) by

5

The magnetic field evolution (1) is written in terms of G and S (Reisenegger et al. 2007) as

6

7

where G, S and the spatial length are appropriately normalized. As has been pointed out (e.g. Vainshtein et al. 2000; Pons & Geppert 2007), the evolutionary equation of a purely toroidal magnetic field is very similar to Burgers’ equation, which is a simple example of non-linear propagation with diffusion in one spatial dimension (Whitham 1974). Equation (7) for G= 0 is thus reduced to

8

If the function S depends only on Z, equation (8) is exactly Burgers’ equation.

The functions G and S in equation (2) can be expressed by a sum of Legendre polynomials P_l(θ):

9

10

The functions g_l and s_l evolve independently in the absence of the Hall term (⁠⁠). However, non-linear coupling between g_l and s_l with different indices l becomes important with the increase of ⁠. Most numerical calculations of magnetic decay with the Hall effect have been performed with such an expansion (Naito & Kojima 1994; Hollerbach & Rüdiger 2002, 2004; Pons & Geppert 2007), with one exception (Shalybkov & Urpin 1997). The angular part of the field is expanded by spherical harmonics, but finite differences are used for the radial direction in the quasi-spectral method (Naito & Kojima 1994; Pons & Geppert 2007). Chebyshev polynomials are used for the radial direction in the spectral method (Hollerbach & Rüdiger 2002, 2004). The limitations of spectral or quasi-spectral methods are also discussed, for example by Pons & Geppert (2007). A numerical Gibbs oscillation appears when the function evolves to form a steep gradient caused by advection. This is likely to occur in the large magnetized parameter ⁠. The finite-difference method is used in this paper as an alternative approach. The numerical scheme used is a simple stable one, first-order forward time differencing and second-order centred space (FTCS) differencing (see, for example, Press et al. 1992). There are more sophisticated schemes, but it is not easy to apply them to the non-linear Hall drift term, which is most important here. The grid is a staggered one with equal spacing. The typical number of grid points is 100 × 120. Our numerical results are verified using previous results based on the spectral method in Hollerbach & Rüdiger (2002) and the method works well for most parameters.

2.2 Initial configuration and boundary conditions

The magnetic field is assumed to be located outside the superconducting core. The crust ranges from r₁ to the surface r_s. A typical size in neutron star models is (r_s−r₁)/r_s∼ 0.1; this depends on the equation of state and the stellar mass. A slightly thicker crust model is chosen here, r₁/r_s= 0.75, because this allows an easier demonstration of the numerical results. In an actual implementation it is necessary to use more realistic models that include stratified conductivity, number density and so on, but this simple model is useful for understanding the fundamental dynamics.

The magnetic field cannot penetrate into the core (r < r₁). The condition for the toroidal field is simply S= 0 at r₁. The condition for the poloidal field means that G is a constant at r₁, which is chosen as G= 0. As discussed in Hollerbach & Rüdiger (2002) and Pons & Geppert (2007), the tangential components of the electric field, E_θ and E_φ, should vanish. They are explicitly given by

11

The second term on the right-hand side vanishes because B= 0 at r₁. The first term may be negligible for a large conductivity σ. Otherwise, some conditions should be imposed to cause the tangential current to vanish. The condition j_θ= 0 is satisfied if the function S rapidly approaches 0 as r goes to r₁. The condition j_φ= 0 should be imposed on the function G, more precisely on ⁠, which is not easily treated. In the numerical simulation, the conditions G=S= 0 are used at r₁, assuming a large σ. The current distributions should be set up at the initial time for the conditions to be satisfied.

