1. INTRODUCTION
The ion Bernstein waves (IBWs) play an important role in radio frequency heating of tokamak in the ion cyclotron range of frequencies (ICRF), hence are a subject of continued attention (Bonoli et al., Reference Bonoli, Ò Shea, Brambilla, Golovato, Hubbard, Porkolab, Takase, Boivin, Bombarda, Christensen, Fiore, Garnier, Goetz, Granetz, Greenwald, Horne, Hutchinson, Irby, Jablonski, Labombard, Lipschultz, Marmar, May, Mazurenko, Mccracken, Nachtrieb, Niemczewki, Ohkawa, Pappas, Reardon, Rice, Rost, Schachter, Snipes, Stek, Takase, Terry, Wang, Watterson, Welch and Wolfe1997; Tripathi et al., Reference Tripathi, Liu and Chiu1987; Paoletti et al., Reference Paoletti, Cardinali, Bernabei, Post-Zwicker, Tighe and Von Goeler1999; Brizard & Kaufman, Reference Brizard and Kaufman1996; Sharma & Tripathi, Reference Sharma and Tripathi1988; Myra & D'lppolito, Reference Myra and D'Lppolito1997; Zhao et al., Reference Zhao, Li, Luo, Wan, Mao, Bao, Lin, Gong, Gao, Yang, Jie, Liu, Liu, Xie and Wan2001; Li et al., Reference Li, Wan, Luo, Kuang, Zhao, Zhang, Liu, Fu, Xie, Zhang, Gu, Mao, Shan, Bai, Gentle, Rowan, Philippe, Huang, Lao, Chan, Watari, Seki and Nakamura2003; Sharma et al., Reference Sharma, Kumar, Kumar and Tripathi1994). A large amplitude IBW can strongly couple with low frequency turbulence and influence plasma transport. It may also give rise to parametric instabilities and ponderomotive effects that have been suggested as cause for serious impurity release from the walls in experiment on IBW heating (Li et al., Reference Li, Bao, Zhao, Luo, Wan, Gao, Xie, Wan and Toi2001; Clark & Fisch, Reference Clark and Fisch2000; Kumar & Sharma, Reference Kumar and Sharma1989). The IBW can penetrate the hot plasma core without strong attenuation until approaching the harmonic cyclotron layers where strong ion cyclotron damping occurs (Cardinali, Reference Cardinali1993; Ono, Reference Ono1993; Cardinali et al., Reference Cardinali, Castaldo, Cesario and De Marco1998). Plasma heating by directly launched ion Bernstein waves (IBWH) has been actively investigated in recent years (Sugaya, Reference Sugaya1987; Ono et al., Reference Ono, Beiersdorfer, Bell, Bernabei, Cavallo, Chmyga, Cohen, Colestock, Gammel, Greene, Hosea, Kaita, Lehrman, Mazziteli Mazzucato, Mcneill, Sato, Stevens, Timberlake, Wilson and Wouters1988). As a result of their relatively short wavelength, the IBW can heat the bulk-ion distributions, and the wave polarization, and the relatively wide operating frequency range permit a flexible waveguide launcher design attractive for the compact ignition device. IBWH can also interact nonlinearly with subharmonics of the ion cyclotron frequencies, giving rise to new heating scenarios. Porkolab (Reference Porkolab1985) has shown that nonlinear ion Landau (cyclotron) damping efficiently absorbs the pump-wave power during ion Bernstein wave heating experiments in tokamak and tandem mirrors. The heating experiments on the Frascati Tokamak UP-grade (Cesario et al., Reference Cesario, Cardinali, Castaldo, Leigheb, Marinucci, Pericoli-Ridolfini, Zonca, Apruzzese, Borra, De Angelis, Giovannozzi, Gabellieri, Kroegler, Mazziteli, Micozzi, Panaccione, Papitto, Podda, Ravera, Angelini, Apicella, Barbato, Bertalot, Bertocchi, Buceti, Cascino, Centioli, Chuilon, Ciattaglia, Cocilovo, Crisanti, De Marco, Esposito, Gatti, Gormezano, Grolli, Lannone, Maddaluno, Monari, Orsitto, Pacella, Panella, Pieroni, Righetti, Romanelli, Sternini, Tartoni, Trevisanutto, Tuccillo, Tudisco, Vitale, Vlad and Zerbini2001) have reported efficient ion heating up to the fourth harmonic of hydrogen plasma. There is also a possibility that the IBW could be employed to enhance plasma confinement and drive poloidal current.
