Linköping University Post Print
The magnetic Rayleigh-Taylor instability and
flute waves at the ion Larmor radius scales
O G Onishchenko, O A Pokhotelov, Lennart Stenflo and P K Shukla
N.B.: When citing this work, cite the original article.
Original Publication:
O G Onishchenko, O A Pokhotelov, Lennart Stenflo and P K Shukla, The magnetic
Rayleigh-Taylor instability and flute waves at the ion Larmor radius scales, 2011, PHYSICS OF
PLASMAS, (18), 2, 022106.
http://dx.doi.org/10.1063/1.3554661
Copyright: American Institute of Physics
http://www.aip.org/
Postprint available at: Linköping University Electronic Press
The magnetic Rayleigh–Taylor instability and flute waves at the ion Larmor
radius scales
O. G. Onishchenko,1,a兲 O. A. Pokhotelov,1,b兲 L. Stenflo,2,c兲and P. K. Shukla3,d兲
1
Institute of Physics of the Earth, 10 B. Gruzinskaya, 123995 Moscow, Russia 2
Department of Physics, Linköping University, SE-58183 Linköping, Sweden 3
RUB International Chair, International Centre for Advanced Studies in Physical Sciences, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany
共Received 21 September 2010; accepted 21 January 2011; published online 16 February 2011兲 The theory of flute waves共with arbitrary spatial scales compared to the ion Larmor radius兲 driven by the Rayleigh–Taylor instability 共RTI兲 is developed. Both the kinetic and hydrodynamic models are considered. In this way we have extended the previous analysis of RTI carried out in the long wavelength limit. It is found that complete finite ion Larmor radius stabilization is absent when the ion diamagnetic velocity attains the ion gravitation drift velocity. The hydrodynamic approach allowed us to deduce a new set of nonlinear equations for flute waves with arbitrary spatial scales. It is shown that the previously deduced equations are inadequate when the wavelength becomes of the order of the ion Larmor radius. In the linear limit a Fourier transform of these equations yields the dispersion relation which in the so-called Padé approximation corresponds to the results of the fully kinetic treatment. The development of such a theory gives us enough grounds for an adequate description of the RTI stabilization by the finite ion Larmor radius effect. © 2011 American Institute of Physics. 关doi:10.1063/1.3554661兴
I. INTRODUCTION
The flute waves driven by the magnetic Rayleigh–Taylor instability共RTI兲 are believed to be plausibly responsible for the anomalous transport in magnetic confinement plasmas. This RTI is the counterpart of the classic RTI of an inhomo-geneous fluid in a gravitational field. The gravity force may alternatively be replaced by a force due to the centrifugal acceleration of the particles 共ions and electrons兲 moving along the curved magnetic field lines. In this case the insta-bility is termed magnetic-curvature-driven RTI. It has been recently recognized that coherent large-scale structures such as zonal flows and streamers, arising in small scale flute microturbulence, can play an essential role in plasma transport.1,2The finite ion Larmor radius 共FLR兲 effects sub-stantially influence the evolution of the flute wave dynamics since they lead to the stabilization of the magnetic RTI.3–5
The previous theories of RTI and nonlinear structures of the flute waves1–13 were restricted to the long wavelength limit where the wave spatial scale is much larger than the ion Larmor radius. In what follows, we shall use a local Carte-sian coordinate system where the z-axis is along the external magnetic field B0, the x-axis is along the direction of the plasma inhomogeneity and the direction of the gravitational acceleration g, and where the y-axis completes the triad. In low beta plasmas the dispersion relation and the growth rate of the flute共interchange兲 waves are3–5
共−iD−g兲 + ky2 k⬜2 Ng = 0, 共1兲 ␥ ⌫=
冋
1 − z 4共1 ++兲 2册
1/2, 共2兲whereis the wave frequency, k⬜2= kx2+ ky2, kxand kyare the
components of the wave vector in the x and y directions, iD= kyviD, g= kyvg, viD= −i
2
ci共N+T兲yˆ is the
equilib-rium ion diamagnetic drift velocity, N= −d ln n0i/dx⬎0,
= g/NvTi
2 ⬎0,
T= −d ln T0i/dx, n0iand T0iare the equilib-rium ion number density and temperature, respectively, i=共T0i/mici
2兲1/2 is the ion Larmor radius,
ci= ZeB0/mi is
the ion cyclotron frequency, mi is the ion mass, Z is the ion
charge number, e is the magnitude of the elementary charge, yˆ is the unit vector along the y axis, vg= −共g/ci兲yˆ is the ion
gravitational drift velocity, =T/N, ⌫=兩ky/k⬜兩共gN兲1/2 is
the maximum value of the RTI growth rate,vT=共T0i/mi兲1/2is
the ion thermal velocity, and z = k⬜2i
2 .
