Timescale for the inwards migration of the circumnuclear disk's inner rim -- Marvin
The central stellar cluster's wind extracts angular momentum from the circumnuclear disk's inner rim, therefore the inner rim migrates
inwards. In the following I derive the timescale for the inwards migration of the CND's inner rim. Please see our paper draft for more
details.
We assume that angular momentum is extracted from a ring of radius
and radial extension , therefore it has the mass . This ring has a total angular momentum of with specific angular momentum .After a time
the wind adds the mass to the ring, where is the wind's outflow rate and is thefraction of the wind that interacts with the disk. Thus the ring has the mass
and the specific angular momentum.
Furthermore the ring moves to
, and because and we get
where I used a taylor expansion in the last step.
Using
, and integrating this equation gives the time the inner rim needs tomigrate inwards:
,
where
is the disk's initial inner rim and the location of the rim after inwards migration.In our simulations
, for we get , for we get . The actual time the inner rim needs to move inwards (as observed in the simulations) is and , respectively. For the simulation this is pretty close, for the simulation the deviation is less than a factor of two, which I find an acceptable deviation considering that the above mentioned derivation is only an order of magnitude estimate.Simulations of the circumnuclear disk with a larger inner cavity, Part II -- Marvin
In this post I am presenting 10 simulations of the circumnuclear disk, 5 with an inner cavity of r=1pc (see first figure and first animation below) and 5 with an inner cavity of r=2pc (see second figure and second animation below). The time is given in units of
yrs. The r=1pc simulations cover a time of years, the r=2pc simulations years. The following table shows the details of these simulations. (Note: B=1mG corresponds to a plasma-beta of about 0.1, B=0.1mG to a plasma-beta of about 10.)
Number | properties | linestyle |
---|---|---|
1. | without magn. fields and with outflow | blue, dotted |
2. | with magn. fields (B=1mG) and with outflow | red, dashed |
3. | without magnetic fields and without outflow | turquoise, solid |
4. | like 2., but with B=0.1mG | green, dashdotted |
5. | with magn. fields and without outflow | violet, solid |
First figure: CND with an inner cavity of r=1pc
Second figure: CND with an inner cavity of r=2pc
First animation: CND with an inner cavity of r=1pc
Second animation: CND with an inner cavity of r=2pc
In all these simulations the inner rim moves inwards in one way or another. Different physical processes can be responsible for that, but they all work on different timescales: Firstly, the inner rim moves inwards due to numerical viscosity on a viscous timescale https://astrobear.pas.rochester.edu/trac/blog/mblank05302015). Secondly, gas pressure can cause the inner rim to move inwards, the corresponding timescale is with the speed of sound . Then, magnetic pressure can also cause the inner rim to migrate inwards on a timescale , where is the Alfven speed. Finally, the inner rim moves inwards due to the angular momentum extraction caused by the interaction with the wind, the corresponding formula can be found in our paper draft. All these timescales are listed in the following table in units of yrs. is the timescale for the angular momentum extraction according to the formula presented in our paper draft, and is the time the inner rim actually needs to move inwards (as seen in the simulations).
(on how to determine the numerical viscosity I refer to one of my other posts:…….
timescale | r=1pc | r=2pc |
---|---|---|
4.5 | 12.7 | |
2400 | 21700 | |
37.8 | 113.5 | |
10 | 30 | |
100 | 300 | |
49 | 490 | |
8 | 42 | |
20 | 65 |
Let's first look at the simulations without outflow (number 3 and 5): There is no big difference between these two models. Thus in the absence of the central outflow magnetic fields do not have a large effect on the evolution of the CND, maybe the reason for this is that initially there is no magnetic field present inside the inner cavity. Only numerical viscosity causes an inwards migration of matter, but as the corresponding timescale is much larger than the simulation time not much matter is actually moving inwards.
Examining the simulations without magnetic fields (model 1) and with a low initial magnetic field strength (model 4) shows that initially they do not deviate much from each other (maybe the magnetic field strength that I've chosen for model 4 is a little bit too low). In both simulations the inner rim takes about
yrs to reach the outflow object ( yrs for the r=2pc simulation). The first yrs ( yrs for r=2pc) they look more or less the same, after this time the surface density is larger when no magnetic fields are present.In the simulation with magnetic fields and with outflow (model 2) the inner rim moves inwards very quickly (faster than in the simulations without magnetic fields and with a low magnetic field strength), it reaches the outflow object after about
yrs ( yrs for the r=2 simulation). After that the surface density decreases about a factor of two, thus matter seems to be removed from the inner cavity. Then the simulation reaches a "steady state", where the surface density experiences no significant changes.Conclusions:
- The outflow is causing the inner rim to movs inwards, the higher the magnetic field strength, the faster it is moving.