There is yet another potential source of IBW excitation, viz., the neutral particle beam injected into tokamak for auxiliary heating. The neutral beam quickly gets ionized to convert into a gyrating ion beam and latter can excite IBW. Saha et al. (Reference Saha, Raychaudhuri and Sengupta1988) have experimentally observed the excitation of IBW by an ion beam in a beam created-plasma in an axial magnetic field. Lonnroth et al. (Reference Lonnroth, Heikkinen, Rantamaki and Karttunen2002) have observed ion Bernstein mode excitation in their particle in cell simulations. Kuo et al. (Reference Kuo, Huang and Lee1998) have studied parametric excitation of IBW by parallel-propagating Langmuir wave in a collisional magneto-plasma. Langmuir wave propagating along the geomagnetic field is considered as a pump for the parametric excitation of IBW and daughter Langmuir waves. Itoh et al. (Reference Itoh, Itoh and Fukuyama1984) have obtained the nonlocal eigenmode of ICRF instability in the presence of parallel high energy beam component in toroidal plasma in an inhomogeneous magnetic field. The mode is the combination of the fast wave and IBW and is excited via kinetic interactions with beam particles. When the driving source of high energy particles overcomes the damping due to the bulk plasma and the resistive loss on the wall, the instability can occur. Mikhailenko et al. (Reference Mikhailenko, Mikhailenko and Stepanov2008) have developed linear theory of electrostatic ion cyclotron instabilities (Sperling & Perkins, Reference Sperling and Perkins1976; Chen, Reference Chen2000; Svidzinski & Swanson, Reference Svidzinski and Swanson2000; Utsunomiya et al., Reference Utsunomiya, Iwamoto, Kondo, Sugawa, Maehara and Sugaya2001; Brambilla, Reference Brambilla1999) of the collisional magnetic field-aligned plasma shear flow, which is applicable to the ionospheric F-region. Chibisov et al. (Reference Chibisov, Mikhailenko and Stepanov2009) have investigated the electrostatic ion cyclotron instability of hydrogen plasma driven by an oxygen ion beam and resulting turbulent heating of both the ion species.
In this paper, we study the excitation of ion Bernstein and ion cyclotron instabilities by neutral beam turned gyrating ion beam in a plasma. We use Vlasov theory to obtain the response of gyrating ion beam to the field of the IBW. In Section 2, we study the problem in local approximation and deduce the growth rate. In Section 3, we develop the nonlocal theory using a slab model. The results are discussed in Section 4.
2. BEAM AND PLASMA RESPONSE
Consider plasma with static magnetic field B sẑ, ion density n 0p, ion mass m i, and ion charge e. A gyrating ion beam of charge Z be and mass m b propagates through the plasma with equilibrium distribution function
We perturb this equilibrium by an electrostatic wave in the ion cyclotron range of frequency. In the local approximation, one may write the electrostatic potential as,
The response of the gyrating ion beam is governed by the Vlasov equation
We express f = f 0 + f 1, linearize the Vlasov equation,
and solve it by following the usual procedure of integration along the unperturbed trajectories to obtain (Stix, Reference Stix1962)
![\eqalignno{\,f_1 &=- Z_b e{\rm \phi} \, \sum\limits_{s^{\prime}} {\sum\limits_s {\displaystyle{{J_{s^{\prime}} \lpar k_ \bot {\rm \rho}\rpar J_s \lpar k_ \bot {\rm \rho}\rpar } \over {\lpar {\rm \omega} - k_{\vert \vert } v_{\vert \vert } - s{\rm \omega} _{cb}\rpar }}} }\cr &\quad \times \left[{s\displaystyle{{\partial f_0 } \over {\partial {\rm \mu} }}+\displaystyle{{k_{\vert \vert } } \over {m_b }}\displaystyle{{\partial f_0 } \over {\partial v_{\vert \vert } }}} \right]\exp \lsqb - i\lcub {\rm \omega} t - {\bf k}.{\bf r} - \lpar s - s^{\prime}\rpar {\rm \theta} \rcub \rsqb \comma \;}](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqn5.gif?pub-status=live)
where s and s′ are integers, ρ = v ⊥/ωcb, ωcb = Z beB s/mc, μ = (1/2)m bv ⊥2/ωcb is the magnetic moment, and θ is the gyrophase angle (v⊥ makes with 
). The density perturbation turns out to be
![\eqalign{{\rm \chi} _b &=- \displaystyle{{{\rm \omega} _{\,pb}^2 } \over {{\rm \omega} _{cb}^2 }}\sum\limits_s \left[\displaystyle{{2sk_ \bot } \over {v_{0 \bot } k^2 }}\displaystyle{{J_s \lpar k_ \bot {\rm \rho} _0\rpar J_s ^\prime \lpar k_ \bot {\rm \rho} _0\rpar } \over {\lpar {\rm \omega} - k_{\vert \vert } v_{0\vert \vert } - s{\rm \omega} _{cb}\rpar }}\right.\cr &\left.\quad +\displaystyle{{k_{\vert \vert }^2 } \over {k^2 }}\displaystyle{{{\rm \omega} _{cb}^2 J_s^2 \lpar k_ \bot {\rm \rho} _0\rpar } \over {\lpar {\rm \omega} - k_{\vert \vert } v_{0\vert \vert } - s{\rm \omega} _{cb}\rpar ^2 }}+\displaystyle{{k_{\vert \vert }^2 } \over {k^2 }}\displaystyle{{{\rm \omega} _{cb}^2 J_{s - 1}^2 \lpar k_ \bot {\rm \rho} _0\rpar } \over {\lpar {\rm \omega} - k_{\vert \vert } v_{0\vert \vert }\rpar ^2 }} \right]}](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU1.gif?pub-status=live)
where ρ0 = v 0⊥ωcb, ωpb = (4πZ bn be 2/m b)1/2.
