Dear Gordon,
I don't disagree with your comment. My thought experiment was a bit different: take an isothermal flux tube expanding as r^2, and with a steady mass conserving flow. Now compare the density at the sonic point in such a Parker wind with the density at the same point if the flux tube were static (same density at the bottom). What I found is a difference of about a factor two.
Obviously the momentum equation shows you that the mass flux term scales as (v/c_s)^2 compared to the pressure gradient term, which was my other point, and hence you can solve in a series expansion around the static solution if the velocities involved are observed to be subsonic.
Cheers,
Piet
Gordon Petrie wrote:
Dear Piet,
If I understand the steady isothermal theory correctly, the scale height can become unbounded wherever the flow approaches the sonic point, v -> c_s, because of a v^2-c_s^2 factor. Beyond a (generally different) critical point, v=sqrt(g_s Z_0) where g_s is the solar surface gravitational acceleration and Z_0 a length scale of the flux tube spatial expansion, mass conservation demands that the density invert.
Best regards,
Gordon
On Mon, 07 Dec 2009 10:11:07 -0700 Petrus Martens pmartens@cfa.harvard.edu wrote:
Gordon,
I agree with your comment. In fact, it is fairly easily shown directly from the momentum equation that flows up to the sound speed increase the pressure scale height by at most a factor two, when v~c_s, much less when v<c_s because the flow effect scales as (v/c_s)^2.
A really original solution as to why observed loops can be overdense near their apexes was presented by Craig Deforest in 2007. If the unresolved strands that make up the observed loop increase in cross- section from footpoints to apex an observer would conclude that the loop is overdense and has a scale height much larger than what follows from the loop temperature. In reality there is simply more emitting volume near the apex. Of course one has to explain why strands increase in cross-section, while the loops that they collectively form appear not to, but Craig shows some nice images in his paper that seem to support his suggestion.
There are ways to verify this from observations.
Cheers,
Piet
Gordon Petrie wrote:
Dear All,
In a basic model of steady, isothermal hydrodynamic flow (http://adsabs.harvard.edu/abs/2006ApJ...649.1078P), steady flows can only affect hydrostatic scale heights under exceptional conditions. These states have fast flows approaching the sound speed and form a small part of the solution space right next to unphysical regimes with density inversions. If significantly many loops really are of this kind, it would be an interesting problem explaining why. On the other hand, it's clear from rho*V*A why steady flows decrease densities.
Best regards,
Gordon
On Mon, 07 Dec 2009 09:11:07 -0500 Leon Golub golub@head.cfa.harvard.edu wrote:
Jim,
We were indeed modelling the long, relatively faint loops seen in the plage regions surrounding sunspots, which are exactly where EIS is seeing the flows. It's been known for a long time (the Palermo people did such modelling 20 years ago) that flows in coronal loops drop the density ("When you start a flow going, the loop disappears.") or alternatively, if you see the loop it means the density is enhanced. I'm not sure why your modelling failed, but we were able to reproduce the scale height quite well. Having the flow (in either direction) extends the emission scale height quite a bit beyond the hydrostatic value.
As you know, the AIA on SDO will have far more extensive temperature coverage than TRACE does. There is a puzzle right now in that EIS sees the flows at higher temperatures than we saw in TRACE. I think that this topic is going to be a major one in the coming years.
Leon
Klimchuk, James A. (GSFC-6710) wrote:
Wow, I'm surprised, and pleased, at the interest this paper has generated! Let me first respond to Leon's comment. As Harry said (thanks!), in order to get the extreme excess densities that are observed in most warm loops, the footpoint heating needs to be so concentrated that no equilibrium exists (which strictly speaking is different from an instability). Hence, thermal nonequilibrium. A few years back, Spiros P. and I addressed your suggestion with Amy that asymmetric heating and steady flows might explain the observations. Our modeling showed that the density enhancement was adequate to explain some loops, but the scale height is too small and the filter-ratio temperature profile is far too structured (paper attached). We thus rejected this explanation. Sorry!
Thanks for your comment, Jim
-----Original Message----- From: loops-bounces@solar.physics.montana.edu [mailto:loops- bounces@solar.physics.montana.edu] On Behalf Of Harry Warren Sent: Sunday, December 06, 2009 6:19 PM To: A mailing list for scientists involved in the observation and modeling of solar loop structures Subject: Re: [Loops] loops and thermal nonequilibrium
Leon,
As I recall, your paper with Amy relied on footpoint heating, which does lead to higher apex densities and flatter temperature ratios. The observed densities near 1 MK are so high, however, that the loops become thermodynamically unstable. Also, the high speed EIS flows are typically seen in faint regions and are not associated with the types of loops that Jim is attempting to model.
Harry
On 12/5/09 3:16 PM, "Leon Golub" golub@cfa.harvard.edu wrote:
> Jim, > > Amy and I addressed the issues of excess density, flat temperature > profiles and large scale height about 10 years ago. Having flows of > 30-40 km/sec, as is observed in TRACE and now verified by EIS, > solves > these problems quite nicely. So there is a viable mechanism other > than > nanoflares. > > Leon > > > Klimchuk, James A. (GSFC-6710) wrote: >> Dear Loops Friends, >> >> >> >> If you are interested, the attached paper shows that coronal >> loops >> cannot be explained by thermal nonequilibrium. The results >> appear to >> rule out the widespread existence of coronal heating that is >> both highly >> concentrated low in the corona and steady or quasi-steady (slowly >> varying or impulsive with a rapid cadence). Comments are welcomed. >> >> >> >> Best wishes, >> >> Jim >> >> >> >>
>> *** >> >> James A. Klimchuk >> >> NASA Goddard Space Flight Center >> >> Solar Physics Lab, Code 671 >> >> Bldg. 21, Rm. 158 >> >> Greenbelt, MD 20771 >> >> USA >> >> >> >> Phone: 1-301-286-9060 >> >> Fax: 1-301-286-7194 >> >> E-mail: James.A.Klimchuk@nasa.gov mailto:James.A.Klimchuk@nasa.gov >> Home page: >> http://hsd.gsfc.nasa.gov/staff/bios/cs/James_Klimchuk.html >> >> >> >>
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