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Plasma vortexes induced by an external rotating helical magnetic perturbation in tokamaks
Physics Letters A 343 (2005) 216–223 www.elsevier.com/locate/pla
Plasma vortexes induced by an external rotating helical magnetic perturbation in tokamaks I.M. Pankratov ∗ , A.Ya. Omelchenko, V.V. Olshansky Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics and Technology”, Akademicheskaya str., 1, 61108 Kharkov, Ukraine Received 8 December 2004; received in revised form 22 April 2005; accepted 1 June 2005 Available online 9 June 2005 Communicated by F. Porcelli
Abstract The occurrence of two or four vortexes per one poloidal perturbation period has been found near the resonant surface as a plasma motion response on the penetration of an external low frequency helical magnetic perturbation in tokamaks. The investigation is carried out on the basis of the two-fluid MHD equations in the linear approximation for the cylindrical model. 2005 Elsevier B.V. All rights reserved. PACS: 52.30.-q; 52.35.Vd; 52.35.We; 52.55.Fa Keywords: Plasma vortexes; Helical magnetic perturbation
1. Introduction The creation of magnetic perturbations (force reconnection) on the plasma edge rational magnetic surfaces by an external rotating low frequency helical magnetic field is investigated in the TEXTOR-DED [1] and HYBTOK-II [2] tokamaks. The control of the edge plasma behavior is the main purpose of these experiments. Direct observations of tokamak plasma responses to an externally applied rotating helical magnetic perturbation have been performed only on a small tokamak HYBTOK-II (R = 0.4 m, a = 0.11 m) in order to clarify the process of penetration of this external magnetic perturbation into tokamak plasmas [2]. The radial profiles of the radial and poloidal magnetic components of the penetrating external field were measured using a small magnetic probe inserted into the plasma. A comparison of the theoretically calculated profiles with these HYBTOK-II experiments shows a good qualitative agreement [2,3]. * Corresponding author.
E-mail address: [email protected] (I.M. Pankratov). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.06.002
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That is why in the present Letter a more detailed theoretical study of the HYBTOK-II experiments is made. The motion of plasma response to an external magnetic perturbation is studied. We observe two plasma vortexes or the formation of four ones per one poloidal period of the perturbation. These vortexes are located near the plasma edge and hence they are suitable for direct observations. They may be responsible for the edge plasma transport.
2. Model and basic equations We start from the two-fluid MHD equations for continuity, momentum and energy conservation for plasma ions and electrons and the Maxwell’s equations. The electron–ion collision frequency is higher than the frequency of the external perturbation. The electron inertia and electron stress tensor are neglected. Hence, we use the plasma equation of motion ∂V + (V · ∇)V = −∇p − ∇ · π i + [J × B], ρ (1) ∂t and the generalized Ohm’s law J /σ + J⊥ /σ⊥ = E + [V × B] + ∇pe /ene − [J × B]/en + (0.71/e)∇ Te
(e > 0),
(2)
where ne and ρ are the plasma and plasma mass densities, p = pe + pi is the total pressure, J is the total current density, π i is the ion gyroviscosity tensor, σ and σ⊥ are the parallel and perpendicular (with respect to the magnetic field B) conductivities, respectively. The parallel electron heat-conductivity coefficient is assumed to be a large value. Hence, in our approximation (Te is the electron temperature) B · ∇Te = 0.