The beam coupling to the wave is strong when ω − k ||v 0|| ≅ 0 or ω − k ||v 0|| ≅ ωcb. Thus, retaining only these three terms, we may write χb as (Kumar & Tripathi, Reference Kumar and & Tripathi2004)
![\eqalignno{{\rm \chi} _b &=- \displaystyle{{{\rm \omega} _{\,pb}^2 } \over {{\rm \omega} _{cb}^2 }}\left[\displaystyle{{k_ \bot ^2 } \over {k^2 }}\displaystyle{{{\rm \omega} _{cb} \lcub J_0^2 \lpar k_ \bot {\rm \rho} _0\rpar - J_2^2 \lpar k_ \bot {\rm \rho} _0\rpar \rcub } \over {2\lpar {\rm \omega} - k_{\vert \vert } v_{0\vert \vert } - {\rm \omega} _{cb}\rpar }}\right.\cr &\left. \quad +\displaystyle{{k_{\vert \vert }^2 } \over {k^2 }}\displaystyle{{{\rm \omega} _{cb}^2 J_1^2 \lpar k_ \bot {\rm \rho} _0\rpar } \over {\lpar {\rm \omega} - k_{\vert \vert } v_{0\vert \vert } - {\rm \omega} _{cb}\rpar ^2 }}+\displaystyle{{k_{\vert \vert }^2 } \over {k^2 }}\displaystyle{{{\rm \omega} _{cb}^2 J_0^2 \lpar k_ \bot {\rm \rho} _0\rpar } \over {\lpar {\rm \omega} - k_{\vert \vert } v_{0\vert \vert }\rpar ^2 }} \right].}](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqn7.gif?pub-status=live)
The density perturbation of Maxwellian plasma ions due to ϕ can be written as (Jain & Tripathi, Reference Jain and Tripathi1987)n 1i =−(k 2/4πe)χi ϕ,
![{\rm \chi} _i=\displaystyle{{2{\rm \omega} _{\,pi}^2 } \over {k^2 v_{thi}^2 }}\left[{1 - \sum\limits_n {\displaystyle{{\rm \omega} \over {\lpar {\rm \omega} - n{\rm \omega} _{ci}\rpar }}} I_n \lpar b_i\rpar \exp \lpar \!-\! b_i\rpar } \right]\comma \; \eqno\lpar 8\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqn8.gif?pub-status=live)
where b i = k ⊥2v thi2/2ωci2, ωpi = (4πn 0pie 2/m i)1/2, I n (b i), and v thi are the ion plasma frequency, modified Bessel function and thermal velocity of ions; and we have assumed ω − nωci ≫ k ||v thi.
The density perturbation of plasma electrons due to the wave is
where
corresponding to the ion Bernstein wave,
corresponding to ion cyclotron wave, ωp = (4πn 0pe 2/m)1/2, v th, and c s are the electron plasma frequency, thermal velocity of electrons and ion acoustic speed, respectively.
3. INSTABILITY IN THE LOCAL APPROXIMATION
Using n 1b, n 1p, n 1e in the Poisson's equation ∇2 ϕ = 4πe(n 1b + n 1i + n 1e), we obtain
Where ɛ = 1 + χb + χi + χe. We consider two cases.
