Symmetric eruption

From FLUX

Jump to: navigation, search

1 Kickoff discussion 15-Nov-2007
2 2nd discussion 4-Dec-2007
- 2.1 Goals
- 2.2 Techniques
3 Previous run with ARMS
4 Simulations with FLUX

[edit]

Kickoff discussion 15-Nov-2007

Present: Spiro, Etienne, Craig, Laurel

Demo images from Laurel:

[edit]

2nd discussion 4-Dec-2007

Initial images from Laurel:

[edit]

Goals

Compare ideal behavior of these axisymmetric fields with the reconnective behavior.

[edit]

Techniques

Spiro wants to deliver a background plane with a circular negative spot on it, and a surrounding circular positive spot. We dither field lines onto the spot.

Background field in the Spiro simulations is uniform -- hard to do with current fluxon code. (Additional photospheric condition needs to be spliced into the code).

[edit]

Previous run with ARMS

[edit]

Initial Magnetic configuration

Figure 2.1: Initial axisymmetric topology

The magnetic field configuration is given by a vertical uniform field and the field due to a vertical magnetic dipole located under the bottom boundary. The resulting field is axi-symmetric about the dipole axis and is given by:

$B_x(x,y,z) =\frac{\mu_0 m_0}{4\pi} \frac{3(z-z_0)x}{(x^2 +y^2 + (z-z_0)^2)^{5/2}}$

$B_y(x,y,z) =\frac{\mu_0 m_0}{4\pi} \frac{3(z-z_0)y}{(x^2 +y^2 + (z-z_0)^2)^{5/2}}$

$B_z(x,y,z) =\frac{\mu_0 m_0}{4\pi} \frac{2(z-z_0)^2-(x^2+y^2)}{(x^2 +y^2 + (z-z_0)^2)^{5/2}} - B_v.$

With $B v ( = 1)$ the uniform background field , $m 0 ( = 25)$ the magnetic dipole strength and $z 0 ( = - 1.5)$ the depth of the dipole (located below the bottom surface) and $μ 0 ( = 4π)$ . The corresponding topology is illustrated in Fig. 2.1.

The superposition of the uniform negative vertical magnetic field and the positive dipole field produces a magnetic null point positioned along the symmetry axis at height:

$z_n=(\mu_0 m_0/2\pi B_v)^{1/3}+z_0 \approx 2.2$

For the simple fields assumed here, the fan surface is a section of a sphere. The intersection of the fan with the bottom boundary, therefore, is simply a circle of radius $r F = 3.7$ . Within this circle, the net magnetic flux is precisely zero, since all field lines inside the fan close back to the solar surface. Outside of this circle, all field lines are open.

[edit]

Bottom boundary constraint

Figure 2.2: Bottom boundary cut of the magnetic profile and of the rotation rate

The bottom boundary is assumed to be line-tied, with the tangential velocity $\vec v_\perp(x,y,z=0)$ given by:

$\vec v_\perp = v_0 f(t) k_B \frac{B_r-B_l}{B_z}\tanh \left(k_B\frac{B_z-B_l}{B_r-B_l}\right)\hat z \times \nabla B_z$ with $\quad f(t)=\frac{1}{2}\left[1-\cos\left(2\pi\frac{t-t_l}{t_r-t_l}\right)\right]$

for $t \in [t_l,t_r]$ and $B_z \in [B_l,B_r]$ with $v_0 (=3.7 \times 10^{-5})$ and $k B = 15$ . The magnitude of the twisting motion is assumed to vanish at all other times and surface locations. We set $t l = 100$ and $t r = 1100$ , to allow a brief period of relaxation of the initial state before switching on the motions. The cosine time profile allows a gradual acceleration from rest and deceleration to rest of the imposed flows. We also set $B l = 0.1$ and $B r = 13$ , so that only the positive polarity flux is rotated. The resultant spatial profile has an approximately solid-body rotation of the positive flux near the axis, with a smooth but steep falloff to zero near the polarity inversion line, as shown in Fig. 2.2 . Our objective is to accumulate slowly a large amount of free energy in the magnetic field, while delaying as long as possible the buildup of strong currents near the fan surface. The maximum velocity of the imposed twisting motions is about 1% of the local Alfvén speed.

[edit]

Other boundary conditions

The top boundary is open (no gradients though!) and the sides boundaries are reflecting boundaries with null gradients of the variables.

[edit]

System evolution

Figure 2.3: White field lines: open field. Blue field lines: closed field. Yellow isosurface: density

The magnetic evolution is shown in Fig. 2.3. The temporal variation of the total magnetic energy is presented in Fig. 2.4 .

Figure 2.4: Variation of the magnetic and kinetic energies within the numerical domain.

[edit]

Non-axisymmetric Magnetic configuration

The coronal (open field) is now tilted with an angle $θ$ with the vertical axis $z$

$B_x(x,y,z) =\frac{\mu_0 m_0}{4\pi} \frac{3(z-z_0)x}{(x^2 +y^2 + (z-z_0)^2)^{5/2}} + B_v \sin(\theta)$

$B_y(x,y,z) =\frac{\mu_0 m_0}{4\pi} \frac{3(z-z_0)y}{(x^2 +y^2 + (z-z_0)^2)^{5/2}}$

$B_z(x,y,z) =\frac{\mu_0 m_0}{4\pi} \frac{2(z-z_0)^2-(x^2+y^2)}{(x^2 +y^2 + (z-z_0)^2)^{5/2}} - B_v\cos(\theta).$

I have used differnt values for $\theta \in [10;30]$ . A good start would be $θ = 15$ .

[edit]

Simulations with FLUX

[edit]

Progress

LR 1/9/08: So far I have been able to set up and relax the system. I have tried it with a system of radius 8 with 100 and 500 fluxons. In each case the result is essentially the same and looks fine (image below for the 500 case). I was a little worried about the open field though. It looked rather splayed for a field that was supposed to be straight. So I tried a system of uniform field, the same size, with 200 fluxons. Indeed the field is splayed. I am not sure if this splay is too great for the purposes of this project. If it is, we will need to modify the code to support two closed boundaries and an open boundary (closed on the bottom and sides of a cylinder, open on the top). This might help to straighten the external field if we need it.

LR 1/15/08: Rotation code works, need to do some long-run tests.

[edit]

Hemispherical Simulation

For this simulation, I used 600 fluxons. The base was 15 units in radius and the height was also 15 units. The system rotated until it opened through the boundary. It stayed essentially symmetrical the whole time. Note that this was before we figured out the time glitch, so the times here don't correspond to the same as in the ARMS simulations.

conclusion: the hemispherical boundary did not provide enough of a vertical field to initiate symmetry breaking, try a cylindrical boundary.

Media:hemisphere_top.mov

Media:hemisphere_side.mov

[edit]

Vertical Cylinder

radius=12

height=50

number of fluxons=250

The times do correspond to the ARMS simulations. The simulation appears to be on the verge of 'kinking', but it is at time=1150 so it is at a lull in turning, so it may take another day to see the results.