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Ion-acoustic shock waves in multi-electron temperature collisional plasma
Volume 59A, number 1
PHYSICS LETTERS
1 November 1976
ION-ACOUSTIC SHOCK WAVES IN MULTI-ELECTRON TEMPERATURE COLLISIONAL PLASMA P.K. SHUKLA and S.G. TAGARE Inat itut für Theoretische Physik, Ruhr-Universirat Bochum, 4630 Bochum, F.R. Germany Received 8 August 1976 The nonlinear propagation of ion-acoustic waves in a collision-dominated double electron temperature plasma is considered. Accounting for the ion viscosity and the ion heat conductivity, it is shown by means of two-warm fluid equations that the nonlinear evolution of the ion-acoustic waves is governed by the Korteweg—de Vries—Burgers equation. Stationary shock solution of the KdV—Burgers equation is presented.
Recently, Jones et al. [1] investigated the propagation of ion-acoustic waves in a collisionless plasma in the presence of two electron components. The linear dispersion relation of the ion-acoustic waves in such a plasma is given by w=kcSftl_k2X~f/2]
(1) 2and XDf = T~ff/47rN 2, w ith Tef = TehTeQ/(NehTeg + NeQTeh). Here 1 = Neh + N~g is the where Csf = (TefImi)’~’ total electron density (normalized with respect0eto background plasma density) and the subscripts 2 and h referring to low and high-temperature components respectively. Goswami and Buti [2] showed that the nonlinear ionacoustic waves obeying the dispersion relation (1) are governed by the well-known Korteweg—de Vries equation [3]. They have remarked that solitary ion-acoustic waves exist only for (NehT~2 +NegTeg)T~ <3 and not for ~ ~ 3. In this note, we investigate the nonlinear evolution of the ion-acoustic waves in a multi-electron temperature collisional plasma. Specifically, we consider the effects of finite ion viscosity and the ion heat conductivity arising from the charged particle collisions in-a plasma. Ion temperature perturbations are also taken into account. The two-warm fluid equations are then solved in the usual manner. It is found that the non-linear ion-acoustic waves are described by the Korteweg—de Vries—Burgers equation. Consider the propagation of finite amplitude ion-acoustic waves in a collision dominated plasma. The propagation is governed by the continuity, momentum and heat transfer equations for the ions and an isothermal equation of state for both low-temperature and high-temperature electrons together with Poisson’s equation. We have ,
~.—
an.
a (2)
au
au
i
a
a
~J2u.
2T.
3 faT ~ 1
aT.\
a
au1
Ne
(3)
4i~. /au.\2 ~-~-~)
(4)
,
1eh Nehexp(Tefø/Teh), 2~P
(Tet4ITe2),
~
=
~~eh+ ~eQ~
~
~
6,7)
ax
where e = (XDf/L)2 ~ 1 is a smallness parameter (square of the ratio of the effective electron Debye length XDf to the typical scale length L of the system). 38
Volume 59A, number 1
PHYSICS LETTERS
1 November 1976
We have defined m = 0.96 EVi/Csf and k~ = 3.9 Vj/Csf corresponding to the coefficient of the ion viscosity and the thermal conductivity, respectively. Here = (XDIL)2 X,~/L.Xm = (v1/v~),v~=(T1/m~)’I2and ‘~is the ion-ion collision frequency. In the above equations the ion density n~,the ion flow velocity u~,the ion temperature T 1, the potential 4), the length x, the time t are normalized with respect to N0, C~,Tef, Tef/e, L, L/csf, respectively. We assume that at x -÷00 the plasma is unperturbed, and as a matter of fact we have the boundary conditions 1, T~=To, u ~~i=j~eh~eQ= 1=4)=O. (8) We introduce the stretched variables ~ and r, as ~=x—X0t, r=et, (9) where X0 = vPh/cSf (vPh is the phase velocity of the ion-acoustic oscillations). The dependent variables may now be expanded as follows:
2)(~, r) +
n1= 1 + n~(~,r)+ 2n~
...
2)(~,r) +
T
u
,
=
u~(~,r)+ 2u~2~(~,r) +
0= e~(’)(~,r) + e2Ø(2)(~,r)+
...,
1= T0+eT~(~,r) + 2T~ 1~eQ=NeQ + en~(~,r) + e2n~(~,r) + ~
...
(10)
,
~eh =Neh+ n~j~(~,r) + e2n(~)(~,r) +
Substituting (9) and (10) into (2)—(7), we have to order set of equations from which one easily finds ~(1)
—
~(l)
—
.
