- Email: [email protected]
Short-wavelength drift vortices
Volume 104A, number 2
PHYSICS LETTERS
13 August 1984
SHORT-WAVELENGTH DRIFT VORTICES A.B. MIKHAILOVSKII L V. Kurchatov Institute of Atomic Energy, Moscow, USSR
G.D. ABURDZHANIYA LN. Vekua Institute of Applied Mathematics, Tbilisi State University, Tbilisi, USSR
O.G. ONISHCHENKO Space Research Institute, Moscow, USSR
and S.E. SHARAPOV Moscow Physical-TechnicalInstitute, Moscow, USSR Received 3 October 1983 Revised manuscript received 5 April 1984
It is shown that in an inhomogeneous plasma immersed in a magnetic field short-wavelength drift vortices can exist, which are similar to the Rossby vortices in a usual fluid.
1. Introduction. Short-wavelength drift oscillations (SDO) were discussed initially in refs. [1,2]. These oscillations are low-frequency ones with respect to electrons (i.e. their typical frequencies are small compared to the electron cyclotron frequency). Transverse electron motion (with respect to the magnetic field) in these oscillations is characterized approximately by the cross-field drift velocity V E = c[E X ez] /B 0 (E is the wave electric field, B011z the equilibrium magnetic field, c the velocity of light and e z the unit vector along z). From a sufficiently small ratio of the plasma pressure to the magnetic pressure, SDO can be treated to be electrostatic, E = -V~o (~0 is the electrostatic potential). Therefore the transverse electron motion has a nonvanishing vorticity, [V X IrE/z = cV2~o/Bo 4= O. Vorticity waves have been studied before in hydrodynamics where it has been shown that Rossby solitary vortices can arise [3]. The aim of the present paper is to analyze the possibility of having SDO solitons similar to the Rossby vortices.
4
2. Statement o f the problem and starting equations. We assume that in the equilibrium plasma the number density n o is inhomogeneous along x, 7n011x. We consider plasma perturbations, assuming them to be quasineutral, designating the perturbed number density of each particle species (electrons and ions) by h. We take into account the short-wavelength character of the perturbations with respect to the ions, i.e. the fact that (~7±ln ¢)2p2 >> 1, where Pi is the ion Larmor radius. Then we can take h in the boltzmannian form (cf. with ref. [4]), h = - eno¢/Ti,
(1)
where Ti is the ion temperature and e the ion charge. In addition, h is related to ¢ by the electron continuity equation which can be written Oh/at + VEVn 0 + n0(div V± + OVz/OZ) = 0.
(2)
Here Vl and V z are the transverse and longitudinal velocities of electrons. It follows from the equation of the transverse electron motion that approximately
Volume 104A, number 2
V±= ge -~ogse [ez x (a/at + v e v ) v e ]
PHYSICS LETTERS
,
(3)
w h e r e 09Be = eBo/mc is the electron cyclotron frequency and m the electron mass. We assume electrons to be cold, T e = 0, for simplicity. The velocity Vz is defined by the equation of the longitudinal electron motion which is taken in the approximate form
(D/at + VEV ) g z = (e/m)a~o/az.
(4)
We look for perturbations which depend on x, r/--
y + az - ut and r - t, where a and u are free parameters. The r plays the role of the "slow time", i.e. a j a r is assumed to be small compared to ua/arl. Then it follows from eqs. ( 1 ) - ( 4 ) that
-- p2(a/aT-)v2~0 + UO2(Cl)V2~O -- Aatp/ar/) = 0.
(5)
Here
(6)
0-2(1 - V,i/u - o~2C2e/U2),
(7)
=
V, i = (cTdeB~)aln no/aXis the ion drift velocity, 02 = Ti[mcog e, c e = T ( m , so that O and c e are the electron Larmor radius and the electron thermal velocity calculated at the ion temperature, respectively.
3. lnvariants ofeq. (5). Multiplying eq. (5) by ~0 and 72~0 and integrating the results over space, we arrive at the conservation laws
aE/ar = O,
aK/ar = 0,
(8)
where
e = 71 f tp2(v, )2 + (1 + c~2c2/u2)~o 21 dr, K=.~I
r 2 = A.
f[o2(V2 0)2 + (1 + a2c2/u2)(Vl~o)2]dr.
