Nov 6, 2004 - and rotational centrifugal instabilities) are carefully revisited and identified in the linear theory. The effect of rigid plasma rotation on the flute ...
Global low-frequency modes in weakly ionized magnetized plasmas : effects of equilibrium plasma rotation Petro Sosenko, Thiery Pierre, Anatoly Zagorodny
To cite this version: Petro Sosenko, Thiery Pierre, Anatoly Zagorodny. Global low-frequency modes in weakly ionized magnetized plasmas : effects of equilibrium plasma rotation. 12th International Congress on Plasma Physics, 25-29 October 2004, Nice (France). 2004.
HAL Id: hal-00002001 https://hal.archives-ouvertes.fr/hal-00002001 Submitted on 6 Nov 2004
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Global Low-Frequency Modes in Weakly Ionized Magnetized Plasmas: Effects of Equilibrium Plasma Rotation Petro Sosenko, Thiery Pierre* and Anatoly Zagorodny
*
LPMIA, UPRES-A 7040, Université Henri Poincaré - Nancy I, 2, Bd des Aiguillettes, BP 239, 54506 Vandoeuvre Cedex, France; also International Centre of Physics, Kyïv, Ukraïna; Laboratoire PIIM - UMR6633 CNRS, Centre Saint Jérôme, case 321 Université Marseille I, F-13397 MARSEILLE Cedex 20 FRANCE
Abstract The linear and non-linear properties of global low-frequency oscillations in cylindrical weakly ionized magnetized plasmas are investigated analytically for the conditions of equilibrium plasma rotation. The theoretical results are compared with the experimental observations of rotating plasmas in laboratory devices, such as Mistral and Mirabelle in France, and KIWI in Germany. Key words: magnetized plasma; rotation; reduced model; global structures Classification: 52.25.-b; 52.30.-q; 52.35.-g; 52.72.+v Comment: 12th International Congress on Plasma Physics, 25-29 October 2004, Nice (France) Introduction ............................................................................................................................................................. 1 Ideal Ion Equilibrium .............................................................................................................................................. 2 Ideal Electron Equilibrium ...................................................................................................................................... 3 Reduced Wave Model ............................................................................................................................................. 4 Electron Fluid Motion ............................................................................................................................................. 6 Dispersion Equation ................................................................................................................................................ 7 Centrifugal Flute Instability..................................................................................................................................... 8 Drift Wave Instability.............................................................................................................................................. 9 Integral Constraints ............................................................................................................................................... 11 Discussion: Theory and Experiment Compared .................................................................................................... 13 References ............................................................................................................................................................. 14
Introduction The important role of low-frequency oscillations is well known, and they continue to be the subject of intense investigation in fusion and astrophysical situations, ionosphere and laboratory applications. Electrostatic low-frequency turbulence and related anomalous phenomena (turbulent transport, transport barriers, zonal flows) attract a special attention in the edge tokamak region, where collisions are essential. Cylindrical laboratory devices containing weakly ionized plasmas can have special importance for controlled fusion studies, or other complicated natural conditions. They allow to test our understanding of basic plasma physics including particle collisions, turbulence, and transport, as well as to develop new reliable methods of plasma diagnostics and control. Sosenko, Zagorodny: NICE - Global/[abstract #313]Topic: C, Fundamental Plasma Physics
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In this paper the linear and non-linear properties of global low-frequency oscillations in cylindrical weakly ionized magnetized plasmas are investigated analytically for the conditions of equilibrium plasma rotation. A new reduced non-linear model for the global oscillations in rotating plasmas is derived and analyzed. The various instabilities in rotating plasmas (current-dissipative and rotational centrifugal instabilities) are carefully revisited and identified in the linear theory. The effect of rigid plasma rotation on the flute and drift modes is investigated, and the relative importance of these modes in the plasma is established. Integral constraints are derived for the general case of arbitrary density and temperature profiles, and eigenfrequencies and instability rates are analyzed on their bases. Our investigation allows for a detailed comparison between the theoretical models and experimental results for eigenfrequencies and instability rates in rotating cylindrical plasmas. The theoretical results are compared with the experimental observations of rotating plasmas in laboratory devices, such as Mistral [Th. Pierre et al. 2004] and Mirabelle [Th. Pierre, G. Leclert, and F. Braun 1987] in France, and KIWI [A. Latten, T. Klinger, A. Piel, Th. Pierre 1995] in Germany. The centrifugal effects can be used as a basis for developing effective methods for turbulence control. Recently, they have been incorporated in new fusion confinement concepts [R. Ellis et al. 2001].