One may be tempted to think that this is due to the magnetic pressure that drives the material inside, as for the simulations with magnetic fields and with outflow (model 2) the timescales
correspond to the times the inner rim needs to move inwards. But even without magnetic fields the inner rim moves inwards, one again could think that this is caused by the gas pressure, because roughly corresponds to the times the inner rim needs to move inwards. However, the simulations without outflow do not show such an inwards movement of the inner rim whatsoever, thus it is the outflow that causes the inner rim to shrink (at least it is the main contributor).- After the inner rim has reached the outflow object, the surface density is lower for higher magnetic field strengths.
But as always, answering some questions leads to a larger number of new questions:
- Why is the collapse faster for higher magnetic field strengths? Magnetic fields seem to accelerate the angular momentum extraction.
- Why is the surface density lower for higher magnetic field strengths? Do magnetic fields prevent the inflow of matter or is there a kind of magnetically driven outflow?
- and fit quite well for the r=1pc simulations, but why is there such a huge difference for the r=2pc simulations?
- Do we have to answer all these questions in our paper? Maybe the first two can be shifted to our future work, but I think the third needs some additional pondering.
Spreading ring calculations, Part II -- Marvin
In one of my last blog posts (https://astrobear.pas.rochester.edu/trac/blog/mblank05302015) I presented some "spreading ring calculations" that allow to estimate the magnitude of numerical viscosity in disk simulations with astroBEAR. There I introduced the parameter that gives the numerical viscosity in units of the maximum -viscosity .
In the following I show a parameter study of these calculations by varying the resolution of the simulations. However, I have performed this study in 2D to save computational time. The simulations from my previous post have a resolution (cell size) of 0.04 pc, here I additionally show simulations with 0.08, 0.02, and 0.01 pc.
The evolution of the spreading ring for all resolution levels is shown in the Figure below (upper left: 0.08 pc, upper right: 0.04 pc, lower left: 0.02 pc, lower right: 0.01 pc).
The qualiative evolution is quite similar for all the simulations, but as expected the spreading of the ring is slower with higher resolution. I also estimated the parameter
following the procedure described in my previous post, and I additionally give the runtime of the simulations, all have been calculated using 64 cores:resolution | runtime | |
---|---|---|
0.08 pc | 11.3 min | |
0.04 pc | 1 h | |
0.02 pc | 5.3 h | |
0.01 pc | 31.1 h |
The upper right figure with a resolution of 0.04 pc can be compared with the corresponding 3D simulation of my previous post, again the qualitative evolution is similar, but the 2D simulation has a higher numerical viscosity than the 3D simulation, which has
.The numerical viscosity decreases with increasing resolution, as expected. Furthermore, it seems to be a linear function of the resolution, which makes sense: Let us consider the following simplified transport equation:
For solving this equation numerically we want to replace the spatial derivative by a difference quotient. Therefore we make a taylor expansion of the function at the point :
Rearranging gives:
And inserting this into the first equation gives:
The term on the right side is responsible for numerical viscosity, and is a linear function of the resolution
.Simulations of the circumnuclear disk with a larger inner cavity -- Marvin
I did some simulations of the circumnuclear disk with a larger inner cavity. My previous simulations, which are shown in the first figure below, have an inner cavity with a radius of 1 pc. The new simulations have an inner cavity of 2 pc and are shown in the second figure.
The plots show radial profiles for the surface density for Model 1 (without magnetic field but with outflow, dotted lines), Model 2 (with magnetic fields and with outflow, dashed lines) and Model 3 (without magnetic fields and without central outflow, solid lines). The vertical line marks the location of the disk's initial inner rim. The time is given in units of the orbital time-scale at 1 pc, which is 4.5e4 yr. The profiles have been calculated by using radial bins of the surface density with a bin size of 0.02 pc.
In the simulations with a small cavity we found that for Model 1 the disk was moving rapidly inwards, and that for Model 2 the inflow was to some extent suppressed by the magnetic field.
