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About blog postsen-USTrac 1.4.1Timescale for the inwards migration of the circumnuclear disk's inner rim -- MarvinmblankWed, 09 Sep 2015 15:41:26 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank09092015
https://bluehound.circ.rochester.edu/astrobear/blog/mblank09092015<p>
The central stellar cluster's wind extracts angular momentum from the circumnuclear disk's inner rim, therefore the inner rim migrates
</p>
<p>
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
</p>
<p>
details.
</p>
<p>
We assume that angular momentum is extracted from a ring of radius <span class="trac-mathjax" style="display:none">s</span> and radial extension <span class="trac-mathjax" style="display:none">\Delta l</span>, therefore it has the mass <span class="trac-mathjax" style="display:none">M_{\text{D}} = 2 \pi \Sigma s \Delta l</span>. This ring has a total angular momentum of <span class="trac-mathjax" style="display:none">J=j M_{\text{D}}</span> with specific angular momentum <span class="trac-mathjax" style="display:none">j</span>.
</p>
<p>
After a time <span class="trac-mathjax" style="display:none">dt</span> the wind adds the mass <span class="trac-mathjax" style="display:none">f \dot{M} dt</span> to the ring, where <span class="trac-mathjax" style="display:none">\dot{M}</span> is the wind's outflow rate and <span class="trac-mathjax" style="display:none">f</span> is the
</p>
<p>
fraction of the wind that interacts with the disk.
Thus the ring has the mass <span class="trac-mathjax" style="display:none">M_{D} + f \dot{M} dt</span> and the specific angular momentum
</p>
<p>
<span class="trac-mathjax" style="display:none">j' = \frac{J}{M_{D} + f \dot{M} dt} = \frac{j}{1+\frac{f \dot{M}}{M_{\text{D}}} dt}</span>.
</p>
<p>
Furthermore the ring moves to <span class="trac-mathjax" style="display:none">s-ds</span>, and because <span class="trac-mathjax" style="display:none">j = \sqrt{GMs}</span> and <span class="trac-mathjax" style="display:none">j' = \sqrt{GM(s-ds)}</span> we get
</p>
<p>
<span class="trac-mathjax" style="display:none">dt = \frac{M_{\text{D}}}{f \dot{M}} \left( \sqrt{\frac{s}{s-ds}} - 1\right) = \frac{M_{\text{D}}}{f \dot{M}} \frac{ds}{2 s}</span>
</p>
<p>
where I used a taylor expansion in the last step.
</p>
<p>
Using <span class="trac-mathjax" style="display:none">M_{\text{D}} = 2 \pi \Sigma s \Delta l</span>, <span class="trac-mathjax" style="display:none">f=\frac{h}{2s}</span> and integrating this equation gives the time the inner rim needs to
</p>
<p>
migrate inwards:
</p>
<p>
<span class="trac-mathjax" style="display:none">\tau = \int dt = \int \frac{M_{\text{D}}}{f \dot{M}} \frac{ds}{2 s} = \frac{2 \pi \Sigma \Delta l}{h \dot{M}} \int s ds = \frac{\pi \Sigma \Delta l}{h \dot{M}} \left( s_0^2 - s_\text{i}^2\right)</span>,
</p>
<p>
where <span class="trac-mathjax" style="display:none">s_0</span> is the disk's initial inner rim and <span class="trac-mathjax" style="display:none">s_\text{i}</span> the location of the rim after inwards migration.
</p>
<p>
In our simulations <span class="trac-mathjax" style="display:none">s_\text{i} = 0.5 \text{pc}</span>, for <span class="trac-mathjax" style="display:none">s_0 = 1 \text{pc}</span> we get <span class="trac-mathjax" style="display:none">\tau = 22 \cdot 10^4 \text{yrs}</span>, for <span class="trac-mathjax" style="display:none">s_0 = 2 \text{pc}</span> we get <span class="trac-mathjax" style="display:none">\tau = 112 \cdot 10^4 \text{yrs}</span>. The actual time the inner rim needs to move inwards (as observed in the simulations) is <span class="trac-mathjax" style="display:none">20 \cdot 10^4 \text{yrs}</span> and <span class="trac-mathjax" style="display:none">65 \cdot 10^4 \text{yrs}</span>, respectively. For the <span class="trac-mathjax" style="display:none">s_0 = 1 \text{pc}</span> simulation this is pretty close, for the <span class="trac-mathjax" style="display:none">s_0 = 2 \text{pc}</span> 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.
</p>
Simulations of the circumnuclear disk with a larger inner cavity, Part II -- MarvinmblankSat, 08 Aug 2015 13:12:22 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank08082015b
https://bluehound.circ.rochester.edu/astrobear/blog/mblank08082015b<p>
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 <span class="trac-mathjax" style="display:none">10^4</span> yrs. The r=1pc simulations cover a time of <span class="trac-mathjax" style="display:none">6 \cdot 10^5</span> years, the r=2pc simulations <span class="trac-mathjax" style="display:none">10^6</span> 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.)
