Proceedings of ITC/ISHW2007

Strong Shear Formation by Poloidal Chain of Magnetic Islands
Vasyl .I. Maslov1, Franco Porcelli2
1

NSC Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine
2

Politecnico di Torino, Turin, Italy(Received day month year / Accepted day month year should be centered 10-point type)We estimate the electron angular velocity shear ∂ r ωθo which can be formed due to plasma heating near the loworder rational surface with poloidal chain of magnetic islands (MIs). We suppose that the electrons are heatedsufficiently that they propagate without collisions around torus. Then the plasma electrons start to miss the MIs
during their radial shift. This provides ion volume charge in some region of MIs. Then the essential shear is formed.
The time of shear formation is small. The conditions on MI width are derived, at which it leads to strong shear. It isshown that even narrow MIs can lead to shear. Shear can damp instabilities with growth rate smaller than ion
cyclotron frequency.
The spatial structures of some convective cells have been described. We derive inverse dependences of radialwidth of excited vortices on ∂ r ωθo and on ∂ r n 0e . Amplitude of the electron radial oscillations is smaller forlarger ∂ r ωθo and ∂ r n 0e . These dependences promote abrupt plasma density profile and internal transport barrier.
Keywords: internal transport barrier, shear, magnetic island, vortex, rational surface, radial electrical field

1. Introduction
Now ITB formation is widely investigated (see [1-3]).
Earlier the effect is considered, that shear of the electron
angle velocity ∂ r ωθo damps anomalous transport,
separating the coherent ordered motion owing the relative
movement of layers. ITB is formed for under-thresholdshear. We consider one more effect of anomalous transportr
suppression in plasma, located in crossed magnetic H 0rand electrical E 0 r fields. Namely, the shape of vortices is
described. The exact nonlinear eq., connecting the electron
vorticity and density, is derived in vector view without any
approximations. We derive inverse dependences of radialwidth of some excited vortices on the shear ∂ r ωθo incrossed fields and on degree of steepness of the plasma
density profile ∂ r n 0e . These dependences promote abrupt
plasma density profile and ITB formation. The amplitudeof the vortex saturation is inversely proportional to theshear. It also promotes ITB formation, suppressing thetransport especially in the case of small magnetic shear. It
is determined by that small magnetic shear leads to large
spatial interval ∆ between rational surfaces [4]. If radial
correlation length of excited perturbations becomes lessthan ∆ the radial transport could be suppressed. The
author’s e-mail: vmaslov@kipt.kharkov.ua

convective-diffusion equation, describing transport ofplasma particles in the field of lattice of overlapped
vortices, is derived.
Formation [5, 6] and role (see [4, 7, 8]) of MIs in
nuclear fusion plasma are investigated now intensively. In
particular, their effect on ITB formation is very important.
It is shown that due to sufficient plasma electron heatingnear the rational surface with the poloidal chain of MIs,
that they become to propagate approximately without
collisions around torus, electrons start to miss MIs and inconsent with [7] the essential shear is formed. The time ofshear formation is derived and shown to be small.
The conditions on MI width are derived, at which itleads to essential shear formation. The condition on MI
width is known, at which it does not lead to anomaloustransport on trapped particles. For that the island should benarrow. It is shown that even narrow island can lead toessential shear. One can note that several poloidal chain ofislands are better for ITB formation as in [7].
It is shown that this shear can damp instabilities withgrowth rate γ smaller than ion cyclotron frequency
γ < ωci .
There are some effects, connected with shear. At
strong anomalous transport, when it is determined by astreamer, formed by single wide vortex, which one