(3)
A model of a current carrying cylindrical plasma, whose axis is taken as the z direction, is used. The external axial magnetic field Bz0 is large in comparison with the poloidal magnetic field Bθ0 produced by the axial current. The perturbation values depend on the azimuthal angle θ , the coordinate z (k = n/R) and the time t as exp[i(mθ − kz − ωt)], m and n are poloidal and toroidal numbers, respectively, R plays the role of the tokamak major radius, ω is the frequency of the external perturbation. In the linear approximation after some algebra we obtain the closed system of the equations for perturbations of radial components of plasma velocity Vr∼ and magnetic field Br∼ (see details in Ref. [3]) 2 d d 2F d r 2 F 2 (r) r ω dF Br∼ rω ρ (rVr∼ ) − ω m2 ρ + i 2 Vr∼ = i 2 F (r) + r 2 2 + 3r (4) , dr dr dr µ0 δ µ0 ω δ ω dr r 2 ω r 2 F (r)Vr∼ d d ∼ 2 r (rBr ) − m − i 2 Br∼ = −i 2 , (5) dr dr ω δ ω δ where √ δ = 1/ µ0 σ ω, ω = ω + (m/r)(Er0 /Bz0 ) + kVz0 , (6) m F (r) = k · B0 = Bθ0 − kBz0 . (7) r In Eqs. (4), (5) the terms kVz∼ and kBz∼ (kr < m) are neglected. We included the poloidal plasma rotation connected with an equilibrium radial electric field Er0 and the toroidal plasma rotation with a homogeneous velocity Vz0 . We assume that the equilibrium quantities are slowly varying. Note, that the right side of Eq. (4) contains the radial derivative of the equilibrium current density dJz0 /dr = (1/µ0 m) r d 2 F /dr 2 + 3 dF /dr . (8)
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The main goal of our Letter is the investigation of plasma motion response induced by an external magnetic perturbation. This study is made on the basis of the solution of the Eq. (4), it takes the next form (see also [4]) √ 0 Vr∼ (r) 3π π |z| 3π (1) (1) 2 R + (u) = √ m−1 H1/4 z exp i du uH1/4 u exp i 0.5(1 + i) VrA 4 4 4 2rres z2 (a)
0 − 0.5(1 − i)
z2 3π 3π (1) (2) − + du uH1/4 u exp i R (u) − du uH1/4 u exp i R (u) 4 4 0
z2 (0)
3π 3π (1) 2 + z exp i R (u) du uH1/4 u exp i 4 4 z2
(2) + H1/4
for r rres ,
(9)
z2 (a)
√ 0 Vr∼ (r) π |z| 3π 3π (1) (1) 2 = √ m−1 H1/4 z exp i du uH1/4 u exp i 0.5(1 + i) R − (u) VrA 4 4 4 2rres z2 (0)
0 − 0.5(1 − i)
(1) du uH1/4
z2 3π 3π (2) + − u exp i R (u) − du uH1/4 u exp i R (u) 4 4 0
z2 (a)
(2) + H1/4
z2 3π 3π (1) z2 exp i R − (u) du uH1/4 u exp i 4 4
for r rres ,
(10)
z2 (0)
where
rres 1/2 |ω |3/4 1 R ± (u) = r m−1 ± δ·Q ω3/4 u3/4
2 1 F F i δ · rres ω 1/2 d 2 d , +√ sgn(ω ) r + 3r |ω | dr µ0 ρω2 u5/4 dr 2 µ0 ρω2 2Q r 2 √ z(r) = (rres Q/2δ)1/2 (r − rres )/rres , Q = nSVzA /ωR, VzA = Bz0 / µ0 ρ, √ (1,2) VrA = Brvac / µ0 ρ r=r , H1/4 (ς) are the Hankel’s functions. S = (rq /q)r=r , res
res
(11)
Here we take into account that in HYBTOK-II case the radial component of vacuum magnetic perturbation is large in comparison with the plasma response one. This means that in this approximation the plasma motion generated by this vacuum perturbation. In this Letter only the main HYBTOK-II resonant mode (m/n = 6/1) is investigated, when the value of F (r) is equal to zero, F (rres ) = 0, on the main resonance surface rres = 8.5 cm, where q(rres ) = 6/1 (q(r) = rBz0 /RBθ0 is the safety factor). The typical HYBTOK-II parameters are used: the toroidal magnetic field Bz0 = 0.27 T, the plasma current Ip = 5 kA, the edge electron density ne = 1.5 × 1018 m−3 and the electron temperature Te = 25 eV.
