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Response to the Comment on: “Theory of vortex flows in partially ionized magnetoplasmas” [Phys. Lett. A 326 (2004) 267]
Physics Letters A 329 (2004) 165–167 www.elsevier.com/locate/pla
Response to comment
Response to the Comment on: “Theory of vortex flows in partially ionized magnetoplasmas” [Phys. Lett. A 326 (2004) 267] D. Jovanovi´c a,∗ , P.K. Shukla b a Institute of Physics, P.O. Box 57, 11001 Belgrade, Serbia and Montenegro b Institut für Theoretische Physik IV, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Received 22 June 2004; accepted 24 June 2004 Available online 6 July 2004 Communicated by F. Porcelli
Abstract It is demonstrated that the comments by Vranješ et al. [Phys. Lett. A, preceding comment paper] on our recently published paper [Phys. Lett. A 326 (2004) 267] are faulty and misleading. 2004 Elsevier B.V. All rights reserved. PACS: 52.35.Kt; 52.35.Fp; 52.35.Mw
In the following, we present the item-by-item response to the comment paper by Vranješ et al. [1]. 1. Two groups of experiments with plasma vortices [2–5] (cited in the comment paper by Vranješ et al.) and [6] (extended version of [2]), are closely related. They were performed in the same experimental device and with similar goals of “studying the effect of dissipation on the property of the vortex” [5], and of the “experimental observation of the tripolar vortex” [5,6] that emerges due to “the existence of neutral particles [which] usually causes a dissipative effect” [6]. The main dissipation mechanisms in these weakly ionized argon and helium magnetoplasmas were the internal viscous friction due to collisions between particles of the same species (with no effective momentum transfer) and the momentum transfer in the mostly charge-exchange collisions between charged particles and neutrals. Changing the experimental conditions, viz. the neutral density and the working gas type (with DOI of original article: 10.1016/j.physleta.2004.04.040. DOI of comment on original article: 10.1016/j.physleta.2004.06.075. * Corresponding author. E-mail addresses: [email protected] (D. Jovanovi´c), [email protected] (P.K. Shukla). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.06.076
166
D. Jovanovi´c, P.K. Shukla / Physics Letters A 329 (2004) 165–167
a different charge-exchange cross-section), the relative importance of these mechanisms was varied yielding different types of plasma vortices, such as a Burger’s vortex/plasma hole [4,5] and a tripole [2,3,6]. As stated in the conclusions of Ref. [7], our comprehensive analytic theory provides “a complete description of weakly zdependent vortices in partially ionized, collision-dominated magnetoplasmas, including the effects of ionization and recombination, axial and radial flows, viscous damping, neutral drag force, and the ion reaction to it”. Thus, in plasmas dominated by ion–neutral collisions, such as dense argon plasmas of [2,3,6], we obtained a tripole, see our Eqs. (10)–(17) and Fig. 1, while in the case when the effects of viscous and ion–neutral collisional damping were the same order, as in the less dense helium plasma experiments [4,5] we obtained a Burger’s vortex, see Eqs. (8) and (9) of Ref. [7]. We strongly believe that such a successful unified theory should be welcomed in the scientific community. Strangely, Vranješ et al. [1] regard it as “mix[ing the] physics of the two independent experiments”. 2. As stated in Ref. [5], “Burger’s vortices have been considered as an elementary vortical structure in turbulence”. In our context, they emerge asymptotically for large times in the presence of an inward radial flow of neutrals (associated with their strong ionization and the axial flow of the newly born plasma), ion–neutral coupling, and of effective viscosities of both ions and neutrals. These effects are present in all experiments reported in Refs. [2–6]. When the ionization and related anomalous ion viscosity are very strong, the Burger’s vortex described by Eqs. (8) and (9) of Ref. [7] develops a deep hole in the neutral density [4,5]. Conversely, in the regime when the ion–neutral coupling dominates the ion viscosity, a tripolar structure emerges [2,3,6] superimposed on a background Burger’s vortex, since even very weak ion viscosity produces effects in a time-asymptotic regime. The existence of the Burger’s vortex in experiments [2,4,5] manifests as the “spatially nonuniform distribution of the neutral gas which could be described by a r-dependent parabolic function” [2], in full agreement with Eq. (8) of Ref. [7]. Actually, a Gaussian neutral density profile characteristic for a Burger’s vortex [see Eq. (9) in Ref. [7]] was used by Vranješ et al. [2] in the construction of their analytic tripolar solution, but their assumed plasma density profile did not have the appropriate Burger’s ∼ log r asymptotic dependence [compare to Eq. (12) in Ref. [7]]. 