(9) (10)
It can be seen that E and K are invariants of eq. (5). Within the accuracy of coefficients, E is the wave energy and K the generalized enstrophy (for ct = 0 the K corresponds to the usual enstrophy, cf. ref. [5]).
4.Solitary SDO vortices. For a/ar
=
0 it follows
from eq. (5) that cbv2~0 = Aa~0/an.
(11)
(12)
Using eq. (7) we find from (12) that (13)
For a = 0 this corresponds to the vortex moving along y with the velocity u = V,i/(1 - K202),
cb = a / a n - (cluBo)tV~ x v ] ~,
A
This equation has the same structure as those studied in refs. [3,5,6]. Therefore, using the results of those papers, we can conclude, without additional analysis, that eq. (11) has a solution in the form of a two-dimensional solitary vortex. The simplest case of such vortices in the dipole vortex characterized by a potential of the form ~0= ~(r)cos 0. Here r = (x 2 + r/2) 1/2, 0 = tg-l(rl/x), ¢ ( r ) is a function decreasing at r -->o o as r-1/2exp(-~r), K a free parameter related to the free parameters u and a by the so-called modified dispersion relation of the form
u2(1 -- K2O2) -- V*iu - a2c 2 = 0.
(1 + C~2C2/U2) a¢/ar
13 August 1984
(14)
which is larger than the ion drift velocity V, i (we assume that K2p 2 < 1). For a ~ 0 two types of vortices with different signs of u / V , i are possible. Eq. (13) remains valid in the case of a plasma with homogeneous number density, V, i = 0. In this case it describes the vortices of the electron sound waves.
5. Discussion o f the results. We have analyzed the role of the vector nonlinearity, characterized by the expression of the form [V~0 X 7] z72S0 in the problem o f SDO. Based on the analogy with hydrodynamics [3], we have shown that this nonlinearity can lead to the solitary vortex of SDO localized in a plane perpendicular to the magnetic field. Such solitary vortices are an alternative to the solitons o f SDO due to the scalar nonlinearity [6]. In this respect, the general situation of the solitons of SDO is similar to that of the solitons of the long-wavelength drift oscillations [5,7,8] : in both cases we have solitons due to both scalar and vector nonlinearities. We believe that the ideas about the vortices of SDO can be useful for the study of high-frequency heating of a plasma in laboratory magnetic confinement systems [9] and in the study of the dynamics of the magnetospheric plasma [10], as well as in other problems where one deals with magnetic confinement o f a plasma with hot ions.
95
Volume 104A, number 2
PHYSICS LETTERS
References [1] A.B. Mikhailovskli, Nucl. Fusion 5 (1965) 125. [2] A.B. Mikhailovskii, Zh. Techn. Fiz. 37 (1967) 1365. [3] V.D. Larichev and G.M. Reznik, Dokl. Akad. Nauk SSSR 231 (1976) 1077. [4] G.D. Aburdzhaniya, A,B. Mikhailovskii and S.E. Sharapov. Phys. Lett. 100A (1984) 134. [5] J.D. Meiss and W. Horton, Phys. Fluids 26 (1983) 990.
96
13 August 1984
[6] A.B. Mikhailovskii,G.D. Aburdzhaniya, O.G. Onishchenko and S.E. Shatapov, Phys. Lett. 100A (1984) 503. [7] V.N. Oraevskii, H. Tasso and H. Wobig, Plasma Phys. Contr. Nuel. Fus. Res. IAEA, Vienna 1 (1969) 671. [8] V.I. Petviashvili, Fiz. Plasmy 3 (1977) 270. [9] J.P. Goeldbloed, A.I. Pyatak and V.L. Sizonenko, Zh. Eksp. Teor. Fiz. 64 (1973) 2084. [ 10] V.I. Sotnikov, V.D. Shapiro and V.I. Shevchenko, Zh. Eksp. Teor. Fiz. 78 (1980) 576.
