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Momentum transport and collective modes in high ion temperature regimes
15 May 1905
PHYSICS
LETTERS
A
Physics Letters A 201 (199.5)66-69
Momentum transport and collective modes in high ion temperature regimes B. Coppi Massachusetts
Institute of Technology. Cambridge, MA 02139, USA
Received 28 February 1995; accepted for publication 4 March 1995 Communicated by M. Porkolab
Abstract Plasma collective modes that are driven by a transverse gradient of the flow velocity in the direction of the confining magnetic field have an important role in explaining the experimentally observed momentum transport in conditions where the ion temperature exceeds the electron temperature. A new relevant regime is identified where modes with the desired
characteristics are found to be excited.
1. General considerations It is well known both from space physics and from laboratory experiments that the rate of transport of momentum in plasmas can be significantly higher than that estimated from viscosity resulting from the collisional transport theory. The explanation for this is then to be found among the collective modes that can induce a transport of momentum. If we consider in particular a plasma that is well confined in a magnetic field whose pressure is considerably larger than the plasma kinetic pressure, a new type of mode can be found in regimes where the ion temperature is significant relative to the electron temperature. This mode is driven unstable by a relatively strong transverse (to the magnetic field) gradient of the longitudinal flow velocity. The mode is “dissipative” in nature in that it depends on the finiteness of the longitudinal ion thermal conductivity or on a zero phase velocity mode-particle resonance. Typically, the mode growth rate y is in the range Y < ‘11V,hi Elsevicr Science B.V. SSDI 0375-9601(95)00217-O
(1)
unlike those ordinarily considered [l-3] and that do not require a finite ion temperature for their existence. Here k,, is the longitudinal (relative to the confining magnetic field) mode number and V,,i is the ion thermal velocity. We consider in particular electrostatic modes whose electric field JjZ
-V&
(2)
and, considering fj=$(x)
a plane geometry,
exp(-iwf+ik,y+ik,,z).
The relevant instability k,. A,----
with
1
kii ‘,i
dV,I” dx
condition
7’, >Tfl, L
(3) is
(4)
where V,,” is the longitudinal plasma flow velocity, Qi is the ion cyclotron frequency, and T/T, is the ratio of the ion to the electron temperature. In order to demonstrate this we note that since w < k,,V,,, the electron longitudinal momentum conservation equation reduces to 0 = - V,,( A,T, + n,fe)
+ enV,, rj
B. Coppi/Physics
from which we have
that gives
(5) The relevant expression for fC can be obtained from the electron thermal energy balance equation where the longitudinal electron thermal conductivity is assumed to be large in the sense that y=
-i(
67
Letters A 201 (1995) 66-69
w - k,,V,“) < k;S,;: >
where S$ is the relevant diffusion coefficient. Then, if we ignore for simplicity the presence of density and temperature gradients, we have
A, = 1 f;
+ 6A, e
and ---
(14)
= ,Ai;. I
Thus, for 9,;: 0 and
the instability
condition
is 6A,
(15) If we consider the limit
Thus
Y < k~~~~~
kiQr$ < Y < kiiV,hi3
(7) We can treat the ion population when considering the limit
in a similar fashion
y < k;“,;;,
(8)
we obtain 3Ti --22Ti
ni n
instead of Eq. (9) and
then we obtain
=A,
(16)
- 1,
(9) On the other hand the relevant longitudinal tum balance equation is of the form mini?,,--
dVI”= dx
-V,,(;liTi
+ nfi)
-
enV,,&,
momen-
(10)
where
which requires A, = 1 + i(T,/T,) to find y > 0. There is a collisionless version of this instability that can be derived from the Vlasov equation after considering a one-dimensional ion equilibrium distribution,
( 2rT,/mi)
(11) 4
k,
I”
X exp[ -mi(
Thus Eq. (10) gives ii -I,_ n
n
F,( L’;,) =
1
r,+---- k,, flci
dyr dx
L$)~/~T~],
(17)
where
e$ T;
(12)
and then
The relevant perturbed
equation e
aFi
(13)
is
*I aFi (18)
mi “dr,, ax+-E-=07 The quasi-neutrality persion equation
condition
fie = iii yields the dis-
g&i-&)=(A,-+&)
where 3Fi -= %
I mi”li
aJ,j
Ti
_
-I
--F,,
ax
aFi a9
-(
dVllF
dx
1
’
68
B. Coppi / Physics Letters A 201 (1995) 66-69
Thus (-iy+
k,,r;,)Gi + qF,k,,r;,(I I
-A,)
and V,’ < V,hi no instability driven by dV,r/dx can be found consistently for 1y 1 < k,,V,,,. Then, we consider the case where A, z+ 1 + T/T, and T, > T,. The condition under which Eq. (20) can yield an instability is k,,yhi < y and this corresponds to the well-known [ 11 result
= 0
and n si = ;,I
-A,)W,
(19)
I
dt 5’
Xexp( -{‘)/(
[‘+
T’),
9~ y/k,,V,,i and Vii = 2Ti/mi. dispersion relation becomes T,
Then
(25)
the relevant where
1 1
Ap=I+E]lVi
w=w-k,,i$
and for 3 < 1
The quasilinear momentum excitation of the considered as = min(L:E,i;;,
+ Q+
,.