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3. Results and discussion In Eqs. (4), (5) we neglected the ion diamagnetic drift velocity (compare with Ref. [3]). Note that Eqs. (4), (5) do not depend on the driving frequency ω because these equations contain ω in the combination ωδ 2 (ω) only. Hence Eqs. (4), (5) contain the Doppler shifted frequency ω as a main parameter. This frequency defines the resonant (resistive) zone width [3,5] r ∼ (2rres δ/Q)1/2 |ω |1/4 /ω1/4 .
(12)
Recall [3] that for the HYBTOK-II and TEXTOR-DED experiments [1,2] the resistive effects dominate in a broader region than that defined by the Alfvén resonances. Results of the calculations depend on the local value of the conductivity σ . For the HYBTOK-II experiments the value of σ depends not only on Zeff , but also on the contribution of neutral particles [6]. First we put the skin depth value δ = 1 cm for f = 30 kHz. For this case in Figs. 1–3 the results of calculations for three Doppler shifted frequencies f = 10, 30 and 40 kHz are presented. ∼ , V ∼ and J ∼ are normalized to the values B vac (r ), V vac √ In the figures the values of Br,θ res rA = Br / µ0 ρ|r=rres z r r,θ vac vac and Br (rres )/µ0 rres , respectively. For estimates we use Br (rres ) = 1 G. ∼ amplitudes and their phases ψ In Figs. 1a, d, 2a, d, 3a, d the radial profiles of Br,θ Br,θ are shown. These calculated results are in a good qualitative agreement with HYBTOK-II experimental measurements (see, e.g., [2, case I]). The gap in the profile of |Bθ∼ | is clearly visible near r ≈ rres . The minimum value of this gap is shifted to the plasma interior from the surface r = rres . ∼ . Because |B ∼ | grows towards the antenna, Figs. 1b, e, 2b, e, 3b, e show the radial profiles of the velocities Vr,θ r ∼ ∼ |Vr | has a finite value at the plasma edge. In Figs. 2b, 3b |Vr | ∼ 0.4 km/s at r > 10 cm. In Figs. 1e, 2e, 3e the maximum values of |Vθ∼ | are max |Vθ∼ | ∼ 2–3 km/s. The value of |Vθ∼ | grows, when f drops. For example, max |Vθ∼ | ∼ 10 km/s for f = 1 kHz.
Fig. 1. Radial profiles of (a) amplitudes and (d) phases of Br∼ , Bθ∼ , (b) a velocity Vr∼ , (e) a velocity Vθ∼ . (c) Plasma fluxes in the plane (r, θ), (f) 2D profile of the perturbed current density. Case f = 10 kHz, δ = 1 cm. Two vortexes are observed per one poloidal period of the magnetic perturbation.
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Fig. 2. Same as in Fig. 1 for f = 30 kHz, δ = 1 cm. Two vortexes are still observed per one poloidal period of the magnetic perturbation.
Fig. 3. Same as in Fig. 1 for f = 40 kHz, δ = 1 cm. A formation of four vortexes is observed per one poloidal period of the magnetic perturbation.