Ideal Ion Equilibrium
(1) (2)
The continuity and momentum balance equations for the ions are r r ∂ t n + ∇ ⋅ ( nV ) = 0 ,
r r r r r q r r (∂ t + V ⋅ ∇ ) V = Ω V × b − ∇ φ − ν V , m
where Ω = qB0 / mc , and the ion index is omitted. In the case of stationary equilibrium, the equations of motion in cylindrical coordinates are (3)
1 q 1 q (Vr ∂ r + Ω z ∂ϑ )Vr − Vϑ2 = E r + ΩVϑ , (Vr ∂ r + Ωz ∂ϑ )Vϑ + VϑVr = Eϑ − ΩVr , r m r m
where the frequency of charge-neutral collisions is neglected in comparison with the cyclotron frequency, and Ωz = Vϑ / r is a rotation frequency. Let us assume a rigid equilibrium rotation: r (4) V0 = V ( r )ϑˆ , Ωz = V / r = const , with axial symmetry ∂ϑ = 0 . Then (5) Vr = 0 , n = n0 ( r ) , Sosenko, Zagorodny: NICE - Global/[abstract #313]Topic: C, Fundamental Plasma Physics
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and the quadratic equation for the rotation frequency follows from the equations of motion (6)
Ω z2 + ΩΩ z − ΩΩ E = 0 ,
where ΩE ≡ − cEr / Br is an electric drift rotation frequency. There are two solutions, (7)
Ωz / Ω =
1 2
(− 1 ±
)
1 + 4ΩE / Ω ,
which can correspond to fast and slow rotation in the clock or counter-clock direction depending on the magnitude and the sign of ΩE / Ω . When ΩE Ω < 0 , the physical solutions exist if only (8)
4 ΩE / Ω < 1 .
For the very slow rotation, with 4 ΩE / Ω > 1 → Z i ni = ne , the eigen-value equation simplifies:
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~ − iν ω r r k2 ω χ e ~ i δ φˆ . ( ∆ ⊥ + κ n 0 ⋅ ∇ ⊥ ) δ φˆ = − ~* + ~//2 − ~ ω ω ω
(22)
Electron Fluid Motion
In the drift approximation, the electron continuity equation is r r ∂ t − ν p + VE ⋅ ∇ ne + ∇ // ( neV// e ) = 0 .
[
(23) The
]
electron
velocity
can
be
expressed
in
terms
of
the
current
density
J // = e( Z i niV//i − neV//e ) ≈ −en eV//e . The parallel velocityV//e is governed by the equation of electron motion r r e 1 ∇ // Φ − ∇ // ( neTe ) , ( d te + Vde ⋅ ∇ + V// e ∇ // + ν) V// e = me me n e r r r r with the convective derivative d t ≡ ∂ t + VE ⋅ ∇ . The term Vde ⋅ ∇V //e in this equation must be
(24)
omitted, since a more consistent treatment reveals the well-known cancellation of this term with the r r contribution from ∇ ⋅ π ⋅ b , the collisionless part of the stress tensor. For sufficiently large collision frequency
ν e , the electron inertia can be disregarded, and the last equation yields
V// e : V// e =
1 (e∇ // Φ − Te ∇ // ln ne ) , me ν e
∇ //Te = 0 . Thus, we can eliminateV// e from the continuity
equation: (25)
d t ne =
1 (Te ∇ // ne − ene ∇ // Φ) . me ν e
If there is an equilibrium electron drift along the magnetic field produced by a constant r electric field E0 , with the velocity U = − eE0 me ν e , then the potential in the basic equations must be
formally
V// e = U +
replaced
according
to
∇ // Φ → ∇ // Φ − E0
and
the
parallel
velocity
is
1 (e∇ // Φ − Te ∇ // ln ne ) . me ν e
In this case, the continuity equation becomes (26)
( d t + U ∇ // − ν p ) ne =
1 (Te ∇ // ne − ene ∇ // Φ) . me ν e
The reduced continuity equation for electrons in the drift approximation can be expressed in dimensionless variables: (27)
(∂
t
)
δn r r δ ne r r δφ , R ≡ ΩR / Ωi = 1 + 2Ωzi / Ωi + U ∇ // + Rv E ⋅ ∇ + Rv * ⋅ ∇δφ = De ∇ 2// e − τ ne 0 ne 0
