<html><head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
</head>
<body>
<font size="+1"><font face="monospace">Andy -<br>
<br>
Perhaps I am still missing something about the overall setup,
but if the domain is periodic in the lateral (vertical)
direction and the initial data is axially symmetric, why does
the noise does not reappear at the bottom (left side in the
movie)? It seems to grow stronger away from the equator.<br>
</font></font><br>
<font size="+1"><font face="monospace"><font size="+1"><font face="monospace">So it seems like the outer boundary
conditions, perhaps? I imagine perturbations are created
from the very beginning. Plotfiles have limited accuracy and
using checkpoint files is preferred for the debugging
purposes. Also, one can always produce specific diagnostic
directly from the code, perhaps something simply such as
output a certain data at certain mesh locations.<br>
<br>
The question then is what is the origin of perturbations and
why they grow.<br>
<br>
As for the source, early versions of the FLASH USM solver
were known to produce spurious, small scale magnetic fields,
which was due to numerical issues involved in spectral
decomposition. I believe that problem was successfully
resolved, at least judging from our simulation results. I
would not completely exclude this possibility, and perhaps a
simplified version of your problem could offer some insight
here.<br>
<br>
</font></font>There are usually two reasons for the observed
zone-to-zone oscillations to appear. Either there is a diffusion
process that is not resolved correctly in time and the solution
blows up, or else small perturbations in one solution component
are not correctly coupled to other components. The former
problem is due to violation of the CFL condition, and the latter
is known as the odd-even decoupling and typically caused by
approximation used in solving the Riemann problem (flux
calculation).<br>
<br>
I do not think you are using any diffusion in the current setup
(I do not know what long wavelength perturbations are and how
you suppress them, though). But you can experiment with various
approximate Riemann solvers provided with the code.<br>
<br>
There was some discussion of modeling strongly stratified
systems with equilibria in the context of solar atmosphere. I
remember there were some stability issues reported.<br>
<br>
Tomek<br>
--</font></font><br>
<div class="moz-cite-prefix">On 4/9/21 11:21 AM, Andy Sha Liao
wrote:<br>
</div>
<blockquote type="cite" cite="mid:CAKfEzWB0DwNavLbad=PPxpZkRE+=JvD1fNAA+e5J+zBTQDvY1Q@mail.gmail.com">
<div dir="ltr">
<div dir="ltr">
<div dir="ltr">Tomek,
<div><br>
</div>
<div><br>
</div>
<div>I suppress the long wave oscillations and plot the
lineout for radial velocity over alfvén velocity along z
(lateral direction) at r={.0200,.0500,.1000} cm in
{cyan,magenta,green}: <a href="https://urldefense.com/v3/__https://drive.google.com/file/d/14T52ybx9ukQDee3zzaHdd-8VRs2Tp-C0/view?usp=sharing__;!!PhOWcWs!mVh2uuX9UIKjIAFE20HKpSEMByw5oYYQd5CayQbNnKO6h7Bs6OnpI7mbzIFWQQ$" target="_blank" moz-do-not-send="true">https://drive.google.com/file/d/14T52ybx9ukQDee3zzaHdd-8VRs2Tp-C0/view?usp=sharing</a></div>
<div><br>
</div>
<div>Zooming in on the green lineout: <a href="https://urldefense.com/v3/__https://drive.google.com/file/d/1wqay_ZmPtpvwuej8C0A9ZhqCdJszy1-l/view?usp=sharing__;!!PhOWcWs!mVh2uuX9UIKjIAFE20HKpSEMByw5oYYQd5CayQbNnKO6h7Bs6OnpI7mG4QBS8w$" target="_blank" moz-do-not-send="true">https://drive.google.com/file/d/1wqay_ZmPtpvwuej8C0A9ZhqCdJszy1-l/view?usp=sharing</a></div>
<div><br>
</div>
<div>The zoomed movie starts at frame 33, but the teeth are
observable much earlier, when their amplitudes are as low
~1 ppm of their centroid value. </div>
<div><br>
</div>
<div>The curious thing is that the teeth grow independently
of the lateral mean value of the quantity, and before they
become large enough to become disruptive, their pattern
stays coherent. </div>
<div><br>
</div>
<div><br>
</div>
<div>Andy</div>
</div>
</div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Thu, Apr 8, 2021 at 4:20 PM
Tomasz Plewa <<a href="mailto:tplewa@fsu.edu" target="_blank" moz-do-not-send="true">tplewa@fsu.edu</a>>
wrote:<br>
</div>
<blockquote class="gmail_quote" style="margin:0px 0px 0px
0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
<div> <font size="+1"><font face="monospace">Andy -<br>
<br>
I would think that with that many cells per scale height
the mesh resolution is adequate.<br>
</font></font><br>
<font size="+1"><font face="monospace"><font size="+1"><font face="monospace">I have a sense something happens
near the lower-left boundary (perhaps across the
equatorial plane), and a perturbation from that
region sweeps through the rest of the domain.<br>
<br>
</font></font>Whatever happens breaks the assumed
radial symmetry and lateral uniformity. I would be more
interested in lateral (momentum, pressure gradients,
field gradients) rather than radial components as the
former should nominally remain zero at all times. You
could try to suppress lateral fluxes and see whether any
particular part of the domain eventually becomes a
source of perturbations. In either case, the simulation
is in trouble as soon as lateral perturbations develop
and their source is not controlled.<br>
<br>
I assume the mesh is uniform, correct?<br>
<br>
In the plots, is the symmetry axis at the left and
equatorial plane at the bottom?<br>
<br>
Not sure if assuming reflecting boundary at Rmax is
justified? Depending how it is implemented in the code,
there might be a jump in some magnetic field components.