Proceedings of ITC/ISHW2007
overlaps all inhomogeneous area, the shear mince vortices.
2π l θ , i.e. δrv (l θ 2π)∂ r ωθ0 r =rv ≤ γ . Here l θ is the
It leads to intermediate ITB formation. But in this caseazimuth wave number of excited vortices. From thissome anomalous transport remains after ITB formation [1].
expression we derive
At not strong anomalous transport, determined by lattice ofvortices, the shear can spatially separate vortices. In thiseφo − δpe 0 n e (rv ) = (γπ l θ )2 rv me ωce 2 ∂ r ωθo r = rv (5)case the strong ITB is formed.
The amplitude of the vortex saturation is inversely
proportional to ∂ r ωθ0 . The decrease of level of
2. Spatial structure of vortical convective cells
Let us describe some chain on azimuth θ of thefluctuations at ITB formation has been observed in [3]. Itrr
promotes ITB formation.
vortices in the plasma, located in crossed H and E0

r0

fields. Neglecting nonstationary and nonlinear on electric
potential φ of vortices members, from electron motion eq.
one can obtain at small δr ≡ r − rv deviations r fromradial position of vortices rv , eq., describing oscillatory
dynamics of electrons in the field of the perturbation(δr )2 + 4(eφ − δp e n e (rv )) rv m e ωce (rv )∂ r ωθo r = r = C . (1)ωθo ≡ Vθo (r ) / r
v

[

]

[

]

r
r r
r r
Vθ0 = − e e z , E ro m e ωce − e z , ∇p 0e n e m e ωce ,δp e is the electron density perturbation. (1) describes theradialelectronoscillations
δr ≡ r − rv
throughdependences of φ(θ, r ) and electron density perturbations
δn e (θ, r ) on θ and r . Let us connect eφ − δp e n e (rv )with characteristic of electron vortical movementrr
α ≡ ez rotVe . From electron eq. of motion we derive
α Vthi ρ ci = e∆ϕ Te − ∆n e n e(2)
For vortex with dimension ρci we have α < ωci andeϕ Te − δn e n e ≈ α ωci . One can see that the amplitudeof the radial oscillations of the electrons for the given
amplitudes of the electrical potential and density
perturbation is smaller for larger shear ∂ r ωθo . The latter
helps the ITB formation.
In crossed fields the vortices can be with phasevelocity Vph ≈ Vθ0 , as well as slow vortices Vph << Vθ0 ,for example of Rossby kind. Let us derive the spatial
structure of slow vortex. For this purpose one can derive
similar to [9] without any approaches general nonlinear
vectorial equation, describing vortical electron dynamicsr r⎛ α − ωce ⎞ 1 r r r r⎟⎟ =⎜(α − ωce )∇ Vdt ⎜(3)
⎝ ne ⎠ nerr
r r r
α = ∇ × V , d t ≡ ∂ t + V∇ . From it one can obtain eq.,
describing the vortex of the small amplitude. Hence wederive for perturbation of electron trajectory in the field of
the vortex1
δr (θ, r ) = −δn e (θ, r ) .
(4)
ωceo ∂ r (n oe ωce )From here one can see that at the same amplitude ofplasma density perturbation δn e the vortex is narrower inradial direction for larger ∂ r (n oe ωce ) , i.e. for more abrupt
plasma density profile. The latter helps ITB formation.
The vortex is excited up to the amplitude, at which thelayers, trapped by it, during excitation time γ −1 are shifted
relative to each other due to shear on the angle not larger

(

[

]

( )

author’s e-mail: vmaslov@kipt.kharkov.ua)