3.1. Ion Bernstein Mode Excitation (ω ≫ k zv the)
In this limit, the second and third terms in χb can be ignored and the dispersion relation Eq. (12) in the low beam density limit (n 0b ≪ n 0p ) can be written as
![{\rm \omega} _R={\rm \omega} _{ci} \left[ 1+\displaystyle{{I_1 \lpar b_i\rpar e^{ - b_i } } \over {\left\{ 1+\left( \displaystyle{{k^2 v_{thi}^2 } \over {2{\rm \omega} _{\,pi}^2 }}\right) \left( 1+\displaystyle{{{\rm \omega} _p^2 } \over {{\rm \omega} _c^2 }}\right) \right\} }}\right],](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU2.gif?pub-status=live)
![\Delta=\displaystyle{{{\rm \omega} _{\,pb}^2 } \over {{\rm \omega} _{cb}^2 }}\left[ \displaystyle{{{\rm \omega} _{ci} {\rm \omega} _{cb} I_1 \lpar b_i\rpar e^{ - b_i } \lcub J_0^2 \lpar k_ \bot {\rm \rho} _0\rpar - J_2^2 \lpar k_ \bot {\rm \rho} _0\rpar \rcub } \over {2\left\{ 1+\displaystyle{{k^2 v_{thi}^2 } \over {2{\rm \omega} _{\,pi}^2 }}\left( 1+\displaystyle{{{\rm \omega} _p^2 } \over {{\rm \omega} _c^2 }}\right) \right\}^2 }}\right].](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU3.gif?pub-status=live)
Strong coupling between the beam and the wave occurs when ωR = k ||v 0|| = ωcb or
where
The last term in Eq. (14) is smaller than ωci. For ωci > ωcb, k ||v 0|| > 0, i.e., the parallel phase velocity of the ion cyclotron wave is parallel to the parallel velocity of the beam, where as for ωci < ωcb, they are antiparallel. Under Eq. (14), we write ω = ωR + iγ = k ||v 0|| + ωcb + iγ in Eq. (13) and obtain the growth rate,
![{\rm \gamma}=\lpar \!-\! \Delta\rpar ^{1/2}=\displaystyle{{{\rm \omega} _{\,pb} } \over {{\rm \omega} _{cb} }}\left[ \displaystyle{{{\rm \omega} _{ci} {\rm \omega} _{cb} I_1 \lpar b_i\rpar e^{ - b_i } \lcub J_0^2 \lpar k_ \bot {\rm \rho} _0\rpar - J_2^2 \lpar k_ \bot {\rm \rho} _0\rpar \rcub } \over {2 \left\{1+\displaystyle{{k^2 v_{thi}^2 } \over {2{\rm \omega} _{\,pi}^2 }}\left( 1+\displaystyle{{{\rm \omega} _p^2 } \over {{\rm \omega} _c^2 }}\right) \right\}^2 }}\right]^{1/2}.\eqno\lpar 15\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqn15.gif?pub-status=live)
The growth rate scales as half power of beam density. The parallel wave number for the maximally growing mode is given by Eq. (14). This k || must also satisfy (1) k || ≪ (ω − ωci)/v thi = ωci δ′/v thi, i.e., v 0|| ≥ 2(ωci − ωcb)v thi/δ′, and (2) k || ≪ ω/v the or (ωci − ωcb)/ωci ≪ v 0||/v the. The second condition is quite restrictive. It is satisfied when either the beam and plasma ions are the same species or the beam energy is in the MeV range when electron temperature is around 1 KeV. We have carried out the calculations of the growth rate for a hydrogen beam in a deuterium plasma with the following parameters: m i/m b = 2, ρ0/ρi = 2.5, ωpb/ωcb = 2, k 2v thi2/ω pi2 = k 2ρ02/1600. In Figure 1, we plot the variation of growth rate as a function of k ⊥ρi. The growth rate is maximum for k ⊥ρi ~ 1.8 and falls off at larger k ⊥ρi.
