J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)35–39
000–000

Relaxation Oscillations and Transport Barrier Dynamics
in Tokamak Edge Plasmas
BENKADDA Sadruddin, BEYER Peter, FUHR-CHAUDIER Guillaume,
GARBET Xavier1, GHENDRIH Philippe1 and SARAZIN Yanick1
Equipe Dynamique des Systèmes Complexes, Laboratoire PIIMCNRS – Université de Provence, Marseille, France
1
Association Euratom – CEA, CEA/DSM/DRFC CEA Cadarache, France(Received: 9 December 2003 / Accepted: 25 February 2004)

Abstract
Oscillations of turbulent transport of particles and energy in magnetically confined plasmas can be easily observedin simulations of a variety of turbulence models. These oscillations typically involve a mechanism of energy exchange
between fluctuations and a poloidal shear flow. This kind of “predator-prey” mechanism is found to be not relevant
for transport barrier relaxations. In RBM simulations of resistive ballooning turbulence with transport barrier, relaxation
oscillations of the latter are observed even in the case of frozen poloidal shear flow. These relaxations are due to atransitory growth of a mode localized at the barrier center. A one-dimensional model for the evolution of such a modein the presence of a shear flow describes a transitory growth of an initial perturbation. Oscillations in the case of afinite steady-state shear flow are possible due to the coupling of the mode to the dynamics of the pressure profile.
Keywords:
relaxation oscillation, transport barrier, ELMs, plasma turbulence

1. Introduction

exchange of energy between the fluctuations and the poloidal
flow [2,3,4]. In this first part of this article, we will show thatsuch oscillations can be observed in simulations of resistive
ballooning mode (RBM) turbulence at the collisional edge ofa tokamak plasma. Furthermore, it is possible to construct
one-dimensional or zero-dimensional models for this kind of
dynamics.
However, as we will show in the second part of thisarticle, relaxation oscillations of transport barriers can beobserved even with frozen poloidal flow. In this case, a steadystate shear flow is imposed externally to generate the barrier,but the flow does not participate in the dynamics. The
relaxations are due to a transitory growth of a mode localized
at the center of the barrier. Such a transitory growth is foundto be consistent with shear flow stabilization.

Relaxation oscillations in fusion plasmas have a strong
impact on energy and particle confinement. The mostpromising operational regime of future reactors ischaracterized by the existence of a transport barrier at theplasma edge. This barrier is not stable but relaxes quasiperiodically. During such fast relaxation events, turbulent
transport through the barrier increases strongly and thepressure inside the barrier drops. Thereafter, the barrier builds
up again on a slow, collisional time scale. These relaxation
oscillations are linked to so called edge localized modes(ELMs) which are believed to be magneto-hydrodynamical(MHD) modes driven by the edge pressure gradient(ballooning) and/or the edge current (peeling) [1].
The basic physical mechanism underlying these kind ofrelaxation oscillations is not yet understood. In particular,
there is no explanation why the plasma, in stead of remaining
in a statistically stationary state close to pressure gradient oredge current stability limits, crosses these limits quasi
periodically.
A possible mechanism for oscillations of turbulent flux
in confined plasmas is based on the dynamics of poloidal
flows that are generated self-consistently by turbulent
fluctuations. These flows stabilize the turbulence. Therefore,
oscillations are in principle possible due to a periodic

2. Model for resistive ballooning
turbulence
Resistive ballooning mode (RBM) turbulence at thecollisional edge of a tokamak plasma is modeled by reduced
resistive MHD equations for the normalized electrostatic
potential φ and pressure p [5,6],∂t∇⊥2 φ + {φ, ∇⊥2 φ} = –∇||2 φ – Gp + ν∇⊥4 φ ,(1)©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: benkadda@up.univ-mrs.fr
1
35

Benkadda S. et al., Relaxation Oscillations and Transport Barrier Dynamics in Tokamak Edge Plasmas∂t p + {φ, p} = δc Gφ + χ||∇||2 p + χ⊥∇⊥2 p + S .