With a larger cavity the situation seems different, now for Model 2 the disk material moves inwards much faster than for Model 1, only after a very long time (note that the last subplot of the second figure corresponds to 22 orbital timescales and not 13 orbital timescales as in the first figure) the surface density of Model 2 is again lower than the surface density of Model 1.
According to these new results, is our previous statement that the magnetic field suppresses the inflow of material towards the black hole still valid?
Spreading ring calculations -- Marvin
To measure the magnitude of numerical viscosity in disk simulations with AstroBEAR I have performed calculations of a spreading ring.
A short reminder on the spreading ring problem:
If we assume an accretion disk that is rotationally symmetric and geometrically thin and that its angular frequency does not change with time we can derive the following equation that describes the time dependent evolution of such an accretion disk:
Here
is the accretion disk's surface density, its angular frequency and the viscosity of the gas. For a Keplerian gravitational potential, constant viscosity and an initial condition in the form of a delta peak at position
this equation has the following analytical solution (eq. 1):
with
(eq. 2)
and the modified Bessel function of the first kind
.As a delta peak is numerically difficult to handle I use this analytical solution as initial condition, with
and initial values for of and , respectively.The following movie shows the general behavior of such a ring. The numerical and physical parameters are the same than those of my simulations of the CND unless stated otherwise, e.g., the resolution at the location of the ring is about 0.04 pc. However, I switched of cooling and magnetic fields for the spreading ring calculations.
First animation: surface density for $\tau_{\text{i}} = 0.001$
The following two figures show the radial surface density profiles, the first one for
and the second one for . For each snapshot of the simulation I fitted a curve according to eq. 1 to the surface density profiles, which are also shown in these figures.This fit allows to determine the corresponding value of
. Thus we have as a function of time, determining the slope of this function gives the viscosity (see eq. 2).I define the parameter
where
is the maximal alpha viscosity, with and I use typical values of the CND.For the
simulation I get a value of
and for the
simulation a value of
Considering that in the literature a value of
is often used to account for viscous disk evolution, this result is not extraordinarily good, but I think it shows that our simulations are not dominated by numerical viscosity.I furthermore did a spreading ring calculation with central outflow, the following movie shows that the material of the ring is slowly blown away by the wind, contrary to the simulations of the CND where the material of the disk was moving towards the central black hole due to its angular momentum loss.
Second animation: surface density for $\tau = 0.001$, with central outflow
The following figure shows the corresponding radial surface density profiles. Because the development of the ring is completely different from the analytical solution of the spreading ring, I do not show any fits to eq. 1 here.
MHD simulations of the circumnuclear disk - Marvin
I have now finished a simulation that contains all the relevant physics, including cooling, MHD and the outflow object.
But first I want to recall two of my previous simulations, which do not include MHD: The first animation shows the surface density of the accretion disk's inner region. The inner black circle marks the outflow object, the outer back circle marks the initial inner rim of the accretion disk. We see that, although no physical viscosity is present and the outflow is interacting with the disk, material is accreted, i.e. the accretion disk's inner rim moves inwards until it reaches the outflow object.
The second animation shows the same but without outflow object, this is just to show that the accretion disk's inner rim finds a stable configuration and the gas inside the inner cavity has a density of about 1000 cm-3, about 30 times lower than the accretion disk's density.
The third animation now includes MHD and the outflow object. The disk has a toroidal initial field configuration with an initial field strength of 1 mG. There are still clumps and streams of matter forming and moving inwards, but besides these features the inner rim seems to be stable with densities of the inner cavity of about 1000 cm-3 as seen in the second animation. So magnetic fields seem to play an important role in forming the inner cavity of the galactic center's accretion disk.
The fourth animation shows the corresponding face-on magnetic field strength.
First animation: surface density, without magn. field
Second animation: surface density, without magn. field and without outflow
Third animation: surface density, with magn. field and with outflow
Fourth animation: face-on magnetic field strength in Gauss
This leads to two important questions:
- Why is the disk collapsing when the outflow is switched on?
- Why and how do magnetic fields prevent this?
