- Email: [email protected]
Temperature anisotropy induced magnetic drift wave instability
Volume 101A, number 2
PHYSICS LETTERS
12 March 1984
TEMPERATURE ANISOTROPY INDUCED MAGNETIC DRIFT WAVE INSTABILITY
M.Y. YU and P.K. SHUKLA lnstimt fffr Theoretische Physik, Ruhr-Universitiit Bochum, D-4630 Bochum, Fed. Rep. Germany Received 13 January 1984
It is shown that magnetic drift wavesare unstable in the presence of electron temperature anisotropy. Saturation by a quasilinear process which reduces the anisotropy is discussed.
Although the magnetic drift mode was first discussed by Chamberlain in 1963 [1,2], it was largely ignored in the literature until recently [3-10]. This mode has been found to be closely related to enhanced magnetic fluctuations and heat transport in magnetically confined plasmas. In fact, the magnetostatic mode [11-14], as well as the micro-tearing instability [4,5] which have also gained much attention recently, are closely associated with the magnetic drift mode. It is therefore of interest to study the basic stability behavior of magnetic drift waves. It is well known that the magnetic drift waves are unstable when a magnetic field gradient as well as resonant ions are present [1,2]. It is also unstable in the presence of a temperature gradient [6]. The latter is also important for the drift tearing instability [4,5]. In this paper, we show that the magnetic drift waves are unstable in the presence of electron temperature anisotropy. A possible saturation mechanism is discussed. We start with the linearized Vlasov equation with a Krook collision term (Ot + O" v -
(1)
where E 1 = -~atA/'c and B 1 = VA X :~ define the polarization of the mode, ~ is the electron gyrofrequency, and u is the rate of relaxation of electrons toward a bi-maxwellian distribution given by ( m ~l/2exp( my2
oo
ao ) , (3)
[coA-co*-iv(l-A)]n~2+w+iv
--oo
= [(e/m)El'~7o+~2(oXB1/Bo)'~7o]f 0 ,
m
_eA ( ( Vz ) - -mcc 1 - A
+~
~2oX i " V O + u ) f 1
Io= o(x)2-TfT,2- ,,1
where no, m, T±, and T~ are the electron number density, mass, perpendicular (to the external magnetic field ~B0) and parallel temperatures, respectively. Temperature anisotropy can exist in a collisional plasma because of the strong external field. In a strongly magnetized plasma, the particle motion (and thus their interaction parameters) across the magnetic field differs considerably from the parallel motion. This leads to different relaxation times for the parallel and perpendicular temperatures. The anisotropic state will eventually become isotropic through collisions, or, as in the present problem, through instabilities. Furthermore, since the magnetic drift waves do not involve density fluctuations, the use of a nonparticle-conserving Krook collision model is adequate for our problem. Eq. (1) can be easily solved by standard methods. One obtains for the parallel velocity moment,
where A = T~/T±, w* = -(cTN/eBo)k . ~, X V In no, A n = In(k2p 2) exp(-k2p2), and p2 = T±/m~22. Here, I n is the modified Bessel function of the second kind. In the low frequency limit, w ,< ~ , COp,where COp is the electron plasma frequency, we readily obtain from Amp~re's law and (3),
mV2z), (2)
2T,,
0375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
89
k2A = -(4rr/c)noe (vz ) = (6o~A/c 2) {A -- 1 -- [o0A - 6o* - i u ( l - A ) ] A / ( o o + i v ) } ,
(4)
where A = A 0. Eq. (4) yields the solution 1 + p - A ( 1 -- A )
'
>A>
1-A
(6)
Clearly, the plasma is stable in the limiting case A = 1, (k = 0). We note that the dispersion curve of the drift waves is separated into two regions, with co -+ + ~ at the value o f k where the denominator of (5) vanishes. Furthermore, the present instability is identical to that of the purely growing collisional Weibel instability [14]. This is understandable since physically the two differ only in the existence of the density gradient here, which leads to a real frequency. Let us now consider the saturation of the instability. It is clear that saturation will occur when A evolves with the fluctuation level until the latter becomes constant, or when the condition (6) is no longer satisfied, whichever at a lower fluctuation level. Since the unstable drift wave spectrum is broad in k-space, the evolution of T Nand Tj_ can be obtained from the quasilinear theory [ 1 5 - 1 7 ] . The background distributionf 0 changes slowly according to the equation