The exterior of the star is assumed to be a vacuum. The toroidal field should vanish, so that the boundary condition is S= 0 at r_s. The vacuum solution of the poloidal field is expressed by a sum of multipole fields,

12

where the coefficients a_l represent the multipole moments. For example, the dipole moment is a₁=μ. The radial component B_r should be continuous to the exterior vacuum solution across the surface through the condition ∇·B= 0, while B_θ may be discontinuous if a surface current is allowed. In this paper, neglecting surface currents, both components are assumed to be continuous and the coefficients a_l are calculated from the interior numerical function G at the surface r=r_s. The surface boundary condition can be expressed by a sum of a_l up to l_max= 20–30. The results change slightly if the truncation is l_max < 10, but do not change even if l_max is further increased. Compared with spectral or quasi-spectral methods, small-scale structure is not evident in our finite-difference method. The initial configuration contains l= 1 or 2 components, so that the truncation is justified. Another approach without multipole expansion is proposed as a non-local boundary condition through Green’s formula (Pons & Geppert 2010), but is not used here. Finally, for the boundary condition on the symmetric axis, θ= 0 and π are the regularities for the functions G and S, respectively. In other words, G= 0 and S= 0 there.

The functions G and S at the initial time of the numerical simulation should also be subject to boundary conditions. The configuration is easily specified by the functions g_l and s_l in equations (9) and (10). The initial current distribution is chosen as j= 0 at both inner and outer boundaries. One simple solution satisfying this is given by

13

where b_l is a constant. Both the function s_l and its derivative ds_l/dr approach 0 as r→r₁ and r→r_s.

For the poloidal field, the function g_l(r, 0) is given numerically by solving

14

with boundary conditions g_l(r₁, 0) = 0 and g_l(r_s, 0) =a_l. The source term in equation (14) comes from j_φ, which is chosen to be localized near the geometrical centre r≈ (r₁+r_s)/2. The function g_l(r, 0) is smoothly connected with the multipole solution a_l/r^l at the surface r_s.

2.3 Energy and helicity

In previous works (e.g. Hollerbach & Rüdiger 2002, 2004), coefficients of Legendre polynomials were utilized to study the system dynamics; these are useful to understand, for example, energy transfer between different wavelengths at a given time. In order to examine the entire dynamics, it is necessary to calculate the coefficients at many times, because they are time-dependent. Our concern in this paper, however, is with global aspects, so that energy E(t) and helicity H(t) are used to represent the magnetic fields. These are indicators of the field strength and twisted structure. Integrating over the entire space, they are given by

15

16

where dV is the three-dimensional volume element and A is a vector potential for B. The energy is divided into poloidal (E_P) and toroidal (E_T) parts, E=E_P+E_T, by which the energy transfer is examined. From equation (2), the explicit forms are given by functions G and S:

17

From equations (1) and (3), the evolution of magnetic energy is given by

18

This is nothing but Ohmic dissipation. The Hall drift is of no concern for the energy dissipation (Goldreich & Reisenegger 1992; Naito & Kojima 1994), but may affect the dissipation rate by modifying the current distribution. Similarly, the evolution of magnetic helicity H is

19

In this way, both energy and helicity decay through Ohmic dissipation. Note that E and its dissipation term, the right-hand side of equation (18), have a definite sign, but H and its dissipation term, the right-hand side of equation (19), do not. The energy should decrease monotonically, but evolution of the helicity is unknown without numerical integration. In the numerical calculations, the energy balance equation (18) is used to check the accuracy. The relative errors in the initial magnetic energy are smaller than 10⁻². They decrease with increase in the grid number. A relatively large numerical error is produced in the simulation with large magnetized parameter ⁠. It is also prominent in the initial evolution, in which steep structure is formed. Functions in the late phase are rather smooth, so that the errors do not accumulate so much even if numerical integration continues to a longer time.

3 EVOLUTION OF PURELY TOROIDAL FIELDS

This section discusses the evolution of purely toroidal magnetic fields. The system is similar to that of Burgers’ equation, as discussed in Section 2. The initial configuration of S is specified by a single component s_l(r, 0) in equation (13). The three models considered here are characterized by l= 1 with b₁ > 0 in Model A, l= 2 with b₂ > 0 in Model B and l= 2 with b₂ < 0 in Model C. The toroidal magnetic field B_φ in Model C has the same amplitude as in Model B, but the opposite direction. These initial configurations are shown, from top to bottom, in Fig. 1. The coefficient b₁ or b₂ is chosen such that the maximum of |B_φ| is equal to 1, since the magnetic fields in the numerical calculation are normalized by the maximum value at the initial state. The magnetized parameter for these models is ⁠.