=
—
A0
3 2 T0
.1~
TeQ
=
NeQ T ef
n~ eQ
-~=
Teh NehTef
~
eh
(11)
,
where A 1/2. 0 = (12,+ 5T0/3) To order we obtain
an~2~au~2~a
an~1~
(12)
au~’~ au~2~ an~2~aT~ an~’~ au~’~aØ(2) — A 1 + + (TU)~~ Tonf’~) + ~(l) 0 + ~‘0 ~ a~ I a~ ‘ + 1~ aT~ ar°~ au~2~ au~’~ a2T~’) _-±_+u(l)__i_-+~.T 0)__!~~~ — 0, aT~ 0 ~ _+~T‘ i a~ a~ *ko
=0,
~
~
(14)
—~
____
a~2 fl~=Neh
(2)
(2),
~(2) +
(.?i~)
(15)
e
rTef LTth
2 0(1)2]
,
n~2~=N [~~2)+(~)2O(l)21 eQ
\Tehl
where k 77~= 0.96 ~ 0 = 3.9 nf2), Ui/Csf~and n~),~(2), u~2~nd~ Eliminating KdV—Burgers equation + ~(‘)
a~(’)
a3~(’)
—~
~2~(1)
e
TeQ
=
[(3—z~)+~ T
0]/2A0,
(16, 17)
\TeQI
-‘
from the above equations, and making use of(l 1), we get the required
=0,
(18)
where p
(13)
T 170
-_
j3 [(3 —z~)+~ T0}
1,
and
p
2
—
[~
1
+_~—koTo]
39
Volume 59A, number 1
PHYSICS LETTERS
1 November 1976
We note that for plasma consisting of only isothermal electrons with one electron temperature, we have Teh= Teg = Te and z~=1. For realistic experiments [1], TehITeQ may lie between 2 to about 5 while Neh!NeQ lies between ~. to 3. These values yield a variation of i~ from 1.033 to 1.875. Hence, for the case of multi-electron temperature plasma the quantities p, jl and p are positive. We now discuss the stationary solution of(18). By letting x = Ur (where U = V0/C5f), in (18) one obtains for 1~(x)the ordinary differential equation. The latter can be integrated once to yield the function 4)~ d24)~ dØ(~+ 1. ~(1)2 (ri) ~(1) = ~ (19) —
—
dX2
dx
p
where the boundary conditions ~(1)
=
d~(1)= d2~(’) 2 = 0 X dx
at x =
-~
±00
have been imposed.
It is easy [3]to show that (19) describes a shock wave whose velocity v 0 in the rest frame is given by
V0 = C~f(1+ pØ/2)
(20)
,
where 0 Ø(1)(oo) ~(1)(oo), and 4)(1)(oo) = 0. In the downstream region (x = _oo), the asymptotic behavior of (19) can easily be investigated [3]. One notes that the shock wave has a monotonic behaviour for ~ > ~ and an oscillatory profile for p < ~c’where 2(0/Xü)’1’2 (21) —
-
Pc(20P0) It is worthwhile to mention that
the critical value TMc is independent of the presence of different types of electrons with different electron temperatures. However, ~cdecreases with the increase of ionic temperature. Shock velocity v 0 depends on both ~ and T0. We find that for prescribed T0 ion-acoustic shock moves slower as ~ increases. For P ~ ~c’ the stationary solution is given by 2~, (22) 01(x) = + c1 exp(px/2f3) cos(p~/23)’/ where c 1 is a constant. In summary, we have shown that the ion-acoustic waves in a collision-dominated plasma, in which the temperature difference between the two electron components is fairly large, evolve to shock like structures. The shock may dissipate the wave energy to plasma particles. Our results are expected to have important implications in hot collisional plasma of thermonuclear interest, in which one often encounters high energy superthermal electron tail. Finally, we mention that we have treated the electron components as two fluids. This assumption is justified because the processes which produce such electron distributions have time scales much shorter than that of the iontime scale. The work of P.K.S. is performed under the auspices of the Sonderforschungsbereich Plasmaphysik Bochum/ JUlich. S.G.T. acknowledges the Alexander von Humboldt Foundation for the award of a Senior Fellowship.
References [11 W.D. Jones et al., Phys. Rev. Lett. 35 (1975) 1349. [2] B.N. Goswami and B. Buti, Phys. Lett. 57A (1976) 149. 3] V.!. Karpman, Nonlinear waves in dispersive media (Pergamon Press, 1975).