Some time ago we discussed a simple model for the extraction of angular momentum from the inner accretion disk:
Lets assume the accretion disk has an inner rim
, and the inner part of the accretion disk (a ring with mass M) fully interacts with the outflow. The outflow has a massflow , so after a time the mass has been added to the ring. That means that the mass of the ring increases, but its angular momentum does not change because the wind does not have any angular momentum. So the specitic angular momentum is decreasing, leading to the accretion of the disk's material. However, there must be a kind of critical outflow rate, because although the wind does not add any angular momentum to the disk, it does add radial momentum to the disk, and at some point the radial momentum will just win against the loss of angular momentum. To test this simple model I did the following very rough estimate: Both, the ring with mass and the wind with mass , have a specific potential energy and a specific kinetic energy (the wind due to its radial momentum and the ring due to its azimuthal velocity ). If we assume perfect mixing of the ring with the wind material, we can calculate the total energy of the "new" ring (with mass ), which is just the sum of the aforementioned:
When we further assume that the "new" ring finds a new orbital configuration (at radius
) we can use the virial theorem, where
to calculate its new orbit:
So if the wind velocity is smaller than the azimuthal velocity we would expect an inflow, otherwise an outflow of disk material. In the GC we have
, , , so we would always expect that the disk is pushed away from the central black hole, contrary to what we see in the simulations. So apparently this simple model does not account for the angular momentum loss.I furthermore made a rough estimate for the code performance on XSEDE's gordon cluster, as shown in the following figure:
Magnetic diffusion into the outflow object - Marvin
In the last weeks we were facing the problem that the magnetic field diffuses into the outflow object and is amplified there to considerable field strengths.
Here I show two more tests, both animations show the face-on view of the absolute value of the magnetic field strength in Gauss. The inner black circle marks the radius of the outflow object, the outer black circle marks the initial inner rim of the accretion disk.
Animation of the face-on magnetic field strength, outflow with increased ram pressure
Animation of the face-on magnetic field strength, outflow with inner radius set to zero
In the first animation I increased the ram pressure of the outflow by a factor of 8, so that the inner rim of the accretion disk does not reach the outflow object anymore. Nevertheless the magnetic field moves into the outflow object and is amplified to field strengths of more than 10 microgauss.
In the second animation I set the inner radius of the outflow object to zero, so there is no longer a central region where the velocity is set to zero. There is still a magnetic field present in the outflow object, but with very low field strengths of about 10-11 Gauss. I don't think that these have an effect on the results, so in future simulations I will just set the inner outflow radius to zero to avoid this problem.
Face-on magnetic field strength, outflow with increased ram pressure:
Face-on magnetic field strength, outflow with inner radius set to zero:
Interaction between outflow object and magnetic field - Marvin
I show here a simulation of the CND with central outflow object and with an initial toroidal magnetic field of 0.1 milligauss and plot the face-on inverse beta-parameter in the first animation and the edge-on inverse beta-parameter in the second animation. The inner black circle marks the outflow object, the outer black circle marks the initial inner rim of the accretion disk.
As we have seen in last weeks simulations, after some time a magnetic field develops in the outflow object, although this shouldn't be possible. The beams that we see in the edge-on view after some time are probably created because the magnetic field that apperas inside the outflow object is carried outwards by the outflow.
The last animation shows the face-on divergence, it is always below 10-15 Gauss/parsec. So there are no magnetic field sources, maybe the magnetic field is transported into the outflow object.
Animation of the inverse beta parameter, face on
Animation of the inverse beta parameter, edge on
Animation of the divergence in Gauss/parsec, face on
Inverse beta parameter, edge-on:
Divergence in Gauss/parsec, face-on:
Simulations with outflow object - Marvin
To investigate why the inner rim of the CND is unstable when adding an outflow object I did two simulations: One with outflow object but a outflow velocity of zero, and one with an increased ram pressure. The inner black circle in my animations marks the outflow object, the outer black circle marks the initial inner rim of the accretion disk.
Animation of the Surface Density, inner 4 pc of the disk, outflow velocity = zero
Animation of the Surface Density, inner 4 pc of the disk, ram pressure increaed by factor 8
The first Animation shows the surface density of the disk in a simulation with a zero velocity outflow object. The inner rim of the disk is stable, as we have seen in the simulations without outflow object.
The second animation shows the surface density of the disk when the outflow object has an increased ram pressure. I increased the density and the velocity of the outflow by a factor of 2, respectively, so the mass flow is increaed by a factor of 4 and the ram pressure by a factor of 8. We see that with these parameters the outflow is strong enough to prevent the inner rim from collapsing, but not strong enough to push the material outwards.