Figure 1

Snapshots of time evolution for function S: Model A (b₁ > 0) at the top, Model B (b₂ > 0) in the middle and Model C (b₂ < 0) at the bottom. Column times are t/τ_d= 0, 6 × 10⁻⁴, 2 × 10⁻³ from left to right. Whole spherical regions are shown by colour shading in the left nine panels, while their close-ups of the steep region are shown by contour lines with an increment of 0.1 in the right nine panels. The region is limited to 0.5 ≤R/r_s≤ 1, −1 ≤Z/r_s≤ 0 in Model A, 0.5 ≤R/r_s≤ 1, −0.5 ≤Z/r_s≤ 0.5 in Model B and 0.5 ≤R/r_s≤ 1, −1 ≤Z/r_s≤ 0 in Model C.

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The top row shows the evolution of l= 1 with b₁ > 0 (Model A). Snapshots of the configuration are shown for the initial state (left), the Hall drift time-scale ∼6 × 10⁻⁴τ_d (centre) and the Ohmic decay time-scale ∼2 × 10⁻³τ_d (right). Note that the amplitude of s₁ decays by exp (− 166t/τ_d) in the absence of the Hall term (Hollerbach & Rüdiger 2002), so the typical decay time is not τ_d but 6 × 10⁻³τ_d. The difference mainly comes from the choice of normalization length. The stellar radius is used in this paper for comparison with previous works, but the crust size is more appropriate. Thus, τ_d is rather large to characterize the actual Ohmic decay. The maximum of S at t= 0 is located at r≈ (r₁+r_s)/2 and θ=π/2 and moves in the negative Z-direction, as shown in close-up in Fig. 1. The drift stops at the outer boundary at several times the drift time-scale. It is clear that the function S at this time contains higher multipole components s_l(r, t) in addition to the initial s₁(r, t) when it is expanded as in equation (10). Subsequently the shape is nearly fixed, but the overall amplitude gradually decreases with longer Ohmic decay time-scale. Some numerical calculations were performed for longer until ∼τ_d, but the behaviour after 6 × 10⁻²τ_d is well described by a simple exponential decay. The results are limited to the early phase, since subsequent evolution is easily inferred from the extrapolation.

The middle row shows the evolution of Model B (b₂ > 0) and the bottom row shows the evolution of Model C (b₂ < 0). The evolution of the magnetic configuration is quite different according to the initial sign of b₂. The positive and negative regions of S‘collide’ at the equatorial plane in Model B, while they ‘repulse’ each other in Model C. This occurs because the drift is negative in the Z-direction for S > 0 but positive in the Z-direction for S < 0. It is also important to note that the shape predominantly moves in not the θ-direction but the Z-direction (see equation 8). The motion in S is clear in close-ups of steep structure in Fig. 1. In any model, the moved shape decays over a longer decay time-scale.

These results show the remarkable nature of Hall drift, where a different initial sign for B_φ leads to a different fate. This leads to an interesting question: do these different configurations lead to significantly different dissipation rates? If so, the direction of B_φ would have a significant effect. Fig. 2 shows magnetic energy time evolution for three models. The difference between Model B and Model C is relatively minor. A more marked difference is shown for the initial multipole: the energies for Models B and C decay slightly faster than that of Model A. The current is swept to the outer boundary in Model C, but the outer boundary is replaced by the equatorial plane in Model B. In Fig. 2, a curve with exp (−332t/τ_d), corresponding to the energy decay in the absence of Hall drift, is also plotted for comparison.

Figure 2

Normalized magnetic energy as a function of time for three models with magnetization parameter ⁠. The free decay curve is also plotted for comparison.

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Fig. 3 shows the dependence of the magnetic parameter ⁠. The initial configuration is the same as that of Model B, the model of l= 2 with b₂ > 0. The same figure also shows the magnetic energy for free decay. The damping is clearly strong for large ⁠. The Ohmic decay is enhanced in the presence of Hall drift, i.e. current is swept into a certain region where it is effectively dissipated.

Figure 3

Normalized magnetic energy as a function of time for ⁠. The free decay curve is also plotted for comparison.