I would think that zero gradient/outflow conditions
might be safer as any perturbations will likely feed
back into the domain, and one would want to avoid that.
(It seems Tummel et al. actually stress that particular
point.)<br>
<br>
Have you tried a 1D version of this setup?<br>
<br>
Which version of FLASH are you using?<br>
<br>
Tomek<br>
--</font></font><br>
<div>On 4/8/21 11:12 AM, Andy Sha Liao wrote:<br>
</div>
<blockquote type="cite">
<div dir="ltr">
<div dir="ltr">
<div dir="ltr">
<div dir="ltr">Tomasz,
<div><br>
</div>
<div><br>
</div>
<div>Thanks for taking interest in our problem.
Let me send you a few movies from my GDrive so
you have a better picture.</div>
<div><br>
</div>
<div>First, the movie from which the stills were
sent to you previously:</div>
<div><br>
</div>
<div><a href="https://urldefense.com/v3/__https://drive.google.com/file/d/1pKKQzVHn8Qm9RVMNpX0CINJlqULoIfzw/view?usp=sharing__;!!PhOWcWs!nWLN84YtVshbYV71fKfEPMDMQ9Jrl6lAUw98Vz8sWBW6dtvRjvzPqMCWBYUCXQ$" target="_blank" moz-do-not-send="true">https://drive.google.com/file/d/1pKKQzVHn8Qm9RVMNpX0CINJlqULoIfzw/view?usp=sharing</a><br>
</div>
<div><br>
</div>
<div>The movie shows density, radial momentum, and
azimuthal magnetic field.</div>
<div><br>
</div>
<div>Next, a movie of the radial lineout near the
equatorial plane of the domain, of radial
velocity as a fraction of the characteristic
alfven speed ~240 km/s. I show results from two
different interpolation orders, olive is 3rd
order, magenta is 2nd order:</div>
<div><br>
</div>
<div><a href="https://urldefense.com/v3/__https://drive.google.com/file/d/1pCh_Lb4P6TbBWxNOItLotb7-SoWLqvmq/view?usp=sharing__;!!PhOWcWs!nWLN84YtVshbYV71fKfEPMDMQ9Jrl6lAUw98Vz8sWBW6dtvRjvzPqMDM5Tj5lA$" target="_blank" moz-do-not-send="true">https://drive.google.com/file/d/1pCh_Lb4P6TbBWxNOItLotb7-SoWLqvmq/view?usp=sharing</a><br>
</div>
<div><br>
</div>
<div>The problem is not the long waves, but the
sawteeth oscillations. In other simulations, I
suppressed the long waves, but the sawteeth
still came in on schedule. </div>
<div><br>
</div>
<div>To answer your questions, the scale height,
or characteristic length of the pinch is 0.0910
cm, as found in the reference in the previous
message, and the resolution is 32 cells per
characteristic length. We also ran up to 256
cells per characteristic length, but the problem
doesn't go away.</div>
<div><br>
</div>
<div><br>
</div>
<div>Andy</div>
</div>
</div>
</div>
</div>
</blockquote>
<br>
</div>
</blockquote>
</div>
</blockquote>
<br>
</body>
</html>