3 and 4. The Vranješ et al. [2] analytic tripole shows a poor agreement with experimental data and has incorrect behavior at the boundary. Its size is much larger than ∼50% of the vessel diameter, observed in [6]. Furthermore, it features a finite θ -dependent electric field at the argon–wall interface that corresponds to the finite plasma and neutral gas velocities, which are mutually coupled by collisions. Such tripole would be strongly damped by surface friction, since no θ -dependent driving force is provided. A possible Stern–Larichev–Reznik-style extension to larger values of kr0 would make the situation even worse, since such tripole would either have unphysical surface charges at the separatrix or diverge for large r, rather than decay. Our tripoles in Ref. [7] are not damped. The convective cell solutions (12) and (13) in Ref. [7] asymptotically behaves as a Burger’s vortex, sustained by the cylindrically symmetric drive of the ECRH ionization and radial inflow. The drift-wave solution (16) and (17) in Ref. [7] decays exponentially, having almost no drag at the wall. 5. Our model in Ref. [7] takes into account many essential experimental features previously disregarded by Vranješ et al. [2]. In the ‘tripole regime’, when the effects of E × B convection and ion–neutral collisions are much larger than those of sources and sinks, we account for them by using in the time-asymptotic limit the unperturbed (i.e., r-asymptotic) plasma and neutral densities corresponding to a Burger’s vortex. The latter was previously shown to emerge for t → ∞ under actual experimental conditions. Obviously, the essential experimental features [2–6], such as ionization, recombination and charge exchange do not play a role in ordinary fluids. However, the small size tripole in laboratory experiments and large structures in oceans are described, respectively, by the Euler and Rossby equations that are identical to our equations for convective-cell and drift-wave tripoles. Thus, there is an analogy between vortex structures in magnetized plasmas and ordinary fluids. 6. Our source term νc nn in the continuity equation is actually in complete agreement with the “exact” term αnn ne − (βrr + β3b ne )ne ni − ne /τ offered by Vranješ et al. [1] in their Comment. As they have shown, under actual experimental conditions the ionization αnn ne is more than three orders of magnitude larger than the recombination
D. Jovanovi´c, P.K. Shukla / Physics Letters A 329 (2004) 165–167
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(second term). Their third term is just a crude model for the variation of the plasma density at a given position z due to the axial flow. In contrast to Vranješ et al. [1], we have fully resolved the parallel dynamics, see Eq. (4) in Ref. [7] and the discussion underneath, and such term does not appear in our continuity equations. Furthermore, in the paragraph preceding Eq. (5) in Ref. [7] we have clearly stated that we work in the regime when the electron density perturbation is small, (ne − ne,0 )/ne,0 1. The source term in Ref. [7] is then readily obtained by using νc = αne,0 . Although small, recombination must be taken into account in a bounded system. It provides a source for neutral atoms, otherwise their number density would be reduced to zero due to permanent ECRH ionization. It was completely disregarded by Vranješ et al. [2]. In Ref. [7], we assume that it takes place outside of the spatial region under study, where the plasma buildup due to the axial flow and the “neutral burnout” by strong ionization (described in some detail in Ref. [3]) produce the reversal of relative populations of charged and neutral particles, yielding enhanced recombination. 