Variation of local growth rate γ/ωci with k ⊥ρi for ion Bernstein wave for ωcb/ωci = 2, ωp/ωc = 1.5, k ||ρ0 = 0.2, ωpb/ωcb = 2, ωp/ωc = 1.5, ρ0/ρi = 2.5, k 2v thi2/ωpi2 = k 2 ρ02 /1600.
3.2 Ion Cyclotron Instability (ω ≪ k zv the)
In this case there are two possibilities, (a) ω ~ k ||v 0|| +ωcb and (b) ω ~ k ||v 0||. In the former case of cyclotron interaction, the second term in χb dominates and the dispersion relation takes the form
![{\rm \omega} _R={\rm \omega} _{ci} \left[{1+\displaystyle{{I_1 \lpar b_i\rpar e^{ - b_i } } \over {\left\{{1+\lpar T_i /T_e\rpar +\lsqb \lpar k^2 v_{thi}^2\rpar /\lpar 2{\rm \omega} _{\,pi}^2\rpar \rsqb } \right\}}}} \right]\comma \;](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU5.gif?pub-status=live)
![\Gamma=\sqrt {\rm \pi} \displaystyle{{T_i } \over {T_e }}\displaystyle{{{\rm \omega} _{ci}^2 I_1 \lpar b_i\rpar e^{ - b_i } } \over {\left[ k_z v_{the} \left( 1+\displaystyle{{T_i } \over {T_e }}+\displaystyle{{k_z^2 v_{thi}^2 } \over {2{\rm \omega} _{\,pi}^2 }}\right) ^2 \right] }}\comma \;](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU6.gif?pub-status=live)
![\Delta ^\prime=\displaystyle{{{\rm \omega} _{\,pb}^2 } \over {{\rm \omega} _{cb}^2 }} \left[ \displaystyle{{k_{\vert \vert }^2 {\rm \omega} _{ci} {\rm \omega} _{cb}^2 I_1 \lpar b_i\rpar e^{ - b_i } J_1^2 \lpar k_ \bot {\rm \rho} _0\rpar } \over {k^2 \left\{ 1+\displaystyle{{k^2 v_{thi}^2 } \over {2{\rm \omega} _{\,pi}^2 }}\left( 1+\displaystyle{{2{\rm \omega} _p^2 } \over {k^2 v_{th}^2 }}\right) \right\}^2 }}\right] \comma \;](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU7.gif?pub-status=live)
Γ is the damping rate of the ion cyclotron wave in the absence of the beam. Maximum growth occurs when ωR = k ||v 0|| + ωcb. Expressing ω = ωR + δ, we obtain
The roots of this equation are complex. Imaginary part of δ gives the growth rate. For Γ ≪ δ, Eq. (17) gives the growth rate
![{\rm \gamma}=\displaystyle{{\sqrt 3 } \over 2}\left[\displaystyle{{k_{\vert \vert }^2 {\rm \omega} _{\,pb}^2 {\rm \omega} _{ci} I_1 \lpar b_i\rpar e^{ - b_i } J_1^2 \lpar k_ \bot {\rm \rho} _0\rpar } \over {k^2 \left\{ 1+\displaystyle{{k^2 v_{thi}^2 } \over {2{\rm \omega} _{\,pi}^2 }}\left( 1+\displaystyle{{2{\rm \omega} _p^2 } \over {k^2 v_{th}^2 }}\right) \right\}^2 }}\right] ^{1/3}.\eqno\lpar 18\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqn18.gif?pub-status=live)
The growth rate scales as one-third power of the beamdensity. In Figure 2, we have plotted the growth rate as a function of k ⊥ρi for ion cyclotron wave for ωcb/ωci = 2, k || ρ0 = 0.2, ωpb/ωcb = 2, ωp/ωc = 1.5, ωpb/ωci = 2, ωp2/k 2v th2 = 1600/k ⊥2ρ02. The growth rate is maximum at k ⊥ρi = 0.7. For these parameters, the electron Landau damping term is weak.
