From kusano@hiroshima-u.ac.jp Tue May 27 17:54:00 2003 Date: Tue, 27 May 2003 11:00:38 +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'" , 'Cristina Mandrini' , 'Lidia Driel-Gesztelyi' , "[iso-2022-jp] ^[$B??1I>k^[(B ^[$BD+90^[(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] Hello Pascal, Thanks for your reply! 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! Let's restart our discussion! This mail consists of 4 parts as following: In Sec.1 (with fig1), I will explain the principle of my method. This is the introduction to the following part, but may be helpful for your understanding. In Sec.2, I will reply your last message, and ask you about some part of your mail, which is unclear for me. In Sec.3 (with fig2), I will discuss about the reason of why the LCT is used in my method. This part was not yet published, but it could be necessary for us to achieve a common understanding. Finally, in Sec.4 (with figs3 to 5), I will demonstrate our method for some typical cases. My conclusion is that the method based on the LCT plus the induction equation is more reliable than the method based on just the LCT. This series of discussion with you really strengthened my conviction. Thanks! Sec.1 Principle of my method The helicity flux (E x A_p) and the energy flux (E X B) are generated by the electric field E. Therefore, the variation which we must find for the measurement of the helicity and energy fluxes is NOT the velocity, BUT the electric field E. Following to the Farady’s law the electric field E is related to the flux change. (This part may be obvious for you, sorry) See the attached figure 'fig1.gif'. This is a schematic diagram showing the field line evolution. Suppose that the magnetic field line B moves to B' by the real plasma motion from F to P, and that the foot-point shifts from F to F'. Here, what we have to find is the magnetic flux Phi between B and B', and your paper (DB2003) pointed out that it may be possibly detected, if we can find the foot-point motion F to F' using the LCT. (Yes, I agree.) However, practically the LCT is not perfect. So, the velocity measured by the LCT is not the vector FF’, but the vector FF''. In the case, some magnetic flux through the line F''F' is missed. However, my paper showed that, if we solve the induction equation inversely, we can derive the velocity F''Q, and thus it is able to recover the correct flux Phi, which passes through FQ, whereas we cannot find the real velocity FP. My method can apply even to an extreme case, in which the LCT does not at all work and the horizontal velocity FF'' = 0. In the case, my method gives you the velocity FR and thus provides the correct flux Phi. I will really demonstrate that in the final section of this mail, using the real data. Here, some question arises (maybe in your mind?). If the induction equation can recover any loss of flux, why we should use the LCT to determine the horizontal velocity? This is an excellent question, and I will show you the answer later in sec.3. (be patient, please) Sec.2 My reply to your question >>>>>> you wrote<<<<<<<<<< > ** YES , your method does not duplicate the helicity and > energy fluxes (you read well :)= ) Let start from the > classical induction where V is the plasma velocity: > dBn/dt = [ nabla x (V x B) ]_n, (1) > = [ nabla x (Vt x Bn + Vn x Bt) ]_n > = [ nabla x (Ut x Bn) ]_n (2) > where Ut is the tangential foot point motion. That's for the > theory. In practice the LCT is not able to recover fully Ut, > but say only Ulct. I then use a "m" in front of quantities > which are missed by the LCT, Ut = Ulct + mUt, and we got: > dBn/dt = [ nabla x (Ulct x Bn + mUt x Bn) ]_n (3) > = [ nabla x (Ulct x Bn + mVt x Bn + mVn x Bt) ]_n (4) > where I rewrite mUt x Bn in the original plasma velocity. > IF Ulct would be perfect so Ulct=Ut, YES , your method does > NOT duplicate the flux deduced from the LCT (if there is no > problem of unicity... see below).... since it is designed to > get ONLY the velocity missed by the LCT. > NOTICE: I still believe that this point is NOT clear from > your ApJ paper (since the induction equation is written there > as in MHD so with plasma velocities, and it was never mention > the velocity of the foot points ! Now, with your e.mails, I > finally understand your point ! But this need a > re-interpretation