(22)
w;i=k.--.
‘eB
dx
the w:, term can be omitted in Eq. (25), as it assumes W2 > kiV$, and the instability condition Avk;rss2 > &IL& implies
. = iyne+/k,,Ti,
(23) Thus we see that when dV,r/dx < 0 and k,/k,, < 0, as required by the instability condition, the flux is outward, that is, in the direction of decreasing momentum.
that is difficult to satisfy. If the ion temperature gradient is much stronger than the density gradient (i.e. d In T/d In n z+ 1) then this factor dominates the resulting instability unless A,>
2. Influence
CT, 1 dn -k,---, eB n dx
flux resulting from the mode can be estimated
Since nlii,, = iyA,/k,,
o,~= c dTi
(21)
&
(24)
where L’\’= Te/mi, that implies A, >> 2Ti/T,. However we notice that, if density and temperature gradients are present with, a stronger temperative gradient the more complete form of the dispersion relation (24) is
where, if we consider y real, Wi = -(J;;))‘/_:-
y’ = k; v,2Av,
of finite ion pressure gradients
It is easy to verify that if the ion pressure gradient is significant, in the sense that,
$[$#!j-.
We conclude by observing that in the presence of significant density and ion temperature gradients the modes that can be driven unstable are those associated with a finite diffusion of the longitudinal momentum [3-51, such as viscosity due to ion-ion collisions or ion-neutral collisions that have been discussed earlier in the literature.
B. Coppi/Physics
Letters A 201 (1995) 66-69
69
Acknowledgement It is a to thank Basu, W. and S. for stimulating on the subject of paper. This was sponin part the US of Energy.
[I] N. D’Angelo, Phys. Fluids 8 (1965) 1748. [2] S. Migliuolo, J. Geophys. Res. 93 (1988) 867. [3] B. Coppi, Massachusetts Institute of Technology Report PTP89/2, Cambridge, MA (1989). [4] B. Coppi and B. Basu, J. Geophys. Res. 94 (1989) 5316. [5] B. Coppi, Plasma Phys. Control Fusion 36 (1994) B107.
PHYSICS
LETTERS
A
Physics Letters A 201 (199.5)66-69
Momentum transport and collective modes in high ion temperature regimes B. Coppi Massachusetts
Institute of Technology. Cambridge, MA 02139, USA
Received 28 February 1995; accepted for publication 4 March 1995 Communicated by M. Porkolab
Abstract Plasma collective modes that are driven by a transverse gradient of the flow velocity in the direction of the confining magnetic field have an important role in explaining the experimentally observed momentum transport in conditions where the ion temperature exceeds the electron temperature. A new relevant regime is identified where modes with the desired
characteristics are found to be excited.
1. General considerations It is well known both from space physics and from laboratory experiments that the rate of transport of momentum in plasmas can be significantly higher than that estimated from viscosity resulting from the collisional transport theory. The explanation for this is then to be found among the collective modes that can induce a transport of momentum. If we consider in particular a plasma that is well confined in a magnetic field whose pressure is considerably larger than the plasma kinetic pressure, a new type of mode can be found in regimes where the ion temperature is significant relative to the electron temperature. This mode is driven unstable by a relatively strong transverse (to the magnetic field) gradient of the longitudinal flow velocity. The mode is “dissipative” in nature in that it depends on the finiteness of the longitudinal ion thermal conductivity or on a zero phase velocity mode-particle resonance. Typically, the mode growth rate y is in the range Y < ‘11V,hi Elsevicr Science B.V. SSDI 0375-9601(95)00217-O
(1)
unlike those ordinarily considered [l-3] and that do not require a finite ion temperature for their existence. Here k,, is the longitudinal (relative to the confining magnetic field) mode number and V,,i is the ion thermal velocity. We consider in particular electrostatic modes whose electric field JjZ
-V&
(2)
and, considering fj=$(x)
a plane geometry,
exp(-iwf+ik,y+ik,,z).