PHYSICS LETTERS
13 August 1984
SHORT-WAVELENGTH DRIFT VORTICES A.B. MIKHAILOVSKII L V. Kurchatov Institute of Atomic Energy, Moscow, USSR
G.D. ABURDZHANIYA LN. Vekua Institute of Applied Mathematics, Tbilisi State University, Tbilisi, USSR
O.G. ONISHCHENKO Space Research Institute, Moscow, USSR
and S.E. SHARAPOV Moscow Physical-TechnicalInstitute, Moscow, USSR Received 3 October 1983 Revised manuscript received 5 April 1984
It is shown that in an inhomogeneous plasma immersed in a magnetic field short-wavelength drift vortices can exist, which are similar to the Rossby vortices in a usual fluid.
1. Introduction. Short-wavelength drift oscillations (SDO) were discussed initially in refs. [1,2]. These oscillations are low-frequency ones with respect to electrons (i.e. their typical frequencies are small compared to the electron cyclotron frequency). Transverse electron motion (with respect to the magnetic field) in these oscillations is characterized approximately by the cross-field drift velocity V E = c[E X ez] /B 0 (E is the wave electric field, B011z the equilibrium magnetic field, c the velocity of light and e z the unit vector along z). From a sufficiently small ratio of the plasma pressure to the magnetic pressure, SDO can be treated to be electrostatic, E = -V~o (~0 is the electrostatic potential). Therefore the transverse electron motion has a nonvanishing vorticity, [V X IrE/z = cV2~o/Bo 4= O. Vorticity waves have been studied before in hydrodynamics where it has been shown that Rossby solitary vortices can arise [3]. The aim of the present paper is to analyze the possibility of having SDO solitons similar to the Rossby vortices.
4
2. Statement o f the problem and starting equations. We assume that in the equilibrium plasma the number density n o is inhomogeneous along x, 7n011x. We consider plasma perturbations, assuming them to be quasineutral, designating the perturbed number density of each particle species (electrons and ions) by h. We take into account the short-wavelength character of the perturbations with respect to the ions, i.e. the fact that (~7±ln ¢)2p2 >> 1, where Pi is the ion Larmor radius. Then we can take h in the boltzmannian form (cf. with ref. [4]), h = - eno¢/Ti,
(1)
where Ti is the ion temperature and e the ion charge. In addition, h is related to ¢ by the electron continuity equation which can be written Oh/at + VEVn 0 + n0(div V± + OVz/OZ) = 0.
(2)
Here Vl and V z are the transverse and longitudinal velocities of electrons. It follows from the equation of the transverse electron motion that approximately
Volume 104A, number 2
V±= ge -~ogse [ez x (a/at + v e v ) v e ]
PHYSICS LETTERS
,
(3)
w h e r e 09Be = eBo/mc is the electron cyclotron frequency and m the electron mass. We assume electrons to be cold, T e = 0, for simplicity. The velocity Vz is defined by the equation of the longitudinal electron motion which is taken in the approximate form
(D/at + VEV ) g z = (e/m)a~o/az.
(4)
We look for perturbations which depend on x, r/--
y + az - ut and r - t, where a and u are free parameters. The r plays the role of the "slow time", i.e. a j a r is assumed to be small compared to ua/arl. Then it follows from eqs. ( 1 ) - ( 4 ) that
-- p2(a/aT-)v2~0 + UO2(Cl)V2~O -- Aatp/ar/) = 0.
(5)
Here
(6)
0-2(1 - V,i/u - o~2C2e/U2),
(7)
=
V, i = (cTdeB~)aln no/aXis the ion drift velocity, 02 = Ti[mcog e, c e = T ( m , so that O and c e are the electron Larmor radius and the electron thermal velocity calculated at the ion temperature, respectively.
3. lnvariants ofeq. (5). Multiplying eq. (5) by ~0 and 72~0 and integrating the results over space, we arrive at the conservation laws
aE/ar = O,
aK/ar = 0,
(8)
where
e = 71 f tp2(v, )2 + (1 + c~2c2/u2)~o 21 dr, K=.~I
r 2 = A.
f[o2(V2 0)2 + (1 + a2c2/u2)(Vl~o)2]dr.
