5.3 Variations of Magnetic Field Parameters
5.3.1 Varying the Overlying Field with γ = 1: The Non-Generic Case
x = 0. We have chosen the parameters of the positive and negative sources such that both sources have the same absolute maximum normal magnetic field strength (Bi) and radius (ri). We then vary the overlying magnetic field strength (α) between−1 and 1, during which we find six different topological states, as shown in Figure 5.1. The first three of these states involve a negative overlying field while the final three involve a positive overlying field.
We start by examining the states (IV–VI), where we have a positive value for the overlying magnetic field strength (α). For low (positive) values of α, we have a state, state IV (see Figure 5.1d), with one X-type null point on each of the periodic boundaries (i.e. at x=±1, 0 ≤ z < ∞). This state has all four types of flux – overlying, closed, negative open and positive open.
5.3. VARIATIONS OF MAGNETIC FIELD PARAMETERS
(a) (b)
(c) (d)
(e) (f)
Overlying
+ve Open –ve Open
Overlying
+ve Open –ve Open
Ov.
+ve Open –ve Open
Closed Ov. Ov. +ve Open –ve Open Closed Overlying Overlying Closed Overlying Closed
Figure 5.1: (Movie) Magnetic fieldlines and skeletons of our model with different strengths of overlying magnetic field, each with two equal and opposite sources sited atx1 =−0.2 and
x2 = 0.2 of strength B1 = +1 and B2 =−1 and radius r1 = r2 = 0.11, respectively. Blue lines represent fieldlines, red lines are sources (left is positive, right is negative), black lines separate fieldlines of different connectivity (e.g. separatrices) and black stars represent null points. The overlying magnetic field strengths (α) are (a)−1.0, (b)−0.4, (c)−0.3, (d) 0.035, (e) 0.2 and (f) 0.7. These chosen strengths of α give representative frames for each of the respective six states.
CHAPTER 5. GENERIC 2D INTERACTION OF DISCRETE SOURCES
(a) (b)
(c) (d)
Figure 5.2: (Movie) Close up of left edge of the positive source near the time of bifurcation between states V and VI, with overlying magnetic field strengths α = 0.328, 0.334, 0.337, 0.343. The notation is as in Figure 5.1
When α is only just greater than zero, we have nulls situated at (±1, z) wherez is large. As we increase the overlying magnetic field strength (α), the null points descend down along the boundaries x = ±1 until they reach the base, before bifurcating at (x, z) = (±1,0). This bifurcation marks the change to a new state, state V, involving two X-type null points which lie on the z= 0 line (see Figure 5.1e). During state V, as the overlying magnetic field strength increases we find that one of these null points moves right from the left side of the box, while the other null point moves left from the right side of the box. Throughout this phase, all flux from each of the two sources is now connected to the other source leaving just two flux domains containing closed or overlying flux.
As we increase the overlying magnetic field strength (α) further, the two null points move closer to the outside edges of the sources. Once a null point has reached the edge of a source, it disappears as the magnetic field nearby starts to curve upwards at the source’s edge (see Figure 5.2a–b) leading to a further bifurcation. In the case of our positive source, the magnetic field is now directed into it along the base from (x, z) = (−1,0). Since magnetic flux cannot enter the positive source it must begin to rise above it, leaving the base of the box. This rising from the base starts with a shallow gradient, before steepening (see Figure 5.2c–
5.3. VARIATIONS OF MAGNETIC FIELD PARAMETERS
(a) (b)
(c)
Figure 5.3: The contribution from both sources for (a) −B˜x(1, z), (b) −B˜x(0, z) and (c)−B˜x(x,0) with two equal and opposite sources of radiusr1 =r2 = 0.11 atx1 =−0.2 and
x2 = 0.2 of strength B1 = +1 and B2 =−1, respectively. The maximum and minimum of graphs (a) and (b) give the range of the overlying magnetic field strength (α) for which a null point can exist. In (c), the aqua coloured areas represent the locations of the two sources, and the red lines are the values ofα where bifurcation occurs.
d). As our field is anti-symmetric, these effects occur concurrently at both sources. This new state, state VI, has the same topology, based on the connectivity of the magnetic field as state V, however the components of the magnetic skeleton are different (no null points, but a separatrix-like curve) (see Figure 5.1f). We shall discuss the implications and details of this new type of topology later in Section 5.4.
With the exception of the bifurcation when the overlying magnetic field strength (α) changes sign, the other topological states (I–III) may be considered similar to states IV–VI (for increasing|α|) except that they show behaviour in the 0≤x≤2 range of these states and the exact values ofα at the bifurcations are different (see Figures 5.1a–c). At the bifurcation at α = 0, the null point on the side boundaries are found at (±1,∞). As α is decreased (fromα >0) a single null point returns down the centre, from (0,∞) along the x= 0 line.