</p>
<p>
</p>
<table class="wiki">
<tr><th> Number </th><th style="text-align: center"> properties </th><th> linestyle
</th></tr><tr><td> 1. </td><td> without magn. fields and with outflow </td><td style="text-align: center"> blue, dotted
</td></tr><tr><td> 2. </td><td> with magn. fields (B=1mG) and with outflow </td><td style="text-align: center"> red, dashed
</td></tr><tr><td> 3. </td><td> without magnetic fields and without outflow </td><td style="text-align: center"> turquoise, solid
</td></tr><tr><td> 4. </td><td> like 2., but with B=0.1mG </td><td> green, dashdotted
</td></tr><tr><td> 5. </td><td> with magn. fields and without outflow </td><td style="text-align: center"> violet, solid
</td></tr></table>
<p>
First figure: CND with an inner cavity of r=1pc
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank08082015b/150807_sigma1D_cav1.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank08082015b/150807_sigma1D_cav1.png" width="700px" /></a>
</p>
<p>
Second figure: CND with an inner cavity of r=2pc
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank08082015b/150807_sigma1D_cav2.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank08082015b/150807_sigma1D_cav2.png" width="700px" /></a>
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/150807_sigma1D_cav1_animation.gif" title="Attachment '150807_sigma1D_cav1_animation.gif' in u/mblank">First animation: CND with an inner cavity of r=1pc</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/150807_sigma1D_cav1_animation.gif" title="Download"></a>
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/150807_sigma1D_cav2_animation.gif" title="Attachment '150807_sigma1D_cav2_animation.gif' in u/mblank">Second animation: CND with an inner cavity of r=2pc</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/150807_sigma1D_cav2_animation.gif" title="Download"></a>
</p>
<p>
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 <span class="trac-mathjax" style="display:none">\tau_{\text{visc}} = \frac{r^2}{\nu}</span> (on how to determine the numerical viscosity I refer to one of my other posts: <a class="ext-link" href="https://astrobear.pas.rochester.edu/trac/blog/mblank05302015"><span class="icon"></span>https://astrobear.pas.rochester.edu/trac/blog/mblank05302015</a>).
Secondly, gas pressure can cause the inner rim to move inwards, the corresponding timescale is <span class="trac-mathjax" style="display:none">\tau_{\text{gas}} = \frac{\Delta r}{c_{\text{s}}}</span> with the speed of sound <span class="trac-mathjax" style="display:none">c_{\text{s}}</span>. Then, magnetic pressure can also cause the inner rim to migrate inwards on a timescale <span class="trac-mathjax" style="display:none">\tau_{\text{B}} = \frac{\Delta r}{v_{\text{A}}}</span>, where <span class="trac-mathjax" style="display:none">v_{\text{A}}</span> 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 <span class="trac-mathjax" style="display:none">10^4</span> yrs. <span class="trac-mathjax" style="display:none">\tau_{\text{wind,ana}}</span> is the timescale for the angular momentum extraction according to the formula presented in our paper draft, and <span class="trac-mathjax" style="display:none">\tau_{\text{wind,obs}}</span> is the time the inner rim actually needs to move inwards (as seen in the simulations).
</p>
<p>
…….
</p>
<table class="wiki">
<tr><th style="text-align: center"> timescale </th><th> r=1pc </th><th style="text-align: center"> r=2pc
</th></tr><tr><td> <span class="trac-mathjax" style="display:none">\tau_{\text{orb}}</span> </td><td style="text-align: center"> 4.5 </td><td style="text-align: center"> 12.7
</td></tr><tr><td> <span class="trac-mathjax" style="display:none">\tau_{\text{visc}}</span> </td><td style="text-align: center"> 2400 </td><td style="text-align: center"> 21700
</td></tr><tr><td> <span class="trac-mathjax" style="display:none">\tau_{\text{gas}}</span> </td><td style="text-align: center"> 37.8 </td><td style="text-align: center"> 113.5
</td></tr><tr><td> <span class="trac-mathjax" style="display:none">\tau_{\text{B=1mG}}</span> </td><td style="text-align: center"> 10 </td><td style="text-align: center"> 30
</td></tr><tr><td> <span class="trac-mathjax" style="display:none">\tau_{\text{B=0.1mG}}</span> </td><td style="text-align: center"> 100 </td><td style="text-align: center"> 300
</td></tr><tr><td> <span class="trac-mathjax" style="display:none">\tau_{\text{wind,ana}}</span> </td><td style="text-align: center"> 49 </td><td style="text-align: center"> 490
</td></tr><tr><td> <span class="trac-mathjax" style="display:none">\tau_{\text{wind,obs,B=1mG}}</span> </td><td style="text-align: center"> 8 </td><td style="text-align: center"> 42
</td></tr><tr><td> <span class="trac-mathjax" style="display:none">\tau_{\text{wind,obs,B=0}}</span> </td><td style="text-align: center"> 20 </td><td style="text-align: center"> 65
</td></tr></table>
<p>
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.
</p>
<p>
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 <span class="trac-mathjax" style="display:none">20 \cdot 10^4</span> yrs to reach the outflow object (<span class="trac-mathjax" style="display:none">65 \cdot 10^4</span> yrs for the r=2pc simulation). The first <span class="trac-mathjax" style="display:none">25 \cdot 10^4</span> yrs (<span class="trac-mathjax" style="display:none">80 \cdot 10^4</span> 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.
