Monday, June 30, 2014

A Cavity Resonator Model Applied to Solar Loops and Flare Triggers (2)


Preparing the Hybrid Model:

We begin with a coronal loop segment of the form depicted  in Fig. 1 (Part 1)  in which capacitative domains apply, such that:

 C = e [ℓ2||    +  ℓ2^ ]   =  [1  +  (i 4p s2)/ wo ]( ℓ2||    +  ℓ2^ )

Where wo  is the associated frequency. The region has associated with it an initial E-field:  E(z)  =   Eo cos i(wo t – k||  z)

For the fast-mode kink waves expected to be generated we have for the wave number vector associated with the resonator of loop BC in AR 2776::

k||  =  2p / ℓ2||    = 6.2 x  10 -7 m-1

Because of the varying E-field, arising from loop oscillations (for which no  tan(noa - p/4)  =    mo ), we expect a B-field to form in the region, leading to:

ò B j (r) dl  = /  t   òc Ez (r)·n dA

Such that:

B j (r)=   iwo r (m 0 e 0)1/2 /2 [Eo cos i(wo t – k||  z)]

With the most critical observation being that we obtain “corrective” functions in E, based on the zeroth order Bessel function, J o (wo r (m 0 e 0)1/2)  

Assuming a precise boundary cut-off at the value J o (ar)   = 2.405 where:

 ar =(wr Öm 0 Öe 0 ) so that the critical radius is:  r =  2.405/ (wo Öm 0 Öe 0) ,  the cavity is resonant at:

wo =  2.405/ (r Öm 0 Öe 0)

Ideally, this should be a harmonic of the kink-mode global oscillation frequency. This is used here to set up the basic initial tests for falsification, and will lead to more complete falsifying tests based upon the twist of the loop and its helicity current density (which ought to be estimated using proxy indicators)

In the treatment that follows we have in the interior of the loop (cf. Edwin and Roberts, 1983)

Ñ·v    =   R(r) exp (wo t + nq + k||  z)

Where R(r) satisfies the Bessel equation:


So:  d2R/dr2  +   1/r (d R / dr  ) -     (mo2  +  n2/r 2) R  = 0

Where:  mo2  =  [(k2 co2 -   wo 2)( k2 vA2 -   wo 2) /   (co2 +  vA 2)( k2 ck2 -   wo 2)

Where  ck was previously defined (Part 1), as was co.

In terms of the solutions as applied to the axis (r= 0) of  typical coronal loops, one has (Edwin and Roberts, ibid.)

R(r )    =  Ao Io(mor)       mo2  >  0

               Ao Jo(nor)        no2  =  - mo2     >  0


Where  Io(mor)  and  Jo(nor)  are Bessel functions

For the conditions in the corona, a fast kink mode will govern such that:

no  tan(noa - p/4)  =    mo

with period t  = 2L/ ck       »  9 s

In the case of loop BC in AR2776, we have a =   5.5 x 10 8 cm, so that a/ L » 0.006

This rules out “sausage-mode” waves for the loop BC, since these propagate only for the condition:  ka > 1.2 and a = L/10 (cf. Edwins and Roberts, 1983)[1]. 

If the model is correct the coronal cavity for loop BC initiates frequency “pumping” at:

wo  = wcav =  2.405/ (a Öm 0 Öe 0)  =  1.3 x 10 2 s-1

This is the theoretical value we expect to obtain.  Its relatively low magnitude is surprising at first glance and in order to see what is happening, one needs to invoke the conductivity and permittivity   tensors, viz.:
No automatic alt text available.

The main interest in terms of the preceding is how the elements are functions of the various frequencies, e.g. Fitzpatrick, 2004)[2]:

e11 = 1  -  we 2  / w 2  { w/ w -  We  }  -  wi 2  / w 2  { w/ w +  Wi  } 

e22 = 1  -  we 2  / w 2  { w/ w -  We  }  -  wi 2  / w 2  { w/ w -  Wi  } 

e22 = 1  -  we 2  / w 2  -  wi 2  / w 2  

Where e11 and e22 denote permittivities for right and left circularly polarized waves. Now, let there be derivative quantities denoted:

S  =  ½  [e11 + e22] 

And:   D =  ½  [e11 - e22] 

Then for low-frequency wave propagation in magnetized plasma one finds:

   D »  0      S    »  1  +  wi 2  / Wi  2    and  e33      »   - we 2  / w 2  


With:   we 2  / w 2   >>  wi 2  / Wi  2      (in low frequency ordering regime)

To find the approximate observed (empirical analog) value for the loop BC’s coronal cavity resonator angular frequency we use the cavity resonator prescription of Federov et al, 2004, Sec. 4) such that:

wAR  =   p VAR/ dr

Where  k AR is the cavity-associated wave –number vector,  k AR =  wAR / vAR

and VAR  is the Alfven velocity in the cavity, with dr a height-scaled quantity for the cavity (e.g. when seen in edge-on dimension and with curvature correction applied). This is taken to be 1.1 x  10 6 m or roughly the minimal value of xi  in the range noted earlier.

