Changes between Version 3 and Version 4 of AstroBearProjects/resistiveMHD

Timestamp:

08/08/11 14:28:44 (13 years ago)

Author:

Shule Li

Comment:

—

AstroBearProjects/resistiveMHD

@@ -35 +35 @@
 So this approximation only works under '''slowly varying temperature''' situation.[[BR]]
 
-
 = The 2.5-D Magnetized Cloud with Resistivity =
 
@@ -46 +45 @@
 (3)the drifting away of the field provides heating which drives up the cavity temperature. This increase in temperature along with the inflow of material will drive the cavity pressure to be higher than the ambient thus drive the flow backwards (outflow).[[BR]]
 
+We use vr, va and cs to identify the speed of the resistive speed, Alfven speed and sound speed respectively. In the discussion we will see that the relative greatness of the three speeds can sometimes define very different evolution behaviors. For sound speed, we have:[[BR]]
+
+[[Image(http://www.pas.rochester.edu/~shuleli/mceq8.png, 6%)]][[BR]]
+[[Image(http://www.pas.rochester.edu/~shuleli/mceq9.png, 10%)]][[BR]]
+
+and for the resistive speed which characterize how fast the field is drifting away from its center, we have: [[BR]]
+
+[[Image(http://www.pas.rochester.edu/~shuleli/mceq10.png, 6%)]]
+
+
+where eta is the constant resistivity as defined in the previous section, l is the length scale of field variation. In the discussion, we will take l as the clump or cavity radius.[[BR]]
+
 To illustrate the above effects, we have the following set of simulation results corresponding to the density, temperature, field strength and velocity vector on each row. [[BR]]
 This simulation is done with beta = 0.5 inside the cavity. The relationship between vr, va and cs is '''vr > va > cs'''. [[BR]]
@@ -53 +64 @@
 [[Image(http://www.pas.rochester.edu/~shuleli/resclump1.png, 80%)]]
 
-As we stated before, the above run is done with beta = 0.5, in other words, the field is supporting the cavity initially. Next let us look at a run with high beta: beta = 100. The relation between vr, va and cs is  '''vr > > cs >> va'''. We compare the high beta run with the first run in terms of the density deposition and field diffusion. [[BR]]
+As we stated before, the above run is done with beta = 0.5, in other words, the field is supporting the cavity initially. Next let us look at a run with high beta: beta = 100. The relation between vr, va and cs is  '''vr > cs >> va'''. We compare the high beta run with the first run in terms of the density deposition and field diffusion. [[BR]]
 As seen in the following picture, the field diffusion occurs at roughly the same rate since we are keeping vr in the two runs. But the mass deposition is much slower, almost unrecognizable in the high beta case, which makes sense: the energy carried away by the field only takes a tiny potion of the total energy. The heating is not plotted here but the same thing happens: the heating is negligible in the high beta case. This result supports our assumption that the '''heating of the cavity is triggered by the energy carried away by the field diffusion'''.[[BR]]
 
 [[Image(http://www.pas.rochester.edu/~shuleli/resclump2.png, 50%)]]
 
-One of our intuition is that the mass deposition observed inside the cavity is caused by the pushing from the ambient material. The simulation under weak conduction case, can give us some insight on this topic. In the following simulation, we compare the case with resistivity ten times greater comparing to figure 1. This situation can be called '''vr > > cs >> va'''. The following graph shows the comparison between field strength, density, and density at different time, for each row. The first row is comparing the field diffusion speed, which does not need more description than that the greater the resistivity, the faster the spreading of the field. The second row compares mass deposition into the cavity at the same time t = 0.1. As we see from the scale, the right plot has a greater deposition rate and more blurred edge. So the weak conduction does help material to deposit into the cavity. So how fast it is? By comparing the third row, we find that the mass deposited into the blue region of the cavity for weak conduction at t = 0.2, is almost identical to the one for strong conduction at t = 2.0. '''The deposited density does show how resistive the material is.''' And it seems to scale to the resistivity.
+One of our intuition is that the mass deposition observed inside the cavity is caused by the pushing from the ambient material. The simulation under weak conduction case, can give us some insight on this topic. In the following simulation, we compare the case with resistivity ten times greater comparing to figure 1. This situation can be called '''vr > > cs > va'''. The following graph shows the comparison between field strength, density, and density at different time, for each row. The first row is comparing the field diffusion speed, which does not need more description than that the greater the resistivity, the faster the spreading of the field. The second row compares mass deposition into the cavity at the same time t = 0.1. As we see from the scale, the right plot has a greater deposition rate and more blurred edge. So the weak conduction does help material to deposit into the cavity. So how fast it is? By comparing the third row, we find that the mass deposited into the blue region of the cavity for weak conduction at t = 0.2, is almost identical to the one for strong conduction at t = 2.0. '''The deposited density does show how resistive the material is.''' And it seems to scale to the resistivity.
 
 [[Image(http://www.pas.rochester.edu/~shuleli/resclump3.png, 50%)]]