</p>
<p>
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 <span class="trac-mathjax" style="display:none">8 \cdot 10^4</span> yrs (<span class="trac-mathjax" style="display:none">42 \cdot 10^4</span> 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.
</p>
<p>
Conclusions:
</p>
<ol><li>The outflow is causing the inner rim to movs inwards, the higher the magnetic field strength, the faster it is moving.
</li></ol><p>
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 <span class="trac-mathjax" style="display:none">\tau_{\text{B=1mG}}</span> 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 <span class="trac-mathjax" style="display:none">\tau_{\text{gas}}</span> 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).
</p>
<ol start="2"><li> After the inner rim has reached the outflow object, the surface density is lower for higher magnetic field strengths.
</li></ol><p>
But as always, answering some questions leads to a larger number of new questions:
</p>
<ol><li>Why is the collapse faster for higher magnetic field strengths?
Magnetic fields seem to accelerate the angular momentum extraction.
</li></ol><ol start="2"><li>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?
</li></ol><ol start="3"><li><span class="trac-mathjax" style="display:none">\tau_{\text{wind,ana}}</span> and <span class="trac-mathjax" style="display:none">\tau_{\text{wind,obs,B=0}}</span> fit quite well for the r=1pc simulations, but why is there such a huge difference for the r=2pc simulations?
</li></ol><ol start="4"><li>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.
</li></ol>Spreading ring calculations, Part II -- MarvinmblankSat, 08 Aug 2015 12:51:10 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank08082015
https://bluehound.circ.rochester.edu/astrobear/blog/mblank08082015<p>
In one of my last blog posts (<a class="ext-link" href="https://astrobear.pas.rochester.edu/trac/blog/mblank05302015"><span class="icon"></span>https://astrobear.pas.rochester.edu/trac/blog/mblank05302015</a>) 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 <span class="trac-mathjax" style="display:none">\alpha_{\text{num}}</span> that gives the numerical viscosity in units of the maximum <span class="trac-mathjax" style="display:none">\alpha</span>-viscosity <span class="trac-mathjax" style="display:none">\nu_{\alpha} = c_{\text{s}} h</span>.
</p>
<p>
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.
</p>
<p>
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).
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank08082015/150807_spreadingring_resolution.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank08082015/150807_spreadingring_resolution.png" width="700px" /></a>
</p>
<p>
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 <span class="trac-mathjax" style="display:none">\alpha_{\text{num}}</span> following the procedure described in my previous post, and I additionally give the runtime of the simulations, all have been calculated using 64 cores:
</p>
<table class="wiki">
<tr><th> resolution </th><th> <span class="trac-mathjax" style="display:none">\alpha_{\text{num}} [10^{-2}]</span> </th><th> runtime
</th></tr><tr><td> 0.08 pc </td><td> <span class="trac-mathjax" style="display:none">8.11 \pm 0.21</span> </td><td> 11.3 min
</td></tr><tr><td> 0.04 pc </td><td> <span class="trac-mathjax" style="display:none">6.22 \pm 0.05</span> </td><td> 1 h
</td></tr><tr><td> 0.02 pc </td><td> <span class="trac-mathjax" style="display:none">2.53 \pm 0.03</span> </td><td> 5.3 h
</td></tr><tr><td> 0.01 pc </td><td> <span class="trac-mathjax" style="display:none">1.14 \pm 0.02</span> </td><td> 31.1 h
</td></tr></table>
<p>
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 <span class="trac-mathjax" style="display:none">\alpha_{\text{num}} = (3.93 \pm 0.13) 10^{-2}</span>.
</p>
<p>
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:
<span class="trac-mathjax" style="display:none">\frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} = 0</span>
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 <span class="trac-mathjax" style="display:none">u(x)</span> at the point <span class="trac-mathjax" style="display:none">x_{i}</span>:
</p>
<p>
<span class="trac-mathjax" style="display:none">u_{i+1} = u_i + \Delta x \left( \frac{\partial u}{\partial x} \right)_i + \frac{1}{2} \left( \Delta x \right)^{2} \left( \frac{\partial^{2} u} {\partial x^{2}} \right)_i + ...</span>
</p>
<p>
Rearranging gives: <span class="trac-mathjax" style="display:none">\left( \frac{\partial u}{\partial x} \right)_i = \frac{u_{i+1}-u_i}{\Delta x} - \frac{1}{2} \Delta x \left( \frac{\partial^2 u}{\partial x^2} \right)_i</span>
</p>
<p>
And inserting this into the first equation gives:
</p>
<p>
<span class="trac-mathjax" style="display:none">\left( \frac{\partial u}{\partial t} \right)_i + \frac{u_{i+1}-u_i}{\Delta x} = \frac{1}{2} \Delta x \left( \frac{\partial^2 u}{\partial x^2} \right)_i</span>
</p>
<p>
The term on the right side is responsible for numerical viscosity, and is a linear function of the resolution <span class="trac-mathjax" style="display:none">\Delta x</span>.