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4 EVOLUTION OF MIXED FIELDS

In this section, field evolution is studied numerically for a mixed magnetic configuration consisting of poloidal and toroidal fields. The initial configuration is given solely by the l= 1 component for both fields, namely equation (13) for s₁ and equation (14) for g₁. The maximum of each field is chosen as the same amplitude and the magnetized parameter is ⁠. Fig. 4 shows snapshots of the evolving fields at representative times. The colour contour represents the function S of the toroidal field, and lines denote the contour of the magnetic flux function G of the poloidal field.

Figure 4

Snapshots of time evolution for functions G and S. Contour lines represent the level of G for ⁠, which outwardly increases from the polar axis. Colour shading represents S normalized by B₀r_s. Note that different colour scales are used, since S becomes very small at the turnings in the second and fourth panels. Those panels use the colour scale on the left, the others the scale on the right.

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Oscillatory behaviour is clearly evident in G. Initially, the function decreases with increasing cylindrical distance and the maximum is located on the equator, θ=π/2. The maximum moves ‘upward’ in the meridian plane, toward θ < π/2, until t/τ_d≈ 1.4 × 10⁻³ (second panel). It then changes direction and goes ‘downward,’ passing through the equator at t/τ_d≈ 3.2 × 10⁻³ (third panel) and reaching a minimum at t/τ_d≈ 5.2 × 10⁻³ (fourth panel), before returning to the initial position at t/τ_d≈ 7.8 × 10⁻³ (fifth panel). During this cycle, the field strength decreases.

The function S is also oscillatory. The initial configuration contains only the l= 1 component in the angular part (S∝ sin ²θ), which is symmetric with respect to θ=π/2. The state at t/τ_d≈ 1.4 × 10⁻³ (second panel) differs markedly from the initial state. The configuration is no longer symmetric and higher multipoles can be seen. The field strength itself is weak around this time. At t/τ_d≈ 3.2 × 10⁻³ (third panel), the configuration again becomes symmetric like the initial state but the sign of S is reversed. The l= 1 component dominates there. After the direction of B_φ(=S/(rsin θ)) again changes, the configuration returns to the initial one at t/τ_d≈ 7.8 × 10⁻³ (fifth panel). The directional change occurs around t/τ_d≈ 1.4 × 10⁻³ (second panel) and 5.2 × 10⁻³ (fourth panel), which corresponds to a local minimum of toroidal field strength. The overall toroidal field strength also decreases during this cycle.

Fig. 5 clearly shows the oscillatory behaviour of the magnetic energy, which is divided into poloidal and toroidal parts, E_P and E_T, respectively. The magnitudes of the amplitudes at t= 0 are nearly the same.1 The curve of toroidal energy E_T represents a damped oscillation. Local minima can be seen at t/τ_d≈ 1.4 × 10⁻³ and 5.2 × 10⁻³, corresponding to the time of the second and fourth panels in Fig. 4. The configurations of the third and fifth panels in Fig. 4 are those of the local maxima at t/τ_d≈ 3.2 × 10⁻³ and 7.8 × 10⁻³. There is remarkable energy transfer between the toroidal and poloidal parts during the initial phase. Initially E_P increases until t/τ_d≈ 1.4 × 10⁻³, although the total energy E_sum=E_P+E_T decreases. The sum always decreases due to Ohmic decay (see equation 18). The initial rapid decay of E_T is thus partially due to this transfer. Energy is subsequently transferred between the two components in turn, but the behaviour becomes less clear. The magnetic field decays on the time-scale t/τ_d≈ 10⁻², so the coupling becomes weak. The oscillation period gradually becomes longer, since the drift time-scale increases. Fig. 5 also shows the evolution of magnetic helicity H, which exhibits oscillatory damping with Ohmic decay time-scale t/τ_d≈ 10⁻². The change of sign in H denotes an inversion of the toroidal field B_φ, which occurs around the local minima of E_T.

Figure 5

Time evolution of energy E, helicity H (left panel) and multipole moments a_l at the surface (right panel). Energy is normalized by the initial total energy and helicity by the initial value. Coefficient a_l of the lth moment is normalized by the initial dipole value ⁠.