This  yields an observed value:

wAR    =  1.6 x 10 2 s-1

This difference  (wAR    -  wAR )  suggests either: 1) the resonant cavity Alfven speed, VAR  is too high or 2) the quantity d r   is underestimated.  Adopting the theoretical value as given (1.3 x 10 2 s-1) one finds the Alfven velocity in the cavity resonator VAR = 4.5 x 10 7 ms-1  whereas if the observed value is used, one finds: V’AR = 5.6 x 10 7 ms-1  .  The cavity wavelength parameter is:  kAR =  k’AR = 2.8 x 10 -6 m-1  applicable to either VAR or  V’AR .  Hence, k AR is a good proxy wavenumber indicator for the cavity. The Alfven wave conductance for the cavity SAR  = (mo VAR) -1  = 0.017 W -1 based on the theoretical value of  wAR. (0.014 W -1  otherwise). The Alfven wave impedances are, respectively:  ZAR  = (SAR) -1  = 57 W, and Z’AR  = (SAR) -1  = 70 W.  The “characteristic  impedance” can also be approximated using:

 Zch »  ZAR ( kBC/ k AR)

This is modified from the auroral cavity version given by Federov et al (2004, op. cit.). In the above, kBC denotes a wavelength number vector applicable to the loop, and we use the observational (cavity) values for  ZAR , since k BC is an observed value.  For the date on which the 1B/M4 event occurred (11- 5- 80) we have k BC = 1.9 x 10 -7 m-1  so that Zch » 4.6 W.

We are now in position to more realistically assess the quality factor (Q) for the date using equation 64(a) of Hollweg (1983):  Q  » k2r/ (2 êk2i ê) where here we have  k2r »  kAR  and k2i »  kBC . From this we obtain:  Q  »7.5. The loop heating rate from the data (EH = 0.60 erg cm-3 s-1)  allows us to obtain the amplitude damping rate (w i) as defined by Hollweg and thence the resonant period: T = 2 p/  (w r). Using the (EH) datum in conjunction with Hollweg’s equation (31) for the wave energy density (assuming vA2 =  VAR  and that the magnetic energy density (B2/8p, in c.g.s.)  is the primary contributor for a cavity between two nodes cf. Zugzda and Locans, 1982, op.cit.), we get:

T » 69 s

The preceding shows the general approach in obtaining cavity parameters for model testing. With sufficient data resolution and discrimination available for other coronal loops, of the type that was available for loop BC in AR 2776, we should be able to obtain a much better idea of how well this cavity resonator model works and whether it can be suitably generalized.

 The reader should be aware of why the period found here  (»  69 s) differs from that estimated in the June 22 post (»  52 s). In particular, note the latter assumed global kink mode oscillations for a total loop length L = 9.3 x 10 9 cm



[1] P.M. Edwin and B. Roberts, Solar Phys., 88, 179, 1983.
[2] Richard Fitzpatrick: 2004, Introduction to Plasma Physics, Lulu Books, p. 109.
[3] Hollweg, op. cit
[4] .Zugzda and Locans, op. cit.

Conservo Supremes Defecate on the Country With Their 'Hobby Lobby' Decision


Supporters of the 5 Supreme Court Justices'  Hobby Lobby ruling cheer from the peanut gallery of a D.C. zoo. That's exactly where these five "justices"  belong!

I never thought it would come to this, that five alleged top Justices of the Supreme Court would actually defecate on the Constitution and settled federal law but alas, they have.  As most tuned-in U.S. citizens would have learned by now, the Supreme Court in a confounding and insane decision,  have asserted that corporations can hold religious objections that allow them to opt out of the new health law requirement that they cover contraceptives for women. This applies specifically to the Plan B and Ella drugs (as well as intrauterine devices)  that have been hysterically described by the plaintiffs (Hobby Lobby et al) at abortifacents.

Of course this is abject ideologically-driven bullshit that one would have thought the five conservo Justices would have had enough intellect to parse. But evidently not!  The insipid claims are not at all based upon medically based ideas about abortion, but on recycled Foxian Hogwash. As Dr. Petra Casey, an obstetrician-gynecologist at the Mayo Clinic, told the New York Times in a recent piece on the science behind emergency contraception:

"These medications are there to prevent or delay ovulation. They don’t act after fertilization."