</p>
Simulations of the circumnuclear disk with a larger inner cavity -- MarvinmblankSat, 30 May 2015 11:33:07 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank05302015b
https://bluehound.circ.rochester.edu/astrobear/blog/mblank05302015b<p>
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.
</p>
<p>
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.
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank05302015b/sigma1D.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank05302015b/sigma1D.png" width="700px" /></a>
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank05302015b/sigma1D_cav2.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank05302015b/sigma1D_cav2.png" width="700px" /></a>
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
According to these new results,
is our previous statement that the magnetic field suppresses
the inflow of material towards the black hole still valid?
</p>
Spreading ring calculations -- MarvinmblankSat, 30 May 2015 11:25:39 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank05302015
https://bluehound.circ.rochester.edu/astrobear/blog/mblank05302015<p>
To measure the magnitude of numerical viscosity in disk simulations with AstroBEAR I have performed calculations of a spreading ring.
</p>
<p>
A short reminder on the spreading ring problem:
</p>
<p>
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:
</p>
<p>
<span class="trac-mathjax" style="display:none">\frac{\partial \Sigma}{\partial t} + \frac{1}{r} \frac{\partial}{\partial r} \left[ \frac{\frac{\partial}{\partial r} \left( \nu \Sigma r^3 \frac{\partial \omega}{\partial r} \right)}{\frac{\partial}{\partial r} \left( r^2 \omega\right)} \right] = 0</span>
</p>
<p>
Here <span class="trac-mathjax" style="display:none">\Sigma</span> is the accretion disk's surface density, <span class="trac-mathjax" style="display:none">\omega</span> its angular frequency and <span class="trac-mathjax" style="display:none">\nu</span> 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 <span class="trac-mathjax" style="display:none">r_0</span>
</p>
<p>
<span class="trac-mathjax" style="display:none">\Sigma \sim \delta (r-r_0)</span>
</p>
<p>
this equation has the following analytical solution (eq. 1):
</p>
<p>
<span class="trac-mathjax" style="display:none">\Sigma (r,t) \sim \frac{1}{12 \pi r_0^2 x^{1/4} \tau} \text{exp} \left[ - \frac{(1+x^2)}{\tau} \right] I_{1/4} (\frac{2 x}{12 \tau})</span>
</p>
<p>
with
</p>
<p>
<span class="trac-mathjax" style="display:none">x=r/r_0</span>
</p>
<p>
<span class="trac-mathjax" style="display:none">\tau=\frac{\nu t}{r_0^2}</span> (eq. 2)
</p>
<p>
and the modified Bessel function of the first kind <span class="trac-mathjax" style="display:none">I_{1/4}</span>.
</p>
<p>
As a delta peak is numerically difficult to handle I use this analytical solution as initial condition, with <span class="trac-mathjax" style="display:none">r_0 = 2\,\text{pc}</span> and initial values for <span class="trac-mathjax" style="display:none">\tau</span> of <span class="trac-mathjax" style="display:none">\tau_{\text{i}}=0.01</span> and <span class="trac-mathjax" style="display:none">\tau_{\text{i}}=0.001</span>, respectively.
</p>
<p>
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.
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/150528_spreadingring_tau001_sigma.gif" title="Attachment '150528_spreadingring_tau001_sigma.gif' in u/mblank">First animation: surface density for $\tau_{\text{i}} = 0.001$</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/150528_spreadingring_tau001_sigma.gif" title="Download"></a>
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank05302015/tau_001_sigma_lin_0050.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank05302015/tau_001_sigma_lin_0050.png" width="700px" /></a>
</p>
<p>
The following two figures show the radial surface density profiles, the first one for <span class="trac-mathjax" style="display:none">\tau_{\text{i}}=0.001</span> and the second one for <span class="trac-mathjax" style="display:none">\tau_{\text{i}}=0.01</span>. 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.
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank05302015/spreadingring_tau001.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank05302015/spreadingring_tau001.png" width="700px" /></a>
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank05302015/spreadingring_tau01.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank05302015/spreadingring_tau01.png" width="700px" /></a>
</p>
<p>
This fit allows to determine the corresponding value of <span class="trac-mathjax" style="display:none">\tau</span>. Thus we have <span class="trac-mathjax" style="display:none">\tau</span> as a function of time, determining the slope of this function gives the viscosity <span class="trac-mathjax" style="display:none">\nu</span> (see eq. 2).
</p>
<p>
I define the parameter
</p>
<p>
<span class="trac-mathjax" style="display:none">\alpha_{\text{num}}=\frac{\nu}{\nu_{\alpha}}</span>
</p>
<p>
where <span class="trac-mathjax" style="display:none">\nu_{\alpha} = h \cdot c_{\text{s}}</span> is the maximal alpha viscosity, with <span class="trac-mathjax" style="display:none">h=0.2 \text{pc}</span> and <span class="trac-mathjax" style="display:none">c_{\text{s}}=1300 \frac{\text{m}}{\text{s}}</span>
I use typical values of the CND.