of the equations of your ApJ paper (as I > started above). > ** OK, no duplication but can we derive the LCT's missed velocities ? > >From Eq. (3) above this missing term is of the form mUt x Bn, NOT of > >the > form Wn x Bt > as you suppose in the ApJ paper ! > (with re-interpretation of your Eq. (18), your Vt becomes > Ulct and I call Wn your Vn). With a term like mUt x Bn, NO > way to get a unique solution to Eq. (3) (all the others > functions being known) ! So, let transform it to Eq. (4). > Here we have a term of the form Wn x Bt.... but still another > one: mVt x Bn .... so the solution for (mVt,mVn) is still NOT > unique !! What you implicitly assume is that mVt is null ! Here, you still misunderstood my method. In my method, we DO NOT need any additional velocity for the horizontal component (mVt in your equation (4)). When the real plasma velocity is denoted by (Vt, Vn), i.e. (Vt, Vn) = FP in fig.1, we have the induction equation dBn/dt = [ nabla x (Vt x Bn + Vn x Bt) ]_n ..........<1> In that case, the equation we should use in my method is not your equation (4), but dBn/dt = [ nabla x (Wt x Bn + Wn x Bt) ]_n ....... <2> (This is the same as (18) in my ApJ paper, though the notation is different.) Although the equation <2> has the same form as the real induction equation <1>, the velocity (Wt, Wn) does not need to be same as the real plasma velocity (Vt, Vn). The theorem, which was proven in Sec.3.2 of my ApJ paper, guarantees that if you put Wt into <2>, then you can get the unique solution of Wn. For instance in fig1 above, if Wt= FF'', then Wn=F''Q, and if Wt=0, then Wn=FR. The induction equation can give us the correct answer automatically, because they are unique solution! > > CONCLUSION: I am ready to re-interpret your ApJ paper, but in > the following way (again not trivial only from your paper) : > Vt is set to the foot point motion detected by the LCT, so Vt > -> Ulct Vn is set to the vertical plasma motion MISSED by > LCT, so Vn ->mVn You IMPLICITLY suppose that LCT has NOT > missed any horizontal plasma motions , so mVt = 0 So, with > your method, you add up one extra term to the flux (complete > it, no redondancy), but this extra term is NOT sufficient to > get the full flux. Nothing is perfect :)= ....but better to > clarify this.... if you agree ! Your understanding is NOT correct. The induction equation CAN recover full flux at least in the discussion of fig.1, and I DO NOT introduce any assumption for the LCT here. Actually, in the situation of Fig.1, we DO NOT NEED the LCT velocity. See Sec.3 for more discussion. > >>>>>>>>>>> you wrote<<<<<<<<<<<<< > ** YES for the uniqueness (section 3.2) but I got a concern > for the equations solved numerically Now I go back STRICTLY > to your ApJ paper (can be translated with the above notation, > but I believe it will be clearer if I use your ApJ paper's > notations, so V (not U....). First I have redone all your > derivation of Sections 3.2 and 3.3 ...and I agree for ALL the > equations. Second, I believe that your argumentation of > uniqueness, in Section 3.2, is strong enough for the solar > case ! Thanks! > (but in applications it will mean mVt=0 implicitly). This is not the assumption about the real velocity, as I wrote above. > 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 ! So let look at > uniqueness starting with two solutions, 1 and 2, of Eqs. (32, > 37). From (37): V1n - V2n =( Bt X (dAt/dt + nabla_t > (Phi1-Phi2) ) / Bt^2 (5) OK, no problem here, we get back to > Eq. (23). From Eq. (32): nabla^2 (Phi1-Phi2) = nabla_t ( (V1n > - V2n) X Bt ) (6) Then I use (5) inside (6), and get: > nabla_t.( [Bt.nabla_t (Phi1-Phi2)] Bt / Bt^2) (8) So (8) is > not exactly Eq. (21) of your paper....since there is multiple > solutions to Eq. (8) ... So it seams that the solution to our > Eqs.(32, 37) is not unique !!! 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! Here, I would like to ask you about your question. 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, really has multiple solutions? Even when a proper boundary condition, for instance (phi_1-phi_2)=0, is imposed, it has other solution rather than (phi_1-phi_2)=0? Is it true? If so, please show me that. This equation is too complicated for me to discuss about the uniqueness. > > Indeed it is because you took the nabla_t. operation to get > the operator nabla_t^2 ....which is nicer to deal with > numerically ! You get a unique solution if you replace > Eq.