3. Convective – diffusion equation
Let us consider the finite amplitudes, when frequencyΩ r of the electron oscillations, forming the vortex, in its
field becomes larger than the growth rate γ of theexcitation Ω r > γ . Then in vicinity of the vortex borders
the jumps on electron density profile ne(r) are formed.
Hence, on these ne jumps new cells with the greatest
growth rates are excited. It results in ordering of
convective cells. Therefore at achievement of the large
amplitudes the instability is developed for ordering of cells
similar investigated in [10]. Inside borders of a vortex
ordered convective movement of the electrons occurs.
However, they are influenced by environmental vortex
fields and fluctuations. Also it is important that amplitudes
of vortexes are not stationary. Instead of average noe(t,r),which does not take into account correlations, we use four
densities of the electrons nke(t,r) average on small-scale
oscillations: n1e(t,r) (n2e(t,r)) is the average density of theelectrons, located in region 1 in depth of a cell on r>rv (inregion 2 in depth of a cell on r<rv); n3e(t,r) (n4e(t,r)) is theaverage density of the electrons, placed in region 3 nearborder of a cell on r>rv (in region 4 near border of a cell onr<rv). The importance of use of different nke(t,r) is also
determined by that angular speeds of electron rotation
inside a cell are different in dependence on distance fromits axis. Also in central area of the convective cell thefollowing processes are still realized: plateau formation onn e (r ) due to difference of angular speeds of electron
rotations; due to jump formation on n e (r ) at the certain
moments of time in the regions 1 and 2 there is anaccelerated diffusion and an exchange by electrons
between regions 1 and 3 (factor α ), and also between
regions 2 and 4. α is the factor of mixing also due toinfluence of fluctuations, growth of amplitudes, differencesof characteristic times of the electrons.
But at ordering the adjacent cells form integrated
border. The particle in space between individual cell borderand integrated border move in radial direction from cell tocell for the distance min{ l cor , δrv τ cor Ω r π }. l cor , τ cor
are the correlation length and time of vortical convective
cell turbulence.
From the above we have approximately
n1 (t + τ, r ) = (1 − α )n 2 (t, r ) + αβn 3 (t, r )n 2 (t + τ, r ) = (1 − α )n1 (t, r ) + αβn 4 (t, r )(6)

Proceedings of ITC/ISHW2007
n 3 (t + τ, r ) = αn 1 (t, r ) + β(1 − α )n 3 (t, r − δrv ) +[7] is considered near this surface. Vthe is the electron+ 0.5(1 − β )[n 3 + n 4 ]

n 4 (t + τ, r ) = αn 2 (t, r ) + β(1 − α )n 4 (t, r + δrv ) ++ 0.5(1 − β )[n 3 + n 4 ]β is the factor of the convective exchange of cells byparticles. The value of β is determined by ratio of the areawith convective electron dynamics, located between
individual cell borders and integrated borders to all area,
located between individual cell borders and integrated
borders of adjacent cells. From these equations, enteringn = (n 3 + n 4 ) 2 , δn = n 3 − n 4 , N = (n1 + n 2 ) 2 ,δN = n1 − n 2 , we derive()τ∂ t n = α N − β n − (β 2 )(1 − α )δrv ∂ r δnτ∂ t δn + [1 − β(1 − α)]δn = αδN − 2β(1 − α)δrv ∂ r n

τ∂ t N = α(β n − N ) , τ∂ t δN + (2 − α)δN = αβδn(7)One can see that introduction n is similar to averagenoe(t,r) but with taking into account correlations. Fromthese equations we have similar to [10] the following
convective – diffusion equation
τ 2 ∂ 2t δn + τ∂ t [(1 − β(1 − α ))δn − αδN ] =
(8)
β
⎡
⎤
= −2β(1 − α )δrv ∂ r ⎢α(N − β n ) − (1 − α )δrv ∂ r δn ⎥2⎣⎦
As β is proportional to (δrv − ∆ ) δrv then at δrv < ∆we have β = 0 and there is no convective radial transportbecause convective cell exchange by particles disappears.

4. Shear formation by magnetic islands
Let us consider the shear formation near the poloidal
chain of narrow magnetic islands. The electrical field E r 0
is approximately equal to zero on the axis of a plasma
column. It receives maximum value inside a plasma
column at r = rm . We suppose, that on a some interval
0 < r < rm (r0-δR<r<r0+∆r+δR around chain of MIs,
located in r=r0+∆r/2 with radial width ∆r, ro is the lower
border of island) on radius r the electrical field E r 0 in thecase of shear absence is proportional to E r 0 ∝ r . Then on
this interval it is possible to present
E r 0 = −2πeN 0 r , 0 < r < rm , N 0 ≡ n e − n i .
(9)It means, that there is no shear ωθ0 ≠ ωθ0 (r ) . One can take
into account effect of oscillations on electron transport byeffective collision frequency νef in diffusion coefficientD ⊥ ∝ (νe + νef ) . ν e is the electron collision frequency
On the plasma cross-section 0 < r < R several chainsof magnetic islands can exist [7]. But we for the simple
case consider influence of one poloidal chain of islands on
shear formation. We consider low order rational magnetic
surface, because important property of this surface isappeared, when plasma is heated sufficiently that its
electrons perform several rotation around toroidal surface
during free pass time. The shear formation at
2 πrtor νe < Vthe
(10)author’s e-mail: vmaslov@kipt.kharkov.ua