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Figs. 1c, 2c, 3c show the magnetic island m/n = 6/1 position and contour plots of the plasma fluxes (the arrows show the direction of motion) Re V∼ (r) exp i(mθ − kz − ωt) (13) in the poloidal cross-section (r, θ ) for one poloidal period of perturbation θ = π/3 near the main resonance surface rres = 8.5 cm for a certain moment of time. For simplicity we put z = 0 and t = 0. In time this picture rotates as a unit. The calculated width of the magnetic island is approximately 0.5 cm. When the Doppler shifted frequency ω increases, the resonant zone (resistive layer) size increases (see Eq. (12)). As a result of the external perturbation effects the plasma vortexes occur. The analysis of the plasma fluxes on the plane (r, θ ) shows that the positions of X- and O-points of plasma vortexes could be found from equations (Vr∼ = |Vr∼ (r)| exp iψVr (r)) cos(mθ + ψVr ) = 0, d r|Vr∼ | = 0. dr 2
(14a) (14b) 2
d d ∼ ∼ For X-points dr 2 (r|Vr |) > 0 and for O-points dr 2 (r|Vr |) < 0. When the value of f is not too high (Figs. 1c, 2c), only two vortexes occur with opposite direction of rotation per one poloidal period of the external perturbation. For sufficiently high frequency f (Fig. 3c) a more complicated behavior of plasma fluxes on the plane (r, θ ) takes place, the formation of four vortexes is observed per one poloidal period of the external perturbation. In the last case, more essential changes of phases ψBr,θ are observed (compare Fig. 3d with Figs. 1d, 2d). The plasma moves across the resonant surface inwards (outwards) of the discharge near O-point (X-point) of the magnetic islands (compare with [7]). The last statement does not concern the plasma inside four vortexes. Recall that four vortexes per one poloidal period of perturbation were also observed during the stability analysis of the resistive tearing eigenmodes (see, e.g., [8]). In HYBTOK-II the sideband modes m/n = 5/1 and m/n = 7/1 are resonant at rres = 7 cm and rres = 9.5 cm, respectively. Hence the m/n = 6/1 vortexes overlap these sideband modes. In this case, a strong coupling between m and m ± 1 modes through the plasma motion is possible (see, e.g., [9]). In Figs. 1f, 2f, 3f the 2D profiles of the perturbed current density Jz∼ are presented. Here Jz∼ ∼ 10–15 kA/m2 . To investigate the influence of the value of conductivity σ on the effect of the external perturbation on the plasma motion response, we consider the second case, where the skin depth value is δ = 2 cm for f = 30 kHz. For this case in Figs. 4, 5 the results of calculations for two Doppler shifted frequencies f = 30 and 50 kHz are presented. The resonant zone width increases and the amplitude of the perturbed current density decreases in comparison with the first case. Changes in phases ψBr,θ in Figs. 4d, 5d are less in comparison with Figs. 2d, 3d. Here only two vortexes occur per one poloidal period of perturbation. ∼ | are observed, In figures the situation f > 0 is presented. For f < 0 (Eq. (6)) the same radial profiles of |Br,θ ∼ ∼ the values of Re Vr and Im Vθ change the sign, and the phases ψBr,θ decrease now towards the plasma interior. The effect on the poloidal rotation profile of an external rotating helical magnetic perturbation was observed near resonant surfaces in the HYBTOK-II experiment [10], but more detailed experiments are needed.
4. Conclusions The present calculations reproduce not only the radial profiles of amplitudes, but also the phase radial profiles of externally induced magnetic perturbations in the HYBTOK-II experiments. The plasma vortexes with opposite direction of rotation are found per one poloidal period of the external perturbation. The cases with two vortexes and the formation of four vortexes per one poloidal period are considered. In the cylindrical geometry the similar modelling of the TEXTOR-DED operation was also made and the similar plasma vortex motions were observed. These vortexes may be responsible for the edge plasma transport.
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Fig. 4. Same as in Fig. 1 for f = 30 kHz, δ = 2 cm. Two vortexes are observed per one poloidal period of the magnetic perturbation.
Fig. 5. Same as in Fig. 1 for f = 50 kHz, δ = 2 cm. Two vortexes are observed per one poloidal period of the magnetic perturbation.
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Note again that a strong coupling between m and m ± 1 poloidal modes through the plasma motion is possible. The calculations for both HYBTOK-II and TEXTOR-DED tokamaks show, that distributions of the perturbed current density and the plasma vortexes are not symmetric with respect to resonant surface.