From Eq. 共2兲 it follows that instability stabilization oc-curs for finite z in a plasma with finitevN/vg=NvT
2/g, where
vN= −i
2
ciNis the ion diamagnetic drift velocity due to the
ion density inhomogeneity. Since Eqs.共1兲 and共2兲 were ob-tained in the long wavelength approximation, zⰆ1, the ques-tion of how the results will change qualitatively and quanti-tatively at z⯝1 still remains open.
The development of kinetic and hydrodynamic theories of flute waves with the spatial scales of the order of the ion Larmor radius propagating in a low- plasmas is the main purpose of the present study. Furthermore, the effects of fi-nite ratiovN/vgare also taken into account.
The manuscript is organized as follows. Section II de-scribes the results of the kinetic approach in the linear ap-proximation. A closed set of equations describing the nonlin-ear dynamics of the flute waves with arbitrary spatial scales is deduced in Sec. III. Our discussion and conclusions are found in Sec. IV.
a兲Electronic mail: onish@ifz.ru. b兲Electronic mail: pokh@ifz.ru.
c兲Electronic mail: lennart.stenflo@physics.umu.se. d兲Electronic mail: profshukla@yahoo.com.
II. KINETIC DESCRIPTION
Using the fully kinetic description one can obtain the linear response of the normalized ion density perturbation to the variation of the electrostatic field as5,14–16
␦n = −N
⬘
⌽ −冋
␣11冉
1 − N ⬘
冊
−␣12 T ⬘
册
⌽, 共3兲where ␦n = n˜i/n0i, n˜i= ni− n0i and n0i are the perturbed and unperturbed ion number densities, N,T= −kyi2ciN,T is
the ion drift frequency due to inhomogeneity of the ion density or temperature,
⬘
=− kyvg, ⌽=Ze/T0i is the normalized electrostatic potential, and E⬜= −ⵜ⬜. Further-more, the subscript ⬜ denotes the vector component perpendicular to the ambient magnetic field, ␣11= 1 −⌫0共z兲,␣12= z⌫0共z兲关1−I1共z兲/I0共z兲兴, ⌫0共z兲⬅exp共−z兲I0共z兲, where I0,1 is the modified Bessel functions of the first kind. In contrast to Refs.5and14–16we have taken into account that due to the ion gravitational drift the wave frequency is Doppler-shifted by the value kyvg.
The variation of the normalized electron number density is
␦ne= −
N
⌽, 共4兲
where␦ne= n˜e/n0e, n˜e= ne− n0eand n0eare the perturbed and unperturbed electron number densities, respectively.
Invoking the quasineutrality,␦ne=␦n, we obtain the
lin-ear dispersion relation for the flute waves,
冉
−N− ␣12 ␣11 T−g冊
+ z ␣11 ky 2 k⬜2Ng = 0, 共5兲 and the expression for the normalized growth rate,␥ ⌫=
冋
z ␣11 − z 4冉
1 + ␣12 ␣11 +冊
2册
1/2 . 共6兲In the long wavelength approximation, zⰆ1, i.e., when ␣12⯝␣11⯝z, the dispersion relation 共5兲and the growth rate
共6兲coincide with those given by Eqs.共1兲 and共2兲. As it was mentioned before our consideration is valid only for low beta plasmas. The effects of finite beta were recently discussed in Refs.12and13. It should be noted that derivation of Eqs.共1兲 and共2兲 requires the use of the ion gyroviscosity or the gen-eralized Ohm’s law in an extended magnetohydrodynamic 共MHD兲 model.4,7
From Eq.共6兲one easily finds that the cut-off wave num-ber共at which␥= 0兲 is defined from the condition
4 1 − e−zI0共z兲 =
冉
1 ++ze −zI 0共z兲共1 − I1共z兲/I2共z兲兲 1 − e−zI0共z兲 冊
2. 共7兲 In a plasma with uniform temperature共= 0兲 it reduces to a very simple form,e−zI0共z兲 =
共1 −兲2
共1 +兲2. 共8兲
Equation共8兲possesses the finite solution for z = zcrexcept for the case when= 1. The latter assumes that the critical wave number for complete FLR stabilization tends toward infinity
and thus such stabilization in this case is absent. The incor-poration of the finite values ofalso does not lead to disap-pearance of singularity and plasma remains unstable when = 1. A similar conclusion about the absence of complete FLR stabilization of RTI has been made also in Ref. 13. However, such a statement has been based in this paper on the extended MHD equations which are valid only in the long wavelength limit.