atfo = ( f f dk'dk ieA(k')mc \
x [(~
- k ' . o ) ~ + Vz k ' ] •
Wfl(k)),
(7)
where the angular brackets denote an ensemble average. The perturbed distribution function f l is, from the linear theory,
fl (k) = [eA(k)vzfo/crN] X {1 - A +
[WkA -- w~ -- i v ( 1 - - A ) ] j 2 / ( W k + iP)},
(8) 90
v
0d = 0,
(9)
(5)
where p = e2k2/¢o 2. &T h e real part of (5) gives the frequency of the magnetic drift waves. The imaginary part shows that the waves are unstable if 1
where the argument of the Bessel junction Jo is kv±/~2. Note that since k z = 0, neither the instability nor the saturation is associated with wave particle resonance. From (7) and (8) one readily obtains ~m
A¢o* + i u [ ( A - 1 ) ( 1 - A ) - p ] co =
12 March 1984
PHYSICS LETTERS
Volume 101A, number 2
~', =$rn' fV2z ~fod =-2 f clk'rkk2[Akl2/8rr , (10) 0
where the dot refers to the time derivative, and where we have made use of the dispersion relation (5), as well as the symmetry relations iA - k 12 = IAk i2, ~°k = --W_k, and 3'k = 3'_k. Thus, T± is constant and there is an exchange between the parallel kinetic energy of the electrons and the magnetic fluctuation energy. This result is not surprising since only the particle motion along the external field is directly involved in the magnetic drift mode. It should, however, be pointed out that the VzB1/B0 particle motion perpendicular to the external field gives rise to a nonvanishing electron heat flux across B 0. The evolution of the magnetic fluctuations is governed by the equation
Ot IAk 12 = 23'k(t) IAk 12 ,
(11)
where 7(t) is the linear growth rate given in (5), but with T n replaced by
T, = T,O - f dk k 2 IA k 12/Srr,
(12)
which can be obtained from (10) and (11). The latter also indicate that energy is conserved, since IE 112 ,~ [B 1 [2 for w* ,~ kc. More correctly, ~/k(t) should be calculated as in the linear theory, but using the f0 given by the solution of (7) with (3) as the initial condition. However, such a procedure is too complicated to yield any useful results without additional drastic simplifications. According to (12), T N decreases as the magnetic fluctuation level increases. Saturation of a particular unstable mode occurs either when the fluctuation level of this mode reaches a constant value (3'k = 0), or when the right inequality in (6) no longer holds. If the former situation occurs, we find that the total magnetic energy density must be
Volume 101A, number 2
f
k2lAk[2 dk
8/r
[sat =
[(A -- 1)(1 -- A ) - p ] T A(1 - A)
PHYSICS LETrERS
12 March 1984
References
l (13)
In the alternative situation, we find the same result, but a factor A must be multiplied to the right hand side of (13). This discrepancy is due to the fact that in obtaining (13), the denominator in (5) also contributes, while the righthand inequality o f (6) is governed only by the numerator o f ( 5 ) . Since A < 1, we see that stabilization is due to the violation o f the marginal instability condition. It is o f interest to point out here that the stabilization process is somewhat different from that o f the collisionless Weibel (T l > Ttl ) instability in an unmagnetized plasma, for which T± decreases and T n increases with the fluctuation level until the plasma becomes isotropic [17]. Here, the parallel and perpendicular directions are with respect to k. If no m o d e - m o d e coupling is present, one can presume that the plasma becomes stable when the fastest growing mode for any given set o f initial conditions is stabilized. In real situations, energy cascading between different modes by m o d e - m o d e coupling can occur. For example, growing modes might transfer their energy to the damped modes. In fact, coupling to other normal modes can also occur [7,18]. In such cases, the saturation level can be less than the one predicted here.