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where ∇ // ne 0 = 0 , τ = Te / T , De = τM Z i ν e is a dimensionless diffusion coefficient ( De = S e2 / ν′ , r r r in dimensional units), M = mi me , v* ≡ ∇ ln ne 0 × b is the density-gradient-driven drift velocity, R ≡ ΩR / Ωi = 1 + 2Ωzi / Ωi is an ion rotation parameter. This is a non-linear equation. From here the χ δ φˆ . density perturbation follows in the linear approximation: δ nˆ n = ~ e
e0
e
In cylindrical geometry, (28)
~ χe =
Rω* + iΓ // /τ ω − ωU − ωE + i Γ //
Here ωU = U k // M Z i is the frequency shift in the presence of axial electron flow, with U = U / S e the ratio of parallel drift velocity and parallel thermal velocity for electrons, Γ // = De k //2 is the rate of electron diffusion along the magnetic field. ωE = lΩE , ΩE is the angular frequency, r which corresponds to the electric drift rotation with the velocity VE ≡ VEϑˆ , ΩE = VE / r = const . In r r r dimensionless variables VE ≡ Rb × ∇φ 0 = VEϑˆ , VE = Rdφ0 / dr , ΩE = VE / r . If the electron inertia is important, then ν e in Γ // is replaced with ν e − i (ω − ωU − RωE ) . r The electron electric susceptibility in the local theory is related to ~ χ e : χ e ( k , ω) = ~ χ e k 2 λ 2e in dimensional units, with λ e2 = Te 4 πe2 ne 0 . A misprint in [R.F. Ellis, E. Marden-Marshall, 1979] is to be noticed in the expression for the electron density response, given for τ = 1 , ωE = 1 : ωU is missing in the denominator. When Γ // is much larger then all other frequencies in ~ χ e , then the electron density perturbation is governed approximately by the linearized Boltzmann's law (“ δ -models”), (29)
1 1 ~ χ e = (1 + iδ), δ = (ω − ωU − ω E − τRω* ) . Γ // τ
Small deviations from the Boltzmann's law, described by δ , determine possible mechanisms of weak instability.
Dispersion Equation
Let us consider the approximation of constant electron temperature, τ = 1 . The Gaussian density profile n0 ~ exp( − r 2 L2N ) is a good approximation for many experimental situations. Then the wave equation simplifies and reduces to the eigen-value problem : r r (∆ ⊥ + κ n 0 ⋅ ∇ ⊥ ) δφˆ = − K 2δφˆ . (1) A real positive constant K 2 governs the eigen-frequencies via the dispersion equation. In cylindrical geometry, the eigen-functions are related to the confluent hypergeometric function M ( a, b, z ) : Sosenko, Zagorodny: NICE - Global/[abstract #313]Topic: C, Fundamental Plasma Physics
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2l − K 2 L2N
. 4 The eigen-values come from the boundary condition δφ (R0 ) = 0 , r2 L2N
(2)
δφˆ ~ r l M
(3)
K 2 L2N = 2l − 4a l n ( R02 L2N ) , n ≥ 0
, l + 1,
,
where al n ( z ) < 0 is the real root number n + 1 of the equation M ( a, l + 1, z ) = 0 . Thus in the case of low-frequency global modes in the rotating plasma, the dispersion equation is (4)
~ − iν k2 ω ω K 2 = ~* + ~//2 − ~ χe ~ i , ω ω ω
where ~ χe =
Rω* + iΓ // /τ for the simple electron fluid model introduced above, ω* = 2l / L2N , ω − ωU − ωE + i Γ // ∆ln ≡ −2al n ( R02 L2N ) / l
K 2 = ω*( 1 + ∆ln ) , and
is a size parameter. In the large-radius
approximation, when R02 L2N >> 1 , and when l is not very large ⇒ al n ≈ − n , K 02 ≈ 2(l + 2n ) / L2N . Then K 2 = ω* for the modes with n = 0 .