Acknowledgements The authors would like to thank Prof. S. Takamura for fruitful discussions.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
M.W. Jakubowski, S.S. Abdullaev, K.H. Finken, Nucl. Fusion 44 (2004) S1. Y. Kikuchi, Y. Uesugi, S. Takamura, A.G. Elfimov, Nucl. Fusion 44 (2004) S28. I.M. Pankratov, A.Ya. Omelchenko, V.V. Olshansky, K.H. Finken, Nucl. Fusion 44 (2004) S37. I.M. Pankratov, A.Ya. Omelchenko, V.V. Olshansky, Problems of Atomic Science and Technology. Series: Plasma Physics (8), (2002) No. 5, p. 3. D.W. Faulconer, R. Koch, Fusion Eng. Des. 37 (1997) 399. S. Takamura, Y. Kikuchi, Y. Uesugi, M. Kobayashi, Nucl. Fusion 43 (2003) 393. T. Tuda, M. Kobayashi, G. Kurita, S. Takamura, Contrib. Plasma Phys. 40 (2000) 256. G. Bateman, MHD Instability, MIT Press, Cambridge, 1979. J.M. Finn, Phys. Fluids 20 (1977) 1749. V.P. Budaev, Y. Kikuchi, M. Toyoda, et al., J. Nucl. Mater. 313–316 (2003) 1309.
Plasma vortexes induced by an external rotating helical magnetic perturbation in tokamaks I.M. Pankratov ∗ , A.Ya. Omelchenko, V.V. Olshansky Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics and Technology”, Akademicheskaya str., 1, 61108 Kharkov, Ukraine Received 8 December 2004; received in revised form 22 April 2005; accepted 1 June 2005 Available online 9 June 2005 Communicated by F. Porcelli
Abstract The occurrence of two or four vortexes per one poloidal perturbation period has been found near the resonant surface as a plasma motion response on the penetration of an external low frequency helical magnetic perturbation in tokamaks. The investigation is carried out on the basis of the two-fluid MHD equations in the linear approximation for the cylindrical model. 2005 Elsevier B.V. All rights reserved. PACS: 52.30.-q; 52.35.Vd; 52.35.We; 52.55.Fa Keywords: Plasma vortexes; Helical magnetic perturbation
1. Introduction The creation of magnetic perturbations (force reconnection) on the plasma edge rational magnetic surfaces by an external rotating low frequency helical magnetic field is investigated in the TEXTOR-DED [1] and HYBTOK-II [2] tokamaks. The control of the edge plasma behavior is the main purpose of these experiments. Direct observations of tokamak plasma responses to an externally applied rotating helical magnetic perturbation have been performed only on a small tokamak HYBTOK-II (R = 0.4 m, a = 0.11 m) in order to clarify the process of penetration of this external magnetic perturbation into tokamak plasmas [2]. The radial profiles of the radial and poloidal magnetic components of the penetrating external field were measured using a small magnetic probe inserted into the plasma. A comparison of the theoretically calculated profiles with these HYBTOK-II experiments shows a good qualitative agreement [2,3]. * Corresponding author.
E-mail address: [email protected] (I.M. Pankratov). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.06.002
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That is why in the present Letter a more detailed theoretical study of the HYBTOK-II experiments is made. The motion of plasma response to an external magnetic perturbation is studied. We observe two plasma vortexes or the formation of four ones per one poloidal period of the perturbation. These vortexes are located near the plasma edge and hence they are suitable for direct observations. They may be responsible for the edge plasma transport.
2. Model and basic equations We start from the two-fluid MHD equations for continuity, momentum and energy conservation for plasma ions and electrons and the Maxwell’s equations. The electron–ion collision frequency is higher than the frequency of the external perturbation. The electron inertia and electron stress tensor are neglected. Hence, we use the plasma equation of motion ∂V + (V · ∇)V = −∇p − ∇ · π i + [J × B], ρ (1) ∂t and the generalized Ohm’s law J /σ + J⊥ /σ⊥ = E + [V × B] + ∇pe /ene − [J × B]/en + (0.71/e)∇ Te
(e > 0),
(2)
where ne and ρ are the plasma and plasma mass densities, p = pe + pi is the total pressure, J is the total current density, π i is the ion gyroviscosity tensor, σ and σ⊥ are the parallel and perpendicular (with respect to the magnetic field B) conductivities, respectively. The parallel electron heat-conductivity coefficient is assumed to be a large value. Hence, in our approximation (Te is the electron temperature) B · ∇Te = 0.