thermal velocity, moving near this surface. According to [4,7] we consider local plasma heating near this surface. The
local heating leads to important effect: inequality (10)becomes to be satisfied more easier.
If condition (10) is satisfied the electrons starts to missthe island. The part of the electrons, located in the island,
escapes its, up to reaching E r to zero inside it, except for
the layer of widthδrsep≈ 2 πnrtor D ⊥ D ≈ ρcenear r=r0. D ⊥ , D(νe + νef ⊥ ) (νe + νef )are the transversal and longitudinal

diffusion coefficients, νef ⊥ and ν ef

are the transversal

and longitudinal effective collision frequencies. When partof the electrons escapes the magnetic island, in the regionof the island the plasma ion volume charge n i are notcompensated by electrons on δn . δn can be determinedfrom observed in experiment condition, that in the island
E r 0 ≈ 0 is approximately established.
Let us consider shear formation near chain more indetail. In the case (10) the electron transport through MIschanges from slow collisional to quick one. Electrons miss
MI. Though the radial size of MI is small the quick
electron transport leads to appearance of strong
uncompensated ion volume charge in MI and to the strong
shear ∂ r ωθo .
That in the region of the magnetic island theuncompensated ion volume charge appears it is necessarythat its width ∆r should be larger ∆r > ρ ce incomparison with the electron cyclotron radius.
Let us consider radial electron dynamics in smallneighborhood of the chain of islands. The electrons mover r
in radial direction in crossed E × H fields with velocity
2. When electron
V0 r = −(eE or + ∂ r p0e n e )(νef + νe ) m e ωcereaches with V0 r island from the side of small r, it
propagates collisionally through island in the case(hot ) . But in the case 2 πr ν << V (hot )2πrtor νe >> Vthe
tor e
the
electron without collision quickly, during the time
(hot ) get on the second boundary of the island.
2πrtor Vthe
After that electron again slow propagates with velocity
V0 r in the direction of large r.
(hot ) near the island the shear isThus at 2πrtor ν e < Vthe
formed. Using approximation of poloidal chain of narrow
MIs as a azimuth symmetrical narrow layer we have for theradial electric field distribution⎧ N a r , 0 ≤ r ≤ r0⎪
E r 0 ≈ −2πe ⎨ N a r02 r − δn r − r02 r , r0 ≤ r ≤ r0 + δrsep (11)⎪⎩0, r ≥ r0 + δrsep()

If the field is small E r 0 ≈ 0 at r = r0 + δrsep , we haveN a ≈ 2δnδrsep r0 . One can see that the density of

uncompensated plasma ion volume charge is relative large
in the island r0 >> ∆r if E r ≈ 0 at r = r0 + δrsep .

Proceedings of ITC/ISHW2007

One can use estimation N a ≈ 4n 0 (e∆ϕ Ti )(rdi L )2 . L
is the width of the region with essential E r , ∆ϕ is thepotential on L. One can conclude that the island can be
narrow for essential shear formation(e∆φ/Ti)(r2di/Lδrsep)<1 .
(12)
This inequality means that in real island the small ion
volume charge uncompensation δn << n 0 is sufficient foressential shear formation.
Let us calculate shear of the electric field andnormalize it on the electric field E r = −2πeN 0 r in shear(

absence S ≡ E r no TB
have) r ∂ (E r ) . If E

(

−1 2

r

r

)(

r r = r0 + δrsep ≈ 0

we)⎧⎪(N a N 0 ) r02 r 2 2 + r0 δrsep , r0 ≤ r ≤ r0 + δrsep(13)
S=⎨⎪⎩0, r ≥ r0 + δrsep
The shear is large for region of narrow magnetic islandsr0 >> ∆r > δrsep()