As the Times  went on to note, emergency contraception like Plan B, Ella and the hormonal IUD do not work by preventing fertilized eggs from implanting in the womb. Instead, these methods of birth control delay ovulation or thicken cervical mucus to prevent sperm from reaching the egg, meaning that fertilization never even occurs.

But the five SC conservos who have deemed corporations, "persons" - were too fucking dumb or ignorant to process that. Maybe they need to remove their black robes and go back to high school to take a course in basic human biology, the reproductive aspect.  Here's more basic biology from Dr. Anne Davis on the difference:
"An important distinction here is that fertilization is not the same thing as pregnancy for the very, very simple reason that these things take time. There are a lot of steps between fertilization and implantation being successful. For the average woman who is not on any kind of birth control, that process of fertilization is occurring over and over and over again and those women are not getting pregnant over and over and over again because many of those fertilizations never, ever implant.
There are all these things that have to happen for pregnancy to occur. I think if people think about the question of when a woman is pregnant, the answer seems pretty intuitive. When I go to somebody and ask, “How many times have you been pregnant?” I’m not asking, “How many times have you had fertilized eggs?”

So now, because of basic biological ignorance, we're saddled with a court decision for corporations that's as daft as it is incompetent.

And we won't even get into the colossal LEGAL ignorance and deliberate flouting of more than 200 years of settled federal and constitutional law that NEVER, EVER before recognized a corporation as a religious being!  Don't these five,  dumbass mutts grasp what the hell they have done here? Do they not comprehend the consequences?

Here's some further perspective: As David H. Gans recently noted at Slate, “Not one Fortune 500 company filed a brief in the case. Apart from a few isolated briefs from companies just like Hobby Lobby and Conestoga Wood, the U.S. business community offered no support for the claim that secular, for-profit corporations are persons that can exercise religion.”
Further, polls on public opinion consistently show most Americans (up to 70%) believe Hobby Lobby and its deluded cohort are way out of line. To the point of being detached from reality.
But what about the consequences now? Given the five c-turds have ruled that corporations can have "sincerely held religious beliefs", this ruling effectively validates the kind of discrimination conservative lawmakers in Arizona tried to pass earlier this year with SB 1062 the law of the land.  As Ian Millhiser of the Center for American Progress wrote earlier this year:
"Denying birth control to your workers because of your own religious objections to it superimposes your own personal beliefs about conscience and faith onto your employees. So does refusing to serve a gay person due to a religious objection to their sexual orientation
Or a black man  is denied service because a fundie fervently believes (according to his pet Bible) that the black man is numbered among the "Sons of Ham" so must be  served with a life of hardship, humiliation and denial.
Frankly, this decision put forth by the five conservo justices amounts to a legalistic defecation on the settled law of the land.  These five clowns have now shown they’re not only incompetent and irrational but have no respect for the rights of the majority of citizens in the country. It also means the Obama administration must search for a different way of providing free contraception to women who are covered under the objecting companies’ health insurance plans. (A Yale law prof interviewed this afternoon on 'The Cycle' believes such a solution might be feasible)

Justice Samuel Alito, who wrote the majority opinion , ought to have been ”borked” from SC nomination when considered back in 2005.  But we know ten pussy-wussy Dems balked, and decided not to reject this repuke punk out of "bipartisan spirit".  See e.g.
http://www.salon.com/2014/07/03/alito_could_have_been_stopped_why_dems_should_have_filibustered_the_radical/
They were led by none other than Ken Salazar who now will live in infamy for his pusillanimous deference to zealots.
This Court stressed that its ruling applies only to corporations that are under the control of just a few people in which there is no essential difference between the business and its owners. So I guess if there's just a few rabid Islamics,  all they need to do is start a business and they can impose Sharia law on their workers.
Hopefully, Obama can find some way to do an end run around these shit bags, for the benefit of the women affected, and the citizens' rights they've trampled on.
See also:

Ruth Bader Ginsburg's blistering response:

http://www.scribd.com/doc/231974154/Ginsburg-Dissent

Excerpt:

"Until this litigation, no decision of this court recognized a for-profit corporation's qualification for a religious exemption from a generally applicable law, whether under the Free Exercise Clause or RFRA.  The absence of such precedent is just what one would expect, for the exercise of religion is characteristic of natural persons, not artificial legal entities.  As Chief Justice Marshall observed nearly two centuries ago:

"A corporation is "an artificial being, invisible, intangible, and existing only in contemplation of law."
------
and:

http://www.salon.com/2014/06/30/there_was_nothing_limited_about_scotus_hobby_lobby_ruling_why_it_matters_for_everyone/

Why Today's Hobby Lobby Ruling Should Be A Slam Dunk for Common Sense

Photo: Cartoon on today's Supreme Court decision (Hobby Lobby)Today the Supreme Court considers the “Hobby Lobby” case and there’s really no excuse for not getting it right – which is to say slapping down Hobby Lobby once and for all – preferably in a 9-0 decision.

