Table Of ContentComment on “The linear instability of magnetic Taylor-Couette flow with Hall effect”
M. Rheinhardt
Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482, Potsdam, Germany, [email protected]
U. Geppert
Max-Planck-Institut fu¨r Extraterrestrische Physik, Gießenbachstraße, 85748 Garching, Germany, [email protected]
(February 2, 2008)
B0
Inthepaperwecommenton(Ru¨diger&Shalybkov,Phys.
Rev. E.69,016303(2004)(RS)),theinstabilityoftheTaylor– ∂B′ ′ ′ ′
Couetteflowinteractingwithahomogeneousbackgroundfield = curl(u0×B +u ×B0)+η∆B
∂t
5
0 subject to Hall effect is studied. We correct a falsely gener- −βcurl(curlB′×B0) (1)
alizing interpretation of results presented there which could
0
be taken to disprove the existence of the Hall–drift induced
2
magnetic instability described in Rheinhardt and Geppert, ∂u′ ′ ′
n Phys. Rev. Lett. 88, 101103. It is shown that in contrast to +(u∇)u0+(u0∇)u =
a ∂t (2)
J wtohaentaisblseugsguecshteadnbiynsRtaSb,inliotyadwdiitthioananlosnh–eparotfleonwtiaislnmeacegsnseatriyc −∇p′/ρ+ν∆u′+curlB′×B0/(µ0ρ)
7
background field, whereas for a curl–free one it is. In the
1 where we used the symbols introduced in RS. Standard
latter case, the instabilities found in RS in situations where
arguments yield the following evolution equation for the
neithera hydrodynamicnora magneto–rotational instability
1 total energy E of the perturbations:
exists are demonstrated to be most likely magnetic instead
v
46 oafremcalagrnifieteodh.ydrodynamic. Further, some minor inaccuracies dE = 1 d B′2/µ0dV + ρu′2dV =
3 dt 2dt(cid:18)ZV′ ZV (cid:19)
1 ′ 2 2 ′ 2
− (curlB ) /(µ σ)+ρν(curlu) dV
0 I. Z 0
5 V (cid:0) (cid:1) (3)
h/0 (fuTrthheermoaninrpefuerrpreodsetoofatshiRsSC)oims mtoenptreovnentthea pinacpoerrre[c1t] +ZVcurlB′·(u0×B′)dV/µ0−ρZVcurlu′·(u0×u′)dV
p ′
conclusion with respect to our work [2,3] which could be with V being the infinite space minus any volume with
-
o drawnfromanincorrectstatementin the discussionsec- infinite conductivity and V the volume of the container.
r tionofRS. There,atthe endofthe third paragraph,the Ofcourse,solutionswithgrowingtotalenergyareimpos-
st authorsconcludefromtheinvarianceoftheirresultswith sible,aslongasu0 =0. Moregenerally,evenifwewould
a respectto simultaneoussign inversionsof shearandHall admit a rigid body motion for u0, growing solutions do
:
v termthat noinstabilities arepossible withoutshear. Al- not exist.
Xi though this conclusion being looked at out of context is The situation changes qualitatively, if B0 is no lo′nger
not comprehensible, it is nevertheless true for the spe- curl–free: The additionalterm−βcurl(curlB0×B )oc-
r
a cial case of a homogeneous (more generally: curl–free) curringinthelinearizedinductionequationresultsinthe
background field B0, but not in general. As the scheme additional energy term
(40)ofRSisvalidfornonpotential(axisymmetric)fields,
too, and the quoted conclusion is drawn completely on −β curlB′·(curlB0×B′)dV/µ0 , (4)
its basis, the reader will be tempted to generalize it. He ZV
or she could then come to the end that the results on
′
a Hall instability without shear reported in [2,3] have to which quite analogously to the term V curlB ·(u0 ×
′
be put in question. (Note, that the term ‘shear’ is used B )dV/µ0 is potentially capable of dRelivering energy.
throughout RS to refer to the macroscopic motion of a Hence, the argument concerning the necessity of shear
forthe occurrenceofaninstability inthe modelofRSin
fluid.) Here, we will show that conclusions on necessary
conditionsforthe instabilitiesinquestioncanreliablybe fact supports our findings in [2,3],when the term ‘shear’
is no longer used to refer to macroscopic motions only,
drawn on the basis of energy considerations. They sup-
port the possibility of a Hall instability without shear. but is extended to the microscopic motions of the carri-
The linearized induction and Navier–Stokes equations ers creating the current curlB0/µ0. If the latter should
describing the evolution of small perturbations B′ and be capable of replacing the shear velocity u0, it must
u′ of the background field B0 and the shear flow (here: not be interpretable as a rigid body motion. Therefore,
differentialrotation)u0,respectively,readforacurl–free a background field exhibiting a sufficiently curved pro-
file, is a necessary condition for the occurrence of the
1
instability we reported on, as we stressed in all our pa- II.