Variation of local growth rate γ/ωci with k ⊥ρi for ion cyclotron wave for ωcb/ωci = 2, ωp/ωc = 1.5, k ||ρ0 = 0.2, ωpb/ωcb = 2, ωpb/ωci = 2, ρ0/ρi = 2.5, k 2v thi2/ωpi2 = k 2 ρ02 /1600, ωp2/k 2v th2 = 1600/k ⊥2 ρ02.
In the other case of ω ~ k ||v 0||, corresponding to Cerenkov interaction the last term in χb dominates, giving the dispersion relation
where Δ″ = Δ′J 02 (k ⊥ρ0)/J 12 (k ⊥ρ0). The maximum growth rate turns out to be
![{\rm \gamma} ^\prime={\rm \gamma} \left[ \displaystyle{{J_0^2 \lpar k_ \bot {\rm \rho} _0\rpar } \over {J_1^2 \lpar k_ \bot {\rm \rho} _0\rpar }}\right] ^{1/3}.\eqno\lpar 20\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqn20.gif?pub-status=live)
4. NONLOCAL EFFECTS ON ION BERNSTEIN MODE EXCITATION (ω ≫ k zv the)
Consider a plasma slab of x — width 2R and electron density n 0p. An ion beam of x— width 2R b, beam density n ob, beam charge Z be, and beam mass m b propagates through it. Other plasma parameters are the same as before. The wave equation governing ϕ inside the plasma can be obtained from Eq. (12) by replacing k ⊥2 by k y2 − ∂2/∂x 2 and expanding ɛ using Taylor expansion as 
Where β = (β1/β2)1/2,
![{\rm \beta} _1=\displaystyle{{{\rm \omega} _{\,pi}^2 } \over {{\rm \omega} _{ci}^2 b_i }}\left[1 - \displaystyle{{\rm \omega} \over {\lpar {\rm \omega} - {\rm \omega} _{ci}\rpar }}I_1 \lpar b_i\rpar e^{ - b_i } - \displaystyle{{m_i } \over m}\displaystyle{{k_{\vert \vert }^2 } \over {k_y^2 }}\displaystyle{{{\rm \omega} _{ci}^2 } \over {{\rm \omega} ^2 }}b_i \right] \comma \;](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU8.gif?pub-status=live)
![{\rm \beta} _2=\displaystyle{{{\rm \omega} _{\,pi}^2 } \over {{\rm \omega} _{ci}^2\, k_y^2 }}\left[ - \displaystyle{m \over {m_i }}\displaystyle{{k_{\vert \vert }^2 } \over {k_y^2 }}+\displaystyle{1 \over {b_i }}+\displaystyle{{{\rm \omega} b_i } \over {\lpar {\rm \omega} - {\rm \omega} _{ci}\rpar }}\displaystyle{\partial \over {\partial b_i }}\left\{ \displaystyle{{I_1 \lpar b_i\rpar e^{ - b_i } } \over {b_i }}\right\} \right] \comma \;](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU9.gif?pub-status=live)
. We solve this equation iteratively. First, we ignore the beam term. Then Eq. (21) gives
The boundary condition, ϕ = 0 at x = R, gives β = βl, where
In the first order perturbation theory, when beam term is nonzero (but small) the Eigen function may be treated to be unmodified, only the Eigen frequency is modified. Thus, using Eq. (22) in Eq. (21), multiplying the resulting equation by ϕ (where * denotes the complex conjugate) and integrating over x, we obtain
Further simplifying Eq. (24) may be written as
![{\rm \omega} _R={\rm \omega} _{ci} \left[ 1+\displaystyle{{I_1 \lpar b_i\rpar e^{ - b_i } } \over {\left( 1 - b_i \displaystyle{{m_i } \over m}\displaystyle{{k_{\vert \vert }^2 } \over {k_y^2 }} - {\rm \beta} _l^2 {\rm \beta} _2 b_i\right) }}\right] \comma \;](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU11.gif?pub-status=live)
![\Delta _1=p_1 b_i \displaystyle{{{\rm \omega} _{ci}^2 } \over {{\rm \omega} _{\,pi}^2 }}\displaystyle{{{\rm \omega} _{\,pb}^2 } \over {{\rm \omega} _{cb}^2 }}\left[ \displaystyle{{{\rm \omega} _{cb} {\rm \omega} _{ci} I_1 \lpar b_i\rpar e^{ - b_i } \lcub J_0^2 \lpar k_ \bot {\rm \rho} _0\rpar - J_2^2 \lpar k_ \bot {\rm \rho} _0\rpar \rcub } \over {2 \left(1 - b_i \displaystyle{{m_i } \over m}\displaystyle{{k_{\vert \vert }^2 } \over {k_y^2 }} - \displaystyle{{{\rm \omega} _{ci}^2 } \over {{\rm \omega} _{\,pi}^2 }}{\rm \beta} _l^2 {\rm \beta} _2 b_i\right)^2 }}\right] \comma \;](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU12.gif?pub-status=live)
. The factor 
characterizes the nonlocal effects.
The instability occurs with large growth rate when
and the maximum growth rate turns out to be
![{\rm \gamma}= \lpar \Delta _1\rpar ^{1/2}= \left[ p_1 b_i \displaystyle{{{\rm \omega} _{ci}^2 } \over {{\rm \omega} _{\,pi}^2 }}\displaystyle{{{\rm \omega} _{\,pb}^2 } \over {{\rm \omega} _{cb}^2 }}\left[ \displaystyle{{{\rm \omega} _{cb} {\rm \omega} _{ci} I_1 \lpar b_i\rpar e^{ - b_i } \lcub J_0^2 \lpar k_ \bot {\rm \rho} _0\rpar - J_2^2 \lpar k_ \bot {\rm \rho} _0\rpar \rcub } \over {2 \left( 1 - b_i \displaystyle{{m_i } \over m}\displaystyle{{k_{\vert \vert }^2 } \over {k_y^2 }} - \displaystyle{{{\rm \omega} _{ci}^2 } \over {{\rm \omega} _{\,pi}^2 }}{\rm \beta} _l^2 {\rm \beta} _2 b_i\right)^2 }}\right]\right]^{1/2}.\eqno\lpar 25\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqn25.gif?pub-status=live)
In Figure 3, we have plotted the nonlocal growth rate γ/ωci with k ⊥ρi for ion Bernstein wave for ωcb/ωci = 2, ωp/ωc = 1.5, k ||ρ0 = 0.2, ωpb/ωcb = 2, ωp/ωc = 1.5, R b = v thi /ωci, R = 2v thi /ωci, ρ0/ρi + 2.5, k 2v thi2/ωpi2 = k 2 ρ02 /1600. The growth rate of IBW is maximum at k ⊥ρi = 1..8 and then falls sharply for larger k ⊥ρi.
Variation of nonlocal growth rate γ/ωci with k ⊥ρi for ion Bernstein wave for ωcb/ωci = 2, ωp/ωc = 1.5, k ||ρ0 = 0.2, ωpb/ωcb = 2, ωp/ωc = 1.5, m i/m = 65, R b = v thi /ωci, ωp/ωpi = 65, R = 2v thi /ωci, ρ0/ρi = 2.5, k 2v thi2/ωpi2 = k 2 ρ02 /1600.