[1] J.C. Chamberlain, J. Geophys. Rev. 68 (1963) 5667. [2] N.A. KraU, in: Advances in plasma physics, Vol. 1, eds. A. Simon and W.B. Thompson (Wiley, New York, 1968) pp. 176-179. [3] T. Ohkawa, Phys. Lett. 67A (1978) 35. [4] J.F. Drake, N.T. Gladd, C.S. Liu and C.L. Chang, Phys. Rev. Lett. 44 (1980) 994. [5] N.T. Gladd, J.F. Drake, C.L. Chang and C.S. Liu, Phys. Fluids 23 (1980) 1182. [6] A.B. Hassam, Phys. Fluids 23 (1980) 2493. [7] K. Nozaki, J. Phys. Soc. Japan 49 (1980) 2326. [8] V.P. Pavlenko and J. Weiland, Phys. Rev. Lett. 46 (1981) 246; Phys. Scr. 26 (1982) 225. [9] P.K. Kaw and L. Chen, Princeton Plasma Physics Laboratory Report PPPL-1897 (1982). [10] P.K. Shukla, M.Y. Yu and H.U. Rahman, Phys. Rev. A27 (1983) 598. [11] C. Chu, M.S. Chu and T. Ohkawa, Phys. Rev. Lett. 41 (1978) 653. [12] P.K. Shukla, M.Y. Yu and K.H. Spatschek, Phys. Rev. A23 (1981) 3247. [13] P.K. Shukla, M.Y. Yu, H.U. Rahman and K.H. Spatschek, Phys. Rev. A24 (1981) 1112. [14] P.K. Shukla and G.S. Lakhina, Phys. Rev. A25 (1982) 3419. [15] A.A. Vedenov, E.P. Velikhov and R.Z. Sagdeev, Nucl. Fusion 1 (1961) 82; Suppl. Part II (1961) 465. [16] W.E. Drummond and D. Pines, Nucl. Fusion Suppl. Part III (1962) 1049. [17] R.C. Davidson, D.A. Hammer, I. Haber and C.E. Wagner, Phys. Fluids 15 (1972) 317. [18] M.Y. Yu and P.K. Shukla, Phys. Fluids 26 (1983) 2983.
This work was supported b y the Sonderforschungsbereich 162 Plasmaphysik Bochum/Jfilich.
91
PHYSICS LETTERS
12 March 1984
TEMPERATURE ANISOTROPY INDUCED MAGNETIC DRIFT WAVE INSTABILITY
M.Y. YU and P.K. SHUKLA lnstimt fffr Theoretische Physik, Ruhr-Universitiit Bochum, D-4630 Bochum, Fed. Rep. Germany Received 13 January 1984
It is shown that magnetic drift wavesare unstable in the presence of electron temperature anisotropy. Saturation by a quasilinear process which reduces the anisotropy is discussed.
Although the magnetic drift mode was first discussed by Chamberlain in 1963 [1,2], it was largely ignored in the literature until recently [3-10]. This mode has been found to be closely related to enhanced magnetic fluctuations and heat transport in magnetically confined plasmas. In fact, the magnetostatic mode [11-14], as well as the micro-tearing instability [4,5] which have also gained much attention recently, are closely associated with the magnetic drift mode. It is therefore of interest to study the basic stability behavior of magnetic drift waves. It is well known that the magnetic drift waves are unstable when a magnetic field gradient as well as resonant ions are present [1,2]. It is also unstable in the presence of a temperature gradient [6]. The latter is also important for the drift tearing instability [4,5]. In this paper, we show that the magnetic drift waves are unstable in the presence of electron temperature anisotropy. A possible saturation mechanism is discussed. We start with the linearized Vlasov equation with a Krook collision term (Ot + O" v -
(1)
where E 1 = -~atA/'c and B 1 = VA X :~ define the polarization of the mode, ~ is the electron gyrofrequency, and u is the rate of relaxation of electrons toward a bi-maxwellian distribution given by ( m ~l/2exp( my2
oo
ao ) , (3)
[coA-co*-iv(l-A)]n~2+w+iv
--oo
= [(e/m)El'~7o+~2(oXB1/Bo)'~7o]f 0 ,
m
_eA ( ( Vz ) - -mcc 1 - A
+~
~2oX i " V O + u ) f 1
Io= o(x)2-TfT,2- ,,1
