From kusano@hiroshima-u.ac.jp Tue May 27 17:53:45 2003 Date: Wed, 21 May 2003 00:40:46 +0900 From: Kanya Kusano To: 'Pascal Demoulin' Cc: 'Jongchul Chae' , 'mitch Berger' , 'Pablo Mininni' , 'Richard Canfield' , 'Alexander Nindos' , 'BC Low' , 'Jim Klimchuk' , 'Marcelo Lopez-Fuentes' , 'Sarah Gibson' , sakurai@solar.mtk.nao.ac.jp, 'Terry Forbes' , "'Brian T. Welsch'" , dana@mithra.physics.montana.edu, magara@mithra.physics.montana.edu, apevtsov@nso.edu, haimin@sundog.caltech.edu, yjmoon@bbso.njit.edu, "[iso-2022-jp] '?^[$B;3^[(B ^[$B1{L@^[(B'" Subject: RE: 2 new papers on magnetic helicity [The following text is in the "iso-2022-jp" character set] [Your display is set for the "US-ASCII" character set] [Some characters may be displayed incorrectly] Dear Pascal, Thanks for your reply. I think that we are now approaching to a final agreement (maybe). Let's try! First, I like to stress the explanation of why my method does not duplicate the helicity and energy fluxes, because it is our original problem, but I was not yet success to convince you (maybe). When you inversely solve the equation dBn/dt = [nabla x (UtxBn -Ft)]_n, (that is the same as you wrote) ......(1) giving dBn/dt, Ut, and Bn, the solution Ft is not unique, as you pointed out at the last mail, because any gauge can be included. However, in the equation dBn/dt = [nabla x (Wt x Bn + Wn x Bt)]_n (that is same as eq.18 of my ApJ paper) ......(2) the solution Wn is unique, if you specify dBn/dt, Wt, Bn, and Bt, in the most cases. Refer to Sec.3.2 of my ApJ paper for the detail proof. (As I wrote in the paper, some exception arises, but it is unlikely in the real data of Bt.) Therefore, an infinite number of the solution (Wt, Wn) are available, and you can pick-up one from them by specifying Wt. Suppose that the LCT perfectly works, and can provide the velocity Ut, which satisfies the equation dBn/dt = [nabla x (UtxBn )]_n. ........(3) It obviously indicates that the velocity (Wt,Wn)=(Ut, 0) is a special solution of (2), so that if you put Wt=Ut into (2), we can get the solution Wn=0. (OK?) Therefore, when we use the induction equation, we do not duplicate the flux included in the term of Ut. I really hope to share this point with you. Secondly, I like to discuss about the strategy, when the LCT does not work perfectly. (This is true in the real world, I guess) ==============You wrote===================== Yes, I agree that this R still need to be computed. It is a measure of how well the LCT works. But what to do next with R large ? Three possible options: - improve the LCT algorithm, - "improve the data": use high resolution white light images rather than magnetograms - use the induction equation (third agreement ! :)= ). But in the last case, I do not fully agree with you (nothing is perfect yet !). ==============You wrote===================== I agree with your opinion. As you wrote, if the LCT cannot make the precise velocity satisfying the equation Ut = Vt - (Bt/Bn)Vn, "nothing is perfect yet", even if we use (or not use) the induction equation. Yes, it is true, I agree. When you use the induction equation dBn/dt = [nabla x (Vt x Bn + Vn x Bt)]_n (here, Vt & Vn are the real velocity) ......(4) if and only if you find the real velocity Vt, we can derive the perfectly correct fluxes. (Vn can be automatically derived, because of the uniqueness of solution, again) It is due to the fact that, even if [nabla x (Vt x Bn + Vn x Bt)]_n = [nabla x (Wt x Bn + Wn x Bt)]_n, ...(5) (Wt x Bn + Wn x Bt) is not (Vt x Bn + Vn x Bt) in general. However, since we cannot directly measure Vt, we have to introduce the assumption to give Vt as written in my paper, so that the answer from my method depends on the assumption. This is a limitation of my method, and in order to improve the reliability we have to investigate the sensitivity to the assumption. (I have done that in part, but not published) On the other hand, the advantage of my method is that the electric field E_t=-(Wt x Bn + Wn x Bt), which is used in the flux measurements, is always consistent with the magnetic evolution. I believe that it is the necessarily condition for the precise measurement of the fluxes, as I wrote in the last mail. The importance of your paper is that you showed the POSSIBILITY that we may measure the helicity and energy flux from the LCT. This is a great finding, and I'm happy if my work more or less contributed to your work! Best my regards, Kanya -------------------------------------------------------- Kanya KUSANO, Graduate School of Advanced Sciences of Matter, Hiroshima University Higashi-Hiroshima, Hiroshima 739-8530, Japan Phone:[81] (824) 24-7016 fax: [81] (824) 24-7014 e-mail: kusano@hiroshima-u.ac.jp home page: http://plasma.sci.hiroshima-u.ac.jp/~kusano/ -------------------------------------------------------- [Part 2, Text/HTML (charset: ISO-2022-JP "Latin & Japanese") 343 lines] [Unable to print this part]