(3)
A model of a current carrying cylindrical plasma, whose axis is taken as the z direction, is used. The external axial magnetic field Bz0 is large in comparison with the poloidal magnetic field Bθ0 produced by the axial current. The perturbation values depend on the azimuthal angle θ , the coordinate z (k = n/R) and the time t as exp[i(mθ − kz − ωt)], m and n are poloidal and toroidal numbers, respectively, R plays the role of the tokamak major radius, ω is the frequency of the external perturbation. In the linear approximation after some algebra we obtain the closed system of the equations for perturbations of radial components of plasma velocity Vr∼ and magnetic field Br∼ (see details in Ref. [3]) 2 d d 2F d r 2 F 2 (r) r ω dF Br∼ rω ρ (rVr∼ ) − ω m2 ρ + i 2 Vr∼ = i 2 F (r) + r 2 2 + 3r (4) , dr dr dr µ0 δ µ0 ω δ ω dr r 2 ω r 2 F (r)Vr∼ d d ∼ 2 r (rBr ) − m − i 2 Br∼ = −i 2 , (5) dr dr ω δ ω δ where √ δ = 1/ µ0 σ ω, ω = ω + (m/r)(Er0 /Bz0 ) + kVz0 , (6) m F (r) = k · B0 = Bθ0 − kBz0 . (7) r In Eqs. (4), (5) the terms kVz∼ and kBz∼ (kr < m) are neglected. We included the poloidal plasma rotation connected with an equilibrium radial electric field Er0 and the toroidal plasma rotation with a homogeneous velocity Vz0 . We assume that the equilibrium quantities are slowly varying. Note, that the right side of Eq. (4) contains the radial derivative of the equilibrium current density dJz0 /dr = (1/µ0 m) r d 2 F /dr 2 + 3 dF /dr . (8)
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The main goal of our Letter is the investigation of plasma motion response induced by an external magnetic perturbation. This study is made on the basis of the solution of the Eq. (4), it takes the next form (see also [4]) √ 0 Vr∼ (r) 3π π |z| 3π (1) (1) 2 R + (u) = √ m−1 H1/4 z exp i du uH1/4 u exp i 0.5(1 + i) VrA 4 4 4 2rres z2 (a)
0 − 0.5(1 − i)
z2 3π 3π (1) (2) − + du uH1/4 u exp i R (u) − du uH1/4 u exp i R (u) 4 4 0
z2 (0)
3π 3π (1) 2 + z exp i R (u) du uH1/4 u exp i 4 4 z2
(2) + H1/4
for r rres ,
(9)
z2 (a)
√ 0 Vr∼ (r) π |z| 3π 3π (1) (1) 2 = √ m−1 H1/4 z exp i du uH1/4 u exp i 0.5(1 + i) R − (u) VrA 4 4 4 2rres z2 (0)
0 − 0.5(1 − i)
(1) du uH1/4
z2 3π 3π (2) + − u exp i R (u) − du uH1/4 u exp i R (u) 4 4 0
z2 (a)
(2) + H1/4
z2 3π 3π (1) z2 exp i R − (u) du uH1/4 u exp i 4 4
for r rres ,
(10)
z2 (0)
where
rres 1/2 |ω |3/4 1 R ± (u) = r m−1 ± δ·Q ω3/4 u3/4
2 1 F F i δ · rres ω 1/2 d 2 d , +√ sgn(ω ) r + 3r |ω | dr µ0 ρω2 u5/4 dr 2 µ0 ρω2 2Q r 2 √ z(r) = (rres Q/2δ)1/2 (r − rres )/rres , Q = nSVzA /ωR, VzA = Bz0 / µ0 ρ, √ (1,2) VrA = Brvac / µ0 ρ r=r , H1/4 (ς) are the Hankel’s functions. S = (rq /q)r=r , res
res
(11)
Here we take into account that in HYBTOK-II case the radial component of vacuum magnetic perturbation is large in comparison with the plasma response one. This means that in this approximation the plasma motion generated by this vacuum perturbation. In this Letter only the main HYBTOK-II resonant mode (m/n = 6/1) is investigated, when the value of F (r) is equal to zero, F (rres ) = 0, on the main resonance surface rres = 8.5 cm, where q(rres ) = 6/1 (q(r) = rBz0 /RBθ0 is the safety factor). The typical HYBTOK-II parameters are used: the toroidal magnetic field Bz0 = 0.27 T, the plasma current Ip = 5 kA, the edge electron density ne = 1.5 × 1018 m−3 and the electron temperature Te = 25 eV.