III. HYDRODYNAMIC DESCRIPTION
On one hand, the description of waves with spatial scales of the order of the ion Larmor radius requires kinetic treat-ment. On the other hand, the hydrodynamic approximation can help us in clarifying the physics of the involved phenom-ena. In order to formulate the relevant hydrodynamic equa-tions we decompose the ion velocity as
vi⯝ vE+ viD+ vg+ vE P + viD P + vg P , 共9兲
where the low frequency limit is assumed. Furthermore, vEis
the E⫻B drift velocity, viD=共n0imici兲−1共zˆ⫻ⵜp兲 is the ion
diamagnetic velocity, zˆ is the unit vector along the external magnetic field B0, p is the ion pressure, and vE
P
, viD P
, and vg P
stand for the polarization parts of the ion velocity related to the drift velocities vE, viD, and vgthrough the relations
vE P = 1 ci
冉
zˆ⫻dvE dt冊
, viD P = 1 ci冉
zˆ⫻ dviD dt冊
, 共10兲 and vg P = 1 ci冉
zˆ⫻dvg dt冊
.Following Refs. 14, 15, and 17 we note that the convec-tive time derivaconvec-tive d/dt in Eq. 共10兲 is d/dt⬅/t +共vE+ vg兲·ⵜ⬜. The absence of the contribution from viD·ⵜ⬜
term in the round brackets reflects the “gyroviscous cancel-lation” effect according to which the correction to the ion diamagnetic velocity due to the magnetic viscosity cancels the contribution from advective diamagnetic part of the ion velocity viD·ⵜ⬜. After substitution of Eqs.共9兲and共10兲into
the ion continuity equation in the dimensionless form, one obtains d d共1 − ⵜ⬜2兲␦n − d dⵜ⬜2␦T +ˆN ⌽ Y − d dⵜ⬜2⌽ +␦n
冉
ˆT ␦n Y −ˆN ␦T Y冊
−兵ⵜ⬜⌽,ⵜ⬜␦p其 = −ˆⵜ⬜ 4⌽, 共11兲where the spatiotemporal variables are normalized toiand
ci −1 , i.e., 共X,Y兲=共x,y兲i −1 , = tci, ˆN=Ni, gˆ = g/共civT兲, d/d=/+兵⌽,...其−gˆ/Y, ␦T =共T−T0兲/T0 is the varia-tion of the normalized ion temperature, ˆ =共3/10兲/ci
is the dimensionless dynamic collision viscosity, and is the ion-ion collision frequency. Furthermore, 兵f ,g其 =共f/X兲g/Y −共f/Y兲g/X denotes the Poisson bracket. Here the collisional term is retained keeping in mind the possible use of these equations for numerical simulations in which the so-called numerical viscosity can be introduced.