(32) as follow. Eq.(29) gives: [dA/dt]t = [VxB]t - nabla_t > Phi Taking simply the scalar product with Bt, you got: > Bt.nabla_t Phi = -Bt.[dA/dt]t + Bt.[Vt x Bn] (9) It has the > advantage of uniqueness (recover easily Eq. (24) ), but also > that the unknown Vn is not present in (9) ....while it was in > Eq. (32). The system to solve for (Vn,Phi) is "less coupled". > 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. 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. Sec.3 Why we use the LCT. I pointed out that the induction equation can recover the full flux in fig.1, even if we assume Wt=0. If so, does the LCT need to be used for the measurement of the helicity and energy fluxes? My answer is YES for the helicity flux, but NO for the energy flux! Let me explain the reason of them. Now, we have two equations <1> and <2>, where Wt is decided by the LCT, i.e. Wt=FF'' (not FF’!!) in fig.1. In that case, our concern is whether Vt x Bn + Vn x Bt = Wt x Bn + Wn x Bt, or not? (Note that Vt and Vn are still unknown.) The answer is of course “NO” in general, because any gauge arises. So, we have the equation (Vt-Wt) x Bn + (Vn-Wn) x Bt = nabla xi, ......... <3> where xi is any scalar field. If you take the scalar product of <3> and the vector Bt, then you get the equation (Bt x Bn) . (Vt-Wt) = Bt.nabla xi. ..........<4> Here, if the left hand side is vanished, then 0 = Bt.nabla xi. It means that xi is uniform in the solar surface connecting by the two-dimensional field lines of Bt, and the gauge (nabla xi) disappears. (That is same logic used in the proof of uniqueness in Sec.3.2 of my ApJ paper, as you know.) Therefore, whether the induction equation can provide the precise answer of the electric field E depends on whether (Bt x Bn) . Wt = (Bt x Bn). Vt. .................<5> See fig.2, that is the three-dimensional version of fig.1, and the foot-point shift is denoted by FF', and the real plasma motion is represented by FP. The equation <5> indicates that, once we can measure the velocity component perpendicular to the field line projection onto the photosphere (y-component in the figure), the solution Wt x Bn + Wn x Bt is same as the real electric field Vt x Bn + Vn x Bt. From three points of view, this result indicates that my method is greatly advantageous for our purpose. First, the perpendicular velocity P''P' is able to be measured by the LCT in principle, because it is same as the perpendicular component (fF') of the foot-point velocity (FF'). This is the real reason of why we use the LCT for the selection of Wt. This fact indicates that the induction equation CAN recover the flux carried by the parallel velocity (Ff), but it CAN NOT do the flux by the perpendicular velocity (fF'). Therefore, we have to measure that using LCT! (I will really demonstrate that later.) The second point is that the measurement of the perpendicular velocity (fF') is much easier than that of the parallel velocity (Ff), because it does not diverge even if the elevation angle of the field line (theta) is small. Therefore, the LCT may provide more precise results for the perpendicular velocity compared to the parallel velocity. Once the perpendicular velocity is measured, the induction equation derives the correct answer, however large the error in the parallel velocity is. The third point is that the energy flux (Poynting flux) [Bx(VxB)]_n = Bt^2 Vn - (Vt.Bt) Bn actually does NOT depend on the perpendicular velocity, but depends only on the parallel velocity. Therefore, we do NOT need to measure the perpendicular velocity in order to derive the energy flux! The induction equation can derive the correct answer of the energy flux without the LCT, but the LCT needs for the helicity flux measurement. (I will clearly demonstrate that too.) These all facts indicate that the method using the LCT plus the induction equation is very advantageous and more reliable for the helicity measurement than the method only using the LCT. Furthermore, our method can derive the energy flux without usage of the LCT. This is my conclusion. Sec.4 Demonstration Let me demonstrate how my method works in three different cases, where the horizontal velocity Wt is