(2)Equation (1) corresponds to the charge balance in the drift
approximation involving the divergences of the polarization
current, the parallel current, and the diamagnetic current, andeffects due to a viscosity ν, respectively. Equation (2)represents the energy balance, where χ|| and χ⊥ are effective
parallel and perpendicular collisional heat diffusivities andS(r) is an energy source. The curvature operator G arises fromthe compressibility of diamagnetic current and E × B drift.
In this MHD model, the diamagnetic velocity is neglectedwith respect to the E × B velocity, and the parallel current isevaluated using a simplified electrostatic Ohm’s law, η||0 j|| =–∇||φ (in dimensional units), where η||0 is a reference value ofthe parallel resistivity.
The system (1, 2) has been normalized using the resistive
interchange time τint and the resistive ballooning length ξ bal
with τ 2int = R0 Lp /(2c2S0) and ξ2bal = mi n0 η||0 L2s /(τint B02 ). Here,R0 , Lp , and Ls are characteristic values of the major radius,
the pressure gradient length, and the magnetic shear length,
respectively, cS0 = [p0 /(mi n0)]1/2 is the sound speed calculatedwith reference values of the pressure p0 and the density n0 ,mi is the ion mass, and δc = (5/3)2Lp /R0 . Time is normalized
to τint and the perpendicular and parallel length scales are
given by ξ bal and Ls , respectively. Electrostatic potential andpressure are normalized to B 0 ξ2bal / τint and ξ bal p 0 /L p ,respectively.
The geometry of magnetic flux surfaces corresponds toa set of circular concentric tori, (r, θ, ϕ) being the labels ofminor radius, poloidal and toroidal angles. The Poissonbracket is {φ, ·} = r–1(∂r φ∂θ – ∂θ φ∂r), the curvature operator
is G = sinθ∂r + cosθr–1∂θ and the gradient along field lines is∇|| = ∂ϕ + q–1∂θ . RBM turbulence is expected to be relevantin collisional edge plasmas. Assuming a monotonicallyincreasing safety factor q(r), the domain chosen for the studyof RBM turbulence typically covers a region between q = 2
and q = 3 at the plasma edge.

Fig. 2Time evolution of two fluctuating modes (a1 , a–1) andthe poloidal flow a0 in an oscillating state.

a· 0 = – γ 0 a0 + γ 0 a1 a1–
a· 1 = γ 1 a1 – a0 a1– – ν1 a 1–2 a1 – ν2 a13
a· – = –γ – a – + a a – ν a2 a –
1

1

1

0 1

1 1 1

(3)

Here, a1 and a¯1 are the amplitudes of the most important➞
spatial structures of the fluctuations ũ , and a0 is the amplitude– γ , γ ¯,of the relevant radial profile of the poloidal flow u.
11
γ 0 > 0 are linear growth rates, ν1 , ν2 are coefficients, andis a small parameter. As can be seen in Fig. 2, the system (3)reproduces qualitatively the oscillations observed in the fullsystem (1, 2). Note that if one freezes the dynamics of theamplitude a0 , i.e. a·0 = 0, the system always converges to astable fixed point. In other words, the dynamics of a0 is
crucial for the oscillating solution of (3) even if theoscillations of the amplitude a0 itself are not very pronounced.

In full three-dimensional numerical simulations of thesystem (1, 2), oscillating states can be found where kinetic
energy is periodically exchanged between fluctuating➞–
components ũ of the E × B velocity and the poloidal flow u.
The latter are defined by➞Time evolution of the kinetic energy of fluctuations andpoloidal flow in a 3D simulation of RBM turbulence,
initialized with an unstable pressure profile and small
fluctuations.

projection then provides a system of ordinary differential
equations for the amplitudes of these structures. Such a
reduced model is given by [2],3. Oscillations with poloidal flow
dynamics➞➞➞
u–θ ê θ = 〈u 〉 , ũ = u – u–θ ê θ , with