Centrifugal Flute Instability
The dispersion equation is especially simple for flute modes, k // → 0 , when ~ χe =
Rω* . ω − ωU − ωE
In what follows the ideal limit is considered, and the frequency is counted from lΩ zi . Then the dispersion equation takes the form (30)
K2 =
ω* ω
−
Rω* . ω − ωU + l (Ω zi − ΩE )
One can clearly see the basic effects responsible for charge separation in wave motion : different equilibrium rotation frequencies for ions and electrons, Ω zi ≠ ΩE , the Coriolis effect R ≠ 1 , and axial electron current, ωU ≠ 0 . The latter factor is omitted for simplicity.
This dispersion equation demonstrates the limitations of the effective gravity model for the instabilities in rotating plasmas. Such a model will miss the Coriolis effect. This equation is quadratic in frequency :
fx 2 + 2 x + 1 = 0 , where
x ≡ ω / lΩ zi ,
f ≡ K 02 l / ω * = l − 2al n ( R02 L2N ) > 1 ,
(31)
x=
(
)
1 −1± 1− f . f
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Thus all the modes are instable. The eigenfrequency and the instability rate are (32)
f −1 , ωi = lΩ zi f
ω r = lΩ zi
f −1 . f
The rate is maximal for the modes with n = 0 . In the large-radius approximation, ω r = Ω zi (l − 1) ,
ω i = Ω zi l − 1 .
Drift Wave Instability
In the non-ideal case, and for non-adiabatic electrons, when δ ≠ 0 , the dispersion relation yields the growth/damping rate ωi , ω ≡ ω r + iω i =
ω* − ν i K 2 1 + K 2 + iδ
. The necessary instability
condition is ω r Reδ < 0 .
(5)
For weakly non-adiabatic electrons, with δ 0 , and to weaker rotation if
ΩE Ωi < 0 . When R 2 > ω*0( 1 + ∆ln ) , the eigen-frequency 1 + ∆ln
decreases when R grows : ωr /Ωi = ω* 0 / R . The instability rate is represented as a sum ω i ≡ γ dis + γ U + γ rot + γ in , (38)
γ dis
2 2 νiK 2 ωU ω r ω r ( R − 1)ω * + RlΩ zi2 1 ωr K , γU = , γ rot = , γ in = − . = Γ// 1 + K 2 Γ// 1 + K 2 Γ// 1+ K2 1+ K2
The part γ dis coincides with the instability rate in the ideal case without axial current and rotation. This part has the following dependence on R : (39)
γ dis 1 + ∆ln ωr3 = . Ωi Γ // Ωi2 R
It is maximal when R 2 = 12 ω* 0( 1 + ∆ln ) . Thus it grows with R , when R 2 < 12 ω* 0( 1 + ∆ln ) : γ dis Ω γ Ω 1 R2 = i . In the opposite limit, dis = (1 + ∆ln )ω*30 i 4 , and it decreases rapidly as 2 Ωi Γ // (1 + ∆ln ) Ωi Γ // R R grows. Let us analyze the part γ U : (40)
γU Ωi
=
ωU Rω r2 . ω *0 Γ// Ωi2
It is maximal when R 2 = 3ω* 0( 1 + ∆ln ) . In the limit R 2 < 3ω* 0( 1 + ∆ln ) , this part grows more rapidly
γU
ωU R3 with R then γ dis : = . In the opposite limit, it decreases much slower then γ dis : Ωi ω *0 Γ// (1 + ∆ln ) 2 Sosenko, Zagorodny: NICE - Global/[abstract #313]Topic: C, Fundamental Plasma Physics
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γU Ωi
=
ωU ω*20 . Thus the current modification of the instability rate becomes more important in ω*0 Γ// R
the rotating plasma. In a similar manner, one finds
γ rot
(41)
Ωi
=
ω r2 Γ// Ωi
( R − 1)(1 + l
R −1
ω *0
) .
The maximum of this part as a function of mode number corresponds to the eigen-frequency maximum. The Coriolis effect is not important when ω*0