decided by three different manners. The results are shown in figs.3 to 5, attached on this mail. (They correspond to Fig.1 of my ApJ paper, and all attached figures are the results of AR8100, which was analyzed in my ApJ paper, too. However, since we improved the pre-process procedure in our data analysis, the results are not exactly same as Fig.1 of ApJ paper. The difference from the published version comes from different reason rather than our current concern.) In each case, we decided Wt as follows: In Case 1 (fig.3), Wt was directly given by the LCT velocity. In Case 2 (fig.4), Wt was decided by the modified velocity from the LCT measurement, based on the equation (38) of my ApJ paper. The perpendicular component of Wt is same as that of the LCT velocity, but only the parallel velocity is changed. In Case 3 (fig.5), we simply assumed that Wt=0. Look at the 3rd panel and the 5th panel from top of each figure, which represent the time evolution of the helicity and energy fluxes, respectively. Here, thin line with small diamonds and black line with open circle indicate the fluxes of horizontal motion (dotH_t and dotE_t, in the notation of my paper) and the fluxes of the vertical motion (dotH_n and dorE_n), respectively, and red thick lines represent the total fluxes (sum of the two). Let us compare cases 1 and 2. You can find that the total flux (red lines) both for helicity and energy are exactly same between them, even though the partition between two fluxes of horizontal motion and vertical motion is changed. It clearly indicates that, as long as the perpendicular velocity ((Bt x Bn).Wt ) is not changed, the induction equation CAN fully recover the flux carried by the parallel velocity. Furthermore, you can find the total energy flux of case 3 is also exactly same as the result of the other two cases, even though the horizontal motion cannot carry any flux in this case! All flux is recovered by the induction equation. It indicates that the LCT does not need for the energy flux measurement. On the other hand, the helicity flux of case 3 is not same as the other cases, because the perpendicular velocity may generate some part of the helicity flux unlike the energy flux. However, the difference between case 3 and the others was not significant at least for this data. It suggests that the major part of helicity may be carried by the vertical motion and the parallel motion, so that the helicity flux is not so sensitive to the selection of the horizontal velocity, if it is calculated using the induction equation. Here, we should mention also that, in case 1, the difference between the total flux (red line) and the flux of horizontal motion (thin with diamond) is not small (in fact, significant in the energy flux). If the LCT perfectly works and if we can detect the foot-point motion precisely, the flux of horizontal motion should match to the total fluxes, as your paper showed. However, my demonstration implies that it does not work well. This is my concern about the LCT Based on the theory and the demonstrations above, I really think that the method of the LCT plus the induction equation is more precise and more reliable than the LCT method. Actually, I strongly believe that we should use the induction equation for the helicity and energy flux measurement, when we have the vector magnetic field data. On the other hand, your method has an advantage that it enables the helicity flux measurement (not energy flux measurement) from the observation only of the longitudinal magnetic field, although the usage of the induction equation may improve the reliability of the measurement. I greatly hope that you correctly understand my method, because you are one of the leaders in this field. And, I appreciate your recommendation about another paper to explain the detail of my method. I also think that it may be necessary. Let me try that! I hope that this discussion is not boring for you (and for other audience!) Any comments from anybody will be welcomed. Thanks a lot! 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, Image/GIF 15KB] [Unable to print this part] [Part 3, Image/GIF 51KB] [Unable to print this part] [Part 4, Image/GIF 47KB] [Unable to print this part] [Part 5, Image/GIF 17KB] [Unable to print this part] [Part 6, Image/GIF 17KB] [Unable to print this part]