Equation 共11兲 should be supplemented by the ion ther-mal balance condition
ni
冉
d
dt+ viD·ⵜ⬜
冊
T + niTⵜ · vi+ⵜ · q⬜= 0, 共12兲 where the ion thermal flux q⬜is given byq⬜= 2pi mici 共zˆ ⫻ ⵜ⬜T兲 − 2 mici 2 d dt共pⵜ⬜T兲 − 2p mici 2ⵜ⬜T. 共13兲
Equations 共12兲 and 共13兲 have been written in a form that takes into account the two-dimensional character of the ion motion in the wave field with the effective ratio of specific heats␥= 2, see Refs.14and15. The first term on the right of Eq. 共13兲 represents the standard thermal flux in Braginskii hydrodynamics, the second one is an addition corresponding to the so-called polarization part of the ion thermal flux in the Grad hydrodynamics. Finally, the third term describes the collisional thermal flux arising due to the ion-ion collisions. In dimensionless form the equation for the variation of the ion temperature is
d d共1 − 3ⵜ⬜ 2兲␦T − d dⵜ⬜ 2␦n − d dⵜ⬜ 2⌽ +ˆ T ⌽ Y +␦p
冉
ˆT ␦n Y −ˆN ␦T Y冊
+兵ⵜ⬜⌽,ⵜ⬜共␦p + 2␦T兲其 =ˆⵜ⬜2␦T, 共14兲where ˆT=Ti, ˆ = 2ii/ci is the normalized ion-ion
fre-quency, and ␦p = p˜/pi0 is the normalized ion pressure perturbation.
Equations共11兲 and共14兲 should be supplemented by the electron continuity equation. For that purpose we decompose the electron velocity as ve= vE+ veD. Here veD= Zii
2
ciNyˆ
is the electron diamagnetic drift velocity, and i= Te/Ti0 is
the ratio of the electron to ion temperature. In the dimension-less form the electron continuity equation is
␦n +ˆN ⌽ Y =兵␦n,⌽其 + Dˆⵜ⬜ 2␦ n, 共15兲
where Dˆ ⬅共me/mi兲Ziˆeeis the dimensionless diffusion
co-efficient,ˆee=ee/ciis the normalized electron-electron
col-lision frequency, and meis the electron mass.
Equations共11兲,共14兲, and共15兲 constitute a closed set of coupled equations for ␦n, ⌽, and ␦T describing nonlinear flute waves with arbitrary spatial scales. These equations can be used for the study of nonlinear dynamics of magnetic RTI. In the general case they are quite cumbersome and can be solved only numerically. We note that a similar set but con-taining only two equations for␦n and⌽ 共the ion and electron continuity equations兲 has been analyzed in Refs.4and6–11. They can be obtained from our equations by setting␦T→0 and rearranging Eq. 共11兲 with the help of Eq. 共15兲. This results in the following equation:
ⵜ⬜2⌽ −共ˆN+ gˆ兲 ⵜ⬜2⌽ Y − gˆ 共1 − ⵜ⬜2兲␦ n Y +ⵜ⬜兵␦n,ⵜ⬜⌽其 + 兵⌽,ⵜ⬜2⌽其 −ˆT␦n ␦n Y =ˆⵜ⬜4⌽ + Dˆⵜ⬜2共1 − ⵜ⬜2兲␦n. 共16兲 Furthermore, if one neglects gˆ relative to ˆN 共this
corre-sponds toⰆ1兲 and scalar nonlinearity 共the last term on the left兲 in the large-scale approximation 共ⵜ⬜2Ⰶ1兲, Eq.共16兲
re-duces to the corresponding equation of Refs. 4 and 6–11. This assumes that the equation for the thermal balance equa-tion 共14兲 remains outside the current consideration. At this point a few comments are in order. In Ref.14 it has noted that such procedure is inadequate since Eqs. 共14兲 and 共16兲 become inconsistent and the ion temperature variations can-not be neglected. Thus, the rigorous consideration of the problem at hand requires consideration of the full system containing three equations for three variables. Summarizing, we note that the detailed picture of the nonlinear dynamics of the magnetic RTI demands quite complicated numerical modeling based on our newly deduced equations共11兲,共14兲, and共15兲.