</p>
<p>
For the <span class="trac-mathjax" style="display:none">\tau_{\text{i}}=0.001</span> simulation I get a value of
</p>
<p>
<span class="trac-mathjax" style="display:none">\alpha_{\text{num}}=(3.93 \pm 0.13) \cdot 10^{-2}</span>
</p>
<p>
and for the <span class="trac-mathjax" style="display:none">\tau_{\text{i}}=0.01</span> simulation a value of
</p>
<p>
<span class="trac-mathjax" style="display:none">\alpha_{\text{num}}=(2.86 \pm 0.08) \cdot 10^{-2}</span>
</p>
<p>
Considering that in the literature a value of <span class="trac-mathjax" style="display:none">\alpha=0.1</span> 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.
</p>
<p>
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.
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/150528_spreadingring_tau001_wind_sigma.gif" title="Attachment '150528_spreadingring_tau001_wind_sigma.gif' in u/mblank">Second animation: surface density for $\tau = 0.001$, with central outflow</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/150528_spreadingring_tau001_wind_sigma.gif" title="Download"></a>
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank05302015/tau_001_wind_sigma_lin.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank05302015/tau_001_wind_sigma_lin.png" width="700px" /></a>
</p>
<p>
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.
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank05302015/spreadingring_tau001_wind.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank05302015/spreadingring_tau001_wind.png" width="700px" /></a>
</p>
MHD simulations of the circumnuclear disk - MarvinmblankTue, 27 May 2014 16:11:53 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank05272014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank05272014<p>
I have now finished a simulation that contains all the relevant physics, including cooling, MHD and the outflow object.
</p>
<p>
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.
</p>
<p>
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<sup>-3</sup>, about 30 times lower than the accretion disk's density.
</p>
<p>
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<sup>-3</sup> 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.
</p>
<p>
The fourth animation shows the corresponding face-on magnetic field strength.
</p>
<p>
<a class="missing attachment">First animation: surface density, without magn. field</a>
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank05272014/A_pics_sigma-log_0010.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank05272014/A_pics_sigma-log_0010.png" width="700px" /></a>
</p>
<p>
<a class="missing attachment">Second animation: surface density, without magn. field and without outflow</a>
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank05272014/B_pics_sigma-log_0020.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank05272014/B_pics_sigma-log_0020.png" width="700px" /></a>
</p>
<p>
<a class="missing attachment">Third animation: surface density, with magn. field and with outflow</a>
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank05272014/C_pics_sigma-log_0040.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank05272014/C_pics_sigma-log_0040.png" width="700px" /></a>
</p>
<p>
<a class="missing attachment">Fourth animation: face-on magnetic field strength in Gauss</a>
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank05272014/D_pics_B-face-zoom_0038.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank05272014/D_pics_B-face-zoom_0038.png" width="700px" /></a>
</p>
<p>
This leads to two important questions:
</p>
<ol><li>Why is the disk collapsing when the outflow is switched on?
</li><li>Why and how do magnetic fields prevent this?
</li></ol><p>
Some time ago we discussed a simple model for the extraction of angular momentum from the inner accretion disk:
</p>
<p>
Lets assume the accretion disk has an inner rim <span class="trac-mathjax" style="display:none">r_0</span>, and the inner part of the accretion disk (a ring with mass M) fully interacts with the outflow. The outflow has a massflow <span class="trac-mathjax" style="display:none">\dot{M}</span>, so after a time <span class="trac-mathjax" style="display:none">\Delta t</span> the mass <span class="trac-mathjax" style="display:none">\Delta m</span> 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 <span class="trac-mathjax" style="display:none">M</span> and the wind with mass <span class="trac-mathjax" style="display:none">\Delta m</span>, have a specific potential energy and a specific kinetic energy (the wind due to its radial momentum <span class="trac-mathjax" style="display:none">\Delta m \, v_{\text{wind}}</span> and the ring due to its azimuthal velocity <span class="trac-mathjax" style="display:none">v_{\phi,0}</span>).
If we assume perfect mixing of the ring with the wind material, we can calculate the total energy of the "new" ring (with mass <span class="trac-mathjax" style="display:none">M+\Delta m</span>), which is just the sum of the aforementioned:
</p>
<p>
<span class="trac-mathjax" style="display:none">E_{\text{total}} = E_{\text{pot,ring}} + E_{\text{pot,wind}} + E_{\text{kin,ring}} + E_{\text{kin,wind}} = - \frac{G M_{\text{BH}}M}{r_0} - \frac{G M_{\text{BH}} \, \Delta m}{r_0} + \frac{1}{2} \, M \, v_{\phi,0}^2 + \frac{1}{2} \, \Delta m \, v_{wind}^2</span>
</p>
<p>
When we further assume that the "new" ring finds a new orbital configuration (at radius <span class="trac-mathjax" style="display:none">r_n</span>) we can use the virial theorem, where
</p>
<p>
<span class="trac-mathjax" style="display:none">E_{\text{vir}}(r_n) = - \frac{G M_{\text{BH}}}{2r_n}(M+\Delta m)</span>
</p>
<p>
to calculate its new orbit:
</p>
<p>
<span class="trac-mathjax" style="display:none">r_{n} = \frac{r_0}{1+\frac{\Delta m}{M+\Delta m}\left( 1- (\frac{v_{\text{wind}}}{v_{\phi,0}})^2 \right)}</span>
</p>
<p>
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 <span class="trac-mathjax" style="display:none">r_{0}=1\,\text{pc}</span>, <span class="trac-mathjax" style="display:none">v_{\text{wind}}=700\,\text{km/s}</span>, <span class="trac-mathjax" style="display:none">v_{\phi,0}=135\,\text{km/s}</span>, 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.