5. NONLOCAL EFFECTS ON ION CYCLOTRON INSTABILITY (ω ≪ k zv the)
Expanding ɛ in a Taylor series around k⊥2 = k y2 and replacing k x by − i(∂/∂x) in Eq. (12), we obtain
Where β′ − (β1′/β2′)1/2,
![{\rm \beta} _1 ^\prime=\displaystyle{{{\rm \omega} _{\,pi}^2 } \over {{\rm \omega} _{ci}^2 b_i }}\left[1 - \displaystyle{{\rm \omega} \over {\lpar {\rm \omega} - {\rm \omega} _{ci}\rpar }}I_1 \lpar b_i\rpar e^{ - b_i }+\displaystyle{{2{\rm \omega} _p^2 } \over {{\rm \omega} _{\,pi}^2 }}\displaystyle{{{\rm \omega} _{ci}^2 } \over {k^2 v_{th}^2 }}b_i \right],](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU14.gif?pub-status=live)
![{\rm \beta} _2 ^\prime=\displaystyle{{{\rm \omega} _{\,pi}^2 } \over {{\rm \omega} _{ci}^2\, k_y^2 }}\left[ \displaystyle{{2m_i } \over m}\displaystyle{{k_y^2 {\rm \omega} _{ci}^2 } \over {k^4 v_{th}^2 }}+\displaystyle{1 \over {b_i }}+\displaystyle{{{\rm \omega} b_i } \over {\lpar {\rm \omega} - {\rm \omega} _{ci}\rpar }}\displaystyle{\partial \over {\partial b_i }}\left\{\displaystyle{{I_1 \lpar b_i\rpar e^{ - b_i } } \over {b_i }}\right\} \right].](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU15.gif?pub-status=live)
Including the nonlocal effects as in Section 4, Eq. (26) gives
The boundary condition,ϕ = 0 at x = R, gives β′ = βl′
Thus, using Eq. (27) in Eq. (26), multiplying the resulting equation by ϕ* and integrating over x, we obtain
Now, there are two possibilities (c) ω ~ k |v 0|| + ωcb (d)ω ~ k ||v 0||. In the former case, the second term in χb dominates and Eq. (29) takes the form
where
![{\rm \omega} _R ^\prime={\rm \omega} _{ci} \left[ 1+\displaystyle{{I_1 \lpar b_i\rpar e^{ - b_i } } \over {\left( 1+b_i \displaystyle{{2m_i } \over m}\displaystyle{{{\rm \omega} _{ci}^2 } \over {k^2 v_{th}^2 }} - \displaystyle{{{\rm \omega} _{ci}^2 } \over {{\rm \omega} _{\,pi}^2 }}{\rm \beta} _l ^{\prime 2} {\rm \beta} _2 ^\prime b_i\right) }}\right],](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqnU16.gif?pub-status=live)
. Here p 1′ characterizes the nonlocal effects.
To obtain the growth rate we write, ω = ωR + iγ = ωcb + k ||v 0|| + iγ and the growth rate turns out to be
![\eqalign{{\rm \gamma} &= \displaystyle{\sqrt {3} \over 2}\lpar \Delta _1^{\prime}\rpar ^{1/3} \cr & = \displaystyle{\sqrt {3} \over 2}\left[ p_1 ^{\prime} b_i \displaystyle{{\rm \omega} _{ci}^2 \over {\rm \omega} _{\,pi}^2 }\displaystyle{{\rm \omega} _{\,pb}^2 \over {\rm \omega} _{cb}^2 }\displaystyle{k_{\vert \vert }^2 \over k^2}{\displaystyle{{{\rm \omega} _{cb}^2 {\rm \omega} _{ci} I_1 \lpar b_i\rpar e^{- b_i} {J_1}^2 \lpar k_ {\bot} {\rm \rho}_0\rpar }\over {\left( 1+b_i \displaystyle{2m_i \over m}\displaystyle{{\rm \omega} _{ci}^2 \over k^2 v_{th}^2} - \displaystyle{{\rm \omega} _{ci}^2 \over {\rm \omega} _{\,pi}^2}{\rm \beta} _l ^{\prime 2} {\rm \beta} _2 b_i\right) ^2}}}\right] ^{1/3}.}\eqno\lpar 31\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqn31.gif?pub-status=live)
In the other case of ω ~ k ||v 0||, the last term in χb dominates and Eq. (29) takes the form
where ![${\rm \omega} _R ^\prime={\rm \omega} _{ci} \left[ 1+\displaystyle{{I_1 \lpar b_i\rpar e^{ - b_i } } \over {\lpar 1+b_i \displaystyle{{2m_i } \over m}\displaystyle{{{\rm \omega} _{ci}^2 } \over {k^2 v_{th}^2 }} - {\rm \beta} _l ^{\prime ^2} {\rm \beta} _2 ^\prime b_i\rpar }}\right] $](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_inline7.gif?pub-status=live)
,
.
The maximum growth rate turns out to be
![{\rm \gamma}^{\prime}={\rm \gamma} \left[ \displaystyle{{J_0^2 \lpar k_ \bot {\rm \rho} _0\rpar } \over {J_1^2 \lpar k_ \bot {\rm \rho} _0\rpar }}\right] ^{1/3}.\eqno \lpar 33\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021095242833-0839:S026303461100053X_eqn33.gif?pub-status=live)
In Figure 4, we have plotted the nonlocal growth rate γ/ωci with k ⊥ρi for ion cyclotron wave for ωcb/ωci = 2, ωp/ωc = 1.5, k ||ρ0 = 0.2, ωpb/ωcb = 2, ωpb/ωci = 2, m i/m = 65, R b = v thi/ωci,R = 2v thi/ωci, ρ0/ρi = 2.5, k 2v thi2/ωpi2 = k 2 ρ02/1600, ωp2/k 2v th2 = 1600/k ⊥2 ρ02. The growth rate of the ion cyclotron wave is small for smaller value of k⊥ ρi and is maximum at k ⊥ρ0 = 1.7 and falls for larger k ⊥ρi.