S ≈ (N a N 0 ) r0 δrsep , r0 ≤ r ≤ r0 + δrsep . S >> 1 (14)Really, in experiment [7] the decrease of correlation ofplasma density fluctuations were observed.
Let us consider the shear of ωθ0 = Vθ0 r
Vθ0 = (m e ωHe )−1 (− eE r 0 − ∂ r p0e n 0e ) ,(15)
as for ITB formation this shear ∂ r ωθ0 is more important.
We determine the angular velocity shear(Sω ≡ ωθ0 no TB) r∂ ω , ω
−1

r

(

2
θ0 no TB = ωpe

θ0)ωHe (N 0 2n 0 )Then we derive⎧⎪(N a N 0 ) r02 r 2 2 + r0 δrsep , r0 ≤ r ≤ r0 + δrsep
Sω = − ⎨⎪⎩0, r ≥ r0 + δrsep
Absolute value of the relative angular velocity shear is oforder of Sω = −(N a N 0 ) r0 δrsep . But absolute shear can

(

)(

)

be increased. In several experiments the strong localizationof the region with V0θ ≠ 0 has been observed. As inexperiments the radial width ∆rsh of area V0θ ≠ 0
localization ∆rsh = 1cm is observed, the shear (∂ r V0θ )apr
can be increased in existing nuclear fusion installations incomparison with smooth case (∂ r V0θ )smooth ≈ V0θ R
almostin100(∂ r V0θ )apr ≈ V0θ ∆rsh ≈ (∂ r V0θ )smooth R ∆rsh .
The shear is formed during()

2τTB ≈ (νef + νe )−1 ωce
2ω2pe (r0 ∆r )(16)

. rtor is the toroidal radius.

Hence the island should be wider
author’s e-mail: vmaslov@kipt.kharkov.ua

D⊥ D

(17)

) ω <<1. But for ITB(= (νei + νef ⊥ ) νei + νef2
ceformation the island should be narrow [11]∆r << R .
Let during free pass time the electron has time tomake q rotations around the torus. Then()

max{ρce , (nρce q ) (1 + νef ⊥ νei ) 1 + νef

νei } < ∆r . (18)

From obtained expressions and from condition [1-3]L∂ r ωθ0 > γ
(19)
using
∆ϕ ≈ 2πeN a L2(20)
one can show(∆ϕe Ti ) ρci2 Lδrsep > γ ωci (21)()

that shear can damp low-frequency instabilities withgrowth rate γ < ωci .

5. Conclusions
It is shown that the amplitude of the radial electron
oscillations is smaller or the vortex is narrower in radial
direction for larger shear ∂ r ωθo and for more abrupt
plasma density profile in nuclear fusion plasma. The latter
helps ITB formation. The amplitude of the vortex
saturation is inversely proportional to ∂ r ωθo . It also
promotes ITB formation. The convective-diffusion
equation for electron transport has been derived.
It is shown that the value of the shear can be large.
The shear is formed in consent with [7] due to plasma
heating near the low order rational surface with poloidal
chain of narrow magnetic islands. The plasma is heated up
to such electron temperature that the electrons become topropagate approximately without collisions around torus.
Small part of electrons leaves the island and the shear isformed. The time of shear formation is small. The
conditions on island width is derived, at which it leads toshear formation. Even narrow island can lead to shear
formation. The condition on island width is known, at
which it does not lead to anomalous transport. This shear
can damp instabilities with growth rate smaller than ion
cyclotron frequency γ < ωci .

times

the short time for not very narrow MI.
Narrow islands, though provide a fast electrontransport through island dimension ∆r , strong suppress
transport in broad their neighborhood. Really inexperiment [4] ∆r << R .
In the island, placed on rational surface with numbers
n, m the uncompensated ion volume charge has appeared at(∆r )2 D ⊥ > (2πnrtor )2 D

max{ρ ce , 2πnrtor D ⊥ D } < ∆r .

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