The relevant instability k,. A,----
with
1
kii ‘,i
dV,I” dx
condition
7’, >Tfl, L
(3) is
(4)
where V,,” is the longitudinal plasma flow velocity, Qi is the ion cyclotron frequency, and T/T, is the ratio of the ion to the electron temperature. In order to demonstrate this we note that since w < k,,V,,, the electron longitudinal momentum conservation equation reduces to 0 = - V,,( A,T, + n,fe)
+ enV,, rj
B. Coppi/Physics
from which we have
that gives
(5) The relevant expression for fC can be obtained from the electron thermal energy balance equation where the longitudinal electron thermal conductivity is assumed to be large in the sense that y=
-i(
67
Letters A 201 (1995) 66-69
w - k,,V,“) < k;S,;: >
where S$ is the relevant diffusion coefficient. Then, if we ignore for simplicity the presence of density and temperature gradients, we have
A, = 1 f;
+ 6A, e
and ---
(14)
= ,Ai;. I
Thus, for 9,;:
the instability
condition
is 6A,
(15) If we consider the limit
Thus
Y < k~~~~~
kiQr$ < Y < kiiV,hi3
(7) We can treat the ion population when considering the limit
in a similar fashion
y < k;“,;;,
(8)
we obtain 3Ti --22Ti
ni n
instead of Eq. (9) and
then we obtain
=A,
(16)
- 1,
(9) On the other hand the relevant longitudinal tum balance equation is of the form mini?,,--
dVI”= dx
-V,,(;liTi
+ nfi)
-
enV,,&,
momen-
(10)
where
which requires A, = 1 + i(T,/T,) to find y > 0. There is a collisionless version of this instability that can be derived from the Vlasov equation after considering a one-dimensional ion equilibrium distribution,
( 2rT,/mi)
(11) 4
k,
I”
X exp[ -mi(
Thus Eq. (10) gives ii -I,_ n
n
F,( L’;,) =
1
r,+---- k,, flci
dyr dx
L$)~/~T~],
(17)
where
e$ T;
(12)
and then
The relevant perturbed
equation e
aFi
(13)
is
*I aFi (18)
mi “dr,, ax+-E-=07 The quasi-neutrality persion equation
condition
fie = iii yields the dis-
g&i-&)=(A,-+&)
where 3Fi -= %
I mi”li
aJ,j
Ti
_
-I
--F,,
ax
aFi a9
-(
dVllF
dx
1
’
68
B. Coppi / Physics Letters A 201 (1995) 66-69
Thus (-iy+
k,,r;,)Gi + qF,k,,r;,(I I
-A,)
and V,’ < V,hi no instability driven by dV,r/dx can be found consistently for 1y 1 < k,,V,,,. Then, we consider the case where A, z+ 1 + T/T, and T, > T,. The condition under which Eq. (20) can yield an instability is k,,yhi < y and this corresponds to the well-known [ 11 result
= 0
and n si = ;,I
-A,)W,
(19)
I
dt 5’
Xexp( -{‘)/(
[‘+
T’),
9~ y/k,,V,,i and Vii = 2Ti/mi. dispersion relation becomes T,
Then
(25)
the relevant where
1 1
Ap=I+E]lVi
w=w-k,,i$
and for 3 < 1
The quasilinear momentum excitation of the considered as = min(L:E,i;;,
+ Q+
,.
(22)
w;i=k.--.
‘eB
dx
the w:, term can be omitted in Eq. (25), as it assumes W2 > kiV$, and the instability condition Avk;rss2 > &IL& implies
. = iyne+/k,,Ti,
(23) Thus we see that when dV,r/dx < 0 and k,/k,, < 0, as required by the instability condition, the flux is outward, that is, in the direction of decreasing momentum.
that is difficult to satisfy. If the ion temperature gradient is much stronger than the density gradient (i.e. d In T/d In n z+ 1) then this factor dominates the resulting instability unless A,>
2. Influence
CT, 1 dn -k,---, eB n dx
flux resulting from the mode can be estimated
Since nlii,, = iyA,/k,,
o,~= c dTi
(21)
&
(24)
where L’\’= Te/mi, that implies A, >> 2Ti/T,. However we notice that, if density and temperature gradients are present with, a stronger temperative gradient the more complete form of the dispersion relation (24) is
where, if we consider y real, Wi = -(J;;))‘/_:-
y’ = k; v,2Av,
of finite ion pressure gradients
It is easy to verify that if the ion pressure gradient is significant, in the sense that,
$[$#!j-.
We conclude by observing that in the presence of significant density and ion temperature gradients the modes that can be driven unstable are those associated with a finite diffusion of the longitudinal momentum [3-51, such as viscosity due to ion-ion collisions or ion-neutral collisions that have been discussed earlier in the literature.
B. Coppi/Physics
Letters A 201 (1995) 66-69
69
Acknowledgement It is a to thank Basu, W. and S. for stimulating on the subject of paper. This was sponin part the US of Energy.
[I] N. D’Angelo, Phys. Fluids 8 (1965) 1748. [2] S. Migliuolo, J. Geophys. Res. 93 (1988) 867. [3] B. Coppi, Massachusetts Institute of Technology Report PTP89/2, Cambridge, MA (1989). [4] B. Coppi and B. Basu, J. Geophys. Res. 94 (1989) 5316. [5] B. Coppi, Plasma Phys. Control Fusion 36 (1994) B107.