Variation of nonlocal growth rate γ/ωci with k ⊥ρi for ion cyclotron wave for ωcb/ωci = 2, ωp/ωc = 1.5, k ||ρ0 = 0.2, ωpb/ωcb = 2, ωpb/ωci = 2, m i/m = 65, R b = v thi /ωci, R = 2v thi /ωci, ρ0/ρi = 2.5, k 2v thi2/ωpi2 = k 2 ρ02 /1600, ωp2/k 2v th2 = 1600/k ⊥2 ρ02.
6. DISCUSSION
A gyrating ion beam with delta function distributions in v ⊥ and v ||, like the one produced by the ionization of neutral beam atoms during neutral beam heating of tokamak, can excite ion Bernstein and ion cyclotron waves of long and short parallel wavelengths. These waves propagate nearly perpendicular to the axial magnetic field. The growth rate of the IBW scales as one-third power of the ion beam density and is large for frequencies close to cyclotron harmonics ω ~ nωci. It is maximum for ω ~ ωci (n = 1) and decreases for higher values of n. When the region of destabilization is limited and x extent of the mode is relatively larger, then the growth rate of the instability is reduced. The total energy content of the mode is determined by its radial extent whereas the rate of energy supplied to its growth is determined by the size of the beam and also by the location of the beam with respect to the field structure of the mode.
In BWI, a beam of energetic neutral atoms is injected into the plasma. As the beam penetrates the plasma an increasing fraction of the beam atoms become ionized and trapped in the magnetic field of the tokamak. The deposition of the beam ions on available range of particle orbits is sensitive to the geometry of the beam injectors as well as the magnetic geometry. Ion cyclotron harmonic damping and the collisional energy exchange between the ions and electrons are main ion heating mechanisms for IBW heating. It can be used not only for ion heating, but also for electron heating via electron Landau damping. IBW experiments have been carried out in the HT-7 tokamak for several topics, such as heating, pressure profile and transport control and instability stabilization.
PIBW's are undamped plasma modes with k ⊥ Bs and become Landau damped by electrons when the direction of propagation deviates by a small amount from exact perpendicularity. In an experiment for P (pressure of the interaction chamber) = 3 − 10 × 10−5 torr, n e = 2 − 15 × 107 cm−3, T e ≈ 1.2 eV, T i ~ 0.5 eV, B s = (600–800 G), (ωpi2 + ωpb2)/ωci2 = 14.0, n pi/n b = 0.4, Saha et al. (Reference Saha, Raychaudhuri and Sengupta1988) found in most situations that k z = 0 and in some cases, the weak axial phase shift observed indicates a maximum value of 7.4 × 10−3 cm−1 for k z. Taking k ⊥= 6.7 cm−1, the measured value, Saha et al. (Reference Saha, Raychaudhuri and Sengupta1988) find that the propagation direction deviates from exact perpendicularity to B s by 0.06° maximum. Also, the frequency of PIBW increases slightly with increase in n pi/n b, as observed experimentally. The value of the instability frequency found experimentally is 21% below the value of the instability frequency for the parameters:B s = 720 G, n pi = 2.1 × 107 cm−3, n b = 5.3 × 107 cm−3, v thi = 0.49 × 106 cms−1, v 0⊥ = 1.45 × 106 cms−1, and (ωpi2 + ωpb2)/ωci2 = 11.05. The instability is a strong one as evidenced by the large growth rate of Im (ω/ωci) ≈ 0.2. The value of k⊥ found theoretically for these parameters is 4.9, which is below the value found experimentally by 34%. Since the experimental value of k ⊥ is determined (for constant mode number) by the radius at which the wave amplitude is maximum, it is subjected to some uncertainty. For the parameters, B s = 600 G, n pi = 2.1 × 107 cm−3, n b = 5.3 × 107 cm−3, v thi = 0.49 × 106 cms−1, v 0⊥ = 1.45 × 106 cms−1, (ωpi2 + ωpb2)/ωci2 = 11.05; the instability is stronger [Im (ω /ωci) ≈ 0.3] than before.
IBW heating in tokomaks is based on the fact that the finite Larmor radius waves in the range of the ion cyclotron frequency excited from the low-field side of machine can penetrate into the hot plasma core without strong attenuation until the waves approach the harmonic cyclotron layers. The superconducting tokama (HT-7) has major radius 122 cm and minor radius 27.5 cm. The plasma current is about 120–170 kA. The toroidal magnetic field is in the range of (1.5 – 5) × 1013 cm−3. The electron and ion temperatures are about 700 eV and 400 eV. The RF frequency is 24 – 30 MHz and the RF power of the generator can reach 300 kW. The IBW frequency is 24 – 30 MHz. For the plasma parameters of HT-7 with two species (hydrogen and deuterium) of working gas and assuming k ⊥0 ρi ≪ 1, k ⊥0 is the perpendicular wave number of the pump and ρi is the ion Larmor radius; the two parametric decay processes are: (1) decay into an IBW and ion cyclotron quasi mode where the quasi mode are characterized by ω = nωci and (2) decay into an IBW and a low-frequency electron Landau damped quasi-mode characterized by ω = k ||v the.