Look, the aspects of this case are pretty basic, so much so that even a non-legal person can grasp them. On the surface the Justices have to weigh the right of female employees of corporations – companies to the birth control of their choice (delivered via Obamacare), versus the “religious rights” of employers – but which are in fact corporations.

Here's a brief summation of the case, which involves contraceptive coverage under the Affordable Care Act: Employers must cover contraception at no extra charge for women - among a range of preventive benefits in employee health plans.  Dozens of companies, including Oklahoma City based Hobby Lobby claim religious objection to covering some or all of .the contraceptives. They insist it's a violation of their "religious rights".

(I already dealt with the fallacious arguments advanced by Catholic organizations here: http://brane-space.blogspot.com/2013/12/fellowship-of-catholic-university.html )

The methods and devices at issue before the Court are those that Hobby Lobby and furniture maker Conestoga Wood Specialties Corp. of East Earl, PA say can work after contraception - which is to say, are abortifacents.  These include the emergency contraceptives 'Plan B' and 'Ella'. Which is total nonsense since none of the drugs works that way, i.e. they are not "abortifacents" - another scare word dreamed up by the zealots. The devices also include intra-uterine devices, which can cost up to $1,000.

As one recent editorial (Dallas Morning News. March 31) framed it:

 the deep-seated personal convictions of  Hobby Lobby’s executive and Conestoga’s Mennonite owners are not in question. What is at issue is whether a private, profit-making business can dictate religious beliefs on employees.”

In other words, an SC affirmative verdict would de facto extend the religious rights of individuals to corporations that are legal artifacts – not persons – and exhibit no explicit religious missions!

Moreover, as I noted before, legal artifacts – whether corporations, municipalities, counties or states do not have rights!  Only flesh and blood people can have rights – as per Prof. Garry Wills famous assertion ( A Necessary Evil: A History Of American Distrust Of Government, Simon & Schuster, 1999, observes, p. 108) that legal artifacts, including states – retain prerogatives which are artificial not natural - since they are things made by contract.  Thus:

Governments and legal entities  have prerogatives, people have rights

If the Justices were to find for Hobby Lobby, they would be torching over 200 years of how rights are defined, and instead conflate prerogatives of a contractual, legal entity with personal rights. The result would be such a legal atrocity that Citizens United would pale beside it.
 
It would also open a ‘Pandora’s box’ allowing corporations – businesses to pell mell deny employees a host of other previously assumed benefits or rights- OR demand behaviors comporting with the owners' religious beliefs.. For example, a Mennonite owned business might one day then decide to order all its female employees to wear Mennonite head gear ('bonnets') at work. They will cite the pro ‘Hobby Lobby’ decision and bellow that it is their “religious right’ to demand their female employees heel to their precious beliefs and this is their "religious liberty" in action.  Or, a fundie owner of a business may demand all employees read the Bible (KJV) for one hour each day during lunch time – while eating their lunch. (Talk about indigestion!)

The point is any conflation of people with legal artifacts forged by contract,  is a dangerous path to tread because businesses and corporations – as legal entities – are distinct from the individuals who created them. As the Obama Administration has persuasively argued:


The owners of Hobby Lobby aren’t entitled to exemption for their business based on their individual religious beliefs

Or, as the Philly-based federal appeals court pithily put it:

Businesses do not pray, worship, observe sacraments or take other religiously motivated actions.”

It is also important to note the Court has never recognized a for profit corporation's "religious rights" under federal law or the Constitution. 

If the Supreme Court changes course, violating all previous rulings and precedents, then this country is totally fucked – as its judiciary has taken flight into the realm of unreality and fantasy.  We will then be only one small jump from becoming a plutocratic theocracy. Worse, any Justices so voting will have shown it isn't  the law they respect or revere, but the  pre-meditated debasement of reality via political chicanery and PR. Indeed, such a ruling would be as vile and debased as the revised FISA law passed by congress (which made the illegal wiretaps of Bush Co. legal, paving the way for the mass surveillance transgressions of the NSA.)

 The country is already headed into the crapper after Citizens United and the subsequent decision to enable unlimited diffusion of funds into the political process.
 
Any pro-Hobby Lobby, pro- religious business decision will finish us off.