From Pascal.Demoulin@obspm.fr Wed May 28 10:37:01 2003 Date: Wed, 28 May 2003 18:08:28 +0200 From: Pascal Demoulin To: Kanya Kusano 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'\"" , 'Cristina Mandrini' , 'Lidia Driel-Gesztelyi' , "[ISO-2022-JP] \"^[$B??1I>k^[(B ^[$BD+90^[(B\"" Subject: Re: 2 new papers on magnetic helicity > Dear Kanya, Thanks for your long detailed e.mail !! > I hope that this discussion is not boring for you (and for other > audience!) At least NOT for me !!!! > Any comments from anybody will be welcomed. Yes ! > I think that the distance between us becomes shorter and shorter > through > this discussion, and I really believe that we are going to a same > destination! YES. So let continue ! Below, I will only answer to the uniqueness issue since I need more time for you other part ! :)= > Sorry, Pascal, I cannot understand what you still wonder about, > although you > have been convinced for the proof of uniqueness. Tell me your concern > more > clearly, please! Sorry, my equation (5) was NOt correct: I should not have put dAt/dt there ! But it is only a typo in writing Eq.(5)... the following is not affected.... except that I go farther below.... > Does your equation (8), > the correct form of which may be > 0 = nabla_t . [Bt.(dAt/dt + nabla_t (phi_1-phi_2)) Bt/Bt^2 No, dAt/dt is not there Below, I am putting the corrected text and explain more: Finally I still have a concern with uniqueness (I am a difficult person ! :)= ) .... because the equations that you numerically solve (32, 37) are not exactly the ones you use in Section 3.2 to derive uniqueness : this is so because you took the horizontal divergence to get your Eq. (31) and so I wonder at the uniqueness of solution for your Eq. (32) (coupled to Eq. 37). So let look at uniqueness starting with two solutions, 1 and 2, of Eqs. (32, 37) and see if the solution of Eqs. (32, 37) is unique or not. That is, a priori, not equivalent to look at uniqueness of solution for Eq. (20) as you do in section 3.2, OK ? From (37): V1n - V2n =( Bt X (nabla_t (Phi1-Phi2) ) / Bt^2 (5) (dAt/dt is not in (5) because it is the same for solutions, 1 and 2, since it is derived from Bn) OK, no problem here, we get back to Eq. (23) with v= V1n - V2n , Psi= Phi1-Phi2 . Next, from the difference of your Eq. (32) between solutions 1 and 2 : nabla^2 (Phi1-Phi2) = nabla_t ( (V1n - V2n) X Bt ) (6) Then I use (5) inside (6), I get: nabla_t.( [Bt.nabla_t (Phi1-Phi2)] Bt / Bt^2) = 0 (8) or nabla_t.( [Bt.nabla_t (Psi)] Bt / Bt^2) = 0 So (8) is not exactly Eq. (21) of your paper, ....since there is multiple solutions to Eq. (8) due to the nabla_t (so horizontal divergence). A general solution of (8) is up to an arbitrary scalar function Eta: [Bt.nabla_t (Phi1-Phi2)] Bt / Bt^2 = (nabla_t Eta) x n or [Bt.nabla_t (Phi1-Phi2)] Bt = (nabla_t Eta) x n Bt^2 (8') (n: normal versor). Can we remove the Eta ambiguity ? I think further than in my previous e.mail.... The vector product of (8') with Bt gives: 0 = Bt x (nabla_t Eta x n) or 0 = - (Bt.nabla_t Eta) .n so 0 = Bt.nabla_t Eta Then the same type of equation than Eq. (24), and your argumentation then apply to show that Eta is indeed a constant ! So Eq. (8') indeed writes: [Bt.nabla_t (Phi1-Phi2)] Bt = 0 or [Bt.nabla_t (Psi)] Bt = 0, just your Eq. (24) ! SO, FINALLY GOOD NEWS: I agree that the solution to Eqs. (32, 37) is unique !!! (but as you have seen, it was not obvious to me from the beginning ! :)= ) > Also here, I do not understand what you meant by the word "less > coupled". > Your equation (9) is correct, and it indicates that we can solve Phi in > principle by integration of (9), and then we can derive Vn from (37) > of ApJ > paper. By "less coupled", I just mean that Vn is NOT present in Eq. (9) so that you can solve it directly for Phi, then you can derive Vn from your Eq. (37)... so just what you wrote in your last e.mail ! Presently the equations that your are solving (Eqs. 32, 37) are "coupled" since both Phi and Vn are present in both equations. > Therefore, I do not think that there is any problem of the uniqueness > of the solution in your argument. Anyway, tell me more for this part, > please. As I said above, I have no longer a problem of uniqueness with the solution to your Eqs. (32, 37) ! I was deriving Eq. (9) to replace your Eq. (32) and solve the uniqueness problem....but indeed there is none ! So the only utility of Eq.(9) is that you can solve Phi directly by integration of (9), and then you can derive Vn from (37). So we really start converging !! Now I need to look to your next part more seriously than I was able to do so far.... My best wishes, Pascal *====================================================================* Pascal Demoulin Phone: 33 1 45 07 78 16 Observatoire de Paris Fax: 33 1 45 07 79 59 section Meudon, LESIA, Bat. 14 http://www.solaire.obspm.fr/demoulin/ F-92195 Meudon Principal Cedex Pascal.Demoulin@obspm.fr France *====================================================================*