Surface Density, inner 4 pc of the disk, outflow velocity = zero:
Surface Density, inner 4 pc of the disk, ram pressure increaed by factor 8:
Interaction between outflow object and magnetic field - Marvin
I show here a simulation of the CND with central outflow object and with an initial vertical magnetic field of 0.1 milligauss. Initially I remove the magnetic field from a cylindrical region around the center.
The following animation shows the face-on view of the inverse beta-parameter with streamlines of the magnetic field lines in black. The small light blue circle indicates the outflow object, the large light blue circle the initial inner rim of the accretion disk. At first the area that is occupied by the outflow object stays field free, but after some time a magnetic field develops (or moves inwards). How is this possible? The magnetic field is frozen into the gas, so it should never be possible for the field to enter the outflow object. Unfortunately I don't have data about the divergence of the magnetic field, so I can not yet tell if the magnetic field moves there or is created there.
Animation of the inverse beta parameter, inner 4 pc of the disk
Simulations with a small outflow object - Marvin
In my previous post I have shown that the inner rim of the CND is stable when a resolution level of 4 is used (0.04 pc). I show here a similar simulation, but now with a central outflow object. The following animaion shows the surface density of the disk, the small black circle marks the outflow object, the large black circle the initial inner rim of the accretion disk. Surprisingly the disk's inner rim moves inwards until it hits the outflow object. Maybe the large velocity gradients/fluctuations increase the numerical viscosity in these simulations?
Numerical Viscosity - Marvin
I did some simulations with different resolutions to investigate the effects of numerical viscosity at the inner rim of the accretion disk. The simulations show the surface density of the disk and are without magnetic field and outflow. The black circle shows the initial inner rim of the accretion disk (1 pc). I run these simulations for about 10 orbital timescales (at 1 pc).
Animation of the Surface Density, inner 6 pc of the disk, level 2
Animation of the Surface Density, inner 6 pc of the disk, level 4
Animation of the Surface Density, inner 6 pc of the disk, level 5
The first simulation has a resolution of 0.16 pc (level 2) at the location of the inner rim and 0.04 pc (level 4) around the center (approx. the inner 0.2 pc). The disk material moves inwards quickly, probably due to numerical viscosity. At the end of the simulation the inner cavity has almost vanished.
In the second simulation the resolution at the location of the inner rim has changed to 0.04 pc (level 4). The inner rim still moves inwards, but only from 1 pc to 0.8 pc. We also have to consider that the disk and its inner rim have to relax from its initial conditions, as there are large density and pressure gradients at the beginning of the simulation.
The third simulation has a resolution of 0.02 pc (level 5) everywhere in the disk. This looks slightly better than the previous simulation, but qualitatively there is no big difference.
So I think level 4 is an adequate resolution to avoid too large effects due to numerical viscosity.
Surface Density, inner 6 pc of the disk, level 2:
Surface Density, inner 6 pc of the disk, level 4:
Surface Density, inner 6 pc of the disk, level 5:
CND with outflow and magnetic field
This simulation includes the central outflow with the usual parameters, and an initially vertical magnetic field with a field strength of 10 microgauss. The first animation shows the density of the central 4 pc, the interaction of the outflow with the inner rim of the accretion disk causes a clumpy structure, as it does in the simulation without magnetic field that I have shown last week. The second animation shows the face-on beta parameter of the disk's inner region. As it is expected with such a weak magnetic field, the beta parameter is very high, so the dynamics are completely dominated by the gas and not by the magnetic field. The third animation shows the edge-on beta parameter, the outflow shapes the magnetic field into a more or less radial configuration.
Animation of the density, face-on, inner 4 pc of the disk
Animation of the beta-parameter, edge-on, with streamlines of the magnetic Field
Density, face-on, inner 4 pc of the disk:
Beta-parameter, face-on, inner 4 pc of the disk, with streamlines of the magnetic Field:
Beta-parameter, edge-on, with streamlines of the magnetic Field:
CND with outflow - Marvin
The following animations show simulations with outflow object, but without magnetic field. The outflow has a radius of 0.8 pc, a velocity of 700 km/s and a density of 100 cm-3. I calculated the evolution for 5e4 years (my previous simulations were 1e6 years). As initial conditions I used data from one of my previous simulations, so we don't have to wait for the disk to relax from its initial conditions. I increased the resolution of the disk from level 2 to level 4 (now we have a resolution of 0.04 pc), so the disk still has to adjust to the new resolution. The first animation shows the edge-on view of the disk's density, I show only the inner 4 pc. Due to the increased resolution the disk becomes thinner, furthermore we see a "bow shock" that is situated at the inner rim of the accretion disk. The second animation shows the corresponding face-on view, like in the 2D simulation that I showed last week clumps are produced, but the density contrast is about a factor of 2 and thus much lower than in the 2D simulation (there the density contrast was 3 orders of magnitude).