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3. Results and discussion In Eqs. (4), (5) we neglected the ion diamagnetic drift velocity (compare with Ref. [3]). Note that Eqs. (4), (5) do not depend on the driving frequency ω because these equations contain ω in the combination ωδ 2 (ω) only. Hence Eqs. (4), (5) contain the Doppler shifted frequency ω as a main parameter. This frequency defines the resonant (resistive) zone width [3,5] r ∼ (2rres δ/Q)1/2 |ω |1/4 /ω1/4 .
(12)
Recall [3] that for the HYBTOK-II and TEXTOR-DED experiments [1,2] the resistive effects dominate in a broader region than that defined by the Alfvén resonances. Results of the calculations depend on the local value of the conductivity σ . For the HYBTOK-II experiments the value of σ depends not only on Zeff , but also on the contribution of neutral particles [6]. First we put the skin depth value δ = 1 cm for f = 30 kHz. For this case in Figs. 1–3 the results of calculations for three Doppler shifted frequencies f = 10, 30 and 40 kHz are presented. ∼ , V ∼ and J ∼ are normalized to the values B vac (r ), V vac √ In the figures the values of Br,θ res rA = Br / µ0 ρ|r=rres z r r,θ vac vac and Br (rres )/µ0 rres , respectively. For estimates we use Br (rres ) = 1 G. ∼ amplitudes and their phases ψ In Figs. 1a, d, 2a, d, 3a, d the radial profiles of Br,θ Br,θ are shown. These calculated results are in a good qualitative agreement with HYBTOK-II experimental measurements (see, e.g., [2, case I]). The gap in the profile of |Bθ∼ | is clearly visible near r ≈ rres . The minimum value of this gap is shifted to the plasma interior from the surface r = rres . ∼ . Because |B ∼ | grows towards the antenna, Figs. 1b, e, 2b, e, 3b, e show the radial profiles of the velocities Vr,θ r ∼ ∼ |Vr | has a finite value at the plasma edge. In Figs. 2b, 3b |Vr | ∼ 0.4 km/s at r > 10 cm. In Figs. 1e, 2e, 3e the maximum values of |Vθ∼ | are max |Vθ∼ | ∼ 2–3 km/s. The value of |Vθ∼ | grows, when f drops. For example, max |Vθ∼ | ∼ 10 km/s for f = 1 kHz.
Fig. 1. Radial profiles of (a) amplitudes and (d) phases of Br∼ , Bθ∼ , (b) a velocity Vr∼ , (e) a velocity Vθ∼ . (c) Plasma fluxes in the plane (r, θ), (f) 2D profile of the perturbed current density. Case f = 10 kHz, δ = 1 cm. Two vortexes are observed per one poloidal period of the magnetic perturbation.
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Fig. 2. Same as in Fig. 1 for f = 30 kHz, δ = 1 cm. Two vortexes are still observed per one poloidal period of the magnetic perturbation.
Fig. 3. Same as in Fig. 1 for f = 40 kHz, δ = 1 cm. A formation of four vortexes is observed per one poloidal period of the magnetic perturbation.