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The coefficients a_l of the multi-moments in equation (12) describe the exterior poloidal field. The right panel of Fig. 5 shows the evolution of a few of the lowest values. Higher multipoles are induced until t/τ_d≈ 1.4 × 10⁻³, the configuration shown in the second panel of Fig. 4. The poloidal field is no longer dipole at this time. Interestingly, the coefficient a₁ is significantly decreased, compared with the initial value, although a large amount of energy is stored in the poloidal part, as shown in the left panel of Fig. 5. The dipole field strength at this time is not a good indicator of the overall magnetic energy. The poloidal field returns back to the dipole at t/τ_d≈ 3.2 × 10⁻³ (the time of the third panel in Fig. 4), at which higher multipoles are temporarily zero. After sinusoidal oscillation, higher multipoles decay rather rapidly, so the polar dipole remains at the later time t/τ_d > 10⁻². The behaviour after 10⁻²τ_d is rather simple. The magnetic energy is significantly dissipated, so that Hall drift becomes less important. Subsequent evolution in the late phase is described by free decay due to Ohmic dissipation and is omitted here.

Figs 6–9 compare the evolution of energy, helicity and polar multipole coefficients for four models. The initial magnetic configuration is the same as that of Figs 4 and 5, but the strength and magnetized parameter vary. The ratio of poloidal to toroidal fields is 1/4 or 4 in amplitude, so 16(=4²) or 1/16 in energy, and is or 100. The right panel shows energy plotted on a logarithmic scale, log₁₀(E), and the left shows helicity H and a few of the lowest multipole coefficients a_l. Fig. 6 shows the results of a model with ⁠, in which the energy is initially dominated by the poloidal part. Polar multipole components are induced through the coupling to the toroidal part, but the dipole field is barely affected. The decay curve is well described by exp (− 55t/τ_d), which coincides with free decay of the dipole. The energy is always dominated by the polar dipole, and the evolution is described by exp(−110t/τ_d), a line on the logarithmic scale. The curve of toroidal energy is non-linear on the logarithmic scale but is oscillatory. The minor component is highly affected by the poloidal one but is almost neglected in the dynamics of the whole system. No consequence of the toroidal field comes from the fast decay rate exp(−166t/τ_d) in the amplitude as seen in Section 4, or from the initial small strength.

Figure 6

Time evolution of energy on a logarithmic scale log₁₀(E) (left panel) and helicity H and multipole moments a_l at the surface (right panel) for an initially large poloidal field with ⁠. The normalizations are the same as in Fig. 5.

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Figure 7

Same as Fig. 6, but for an initially large toroidal field with ⁠.

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Figure 8

Same as Fig. 6, but for an initially large poloidal field with ⁠.

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Figure 9

Same as Fig. 6, but for an initially large toroidal field with ⁠.

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Fig. 7 shows the results of evolution in which the toroidal component dominates at the initial state. Compared with the decay curve of total energy in Fig. 6, the damping is much faster. Most of the magnetic energy is initially stored in the toroidal part but decays rapidly. The poloidal component decays rather slowly and dominates at later times. The higher multipole moments are induced and have larger amplitudes than those of Fig. 6. The overall dipole decay curve is nonetheless very similar to that of the large poloidal case.

Fig. 8 shows the results for a large magnetization parameter and an initially large poloidal field. The evolution of the total energy is very similar to that of Fig. 6, although oscillatory behaviour in the toroidal energy is evident. The time-scale determined by the Hall drift is approximately a fifth that of Fig. 6, in which ⁠. The oscillatory behaviour is also clear in the coefficients a_l and helicity H. However, except for the initial wavy structure seen at the Hall drift time-scale, the decay of the dipole field is similar to that of Fig. 6. The dominant component, the polar dipole, is thus not affected by the toroidal field, which never plays an important role because of its rapid decay.

Fig. 9 shows the results for a large magnetization parameter and an initially large toroidal field. In this case, a large amount of energy is transferred from toroidal to poloidal components. Higher multipoles in the poloidal field are induced around t/τ_d≈ 10⁻³. The configuration of the poloidal field at this time is significantly distorted, as in the second panel in Fig. 4. The toroidal energy becomes much smaller than the poloidal energy around t/τ_d≈ 10⁻² and both components are subsequently decoupled. After that, the dipole field evolution is determined by free decay. Despite the disorder at t/τ_d≈ 10⁻³, the amplitude of the free decay phase, for example at t/τ_d= 2 × 10⁻², does not differ so much from that of Figs 6–8.