CHAPTER 5. GENERIC 2D INTERACTION OF DISCRETE SOURCES
α
−0.499 −0.354 0 0.036 0.331
State I II III IV V VI
Figure 5.4: Bifurcation diagram for the potential 2D model with two equal and opposite sources (as shown in Figure 5.1) in a varying overlying magnetic field of strength α. The vertical blue lines give the locations (inα) of bifurcation between the six states.
This creates state III (Figure 5.1c) which involves all four types of flux in four flux domains. When the null point reaches (0,0) another bifurcation occurs which creates two X- type null points on thez= 0 line between the sources. In this new state, state II (Figure 5.1b) there are just three flux domains containing positive open, negative open and overlying flux. The bifurcation between states II and I is similar to that between states V and VI. It results in a loss of both null points, but does not change the topology or number of separatrix/separatrix-like curves (Figure 5.1a).
An obvious question to ask is when are the exact times of our five bifurcations in this model? The bifurcation when the overlying magnetic field strength (α) changes sign is obvious, and must occur at α= 0. State IV may only exist while the null points are on the periodic sides of our box (i.e. each null point is located with x = ±1). The minimum and maximum values ofαto sustain state IV therefore must be calculated. As our field within the box is anti-symmetric due to our choice of parameters, then during state IV, Bz(±1, z) = 0 for all z≥0 along the linex=±1.
So the null points are obviously located at the point where Bx(±1, z) = 0 with z ≥ 0. We now write Bx(±1, z) as (c.f. Equation 5.14 with x1 =−x2 = x0, B1 =−B2 =B0 and
r1 =r2=r) Bx(±1, z) =α+ ˜Bx(±1, z) =α+ 2B0 ∞ X n=1 (−1)nsin(nπr) sin(nπx 0) nπ(1−n2r2) e− nπz,
where B0 = 1, x0 = 0.2, r = 0.11 and ˜Bx is the contribution to Bx from the two sources alone. By using the graph of ˜Bx in Figure 5.3a, we see that the minimum allowed value of α is zero and occurs when the null points are at (±1,+∞). The maximum allowed value of α
occurs when the null points bifurcate at (±1,0). For this bifurcation at (±1,0), the overlying magnetic field strength must be
α=−B˜x(±1,0) =−2B0 ∞ X n=1 (−1)nsin(nπr) sin(nπx 0) nπ(1−n2r2) ≈0.0359. (5.17) A similar argument may be used for finding the limits ofαfor which state III exists. In this
5.3. VARIATIONS OF MAGNETIC FIELD PARAMETERS state, we know it bifurcates into state IV atα = 0. This state has the null point at x = 0 descending fromz=∞toz= 0 (see Figure 5.3b). At (x, z) = (0,0) a bifurcation occurs. So the value ofα at the bifurcation from state III to state II (when the null point is at (0,0)) is
α=−B˜x(0,0) =−2B0 ∞ X n=1 sin (nπr) sin (nπx0) nπ(1−n2r2) ≈ −0.354. (5.18) We now investigate the ranges of the overlying magnetic field strength (α) for which state II and V exist (see Figure 5.3c). Here the null point is on thez= 0 line. ClearlyBz(x,0) = 0 everywhere except in the sources so we only need consider ˜Bx(x,0). Figure 5.3c shows a plot of−B˜(x,0) versusxand consists of three non-shaded sections: the first section on the left of the negative source (and the last section on the right of the positive source) represent values of α where state V exists. State V exists for values of α which satisfy α+ ˜Bx(x,0) = 0 for values ofxlying within the above sections. The minimum value ofα in state V is obviously equal to the value of α at the bifurcation between states IV and V (i.e. α = 0.0359). The maximum value ofα in state V occurs where the null point touches the edge of the source, and thus is calculated using the series given in Equation 5.14 as
α = −B˜x(−x0−r,0) = B0
∞
X
n=1
sin2(nπr) (1−cos(2nπx0))−sin(nπr) cos(nπr) sin(2nπx0)
nπ(1−n2r2)
≈ 0.331.
To determine the value of α at which the bifurcation between states II and III occurs, we use a similar argument to that used to find theα at which state IV bifurcates into state V. However, at this bifurcation the null point is at (0,0) and thus α = −B˜x(0,0) = −0.354. Using similar arguments, as those used to determine theαat which state V bifurcated from state VI we look for when the null point touches the inner edge of the source to determine for what value of α state I bifurcated to state II. Here we have α=−B˜x(−x0+r,0) which gives a value ofα=−0.499. We have drawn a pictorial diagram of the times of bifurcation between states in Figure 5.4.