</p>
<p>
I furthermore made a rough estimate for the code performance on XSEDE's gordon cluster, as shown in the following figure:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank05272014/E_scaling.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank05272014/E_scaling.png" width="700px" /></a>
</p>
Magnetic diffusion into the outflow object - Marvin mblankMon, 31 Mar 2014 16:01:04 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank03312014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank03312014<p>
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.
</p>
<p>
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.
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/140329_B-face_fast-of.gif" title="Attachment '140329_B-face_fast-of.gif' in u/mblank">Animation of the face-on magnetic field strength, outflow with increased ram pressure</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/140329_B-face_fast-of.gif" title="Download"></a>
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/140329_B-face_thick-of.gif" title="Attachment '140329_B-face_thick-of.gif' in u/mblank">Animation of the face-on magnetic field strength, outflow with inner radius set to zero</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/140329_B-face_thick-of.gif" title="Download"></a>
</p>
<p>
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.
</p>
<p>
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<sup>-11</sup> 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.
</p>
<p>
Face-on magnetic field strength, outflow with increased ram pressure:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank03312014/pics_B-face_0024.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank03312014/pics_B-face_0024.png" width="700px" /></a>
</p>
<p>
Face-on magnetic field strength, outflow with inner radius set to zero:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank03312014/pics_B-face_0025.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank03312014/pics_B-face_0025.png" width="700px" /></a>
</p>
Interaction between outflow object and magnetic field - MarvinmblankMon, 24 Mar 2014 17:00:36 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank03242014b
https://bluehound.circ.rochester.edu/astrobear/blog/mblank03242014b<p>
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.
</p>
<p>
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.
</p>
<p>
The last animation shows the face-on divergence, it is always below 10<sup>-15</sup> Gauss/parsec. So there are no magnetic field sources, maybe the magnetic field is transported into the outflow object.
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/140323_CND_beta-inv-face.gif" title="Attachment '140323_CND_beta-inv-face.gif' in u/mblank">Animation of the inverse beta parameter, face on</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/140323_CND_beta-inv-face.gif" title="Download"></a>
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/140323_CND_beta-inv-edge.gif" title="Attachment '140323_CND_beta-inv-edge.gif' in u/mblank">Animation of the inverse beta parameter, edge on</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/140323_CND_beta-inv-edge.gif" title="Download"></a>
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/140323_CND_div-face.gif" title="Attachment '140323_CND_div-face.gif' in u/mblank">Animation of the divergence in Gauss/parsec, face on</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/140323_CND_div-face.gif" title="Download"></a>
</p>
<p>
Inverse beta parameter, edge-on:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank03242014b/pics_beta-inv-edge_0020.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank03242014b/pics_beta-inv-edge_0020.png" width="700px" /></a>
</p>
<p>
Divergence in Gauss/parsec, face-on:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank03242014b/pics_div-face_0025.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank03242014b/pics_div-face_0025.png" width="700px" /></a>
</p>
Simulations with outflow object - MarvinmblankMon, 24 Mar 2014 16:54:24 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank03242014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank03242014<p>
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.
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/140320_CND_sigma_outf-v0.gif" title="Attachment '140320_CND_sigma_outf-v0.gif' in u/mblank">Animation of the Surface Density, inner 4 pc of the disk, outflow velocity = zero</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/140320_CND_sigma_outf-v0.gif" title="Download"></a>
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/140323_CND_sigma_Mx4.gif" title="Attachment '140323_CND_sigma_Mx4.gif' in u/mblank">Animation of the Surface Density, inner 4 pc of the disk, ram pressure increaed by factor 8</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/140323_CND_sigma_Mx4.gif" title="Download"></a>
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
Surface Density, inner 4 pc of the disk, outflow velocity = zero:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank03242014/pics_sigma_0020.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank03242014/pics_sigma_0020.png" width="700px" /></a>
</p>
<p>
Surface Density, inner 4 pc of the disk, ram pressure increaed by factor 8:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank03242014/pics_sigma_0030.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank03242014/pics_sigma_0030.png" width="700px" /></a>
</p>
Interaction between outflow object and magnetic field - Marvin mblankTue, 18 Mar 2014 17:34:42 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank03182014b
https://bluehound.circ.rochester.edu/astrobear/blog/mblank03182014b<p>
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.
</p>
<p>
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.
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/140316_beta-inv-face.gif" title="Attachment '140316_beta-inv-face.gif' in u/mblank">Animation of the inverse beta parameter, inner 4 pc of the disk</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/140316_beta-inv-face.gif" title="Download"></a>
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank03182014b/pics_beta-inv-face_0015.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank03182014b/pics_beta-inv-face_0015.png" width="700px" /></a>
</p>
Simulations with a small outflow object - MarvinmblankTue, 18 Mar 2014 17:31:45 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank03182014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank03182014<p>
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?