where no, m, T±, and T~ are the electron number density, mass, perpendicular (to the external magnetic field ~B0) and parallel temperatures, respectively. Temperature anisotropy can exist in a collisional plasma because of the strong external field. In a strongly magnetized plasma, the particle motion (and thus their interaction parameters) across the magnetic field differs considerably from the parallel motion. This leads to different relaxation times for the parallel and perpendicular temperatures. The anisotropic state will eventually become isotropic through collisions, or, as in the present problem, through instabilities. Furthermore, since the magnetic drift waves do not involve density fluctuations, the use of a nonparticle-conserving Krook collision model is adequate for our problem. Eq. (1) can be easily solved by standard methods. One obtains for the parallel velocity moment,
where A = T~/T±, w* = -(cTN/eBo)k . ~, X V In no, A n = In(k2p 2) exp(-k2p2), and p2 = T±/m~22. Here, I n is the modified Bessel function of the second kind. In the low frequency limit, w ,< ~ , COp,where COp is the electron plasma frequency, we readily obtain from Amp~re's law and (3),
mV2z), (2)
2T,,
0375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
89
k2A = -(4rr/c)noe (vz ) = (6o~A/c 2) {A -- 1 -- [o0A - 6o* - i u ( l - A ) ] A / ( o o + i v ) } ,
(4)
where A = A 0. Eq. (4) yields the solution 1 + p - A ( 1 -- A )
'
>A>
1-A
(6)
Clearly, the plasma is stable in the limiting case A = 1, (k = 0). We note that the dispersion curve of the drift waves is separated into two regions, with co -+ + ~ at the value o f k where the denominator of (5) vanishes. Furthermore, the present instability is identical to that of the purely growing collisional Weibel instability [14]. This is understandable since physically the two differ only in the existence of the density gradient here, which leads to a real frequency. Let us now consider the saturation of the instability. It is clear that saturation will occur when A evolves with the fluctuation level until the latter becomes constant, or when the condition (6) is no longer satisfied, whichever at a lower fluctuation level. Since the unstable drift wave spectrum is broad in k-space, the evolution of T Nand Tj_ can be obtained from the quasilinear theory [ 1 5 - 1 7 ] . The background distributionf 0 changes slowly according to the equation
atfo = ( f f dk'dk ieA(k')mc \
x [(~
- k ' . o ) ~ + Vz k ' ] •
Wfl(k)),
(7)
where the angular brackets denote an ensemble average. The perturbed distribution function f l is, from the linear theory,
fl (k) = [eA(k)vzfo/crN] X {1 - A +
[WkA -- w~ -- i v ( 1 - - A ) ] j 2 / ( W k + iP)},
(8) 90
v
0d = 0,
(9)
(5)
where p = e2k2/¢o 2. &T h e real part of (5) gives the frequency of the magnetic drift waves. The imaginary part shows that the waves are unstable if 1
where the argument of the Bessel junction Jo is kv±/~2. Note that since k z = 0, neither the instability nor the saturation is associated with wave particle resonance. From (7) and (8) one readily obtains ~m
A¢o* + i u [ ( A - 1 ) ( 1 - A ) - p ] co =
12 March 1984
PHYSICS LETTERS
Volume 101A, number 2
~', =$rn' fV2z ~fod =-2 f clk'rkk2[Akl2/8rr , (10) 0
where the dot refers to the time derivative, and where we have made use of the dispersion relation (5), as well as the symmetry relations iA - k 12 = IAk i2, ~°k = --W_k, and 3'k = 3'_k. Thus, T± is constant and there is an exchange between the parallel kinetic energy of the electrons and the magnetic fluctuation energy. This result is not surprising since only the particle motion along the external field is directly involved in the magnetic drift mode. It should, however, be pointed out that the VzB1/B0 particle motion perpendicular to the external field gives rise to a nonvanishing electron heat flux across B 0. The evolution of the magnetic fluctuations is governed by the equation
Ot IAk 12 = 23'k(t) IAk 12 ,
(11)
where 7(t) is the linear growth rate given in (5), but with T n replaced by