40
PHYSICS LETTERS
1 November 1976
ION-ACOUSTIC SHOCK WAVES IN MULTI-ELECTRON TEMPERATURE COLLISIONAL PLASMA P.K. SHUKLA and S.G. TAGARE Inat itut für Theoretische Physik, Ruhr-Universirat Bochum, 4630 Bochum, F.R. Germany Received 8 August 1976 The nonlinear propagation of ion-acoustic waves in a collision-dominated double electron temperature plasma is considered. Accounting for the ion viscosity and the ion heat conductivity, it is shown by means of two-warm fluid equations that the nonlinear evolution of the ion-acoustic waves is governed by the Korteweg—de Vries—Burgers equation. Stationary shock solution of the KdV—Burgers equation is presented.
Recently, Jones et al. [1] investigated the propagation of ion-acoustic waves in a collisionless plasma in the presence of two electron components. The linear dispersion relation of the ion-acoustic waves in such a plasma is given by w=kcSftl_k2X~f/2]
(1) 2and XDf = T~ff/47rN 2, w ith Tef = TehTeQ/(NehTeg + NeQTeh). Here 1 = Neh + N~g is the where Csf = (TefImi)’~’ total electron density (normalized with respect0eto background plasma density) and the subscripts 2 and h referring to low and high-temperature components respectively. Goswami and Buti [2] showed that the nonlinear ionacoustic waves obeying the dispersion relation (1) are governed by the well-known Korteweg—de Vries equation [3]. They have remarked that solitary ion-acoustic waves exist only for (NehT~2 +NegTeg)T~ <3 and not for ~ ~ 3. In this note, we investigate the nonlinear evolution of the ion-acoustic waves in a multi-electron temperature collisional plasma. Specifically, we consider the effects of finite ion viscosity and the ion heat conductivity arising from the charged particle collisions in-a plasma. Ion temperature perturbations are also taken into account. The two-warm fluid equations are then solved in the usual manner. It is found that the non-linear ion-acoustic waves are described by the Korteweg—de Vries—Burgers equation. Consider the propagation of finite amplitude ion-acoustic waves in a collision dominated plasma. The propagation is governed by the continuity, momentum and heat transfer equations for the ions and an isothermal equation of state for both low-temperature and high-temperature electrons together with Poisson’s equation. We have ,
~.—
an.
a (2)
au
au
i
a
a
~J2u.
2T.
3 faT ~ 1
aT.\
a
au1
Ne
(3)
4i~. /au.\2 ~-~-~)
(4)
,
1eh Nehexp(Tefø/Teh), 2~P
(Tet4ITe2),
~
=
~~eh+ ~eQ~
~
~
6,7)
ax
where e = (XDf/L)2 ~ 1 is a smallness parameter (square of the ratio of the effective electron Debye length XDf to the typical scale length L of the system). 38
Volume 59A, number 1
PHYSICS LETTERS
1 November 1976
We have defined m = 0.96 EVi/Csf and k~ = 3.9 Vj/Csf corresponding to the coefficient of the ion viscosity and the thermal conductivity, respectively. Here = (XDIL)2 X,~/L.Xm = (v1/v~),v~=(T1/m~)’I2and ‘~is the ion-ion collision frequency. In the above equations the ion density n~,the ion flow velocity u~,the ion temperature T 1, the potential 4), the length x, the time t are normalized with respect to N0, C~,Tef, Tef/e, L, L/csf, respectively. We assume that at x -÷00 the plasma is unperturbed, and as a matter of fact we have the boundary conditions 1, T~=To, u ~~i=j~eh~eQ= 1=4)=O. (8) We introduce the stretched variables ~ and r, as ~=x—X0t, r=et, (9) where X0 = vPh/cSf (vPh is the phase velocity of the ion-acoustic oscillations). The dependent variables may now be expanded as follows:
2)(~, r) +
n1= 1 + n~(~,r)+ 2n~
...
2)(~,r) +
T
u
,
=
u~(~,r)+ 2u~2~(~,r) +
0= e~(’)(~,r) + e2Ø(2)(~,r)+
...,
1= T0+eT~(~,r) + 2T~ 1~eQ=NeQ + en~(~,r) + e2n~(~,r) + ~
...
(10)
,
~eh =Neh+ n~j~(~,r) + e2n(~)(~,r) +
Substituting (9) and (10) into (2)—(7), we have to order set of equations from which one easily finds ~(1)
—
~(l)
—
.