pers on this issue. (A suitable profile for a plane slab
−1 ≤ z ≤ 1 with its normal in z–direction is, for in- Anotherremark,connectedwiththeaboveseemstobe
stance, B0 =Bˆ0(1−z2)ex as used in [2].) appropriate. IntheResultsandDiscussionsectionsofRS
As the energy term (4) contains only magnetic fields, the impression is given, that the reported instability is
the possibility exists, that even in the absence of any primarily one of the flow. In our opinion, there are good
macroscopic motions (u′ = u0 = 0), say in a crys- reasons, and moreover even evidences provided by RS
tallized neutron star crust, nevertheless an instability itselfto interpretthis instability as a primarilymagnetic
may occur. For plane geometry we demonstrated that (and not MHD) one for conditions, in which the flow
this possibility is real both for a uniform [2] and strat- wouldbestableotherwise(thatis,for‘positiveshear’and
ified slab [4]. In the cylindrical geometry considered in for ‘negative shear’ with subcritical Reynolds numbers,
RS the instability occurs as well. Figure 1 shows nor- i.e., for the part of the Re–Ha plane beneath the dashed
malized growth rates and wave numbers of the most line in Fig. 6 of RS) .
rapidly growing axisymmetric modes vs. the normalized Considering the induction equation including differen-
strengthofthe backgroundfield. Itsprofilewasspecified tial rotation and Hall effect with the velocity perturba-
as B0(R)∝(R−Rin)2(R−Rout)2ez. tions suppressed (i.e., the kinematic case), one has for-
mally the same equation as that which describes mean–
field dynamos due to differential rotation and the so–
calledω×j–effect,see[6,7]. Fromthesecalculationsand
from qualitative considerations, too, it follows, that the
sign relation between Hartmann number Ha and dΩ/dR
reported in RS, (see, e.g., Sect. III) is just the one nec-
essaryfor dynamo action, that is, a magnetic instability.
(Note, that Cowling’s theorem does not apply.) In Sect.
III B of RS a marginal curve in the Re−Ha plane is
given for the kinematic case u′ = 0. In that part of the
plane where no hydrodynamic or magneto–rotationalin-
stability (MRI) exists, it practically coincides with the
marginalcurveof the full system’s instability. Thus, one
may suppose, that the velocity perturbations are simply
“enslaved”bythemagneticonesincases,inwhichnoin-
stabilitiesoccurwithoutHalleffect. Sinceanenslavedu′
FIG. 1. Growth rates (thick) and wave numbers (thin) of givesrisetoadditionaldissipation,thefullsystemshould
the most unstable axisymmetric magnetic field modes (kine- exhibit smaller growth rates compared with those of the
matic case: u′ = 0) as a function of the background field kinematiccase. AhintonthisisprovidedbyFig.6ofRS
strength. Length, time and magnetic field are normalized by showing that for 1.Ha.7 in the full system a slightly
Rout,Ro2ut/ηandη/β,respectively. Bmaxrepresentsthemax- stronger differential rotation is needed for marginal sta-
imum of the background field profile. Rin/Rout = ηˆ = 0.5. bility than in the kinematic case. To judge the nature of
Boundary conditions as defined in RS; vacuum – vacuum: the instability the signs of those integrals in (3) result-
solid,innerperfectconductor–outervacuum: dashed,perfect ingfromthepotentiallyenergy–deliveringtermswiththe
conductor– perfect conductor: dot–dashed. The eigenmodes calculated eigensolutions inserted could be inspected.
are non–oscillatory in the first two cases, but oscillatory in When assuming the primarily magnetic character of
the third. Interestingly, for the vacuum – vacuum boundary
the instability, its suppression with growing (absolute
condition the instability emerges roughly at Bmax & 3 as in valueofthe)Hartmannnumber(cf. Figs. 2–4,7and8of
theplane model.