7. It is not true that we used Braginskii expressions to describe inelastic processes. As stated in Ref. [7], they were used only for the Coulomb collisions among charged particles, yielding the expressions for ion viscosity. For inelastic collisions we used complete collision integrals with proper cross-sections for charge-exchange and vn − vk ), k = e, i were obtained as their first moments, for ionization. The momentum-transfer terms ν(nn /nk )( small perturbations of electron and ion densities and weak velocity dependence of the cross-section. 8. It is not true that our perpendicular ion dynamics is unaffected by ion–neutral collisions. It is derived from the coupled ion and neutral momentum equations (∂t + v⊥i ∇) (1) v⊥i = (e/mi ) v⊥i × B − ∇⊥ φ − ∇pi /mi − νin ( v⊥i − v⊥n ), v⊥n = −∇pn /mn − νni ( v⊥n − v⊥i ), (∂t + v⊥n ∇)
(2)
where the last terms in the right-hand side represent the momentum transfer between ions and neutrals in the charge-exchange collisions. Setting the left-hand side to zero in the regime ∂t ∼ v⊥i ∇ ∼ v⊥n ∇ eB/mi and using me = mi and νin ni = νni nn , we readily recover the ion and neutral velocities in Ref. [7]. It should be stressed that Vranješ et al. [2,6] used the neutral fluid velocity vn = −Dn ∇nn (Dn is the effective diffusion constant of neutrals), assuming that the ion velocity is completely disregarded in Eq. (2), which clearly violates the momentum conservation in ion–neutral collisions. Hence, the expression (1) for the ion fluid velocity as well as the tripole vortex in Ref. [2] are faulty. In conclusion, we found the criticisms by Vranješ et al. to be physically unsubstantiated. The erroneous statements expressed in their Comment [1] indicate a poor understanding of our work.
Acknowledgements This research was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 591, as well as the European Commission through contract No. HPRN-CT-2001-00314.
References [1] [2] [3] [4] [5] [6] [7]
J. Vranješ et al., Phys. Lett. A, preceding comment paper. J. Vranješ, et al., Phys. Rev. Lett. 89 (2002) 265002. A. Okamoto, et al., J. Plasma Fusion Res. 78 (2002) 1143. K. Nagaoka, et al., J. Plasma Fusion Res. Ser. 4 (2001) 359. K. Nagaoka, et al., Phys. Rev. Lett. 89 (2002) 75001. A. Okamoto, et al., Phys. Plasmas 10 (2003) 2211. D. Jovanovi´c, P.K. Shukla, Phys. Lett. A 326 (2004) 267.
Response to comment
Response to the Comment on: “Theory of vortex flows in partially ionized magnetoplasmas” [Phys. Lett. A 326 (2004) 267] D. Jovanovi´c a,∗ , P.K. Shukla b a Institute of Physics, P.O. Box 57, 11001 Belgrade, Serbia and Montenegro b Institut für Theoretische Physik IV, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Received 22 June 2004; accepted 24 June 2004 Available online 6 July 2004 Communicated by F. Porcelli
Abstract It is demonstrated that the comments by Vranješ et al. [Phys. Lett. A, preceding comment paper] on our recently published paper [Phys. Lett. A 326 (2004) 267] are faulty and misleading. 2004 Elsevier B.V. All rights reserved. PACS: 52.35.Kt; 52.35.Fp; 52.35.Mw
In the following, we present the item-by-item response to the comment paper by Vranješ et al. [1]. 1. Two groups of experiments with plasma vortices [2–5] (cited in the comment paper by Vranješ et al.) and [6] (extended version of [2]), are closely related. They were performed in the same experimental device and with similar goals of “studying the effect of dissipation on the property of the vortex” [5], and of the “experimental observation of the tripolar vortex” [5,6] that emerges due to “the existence of neutral particles [which] usually causes a dissipative effect” [6]. The main dissipation mechanisms in these weakly ionized argon and helium magnetoplasmas were the internal viscous friction due to collisions between particles of the same species (with no effective momentum transfer) and the momentum transfer in the mostly charge-exchange collisions between charged particles and neutrals. Changing the experimental conditions, viz. the neutral density and the working gas type (with DOI of original article: 10.1016/j.physleta.2004.04.040. DOI of comment on original article: 10.1016/j.physleta.2004.06.075. * Corresponding author. E-mail addresses: [email protected] (D. Jovanovi´c), [email protected] (P.K. Shukla). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.06.076
166