Fig. 1➞u = u E×B

and

〈⋅〉 =

1
4π

2π

2

∫ ∫
0

0

2π

4. Dynamics of transport barriersdθ dϕ ,In the RBM turbulence model, a transport barrier can begenerated by externally imposing a locally sheared E × B
flow. This flow locally reduces turbulent transport [9,10,11,
12,13,14]. As a consequence of the total energy flux
conservation, the pressure gradient steepens in the shear layer.
This mechanism is similar to the one observed experimentallywhen generating a transport barrier by edge biasing

where êθ is the unit vector in the poloidal direction. A typicalexample is shown in Fig. 1. By performing a proper
orthogonal decomposition [7,8] of the spatio-temporal fields
of velocity and pressure, it is possible to determine the spatial
structures relevant for the dynamics. A subsequent Galerkin236

Benkadda S. et al., Relaxation Oscillations and Transport Barrier Dynamics in Tokamak Edge Plasmas(a)[15,16,17]. The external drive of the shear flow is included
by modifying the equation for the poloidal flow profile, ū,that follows from the magnetic flux surface average of eq.
(1),∂t u θ = −1
1∂ r 2 u˜θ u˜r + ν∂r ∂r r uθ − µ (uθ − U ) , (4)r2 rrwhere the first terms on the right hand side correspond to thedivergences of the Reynolds stress and the viscosity stress,
respectively, and the last term has been added artificially to
account for the friction with an external flux U. The latter ischosen to be strongly sheared at the reference position r0 =r q=2.5 . In case of a large friction coefficient µ → ∞, thepoloidal flow u–θ tends to the external flow U. For finite µ,
the Reynolds stress drive of the poloidal flow [18,19,20] is
important.
In typical simulations of RBM turbulence with atransport barrier, the latter is not steady state but relaxes
intermittently. This can be seen from Fig. 3a where time
evolutions of the edge energy confinement time τEedge (t), theturbulent flux at the barrier center 〈ur p〉(r0 , t) and the poloidal
flow shear at the barrier position ∂r u–θ (r0 , t) are presented ina typical simulation. The edge confinement time is defined as
the ratio of the energy confined in the volume considered andthe total energy flux across a magnetic surface. It represents
a measure for the “strength” of the barrier. As can be seenfrom Fig. 3a, the evolution of the confinement time ischaracterized by phases of a slow increase quasi periodicallyinterrupted by rapid crashes. The latter correspond torelaxations of the barrier and are associated to large peaks ofthe turbulent flux at the barrier center as well as fluctuationsof the velocity shear at the barrier position.
These relaxation oscillations persist even if the poloidal
flow profile is frozen. Fig. 3b shows the corresponding resultsfrom a simulation with the same parameters as in Fig. 3a
except that the friction coefficient is set to µ → ∞. In practice,
this is achieved by suppressing artificially the Reynolds stressterm in eq. (4). In this case, the velocity shear profile isconstant in time but intermittent flux peaks with relaxation of
the barrier do appear.

(b)

Fig. 3Time evolution of confinement time, turbulent flux atthe barrier center and poloidal velocity shear at thebarrier center. In case (a), the poloidal flow fluctuatesaround the imposed flow due to the Reynolds stress. In
case (b), the latter is suppressed artificially and thepoloidal flow is frozen.

• The strong velocity shear at the barrier center can inprinciple generate a Kelvin-Helmholtz instability. In
contrast to the resistive ballooning instability, this typeof instability is independent of the magnetic curvature.
However, if the magnetic curvature is put to zero
during our simulation, all turbulent fluctuations rapidly
die out.