=
—
A0
3 2 T0
.1~
TeQ
=
NeQ T ef
n~ eQ
-~=
Teh NehTef
~
eh
(11)
,
where A 1/2. 0 = (12,+ 5T0/3) To order we obtain
an~2~au~2~a
an~1~
(12)
au~’~ au~2~ an~2~aT~ an~’~ au~’~aØ(2) — A 1 + + (TU)~~ Tonf’~) + ~(l) 0 + ~‘0 ~ a~ I a~ ‘ + 1~ aT~ ar°~ au~2~ au~’~ a2T~’) _-±_+u(l)__i_-+~.T 0)__!~~~ — 0, aT~ 0 ~ _+~T‘ i a~ a~ *ko
=0,
~
~
(14)
—~
____
a~2 fl~=Neh
(2)
(2),
~(2) +
(.?i~)
(15)
e
rTef LTth
2 0(1)2]
,
n~2~=N [~~2)+(~)2O(l)21 eQ
\Tehl
where k 77~= 0.96 ~ 0 = 3.9 nf2), Ui/Csf~and n~),~(2), u~2~nd~ Eliminating KdV—Burgers equation + ~(‘)
a~(’)
a3~(’)
—~
~2~(1)
e
TeQ
=
[(3—z~)+~ T
0]/2A0,
(16, 17)
\TeQI
-‘
from the above equations, and making use of(l 1), we get the required
=0,
(18)
where p
(13)
T 170
-_
j3 [(3 —z~)+~ T0}
1,
and
p
2
—
[~
1
+_~—koTo]
39
Volume 59A, number 1
PHYSICS LETTERS
1 November 1976
We note that for plasma consisting of only isothermal electrons with one electron temperature, we have Teh= Teg = Te and z~=1. For realistic experiments [1], TehITeQ may lie between 2 to about 5 while Neh!NeQ lies between ~. to 3. These values yield a variation of i~ from 1.033 to 1.875. Hence, for the case of multi-electron temperature plasma the quantities p, jl and p are positive. We now discuss the stationary solution of(18). By letting x = Ur (where U = V0/C5f), in (18) one obtains for 1~(x)the ordinary differential equation. The latter can be integrated once to yield the function 4)~ d24)~ dØ(~+ 1. ~(1)2 (ri) ~(1) = ~ (19) —
—
dX2
dx
p
where the boundary conditions ~(1)
=
d~(1)= d2~(’) 2 = 0 X dx
at x =
-~
±00
have been imposed.
It is easy [3]to show that (19) describes a shock wave whose velocity v 0 in the rest frame is given by
V0 = C~f(1+ pØ/2)
(20)
,
where 0 Ø(1)(oo) ~(1)(oo), and 4)(1)(oo) = 0. In the downstream region (x = _oo), the asymptotic behavior of (19) can easily be investigated [3]. One notes that the shock wave has a monotonic behaviour for ~ > ~ and an oscillatory profile for p < ~c’where 2(0/Xü)’1’2 (21) —
-
Pc(20P0) It is worthwhile to mention that
the critical value TMc is independent of the presence of different types of electrons with different electron temperatures. However, ~cdecreases with the increase of ionic temperature. Shock velocity v 0 depends on both ~ and T0. We find that for prescribed T0 ion-acoustic shock moves slower as ~ increases. For P ~ ~c’ the stationary solution is given by 2~, (22) 01(x) = + c1 exp(px/2f3) cos(p~/23)’/ where c 1 is a constant. In summary, we have shown that the ion-acoustic waves in a collision-dominated plasma, in which the temperature difference between the two electron components is fairly large, evolve to shock like structures. The shock may dissipate the wave energy to plasma particles. Our results are expected to have important implications in hot collisional plasma of thermonuclear interest, in which one often encounters high energy superthermal electron tail. Finally, we mention that we have treated the electron components as two fluids. This assumption is justified because the processes which produce such electron distributions have time scales much shorter than that of the iontime scale. The work of P.K.S. is performed under the auspices of the Sonderforschungsbereich Plasmaphysik Bochum/ JUlich. S.G.T. acknowledges the Alexander von Humboldt Foundation for the award of a Senior Fellowship.
References [11 W.D. Jones et al., Phys. Rev. Lett. 35 (1975) 1349. [2] B.N. Goswami and B. Buti, Phys. Lett. 57A (1976) 149. 3] V.!. Karpman, Nonlinear waves in dispersive media (Pergamon Press, 1975).
40