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Figs. 1c, 2c, 3c show the magnetic island m/n = 6/1 position and contour plots of the plasma fluxes (the arrows show the direction of motion) Re V∼ (r) exp i(mθ − kz − ωt) (13) in the poloidal cross-section (r, θ ) for one poloidal period of perturbation θ = π/3 near the main resonance surface rres = 8.5 cm for a certain moment of time. For simplicity we put z = 0 and t = 0. In time this picture rotates as a unit. The calculated width of the magnetic island is approximately 0.5 cm. When the Doppler shifted frequency ω increases, the resonant zone (resistive layer) size increases (see Eq. (12)). As a result of the external perturbation effects the plasma vortexes occur. The analysis of the plasma fluxes on the plane (r, θ ) shows that the positions of X- and O-points of plasma vortexes could be found from equations (Vr∼ = |Vr∼ (r)| exp iψVr (r)) cos(mθ + ψVr ) = 0, d r|Vr∼ | = 0. dr 2
(14a) (14b) 2
d d ∼ ∼ For X-points dr 2 (r|Vr |) > 0 and for O-points dr 2 (r|Vr |) < 0. When the value of f is not too high (Figs. 1c, 2c), only two vortexes occur with opposite direction of rotation per one poloidal period of the external perturbation. For sufficiently high frequency f (Fig. 3c) a more complicated behavior of plasma fluxes on the plane (r, θ ) takes place, the formation of four vortexes is observed per one poloidal period of the external perturbation. In the last case, more essential changes of phases ψBr,θ are observed (compare Fig. 3d with Figs. 1d, 2d). The plasma moves across the resonant surface inwards (outwards) of the discharge near O-point (X-point) of the magnetic islands (compare with [7]). The last statement does not concern the plasma inside four vortexes. Recall that four vortexes per one poloidal period of perturbation were also observed during the stability analysis of the resistive tearing eigenmodes (see, e.g., [8]). In HYBTOK-II the sideband modes m/n = 5/1 and m/n = 7/1 are resonant at rres = 7 cm and rres = 9.5 cm, respectively. Hence the m/n = 6/1 vortexes overlap these sideband modes. In this case, a strong coupling between m and m ± 1 modes through the plasma motion is possible (see, e.g., [9]). In Figs. 1f, 2f, 3f the 2D profiles of the perturbed current density Jz∼ are presented. Here Jz∼ ∼ 10–15 kA/m2 . To investigate the influence of the value of conductivity σ on the effect of the external perturbation on the plasma motion response, we consider the second case, where the skin depth value is δ = 2 cm for f = 30 kHz. For this case in Figs. 4, 5 the results of calculations for two Doppler shifted frequencies f = 30 and 50 kHz are presented. The resonant zone width increases and the amplitude of the perturbed current density decreases in comparison with the first case. Changes in phases ψBr,θ in Figs. 4d, 5d are less in comparison with Figs. 2d, 3d. Here only two vortexes occur per one poloidal period of perturbation. ∼ | are observed, In figures the situation f > 0 is presented. For f < 0 (Eq. (6)) the same radial profiles of |Br,θ ∼ ∼ the values of Re Vr and Im Vθ change the sign, and the phases ψBr,θ decrease now towards the plasma interior. The effect on the poloidal rotation profile of an external rotating helical magnetic perturbation was observed near resonant surfaces in the HYBTOK-II experiment [10], but more detailed experiments are needed.
4. Conclusions The present calculations reproduce not only the radial profiles of amplitudes, but also the phase radial profiles of externally induced magnetic perturbations in the HYBTOK-II experiments. The plasma vortexes with opposite direction of rotation are found per one poloidal period of the external perturbation. The cases with two vortexes and the formation of four vortexes per one poloidal period are considered. In the cylindrical geometry the similar modelling of the TEXTOR-DED operation was also made and the similar plasma vortex motions were observed. These vortexes may be responsible for the edge plasma transport.
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Fig. 4. Same as in Fig. 1 for f = 30 kHz, δ = 2 cm. Two vortexes are observed per one poloidal period of the magnetic perturbation.
Fig. 5. Same as in Fig. 1 for f = 50 kHz, δ = 2 cm. Two vortexes are observed per one poloidal period of the magnetic perturbation.
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Note again that a strong coupling between m and m ± 1 poloidal modes through the plasma motion is possible. The calculations for both HYBTOK-II and TEXTOR-DED tokamaks show, that distributions of the perturbed current density and the plasma vortexes are not symmetric with respect to resonant surface.
Acknowledgements The authors would like to thank Prof. S. Takamura for fruitful discussions.
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