Animation of the inner disk with central outflow - edge-on
Animation of the inner disk with central outflow - face-on
Disk with central outflow - edge-on:
Disk with central outflow - face-on:
CND with outflow - Marvin
To achieve a proper resolution of the central outflow I increased the refinement level in the center from 4 to 5 (corresponding to a resolution of 0.02 pc) and the thickness of the outflow to 0.3 pc (inner radius: 0.5 pc, outer radius 0.8 pc), corresponding to 15 cells. The grid layout is shown in the first picture. The first two animations (first: density, second: velocity) show a simulation with outflow object and gravity object embedded in an ambient medium (in 2D without disk and magnetic fields) and shows that the outflow is well resolved.
I did another 2D simulation with disk, cooling, outflow and gravity object, the last animation shows the inner 4 pc of the face-on density of the disk. Initially the disk has an inner rim of 1 pc, the outflow causes instabilities along the inner rim that lead to the formation of clumps. The clumps have sizes of approx 0.1-0.2 pc and masses of approx. 50 solar masses.
Outflow embedded in ambient medium: animation of the density
Outflow embedded in ambient medium: animation of the velocity
Outflow with disk in 2D: animation of the face-on density
Grid Layout for outflow simulations:
Clumping due to disk-outflow interaction:
Including an outflow in the CND - Marvin
I am currently working on including the outflow from the central stellar cluster in my simulations. The standard setup that came along with the module that I am using is shown in the first animation (edge-on view, the color code shows the absolute value of the velocity in km/s), we see that we have an inflow rather than an outflow, it seems that the outflow is not able to escape from the central region. The outflow velocity is set to 700 km/s, the escape speed is 200 km/s.
Then I did some test runs that consist only of an outflow object embedded in an ambient medium. The second animation shows a run with the standard parameters (the outflow has a thickness of zero). The outflow reaches a velocity of 140 km/s, far below the chosen 700 km/s. Increasing the thickness to 0.1 pc (third animation) leads to outflow velocities of around 700 km/s, but there are still some numerical artifacts, probably due to an unsufficient resolution.
So I am currently trying to find adequate values for the resolution and for the thickness of the outflow that yield the correct outflow velocity and decrease numerical artifacts to a reasonable level.
Animation of edge-on view of the absolute velocity
Animation of an outflow with thickness 0
Animation of an outflow with thickness 0.1 pc
Edge-on view of the absolute velocity:
CND with cooling and B-field - Marvin
This simulation of the CND was done with a vertical initial magnetic field with 100 microgauss. The first Animation shows the 3D mass density, after approx. half of the time the disk (at least the inner part) reaches a more or less settled state. However, there is a warping of the disk, but we have already seen this in the simulations without magnetic field. The second animation shows a face-on view of the mass density, a spiral pattern develops, but this feature is also present in simulations without magnetic field, as is the feature in the third animation, which shows the edge-on view of the gas velocity and the development of an outflow. The fourth and fifth Animations show edge-on and face-on views of the magnetic energy density with streamlines. The initially vertical field gets transformed into a toroidal field, the ambient medium hosts a chaotic magnetic field. The field strength in the disk reaches values of about 1 mG, which is indeed observed in the CND. So I think an initial field strength of 100 microgauss is a good choice. The magnetic field does not have a large effect on the gas density, as we see in the last Animation, which shows a face-on view of the beta parameter. It is almost always greater than one.
Animation of the 3D mass density
Animation of the mass density (face-on)
Animation of the gas velocity (edge-on)
Animation of the magnetic field (face-on)
Animation of the magnetic field (edge-on)
Updated cooling simulation - Marvin
I repeated the cooling simulation, but I changed some parameters: Gamma is now 1.4 (instead of 5/3) as it is expected for a diatomic molecule and the initial temperature is 200 K (instead of 300 K). The 3D mass density looks the same as last week, so I don't show it here. The Animation below shows the face-on view of the Temperature. It is roughly between 200 and 300 K, as it is expected for the CND. So I think our treatment of cooling is appropriate.
