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Radial plasma response during intense beam heating
Volume 39A, number 3
PHYSICS LETTERS
8 May 1972
RADIAL PLASMA RESPONSE DURING INTENSE BEAM HEATING* G. BENFORD
Department of Physics, University of California, Irvine, Calif. 92664, USA Received 6 March 1972 Heating by microscopic beam mechanisms proceeds until a longitudinal density gradient cuts off phase coherence between the beam and excited plasma waves. A thin plasma column produces this inhomogeneity by radial transport. We find the maximum temperatures which result from several different regimes of the beam-plasma instability.
Current interest in plasma heating by intense relativistic beams has resulted in a number of detailed studies of the beam penetration length into the dense plasma target [ 1 - 3 ] . These models consider a halfspace geometry in which radial plasma transport and temperature gradients are unimportant compared to the same quantities along z, the beam injection axis. Yet heating may be magnified if a thin plasma column is used, so that heat is then not convected readily away from the beam axis. This note considers plasma response when the plasma column radius R ~
m ~Or/at = - k aT/Or,
(1)
anp/at + n o aOr/ar = O,
(2)
Snom aT~at = Enu/L(T)
(3)
where m, np, Or, S are plasma mass, perturbed density, velocity and specific heat at const, volume, respectively; E, n and u are beam energy, density and velocity, re*Work supported by the National Science Foundation under Grant No. CP 28656.
spectively. Plasma equilibrium density is n 0. The Boltzmann const, is k. Eqs. (1), (2), (3) represent radial expansion while heating, if plasma thermal conductivity o is low. If thermal conductivity is not low, heat will diffuse into the column in the usual manner and no density gradient can arise, d = 0. In this case there will be no limitation upon heating rate, because the beam two-stream mechanism will not be disrupted by a density gradient. This state of affairs holds only so long as o >>noL2(T)/r, where r is the time required for heating to produce a density gradient which shuts off the heating. We shall thus treat situations in which o is small (the above inequality is not satisfied), and the density gradient along z is produced by radial excursion of the plasma column. Since L = L(T), we must find an integrated rate equation for the maximum temperature attainable. Eqs. (1)-(3) readily yield the time r, r 3 = noR2m2SL(T)d/Enuk.
(4)
Defining T ' = T/mu 2 , and
D = (k EnuR/noS)2/3 (d/m) 1/3 and using eq. (3) again, we find
r(T') jr'L(r')dr'= f dt. T~ ~) nom~
(5)
This gives our desired result, once L ( T ' ) is specified. The picture of beam heating represented by eq. (3) is quite qualitative, since the view of beam deposition given by even approximate descriptions such as ref. [ 1] displays both a region of rapid deposition and a much longer length over which heating will be slower. It is this first region we describe by eq. (3). We picture rapid 247
Volume 39A, number 3
PHYSICS LETTERS
heating in a length LfT'), shut off when radial expansion of the plasma column creates a density inhomogeneity along z. A beam passing through the plasma heats until the critical d is reached. Thereafter, heating in the lenght L(T') must proceed solely by Joule processes associated with the induced return currents in the plasma, which find their energy absorbed through excitation of the ion-acoustic instability [4,5]. We shall calculate the maximum temperatures which can be reached before the generation of plasma oscillations by the beam can be shut off. The mechanism produces a high-velocity "wave" behind which the plasma has a high temperature, but microscopic beam processes cannot heat any longer. In ref. [1] several mechanisms were proposed for such microscopic heating. Scattering of plasma oscillations by ions leads to a characteristic length
L(T) =uno
.m 1/2 3/2 E 3 -3/2 nCOPe ('--~) T(~--~u2) = L o T
where we have dropped the prime on T'. From eq. (5) we find
where TO is the initial plasma temperature. (Note that the length ratio R/L 0 enters as (R/Lo)2/3.) This mechanism delivers heat through ion-acoustic excitation. The same method in a confining magnetic field for which COce~ COpeleads to the length LH(T } = L' T - 1/2 = LoT(COpe/COce)2 and we find
V~--- V~O = 1 D T - I/6/L' ~ 1DT 0 1/6/L'. Following ref. [1], we consider a dense beam for which
248
8 May 1972
a weak turbulence limit is not valid. Then the deposition length and final temperature are
L*(T) = (///COpe) (E/T) 2 T(COpe/COce)2 - l i T T = T O exp[~ D(T/l)2/31. Similar forms may be found using the more complete recent analysis of Rudakov [2]. The critical density perturbation d which cuts off the instability, due to a difference in phase resonance between the beam electrons and plasma oscillations, is approximately [3] d ~ L (COpe/U)(n/n07) where L is taken to represent either LO, L' or l, depending on which regime is treated. Consideration of particular intense beam experiments show that plasma temperatures ~ 10 keV may be obtained in this first flush of the interaction, before Joule processes begin. It is interesting to note that often the ratio (R/L)2~ 3 appears, indicating that the fact that a plasma column is thin (R < L) is not a terribly strong contributor to the final temperature obtained, T. I would like to thank J. Guillory for illuminating discussions.
References [1 ] A.T. Altyntsev et al., Intern. Atomic Energy Agency Conf. on Plasma physics and CTR, Madison, Wisconsin, 1971 (IAEA/CN-28) Paper E20. [2] L.I. Rudakov, Soy. Phys. JETP 32 (1971) 1134. [3] B.N. Breizman and D.D. Ryutov, JETP Letters 11 (1970) 421. [4] J. Guillory and G. Benford, Plasma Physics, to be published. [5] R. Lovelace and R.N. Sudan, Phys. Rev. Lett. 27 (1971) 1256.