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/140316_sigma-log.gif" title="Attachment '140316_sigma-log.gif' in u/mblank">Animation of the Surface Density, inner 4 pc of the disk</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/140316_sigma-log.gif" title="Download"></a>
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank03182014/pics_sigma-log_0020.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank03182014/pics_sigma-log_0020.png" width="700px" /></a>
</p>
Numerical Viscosity - MarvinmblankMon, 10 Mar 2014 17:01:57 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank03102014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank03102014<p>
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).
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/140309_numvisc_sigma_level2.gif" title="Attachment '140309_numvisc_sigma_level2.gif' in u/mblank">Animation of the Surface Density, inner 6 pc of the disk, level 2</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/140309_numvisc_sigma_level2.gif" title="Download"></a>
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/140309_numvisc_sigma_level4.gif" title="Attachment '140309_numvisc_sigma_level4.gif' in u/mblank">Animation of the Surface Density, inner 6 pc of the disk, level 4</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/140309_numvisc_sigma_level4.gif" title="Download"></a>
</p>
<p>
<a class="attachment" href="https://bluehound.circ.rochester.edu/astrobear/attachment/wiki/u/mblank/140309_numvisc_sigma_level5.gif" title="Attachment '140309_numvisc_sigma_level5.gif' in u/mblank">Animation of the Surface Density, inner 6 pc of the disk, level 5</a><a class="trac-rawlink" href="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/wiki/u/mblank/140309_numvisc_sigma_level5.gif" title="Download"></a>
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
So I think level 4 is an adequate resolution to avoid too large effects due to numerical viscosity.
</p>
<p>
Surface Density, inner 6 pc of the disk, level 2:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank03102014/sigma_0020.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank03102014/sigma_0020.png" width="700px" /></a>
</p>
<p>
Surface Density, inner 6 pc of the disk, level 4:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank03102014/sigma_0021.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank03102014/sigma_0021.png" width="700px" /></a>
</p>
<p>
Surface Density, inner 6 pc of the disk, level 5:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank03102014/sigma_0022.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank03102014/sigma_0022.png" width="700px" /></a>
</p>
CND with outflow and magnetic fieldmblankMon, 24 Feb 2014 17:31:05 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank02242014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank02242014<p>
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.
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140223_CND_rho-face-zoom-lin.gif"><span class="icon"></span>Animation of the density, face-on, inner 4 pc of the disk</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140223_CND_beta-face-stream.gif"><span class="icon"></span>Animation of the beta-parameter, face-on, inner 4 pc of the disk, with streamlines of the magnetic Field</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140223_CND_beta-edge-stream.gif"><span class="icon"></span>Animation of the beta-parameter, edge-on, with streamlines of the magnetic Field</a>
</p>
<p>
Density, face-on, inner 4 pc of the disk:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank02242014/pics_rho-face-zoom-lin_0016.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank02242014/pics_rho-face-zoom-lin_0016.png" width="700px" /></a>
</p>
<p>
Beta-parameter, face-on, inner 4 pc of the disk, with streamlines of the magnetic Field:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank02242014/pics_beta-face-stream_0017.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank02242014/pics_beta-face-stream_0017.png" width="700px" /></a>
</p>
<p>
Beta-parameter, edge-on, with streamlines of the magnetic Field:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank02242014/pics_beta-edge-stream_0017.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank02242014/pics_beta-edge-stream_0017.png" width="700px" /></a>
</p>
CND with outflow - MarvinmblankMon, 17 Feb 2014 17:59:10 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank02172014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank02172014<p>
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<sup>-3</sup>. 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).
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140214_CND_rho-edge-zoom.gif"><span class="icon"></span>Animation of the inner disk with central outflow - edge-on</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140214_CND_rho-face-zoom-lin.gif"><span class="icon"></span>Animation of the inner disk with central outflow - face-on</a>
</p>
<p>
Disk with central outflow - edge-on:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank02172014/pics_rho-edge-zoom_0020.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank02172014/pics_rho-edge-zoom_0020.png" width="700px" /></a>
</p>
<p>
Disk with central outflow - face-on:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank02172014/pics_rho-face-zoom-lin_0010.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank02172014/pics_rho-face-zoom-lin_0010.png" width="700px" /></a>
</p>
CND with outflow - MarvinmblankMon, 10 Feb 2014 18:05:27 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank02102014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank02102014<p>
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.
</p>
<p>
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.