In the case of collisionless plasmas and in the linear approximation using a Fourier transform of these equations, one can obtain the linear response of the ion number density as ␦n = −N
⬘
⌽ −冋
␣11 H冉
1 −N ⬘
冊
−␣12 HT ⬘
册
⌽, 共17兲 where␣11H= z共1+2z兲共1+4z+2z2兲−1,␣12H= z共1+4z+2z2兲−1, and the superscript H denotes the quantities obtained in the hy-drodynamic approximation. In the Padé approximation the expression for ␦n coincides with that obtained in the framework of the fully kinetic treatment equation共3兲 when ␣11⯝␣11H
and␣12⯝␣12
H
. Therefore, the dispersion relation of the flute waves and the expression for the RTI normalized growth rate are
2−
冉
N+ ␣12 H ␣11 HT+g冊
+ k2y k⬜2 z ␣11 HNg = 0 共18兲 and ␥ ⌫=冋
z ␣11 H − z 4冉
1 + ␣12 H ␣11 H+冊
2册
1/2 . 共19兲Figures1–4 show the dependence of the normalized growth rates ␥/⌫ versus k⬜i. The parameter =g/N= g/NvT2
takes the values 1/5 and 1/2. The solid, dashed, and dashed-dotted lines correspond to classic 共large scale兲, kinetic, and hydrodynamic 共Padé approximation兲 approaches, respec-tively. Figures1 and3 correspond to plasmas with uniform ion temperature,=T/N= 0, and Figs.2and4to plasmas
with= 0.5. It is seen that the normalized growth rates ob-tained in the kinetic and model hydrodynamic approaches have qualitatively similar dependencies on k⬜ias compared
to the classical results obtained in the long wavelength approximation.3–5They differ only quantitatively. The
insta-bility region in the k space defined in kinetics or model hydrodynamics is wider than that obtained in Refs.3–5. One can see that the results of the model hydrodynamic descrip-tion are in reasonable agreement with the kinetics in a plasma with = 1/5–1/2. Comparing Figs. 1 and 3 with Figs. 2 and 4 one sees that the instability regions become smaller with an increase in.
IV. SUMMARY
The present analysis is an extension of the previous study of nonlinear flute waves in a low-plasma with non-zero ion temperature gradient that was limited to consider-ation of waves with spatial scales larger than the ion Larmor radius i. The results of a kinetic description of the flute
waves and RTI have been generalized to the case of arbitrary
wave scales. We have extended the previous analysis of non-linear flute waves6–13using two-fluid hydrodynamics that de-scribes waves with arbitrary spatial scales in the so-called Padé approximation. A new closed set of Eqs.共11兲,共14兲, and
共15兲describing the nonlinear dynamics of flute waves with arbitrary spatial scales and incorporating the ion temperature perturbations has been derived. Particular attention has been paid to waves with spatial scales of the order of the ion Larmor radius, k⬜i⯝1. Furthermore, it was shown that
magnetic RTI is not stabilized in a low  plasma when = 1.
Our two-fluid magnetohydrodynamics which adequately describes the ion perturbations in low beta plasmas with ar-bitrary ratio of the characteristic spatial scales to the ion Larmor radius can give us a possibility for numerical simu-lation of nonlinear stage of the magnetic RTI.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 kρi γ/ Γ classic kinetic hydrod
FIG. 1. The normalized RTI growth rate as a function of k⬜iin plasmas
with homogeneous ion temperature共= 0兲 at= g/NvTi2 equal to 1/5. The
solid, dashed, and dashed-dotted lines correspond to the classic共long-scale approximation兲, kinetic, and model hydrodynamic 共Padé approximation兲 ap-proximations, respectively. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 kρi γ/ Γ classic kinetic hydrod
FIG. 2. The same as in Fig.1but for a plasma with inhomogeneous equi-librium ion temperature when= 0.5.
0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 kρi γ/ Γ klassic kinetic hydrod
FIG. 3. The normalized RTI growth rate as a function of k⬜iin plasmas
with homogeneous ion temperature共= 0兲 at= 1/2. The solid, dashed, and dashed-dotted lines describe the results obtained in the classic, kinetic, and model hydrodynamic approximations, respectively.
0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 kρi γ/ Γ klassic kinetic hydrod
FIG. 4. The same as in Fig.3but in a plasma with inhomogeneous equilib-rium ion temperature when= 0.5.
ACKNOWLEDGMENTS
This research was partially supported by the Russian Fund for Basic Research 共Grant Nos. 10-05-00376 and 11-05-00920兲 and the Programs of the Russian Academy of Sciences Nos. 4 and 7, as well as by the Deutsche Forschungsgemeinschaft 共Bonn兲 through Project No. SH21/3-1 of the Research Unit 1048.
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