PHYSICS LETTERS
8 May 1972
RADIAL PLASMA RESPONSE DURING INTENSE BEAM HEATING* G. BENFORD
Department of Physics, University of California, Irvine, Calif. 92664, USA Received 6 March 1972 Heating by microscopic beam mechanisms proceeds until a longitudinal density gradient cuts off phase coherence between the beam and excited plasma waves. A thin plasma column produces this inhomogeneity by radial transport. We find the maximum temperatures which result from several different regimes of the beam-plasma instability.
Current interest in plasma heating by intense relativistic beams has resulted in a number of detailed studies of the beam penetration length into the dense plasma target [ 1 - 3 ] . These models consider a halfspace geometry in which radial plasma transport and temperature gradients are unimportant compared to the same quantities along z, the beam injection axis. Yet heating may be magnified if a thin plasma column is used, so that heat is then not convected readily away from the beam axis. This note considers plasma response when the plasma column radius R ~
m ~Or/at = - k aT/Or,
(1)
anp/at + n o aOr/ar = O,
(2)
Snom aT~at = Enu/L(T)
(3)
where m, np, Or, S are plasma mass, perturbed density, velocity and specific heat at const, volume, respectively; E, n and u are beam energy, density and velocity, re*Work supported by the National Science Foundation under Grant No. CP 28656.
spectively. Plasma equilibrium density is n 0. The Boltzmann const, is k. Eqs. (1), (2), (3) represent radial expansion while heating, if plasma thermal conductivity o is low. If thermal conductivity is not low, heat will diffuse into the column in the usual manner and no density gradient can arise, d = 0. In this case there will be no limitation upon heating rate, because the beam two-stream mechanism will not be disrupted by a density gradient. This state of affairs holds only so long as o >>noL2(T)/r, where r is the time required for heating to produce a density gradient which shuts off the heating. We shall thus treat situations in which o is small (the above inequality is not satisfied), and the density gradient along z is produced by radial excursion of the plasma column. Since L = L(T), we must find an integrated rate equation for the maximum temperature attainable. Eqs. (1)-(3) readily yield the time r, r 3 = noR2m2SL(T)d/Enuk.
(4)
Defining T ' = T/mu 2 , and
D = (k EnuR/noS)2/3 (d/m) 1/3 and using eq. (3) again, we find
r(T') jr'L(r')dr'= f dt. T~ ~) nom~
(5)
This gives our desired result, once L ( T ' ) is specified. The picture of beam heating represented by eq. (3) is quite qualitative, since the view of beam deposition given by even approximate descriptions such as ref. [ 1] displays both a region of rapid deposition and a much longer length over which heating will be slower. It is this first region we describe by eq. (3). We picture rapid 247
Volume 39A, number 3
PHYSICS LETTERS
heating in a length LfT'), shut off when radial expansion of the plasma column creates a density inhomogeneity along z. A beam passing through the plasma heats until the critical d is reached. Thereafter, heating in the lenght L(T') must proceed solely by Joule processes associated with the induced return currents in the plasma, which find their energy absorbed through excitation of the ion-acoustic instability [4,5]. We shall calculate the maximum temperatures which can be reached before the generation of plasma oscillations by the beam can be shut off. The mechanism produces a high-velocity "wave" behind which the plasma has a high temperature, but microscopic beam processes cannot heat any longer. In ref. [1] several mechanisms were proposed for such microscopic heating. Scattering of plasma oscillations by ions leads to a characteristic length
L(T) =uno
.m 1/2 3/2 E 3 -3/2 nCOPe ('--~) T(~--~u2) = L o T
where we have dropped the prime on T'. From eq. (5) we find
where TO is the initial plasma temperature. (Note that the length ratio R/L 0 enters as (R/Lo)2/3.) This mechanism delivers heat through ion-acoustic excitation. The same method in a confining magnetic field for which COce~ COpeleads to the length LH(T } = L' T - 1/2 = LoT(COpe/COce)2 and we find
V~--- V~O = 1 D T - I/6/L' ~ 1DT 0 1/6/L'. Following ref. [1], we consider a dense beam for which
248
8 May 1972
a weak turbulence limit is not valid. Then the deposition length and final temperature are
L*(T) = (///COpe) (E/T) 2 T(COpe/COce)2 - l i T T = T O exp[~ D(T/l)2/31. Similar forms may be found using the more complete recent analysis of Rudakov [2]. The critical density perturbation d which cuts off the instability, due to a difference in phase resonance between the beam electrons and plasma oscillations, is approximately [3] d ~ L (COpe/U)(n/n07) where L is taken to represent either LO, L' or l, depending on which regime is treated. Consideration of particular intense beam experiments show that plasma temperatures ~ 10 keV may be obtained in this first flush of the interaction, before Joule processes begin. It is interesting to note that often the ratio (R/L)2~ 3 appears, indicating that the fact that a plasma column is thin (R < L) is not a terribly strong contributor to the final temperature obtained, T. I would like to thank J. Guillory for illuminating discussions.
References [1 ] A.T. Altyntsev et al., Intern. Atomic Energy Agency Conf. on Plasma physics and CTR, Madison, Wisconsin, 1971 (IAEA/CN-28) Paper E20. [2] L.I. Rudakov, Soy. Phys. JETP 32 (1971) 1134. [3] B.N. Breizman and D.D. Ryutov, JETP Letters 11 (1970) 421. [4] J. Guillory and G. Benford, Plasma Physics, to be published. [5] R. Lovelace and R.N. Sudan, Phys. Rev. Lett. 27 (1971) 1256.