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140204_outflow_rho.gif"><span class="icon"></span>Outflow embedded in ambient medium: animation of the density</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140204_outflow_vel.gif"><span class="icon"></span>Outflow embedded in ambient medium: animation of the velocity</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140209_CND_2D_rho_zoom.gif"><span class="icon"></span>Outflow with disk in 2D: animation of the face-on density</a>
</p>
<p>
Grid Layout for outflow simulations:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank02102014/pics_grid.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank02102014/pics_grid.png" width="900px" /></a>
</p>
<p>
Clumping due to disk-outflow interaction:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank02102014/pics_rho_zoom_0040.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank02102014/pics_rho_zoom_0040.png" width="700px" /></a>
</p>
Including an outflow in the CND - MarvinmblankMon, 03 Feb 2014 17:28:39 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank02032014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank02032014<p>
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.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140126_CND_vel-ges.gif"><span class="icon"></span>Animation of edge-on view of the absolute velocity</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140202_outflow_v-ges_thick0.gif"><span class="icon"></span>Animation of an outflow with thickness 0</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140202_outflow_v-ges_thick1.gif"><span class="icon"></span>Animation of an outflow with thickness 0.1 pc</a>
</p>
<p>
Edge-on view of the absolute velocity:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank02032014/pics_v-ges_0025.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank02032014/pics_v-ges_0025.png" width="700px" /></a>
</p>
CND with cooling and B-field - MarvinmblankMon, 27 Jan 2014 17:37:18 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank01272014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank01272014<p>
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.
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140126_CND_rho3D.gif"><span class="icon"></span>Animation of the 3D mass density</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140126_CND_rho-face.gif"><span class="icon"></span>Animation of the mass density (face-on)</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140126_CND_vel-edge.gif"><span class="icon"></span>Animation of the gas velocity (edge-on)</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140126_CND_B-face.gif"><span class="icon"></span>Animation of the magnetic field (face-on)</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140126_CND_B-edge.gif"><span class="icon"></span>Animation of the magnetic field (edge-on)</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140126_CND_beta_face.gif"><span class="icon"></span>Animation of the beta-parameter (face-on)</a>
</p>
<p>
3D gas density:
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank01272014/pics_rho3D_0090.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank01272014/pics_rho3D_0090.png" width="700px" /></a>
</p>
<p>
Gas velocity (edge-on):
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank01272014/pics_vel-edge_0084.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank01272014/pics_vel-edge_0084.png" width="700px" /></a>
</p>
<p>
Beta parameter (face-on):
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank01272014/pics_beta_edge_0060.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank01272014/pics_beta_edge_0060.png" width="700px" /></a>
</p>
Updated cooling simulation - MarvinmblankMon, 20 Jan 2014 23:15:49 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank01202014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank01202014<p>
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.
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140119_CND_Temp-face.gif"><span class="icon"></span>Animation of edge-on view of the Temperature</a>
</p>
<p>
Face-on view of the Temperature:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank01202014/pics_Temp-face_0098.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank01202014/pics_Temp-face_0098.png" width="700px" /></a>
</p>
Cooling in the CND - MarvinmblankMon, 13 Jan 2014 18:05:48 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank01132014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank01132014<p>
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.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140108_CND_rho3D.gif"><span class="icon"></span>Animation of 3D gas density, without cooling</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140106_CND_rho3D.gif"><span class="icon"></span>Animation of 3D gas density, with cooling</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140108_CND_Temp.gif"><span class="icon"></span>Animation of edge-on view of the Temperature, without cooling</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/140106_CND_Temp.gif"><span class="icon"></span>Animation of edge-on view of the Temperature, with cooling</a>
</p>
<p>
3D gas density, with cooling:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank01132014/pics_rho3D_0060.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank01132014/pics_rho3D_0060.png" width="700px" /></a>
</p>
<p>
Edge-on view of the Temperature, with cooling:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank01132014/pics_T_edge_0060.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank01132014/pics_T_edge_0060.png" width="700px" /></a>
</p>
CND with a very strong initial magnetic field - MarvinmblankTue, 07 Jan 2014 23:39:00 GMT
https://bluehound.circ.rochester.edu/astrobear/blog/mblank01072014
https://bluehound.circ.rochester.edu/astrobear/blog/mblank01072014<p>
These simulations have a strong (10<sup>-4</sup> 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.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
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<sup>-2</sup> M_sun/yr. The total simulation time is 10<sup>6</sup> yr, so the huge mass loss is probably due to the conical outflow (total initial disk mass is 4x10<sup>4</sup> M_sun).
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/131219_cnd_rho3D.gif"><span class="icon"></span>Animation of 3D gas density</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/131219_cnd_beta.gif"><span class="icon"></span>Animation of face-on view of the plasma beta</a>
</p>
<p>
<a class="ext-link" href="http://astrobear.pas.rochester.edu/trac/astrobear/attachment/wiki/u/mblank/131219_cnd_vr.gif"><span class="icon"></span>Animation of edge-on view of the radial velocity</a>
</p>
<p>
3D gas density:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank01072014/pics_rho3D_0035.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank01072014/pics_rho3D_0035.png" width="700px" /></a>
</p>
<p>
Face-on view of the plasma beta:
</p>
<p>
<a href="https://bluehound.circ.rochester.edu/astrobear/attachment/blog/mblank01072014/pics_beta_0035.png" style="padding:0; border:none"><img crossorigin="anonymous" src="https://bluehound.circ.rochester.edu/astrobear/raw-attachment/blog/mblank01072014/pics_beta_0035.png" width="700px" /></a>
</p>
<p>
Edge-on view of the radial velocity:
</p>
<p>
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