Animation of edge-on view of the Temperature
Face-on view of the Temperature:
Cooling in the CND - Marvin
I am using the H2-cooling function that is part of the NEQ-cooling module. This cooling function considers rotational and vibrational lines for several wavelength. I assume that the disk consists to 100% of H2 throughout the simulation.
The first two Animations show the 3D mass density of the CND, once with and once without cooling enabled. With cooling the CND is much thinner and the central densities are larger by two orders of magnitude.
The next two Animations show an edge-on view of the disk temperature. Without cooling the disk heats up to about 80000 K (initial disk temperature: 300 K) and thus is indistinguishable from the ambient medium. With cooling the disk reaches a temperature of about 800 K.
Animation of 3D gas density, without cooling
Animation of 3D gas density, with cooling
Animation of edge-on view of the Temperature, without cooling
Animation of edge-on view of the Temperature, with cooling
3D gas density, with cooling:
Edge-on view of the Temperature, with cooling:
CND with a very strong initial magnetic field - Marvin
These simulations have a strong (10-4 G) vertical initial magnetic field. Unlike my previous simulations, where the properties of the gas (e.g. gas density) were not that much affected by the magnetic field, here the development of the accretion disk seems to be dominated by the magnetic field.
The first animation shows the 3D mass density, after the first 20% of the simulation the disk starts to show strong spatial and temporal variations. Note also the density scale: The disk seems to be affected by a high mass loss.
The second animation shows the face-on view of the disk's plasma beta. The disk has regions with a high beta of 1000 or higher and regions with a beta near equipartition, but beta almost never falls below 1.
The last animation shows an edge-on view of the radial velocity (positive values only). The blue areas indicate inflowing material. We see that a conical outflow develops with speeds of about 700 km/s (these simulations were done without an outflow object). A very rough estimation gives outflow rates of 10-2 M_sun/yr. The total simulation time is 106 yr, so the huge mass loss is probably due to the conical outflow (total initial disk mass is 4x104 M_sun).
Animation of face-on view of the plasma beta
Animation of edge-on view of the radial velocity
3D gas density:
Face-on view of the plasma beta:
Edge-on view of the radial velocity:
Vertical Magnetic Fields in the CND - Marvin
I simulated the CND with a purely vertical initial magnetic field with a field strength of 0.01 mG. The first animation shows a face-on view of the disks magnetic energy density with streamlines and shows that the vertical field is converted quickly into a torodial field. In the corresponding animation of the edge-on view we see that the vertical Field completely vanishes after a short period of time. The third animation shows the magnetic streamlines within a quarter of the disk: The initially vertical field gets dragged around and transforms into a torodial field.
Interestingly the late stage of these simulations resembles the simulations that I posted last week, although they start with a completely different initial field configuration. Even the magnetic energy density inside the disk reaches values of approx 10-6 G2 (corresponding to the observed values) although the simulation starts with a uniform energy density of 10-10 G2.
Animation of face-on view of the magn. energy density
Animation of edge-on view of the magn. energy density
Animation of the magnetic stream lines in a quarter of the disk
Magnetic Fields (face-on):
Magnetic Fields (edge-on):
Magnetic Streamlines in one quarter of the disk:
Update: Bz component of magnetized disk
As you suggested I plotted the Bz component of the disk's magnetic field in units of 1mG (this is the initial field strength of the toroidal field). In the ambient medium the Bz component can take up to 1% of the initial field, these are probably just random variations/noise. Along the z-axis of the disk the Bz component takes up to 10% of the initial field strength
Magnetized disk without outflow object
I simulated the circumnuclear disk without outflow object and extended the simulation time from 105 to 106 yr (corresponds to 3 orbital timescales at the accretion disks outer rim (4 pc) and to 100 orbital timescales at the "inner rim" 0.4 pc).
The first animation shows the 3D mass density. After ca. 40% of the simulation time the disk reaches a more or less settled state (i.e. its behaviour is not dominated by the initial conditions anymore). So I think 106 yr is an adequate simulation time.
The second animation shows egde-on and face-on views of the magnetic energy density with streamlines. We see that the ambient medium is in a quite turbulent state, with field strengths that vary by two orders of magnitude and magnetic loops that move away from the disk.
It seems that even without outflow object the magnetic field at the z-axis is converted into a vertical field. This can be seen best in the third Animation, where I plot the normalized z-component of the magnetic field.