D. Jovanovi´c, P.K. Shukla / Physics Letters A 329 (2004) 165–167
a different charge-exchange cross-section), the relative importance of these mechanisms was varied yielding different types of plasma vortices, such as a Burger’s vortex/plasma hole [4,5] and a tripole [2,3,6]. As stated in the conclusions of Ref. [7], our comprehensive analytic theory provides “a complete description of weakly zdependent vortices in partially ionized, collision-dominated magnetoplasmas, including the effects of ionization and recombination, axial and radial flows, viscous damping, neutral drag force, and the ion reaction to it”. Thus, in plasmas dominated by ion–neutral collisions, such as dense argon plasmas of [2,3,6], we obtained a tripole, see our Eqs. (10)–(17) and Fig. 1, while in the case when the effects of viscous and ion–neutral collisional damping were the same order, as in the less dense helium plasma experiments [4,5] we obtained a Burger’s vortex, see Eqs. (8) and (9) of Ref. [7]. We strongly believe that such a successful unified theory should be welcomed in the scientific community. Strangely, Vranješ et al. [1] regard it as “mix[ing the] physics of the two independent experiments”. 2. As stated in Ref. [5], “Burger’s vortices have been considered as an elementary vortical structure in turbulence”. In our context, they emerge asymptotically for large times in the presence of an inward radial flow of neutrals (associated with their strong ionization and the axial flow of the newly born plasma), ion–neutral coupling, and of effective viscosities of both ions and neutrals. These effects are present in all experiments reported in Refs. [2–6]. When the ionization and related anomalous ion viscosity are very strong, the Burger’s vortex described by Eqs. (8) and (9) of Ref. [7] develops a deep hole in the neutral density [4,5]. Conversely, in the regime when the ion–neutral coupling dominates the ion viscosity, a tripolar structure emerges [2,3,6] superimposed on a background Burger’s vortex, since even very weak ion viscosity produces effects in a time-asymptotic regime. The existence of the Burger’s vortex in experiments [2,4,5] manifests as the “spatially nonuniform distribution of the neutral gas which could be described by a r-dependent parabolic function” [2], in full agreement with Eq. (8) of Ref. [7]. Actually, a Gaussian neutral density profile characteristic for a Burger’s vortex [see Eq. (9) in Ref. [7]] was used by Vranješ et al. [2] in the construction of their analytic tripolar solution, but their assumed plasma density profile did not have the appropriate Burger’s ∼ log r asymptotic dependence [compare to Eq. (12) in Ref. [7]]. 3 and 4. The Vranješ et al. [2] analytic tripole shows a poor agreement with experimental data and has incorrect behavior at the boundary. Its size is much larger than ∼50% of the vessel diameter, observed in [6]. Furthermore, it features a finite θ -dependent electric field at the argon–wall interface that corresponds to the finite plasma and neutral gas velocities, which are mutually coupled by collisions. Such tripole would be strongly damped by surface friction, since no θ -dependent driving force is provided. A possible Stern–Larichev–Reznik-style extension to larger values of kr0 would make the situation even worse, since such tripole would either have unphysical surface charges at the separatrix or diverge for large r, rather than decay. Our tripoles in Ref. [7] are not damped. The convective cell solutions (12) and (13) in Ref. [7] asymptotically behaves as a Burger’s vortex, sustained by the cylindrically symmetric drive of the ECRH ionization and radial inflow. The drift-wave solution (16) and (17) in Ref. [7] decays exponentially, having almost no drag at the wall. 5. Our model in Ref. [7] takes into account many essential experimental features previously disregarded by Vranješ et al. [2]. In the ‘tripole regime’, when the effects of E × B convection and ion–neutral collisions are much larger than those of sources and sinks, we account for them by using in the time-asymptotic limit the unperturbed (i.e., r-asymptotic) plasma and neutral densities corresponding to a Burger’s vortex. The latter was previously shown to emerge for t → ∞ under actual experimental conditions. Obviously, the essential experimental features [2–6], such as ionization, recombination and charge exchange do not play a role in ordinary fluids. However, the small size tripole in laboratory experiments and large structures in oceans are described, respectively, by the Euler and Rossby equations that are identical to our equations for convective-cell and drift-wave tripoles. Thus, there is an analogy between vortex structures in magnetized plasmas and ordinary fluids. 