5 SUMMARY AND DISCUSSION

Numerical simulations demonstrate how Hall drift changes the current and magnetic configuration. In purely toroidal evolution, Ohmic dissipation is enhanced by accumulated currents elsewhere. The spatial location depends on the initial data, but the energy dissipation rate does not depend so significantly on the accumulation position in our uniform conductivity model. In a realistic case the conductivity will decrease with radius, so most of the magnetic energy may be effectively dissipated near the surface. The results become more sensitive to the initial condition. Energy is, in principle, transferred between poloidal and toroidal components if both are initially involved. It is important for understanding the evolution that the polar dipole is the least decay rate in the absence of the Hall drift. When the polar dipole dominates, it is rarely affected by the toroidal field, which rapidly decays. On the other hand, a significant amount of energy is transferred to the poloidal field until almost equipartition when the toroidal component is initially dominant. Global non-linear coupling is manifest in the Hall drift time-scale only when the corresponding energies are of the same order. Moreover, the poloidal field at the surface and the exterior is highly distorted and is no longer described by a pure dipole field. During this phase, the dipole is not a good indicator of overall field strength. On a longer Ohmic dissipation time-scale the toroidal field decays rapidly, so the coupling vanishes. The polar dipole eventually survives.

The early evolution depends strongly on the choice of initial data, i.e. the configuration and strength. As discussed by Reisenegger et al. (2007), stationary conditions in the presence of Hall drift for an axisymmetric magnetic field mean that the isosurface of S coincides with that of G; in other words, S=S(G), for which the coupling term ∇G×∇S vanishes. This condition is not satisfied in the whole shell region. The outer boundary condition requires that the toroidal component is concentrated in the interior (S= 0), whereas the poloidal one may leak out to the exterior, meaning that G≠ 0 at the surface. The interior poloidal field is described by the value at the surface if the topology is simply connected. The condition S(G) therefore means that S= 0 everywhere, irrespective of G. Our numerical calculation starts with an initial configuration different from ‘Hall equilibrium’ and shows the behaviour up to state S= 0. It may be necessary as a next step to study plausible initial configurations, since little is known about them.

The time-scale is less accurate in our simplified model with uniform density and conductivity. Realistic models are necessary, but the distributions depend on the equation of state and neutron star mass. Assuming that our uniform model is obtained as a result of spatial average of a certain model with stratified number density and conductivity, the time-scale is estimated. The overall normalization constant τ_d is given by ≈4.4 × 10⁹ (r_s/10 km)² yr, where is an averaged value of the conductivity. The decay time-scale of the dipole is 2 × 10⁻²τ_d≈ 9 × 10⁷ yr and that of the toroidal one is 6 × 10⁻³τ_d≈ 3 × 10⁷ yr. These numbers are slightly larger, since the crust size in our present model is thick, L=r_s/4. The time-scale is proportional to L², so that the actual values may be smaller by a factor of ∼10⁻²–10⁻¹. A typical decay time-scale is around 10⁵–10⁷ yr.

The characteristic age of the low-field magnetar SGR 0418+5729 is more than 2.4 × 10⁷ yr (Rea et al. 2010). The dipole field (<7.5 × 10¹² G) may decay or survive within this period. However, the internal toroidal field is likely to dissipate more quickly. It is therefore difficult to understand why the toroidal field has a much larger field strength, >10¹³–10¹⁴ G, which is required for activity. One possible explanation is that current age estimates are inaccurate. The characteristic age is normally estimated by assuming a constant dipole magnetic field. The Hall drift significantly affects the surface value, especially at the initial epoch with strong field. As demonstrated in the second panel of Fig. 4, the surface field is highly distorted from a pure dipole. SGR 0418+5729 may correspond to a young phase of oscillatory evolution in which the surface dipole temporarily decreases, but there is a strong internal toroidal component.

This work was supported in part by a Grant-in-Aid for Scientific Research (No.21540271) from the Japanese Ministry of Education, Culture, Sports, Science and Technology(YK) and from the Japan Society for Promotion of Science (SK).

REFERENCES