RS)canbe explainedbythe competitionoftwocounter-
acting effects: On the one hand, growing |Ha| means
The second additional energy term due to an non–
growing dominance of the energy–delivering advection
potential backgroundfield, ′
termcurl(u0×B )in(1)(byvirtueoftheadmittedlyen-
ergetically neutral, but ‘catalyzing’ Hall–term) over the
′ ′
u ·(curlB0×B )dV/µ0, (5) dissipation term. But on the other hand, it means also
Z
V growingefficiencyoftheLorentzforcein(2)whichcauses
is capable of delivering or consuming energy, too. Cor- growing dissipation due to the enslaved velocity pertur-
responding instabilities exist and are (for u0 = 0) usu- bations which drain their energy by virtue of the second
ally referred to as unstable Alfv´en modes (see, e.g., [5]). advectionterm curl(u′×B0) in (1) out of the magnetic
TheirnatureisobviouslyMHD,as(5)vanishesforu′=0 perturbations. Hence,theoccurrenceofaminimumwith
and/or B′=0 . respect to Ha is quite natural .
2
III. [1] G. Ru¨diger and D. Shalybkov, Phys. Rev. E 69, 016303
(2004).
Within the discussion of Fig. 5 of RS (Sect. III A), [2] M. Rheinhardt and U. Geppert, Phys. Rev. Lett. 88,
101103 (2002).
it is falsely stated, that the existence of the current–free
marginal solution B′ = B′ = 0, B′ ∝ R−1, u′=0 re- [3] U.GeppertandM.Rheinhardt,Astron.Astrophys.392,
R z φ 1015 (2002).
quiresbothboundaryconditionstobethoseoftheperfect
[4] M. Rheinhardt, D. Konenkov, and U. Geppert, Astron.
conductor. Infact,thereisnoreasonwhysuchacurrent–
Astrophys. 420, 631 (2004).
free(orvacuum)solutioncouldnotcontinuefromthein-
[5] R. Cross, An Introduction to Alfv´en Waves (Adam
nerboundaryR=Rin ontoinfinitywhatmeansnothing Hilger, Bristol and Philadelphia, 1988).
morethansatisfyingthecorrespondingvacuumcondition [6] K.-H.R¨adler,Monatsber. Dtsch.Akad.Wiss. Berlin 11,
at the outer rim R=Rout. The necessary and sufficient 194 (1969).
conditionfortheexistenceofthisvacuumsolutionevery-
[7] H.K.MoffattandM.R.E.Proctor,Geophys.Astrophys.
where outside the surface R = Rin is the existence of a Fluid Dyn.21, 265 (1982).
net current in z–direction enclosed by this surface. Be- [8] G.Helmis,Monatsber.Dtsch.Akad.Wiss.Berlin10,280
cause an outer electromotive force is missing, a perfect (1968).
conductor in the interior of the inner cylinder is needed. [9] G. Helmis, Beitr. Plasmaphys. 11, 417 (1971).
Then, e.g., anarbitrarysurface currentcanflow without [10] P. Mininni, D. Gomez, and S. Mahajan, Astrophys.
losses and therefore endlessly. J. Lett. 567, L81 (2002).
However, the dashed and the dot–dashed curves in
Fig. 5 which correspond to the perfectly conducting in-
ner cylinder are anyway incomprehensible as they both
should coincide with the curve k = 0: This Figure
is intended to show the wavenumber belonging to the
marginaleigensolutionwiththeminimumReynoldsnum-
ber for a given Hartmann number. In case the inner
boundary condition is ”perfect conductor”, always the
vacuumsolutionexists,whichismarginalandallowsalso
Re=0 (or, equivalently, u0 =0), i.e., is associated with
the minimum possible Re. This solution is characterized
by k = 0 and the mentioned curves should show that,
except, therewereother marginalsolutions withRe=0,
but k 6= 0. But such solutions do not exist, because for
Re = 0 and any Ha there is no (potentially) energy–
delivering term in the induction equation (see (3)) and
the only possibility for a marginal (i.e., non-decaying)
solution is the vacuum one. Thus it appears, that the
vacuumsolutionwasforunknownreasonsexcludedfrom
the analysiswhichledto Fig. 5andcanthereforenotbe
referred to to explain it.
However again, even then Fig. 5 is in disagreement
with Fig. 2 of RS. The dashed line in the former should
correspond to the solid line in the latter figure which
shows no special behavior at Ha= −2 where k becomes
zero in Fig. 5.
IV.
Considerations of the effect of turbulence on the Hall
coefficient do exist [8,9] (see the last paragraph of RS).
Perhapsitisappropriatetomentionherearecentrevival
ofmean–fieldHall–electrodynamicsin[10]althoughthere
only the effect of the Hall–drift onto the α–coefficient is
considered.
3