Animation of the 3D mass density
Animation of the magnetic energy density (face-on and edge-on)
Animation of the normalized Bz-component
Mass density:
Magnetic Fields (face-on and edge-on):
Magnetic Fields (normalized Bz-component):
Magnetized Cirumnuclear Disk
This is the first simulation of the Circumnuclear Disk with magnetic fields. It includes an initial torodial field configuration with a constant field strength of 1 mG. The first animation shows the mass density, it looks more or less the same than the simulations without mgnetic fields. The next two animations show edge-on and face-on views of the magnetic energy density with fieldlines in black. The edge-on view shows a number of magnetic loops that are created and carried outwards, furthermore the central outflow seems to change the magnetic field in the center from a torodial to a vertical configuration. In the face-on animation the magnetic field maintains more or less its torodial configuration, but later the magnetic field seems to vanish from the inner region.
Animation of the 3D mass density
Animation of the magnetic energy density (edge-on)
Animation of the magnetic energy density (face-on)
Mass density:
Magnetic Fields (edge-on):
Magnetic Fields (face-on):
Error in CND-Simulation & Quasiperiodic Boundary Conditions
Eddie and I worked on finding the error that causes my code-crashes, he found a memory leak in the outflow module, but unfortunately that didn't solve the problem. I was able to install valgrind on one of the clusters that I am using, and found out that the code often tries to use an outfow object that has recently been free'd by the routine SinkParticleRestore (see more details in the ticket 324). So I will now focus my attention on this routine and I am open to suggestions and clues leading to the capture of the culprit.
Furthermore I worked on the quasiperiodic boundary conditions, I already finished parallelisation. I am now working on the MHD-part (see ticket 317), I created a module with a disk and a torodial magnetic field as a test case. Here you can see the movie. The results show on the left side a simulation with QPBC and on the right side a simulation of the full disk (but I only show one quarter here). The arrows show the direction of the magnetic field and the color code the magnetic energy density. In the QPBC simulation the field energy gets concentrated at the center and a kind of striped pattern appears at the lower boundary. I think there is still something going wrong, maybe its the EMF-Synchronisation.
Circumnuclear Disk
Here we see a simulation of the circumnuclear disk with a total time of about 1e6 years. I updated some of the parameters: I decreased the disk mass by one order of magnitude (it has now about 4e4 solar masses), increased the velocity of the outflow from 10 km/s to 700 km/s (speed of sound of the ambient medium: 100 km/s) and changed the outflow density so that the mass outflow is about 4e-3 M_sun/year.
At the beginning the edge-on view shows some Kelvin–Helmholtz-Type instabilities, but after some time the inner region of the disk seem to reach a more or less settled state.
Quasiperiodic Boundary Conditions
This clump that is moving around is a test of the quasiperiodic boundary conditions with a refinement level of 3 but still without magnetic fields and without parallelisation.
Simulation of the circumnuclear disk
This is a simulation of the circumnuclear disk with the same parameters as last week, I only extended the final time by a factor of 2.5 (and worked on a different cluster that uses gcc instead of intel compilers).
So the first 40% of this simulation should look the same as last weeks simulation, but as you can see, there is a kind of wind/outflow from the bottom of the disk. The reason is that the initial conditions are not set properly as you can see in the second picture, where we see a contour plot of the disk at t=0 that shows a kind of patchy structure at one of the quarters.
I'm currently trying to find out what is causing this behaviour.
Half-Plane Simulations
This is the Basic-Disk-Example in 3D, but with 4 refinement levels and with a simulation time t=1.
At first the disks show strong variations, maybe it first has to relax from its initial conditions.
The top figure show the simulation of the full disk, the bottom figure show the simulation of only the upper half-plane of the disk, saving half of the computational time. Reflecting boundary conditions were applied at the disk plane.
As we can see, both simulations yield the same results.
On my page you can see the corresponding face-on views.
Simulation of the Circumnuclear Disk
Parameters and scales:
timescale: 1e7 yr
number density scale: 1 ccm
selfgravitation and MHD are switched off
central mass: 4e6 M_sun
the center contains an outflow object with radius 0.4 pc, density 1e4 ccm and velocity 10 km/s
The inner rim of the disk first develops a 'wedge'-like shape due to the outflow, then two knobs delelop that are carried away by the outflow. Later on the outflows seems to be deflected by the inner rim.