T, = T,O - f dk k 2 IA k 12/Srr,
(12)
which can be obtained from (10) and (11). The latter also indicate that energy is conserved, since IE 112 ,~ [B 1 [2 for w* ,~ kc. More correctly, ~/k(t) should be calculated as in the linear theory, but using the f0 given by the solution of (7) with (3) as the initial condition. However, such a procedure is too complicated to yield any useful results without additional drastic simplifications. According to (12), T N decreases as the magnetic fluctuation level increases. Saturation of a particular unstable mode occurs either when the fluctuation level of this mode reaches a constant value (3'k = 0), or when the right inequality in (6) no longer holds. If the former situation occurs, we find that the total magnetic energy density must be
Volume 101A, number 2
f
k2lAk[2 dk
8/r
[sat =
[(A -- 1)(1 -- A ) - p ] T A(1 - A)
PHYSICS LETrERS
12 March 1984
References
l (13)
In the alternative situation, we find the same result, but a factor A must be multiplied to the right hand side of (13). This discrepancy is due to the fact that in obtaining (13), the denominator in (5) also contributes, while the righthand inequality o f (6) is governed only by the numerator o f ( 5 ) . Since A < 1, we see that stabilization is due to the violation o f the marginal instability condition. It is o f interest to point out here that the stabilization process is somewhat different from that o f the collisionless Weibel (T l > Ttl ) instability in an unmagnetized plasma, for which T± decreases and T n increases with the fluctuation level until the plasma becomes isotropic [17]. Here, the parallel and perpendicular directions are with respect to k. If no m o d e - m o d e coupling is present, one can presume that the plasma becomes stable when the fastest growing mode for any given set o f initial conditions is stabilized. In real situations, energy cascading between different modes by m o d e - m o d e coupling can occur. For example, growing modes might transfer their energy to the damped modes. In fact, coupling to other normal modes can also occur [7,18]. In such cases, the saturation level can be less than the one predicted here.
[1] J.C. Chamberlain, J. Geophys. Rev. 68 (1963) 5667. [2] N.A. KraU, in: Advances in plasma physics, Vol. 1, eds. A. Simon and W.B. Thompson (Wiley, New York, 1968) pp. 176-179. [3] T. Ohkawa, Phys. Lett. 67A (1978) 35. [4] J.F. Drake, N.T. Gladd, C.S. Liu and C.L. Chang, Phys. Rev. Lett. 44 (1980) 994. [5] N.T. Gladd, J.F. Drake, C.L. Chang and C.S. Liu, Phys. Fluids 23 (1980) 1182. [6] A.B. Hassam, Phys. Fluids 23 (1980) 2493. [7] K. Nozaki, J. Phys. Soc. Japan 49 (1980) 2326. [8] V.P. Pavlenko and J. Weiland, Phys. Rev. Lett. 46 (1981) 246; Phys. Scr. 26 (1982) 225. [9] P.K. Kaw and L. Chen, Princeton Plasma Physics Laboratory Report PPPL-1897 (1982). [10] P.K. Shukla, M.Y. Yu and H.U. Rahman, Phys. Rev. A27 (1983) 598. [11] C. Chu, M.S. Chu and T. Ohkawa, Phys. Rev. Lett. 41 (1978) 653. [12] P.K. Shukla, M.Y. Yu and K.H. Spatschek, Phys. Rev. A23 (1981) 3247. [13] P.K. Shukla, M.Y. Yu, H.U. Rahman and K.H. Spatschek, Phys. Rev. A24 (1981) 1112. [14] P.K. Shukla and G.S. Lakhina, Phys. Rev. A25 (1982) 3419. [15] A.A. Vedenov, E.P. Velikhov and R.Z. Sagdeev, Nucl. Fusion 1 (1961) 82; Suppl. Part II (1961) 465. [16] W.E. Drummond and D. Pines, Nucl. Fusion Suppl. Part III (1962) 1049. [17] R.C. Davidson, D.A. Hammer, I. Haber and C.E. Wagner, Phys. Fluids 15 (1972) 317. [18] M.Y. Yu and P.K. Shukla, Phys. Fluids 26 (1983) 2983.
This work was supported b y the Sonderforschungsbereich 162 Plasmaphysik Bochum/Jfilich.
91