6. Our source term νc nn in the continuity equation is actually in complete agreement with the “exact” term αnn ne − (βrr + β3b ne )ne ni − ne /τ offered by Vranješ et al. [1] in their Comment. As they have shown, under actual experimental conditions the ionization αnn ne is more than three orders of magnitude larger than the recombination
D. Jovanovi´c, P.K. Shukla / Physics Letters A 329 (2004) 165–167
167
(second term). Their third term is just a crude model for the variation of the plasma density at a given position z due to the axial flow. In contrast to Vranješ et al. [1], we have fully resolved the parallel dynamics, see Eq. (4) in Ref. [7] and the discussion underneath, and such term does not appear in our continuity equations. Furthermore, in the paragraph preceding Eq. (5) in Ref. [7] we have clearly stated that we work in the regime when the electron density perturbation is small, (ne − ne,0 )/ne,0 1. The source term in Ref. [7] is then readily obtained by using νc = αne,0 . Although small, recombination must be taken into account in a bounded system. It provides a source for neutral atoms, otherwise their number density would be reduced to zero due to permanent ECRH ionization. It was completely disregarded by Vranješ et al. [2]. In Ref. [7], we assume that it takes place outside of the spatial region under study, where the plasma buildup due to the axial flow and the “neutral burnout” by strong ionization (described in some detail in Ref. [3]) produce the reversal of relative populations of charged and neutral particles, yielding enhanced recombination. 7. It is not true that we used Braginskii expressions to describe inelastic processes. As stated in Ref. [7], they were used only for the Coulomb collisions among charged particles, yielding the expressions for ion viscosity. For inelastic collisions we used complete collision integrals with proper cross-sections for charge-exchange and vn − vk ), k = e, i were obtained as their first moments, for ionization. The momentum-transfer terms ν(nn /nk )( small perturbations of electron and ion densities and weak velocity dependence of the cross-section. 8. It is not true that our perpendicular ion dynamics is unaffected by ion–neutral collisions. It is derived from the coupled ion and neutral momentum equations (∂t + v⊥i ∇) (1) v⊥i = (e/mi ) v⊥i × B − ∇⊥ φ − ∇pi /mi − νin ( v⊥i − v⊥n ), v⊥n = −∇pn /mn − νni ( v⊥n − v⊥i ), (∂t + v⊥n ∇)
(2)
where the last terms in the right-hand side represent the momentum transfer between ions and neutrals in the charge-exchange collisions. Setting the left-hand side to zero in the regime ∂t ∼ v⊥i ∇ ∼ v⊥n ∇ eB/mi and using me = mi and νin ni = νni nn , we readily recover the ion and neutral velocities in Ref. [7]. It should be stressed that Vranješ et al. [2,6] used the neutral fluid velocity vn = −Dn ∇nn (Dn is the effective diffusion constant of neutrals), assuming that the ion velocity is completely disregarded in Eq. (2), which clearly violates the momentum conservation in ion–neutral collisions. Hence, the expression (1) for the ion fluid velocity as well as the tripole vortex in Ref. [2] are faulty. In conclusion, we found the criticisms by Vranješ et al. to be physically unsubstantiated. The erroneous statements expressed in their Comment [1] indicate a poor understanding of our work.
Acknowledgements This research was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 591, as well as the European Commission through contract No. HPRN-CT-2001-00314.
References [1] [2] [3] [4] [5] [6] [7]
J. Vranješ et al., Phys. Lett. A, preceding comment paper. J. Vranješ, et al., Phys. Rev. Lett. 89 (2002) 265002. A. Okamoto, et al., J. Plasma Fusion Res. 78 (2002) 1143. K. Nagaoka, et al., J. Plasma Fusion Res. Ser. 4 (2001) 359. K. Nagaoka, et al., Phys. Rev. Lett. 89 (2002) 75001. A. Okamoto, et al., Phys. Plasmas 10 (2003) 2211. D. Jovanovi´c, P.K. Shukla, Phys. Lett. A 326 (2004) 267.