(9) (10)
It can be seen that E and K are invariants of eq. (5). Within the accuracy of coefficients, E is the wave energy and K the generalized enstrophy (for ct = 0 the K corresponds to the usual enstrophy, cf. ref. [5]).
4.Solitary SDO vortices. For a/ar
=
0 it follows
from eq. (5) that cbv2~0 = Aa~0/an.
(11)
(12)
Using eq. (7) we find from (12) that (13)
For a = 0 this corresponds to the vortex moving along y with the velocity u = V,i/(1 - K202),
cb = a / a n - (cluBo)tV~ x v ] ~,
A
This equation has the same structure as those studied in refs. [3,5,6]. Therefore, using the results of those papers, we can conclude, without additional analysis, that eq. (11) has a solution in the form of a two-dimensional solitary vortex. The simplest case of such vortices in the dipole vortex characterized by a potential of the form ~0= ~(r)cos 0. Here r = (x 2 + r/2) 1/2, 0 = tg-l(rl/x), ¢ ( r ) is a function decreasing at r -->o o as r-1/2exp(-~r), K a free parameter related to the free parameters u and a by the so-called modified dispersion relation of the form
u2(1 -- K2O2) -- V*iu - a2c 2 = 0.
(1 + C~2C2/U2) a¢/ar
13 August 1984
(14)
which is larger than the ion drift velocity V, i (we assume that K2p 2 < 1). For a ~ 0 two types of vortices with different signs of u / V , i are possible. Eq. (13) remains valid in the case of a plasma with homogeneous number density, V, i = 0. In this case it describes the vortices of the electron sound waves.
5. Discussion o f the results. We have analyzed the role of the vector nonlinearity, characterized by the expression of the form [V~0 X 7] z72S0 in the problem o f SDO. Based on the analogy with hydrodynamics [3], we have shown that this nonlinearity can lead to the solitary vortex of SDO localized in a plane perpendicular to the magnetic field. Such solitary vortices are an alternative to the solitons o f SDO due to the scalar nonlinearity [6]. In this respect, the general situation of the solitons of SDO is similar to that of the solitons of the long-wavelength drift oscillations [5,7,8] : in both cases we have solitons due to both scalar and vector nonlinearities. We believe that the ideas about the vortices of SDO can be useful for the study of high-frequency heating of a plasma in laboratory magnetic confinement systems [9] and in the study of the dynamics of the magnetospheric plasma [10], as well as in other problems where one deals with magnetic confinement o f a plasma with hot ions.
95
Volume 104A, number 2
PHYSICS LETTERS
References [1] A.B. Mikhailovskli, Nucl. Fusion 5 (1965) 125. [2] A.B. Mikhailovskii, Zh. Techn. Fiz. 37 (1967) 1365. [3] V.D. Larichev and G.M. Reznik, Dokl. Akad. Nauk SSSR 231 (1976) 1077. [4] G.D. Aburdzhaniya, A,B. Mikhailovskii and S.E. Sharapov. Phys. Lett. 100A (1984) 134. [5] J.D. Meiss and W. Horton, Phys. Fluids 26 (1983) 990.
96
13 August 1984
[6] A.B. Mikhailovskii,G.D. Aburdzhaniya, O.G. Onishchenko and S.E. Shatapov, Phys. Lett. 100A (1984) 503. [7] V.N. Oraevskii, H. Tasso and H. Wobig, Plasma Phys. Contr. Nuel. Fus. Res. IAEA, Vienna 1 (1969) 671. [8] V.I. Petviashvili, Fiz. Plasmy 3 (1977) 270. [9] J.P. Goeldbloed, A.I. Pyatak and V.L. Sizonenko, Zh. Eksp. Teor. Fiz. 64 (1973) 2084. [ 10] V.I. Sotnikov, V.D. Shapiro and V.I. Shevchenko, Zh. Eksp. Teor. Fiz. 78 (1980) 576.