5. Possible mechanisms for barrier
relaxations

6. Shear flow stabilization
Linearizing the pressure equation (2) for a perturbationpmn (r, t) eimθ –inϕ, supposing a mean poloidal flow with constantshear, u–θ = ωE (r – r 0) and replacing the coupling to thepotential equation (1) by an instability term characterized bya growth rate γ 0 , one obtains the linear evolution equation,

As the relaxation oscillations shown in Fig. 3b appear in
the case of a frozen poloidal shear flow, the mechanism ofenergy exchange between fluctuations and the poloidal flow
is not relevant for these relaxation oscillations. Other possible
mechanisms can either be excluded or seem to be not relevant:• As the resistive ballooning instability is a thresholdinstability, it could be possible that the growth of acentral Fourier component is triggered when the local
pressure gradient at the barrier position passes athreshold. However, looking at the time evolution ofthis local pressure gradient, it often saturates before
the barrier crash, and no clear sign of passing athreshold before the event can be observed.

∂t pmn + ikθωE xpmn = γ 0 pmn – ωt x2pmn + χ⊥∂x2pmn ,(5)with x = r – r0 , kθ = m/r0 , and ωt = χ|| /(r0 Ls)2, where Ls is theshear length. The parallel gradient has been evaluated for a
Fourier mode localized at the barrier center ∇|| p mn = ix/(r0 Ls) pmn . In the absence of magnetic shear, one can calculate
analytically the time evolution according to eq. (5) of an
initial perturbation infinitely localized in space and time. A
solution of the equation
3
37

Benkadda S. et al., Relaxation Oscillations and Transport Barrier Dynamics in Tokamak Edge Plasmas(a)Fig. 4

Time evolution of the mode (m, n) = (5, 2) at the centerof the barrier in a full 3D simulation started with aperturbation of this mode (full line) and analyticalsolution calculated from (7) (dashed line).

(b)∂t pmn + ikθωE xpmn = γ 0 pmn + χ⊥∂x2pmn + Ŝδ (x)δ (t) , (6)is given by

pmn =

⎛
ik ω x ⎞x2+ θ E t⎟exp ⎜ −
24πχ⊥t⎠
⎝ 4 χ⊥tSˆ

⎛
t3 ⎞× exp ⎜ γ 0 t − 3 ⎟ ,
3τD ⎠
⎝

(7)
Fig.5

where τD = [_41 χ⊥(kθ ωE)2]–1/3. Note that for ωE = γ 0 = 0, theusual solution of the diffusion equation is recovered. The
solution (7) describes an initial transient growth of theperturbation for t < τD before the cubic term in the last
exponential takes over the linear term, leading to astabilization. The characteristic time for the transient growthis large for small values of the perpendicular diffusivity χ⊥(close to the collisional value at the barrier center) and low
poloidal wave numbers kθ .
For a typical simulation such as the one shown in Fig. 3,
we have τD ~ 10 in normalized units for the central (m, n) =(5, 2) mode. A transient growth of this mode is found whenstarting a simulation with the corresponding pressure andvelocity profiles, a perturbation of the (m, n) = (5, 2)
component and small noise on all other components (Fig. 4).
The peak of the mode amplitude is well reproduced by theexpression (7). The dashed curve in Fig. 4 shows the graphof pmn = p̂ exp[γ 0 t – t 3/(3τD3)], with p̂ = 1.2, γ 0 = 0.42, and τD= 10.

Time evolution of the rms pressure fluctuations( ∫ p̃ 2dr )1/2 (a) and the (squared) mean pressure ∫ p̄ 2dr (b)according to eqs. (9, 10) with and without imposed
shear flow.

⎛ φ˜ ⎞⎛φ ⎞ ⎛ φ ⎞⎜ ⎟ = ⎜ ⎟ ( r , t ) + ⎜ ⎟ ( r , t ) exp (i mθ − i nϕ ) (8)⎝ p˜ ⎠⎝ p⎠ ⎝ p⎠
and a subsequent Galerkin projection of the evolution
equations (1, 2). The resulting system for the fields φ̄, φ̃, p̄, p̃
is simplified further by assuming a fixed relation between φ̃and p̃ given by the linear mode structure: φ̃ = i(γ 0 /kθ )p̃.
Imposing a poloidal shear flow of the form ∂r φ̃ = u–θ = ωE x, a
1D model is obtained that consists of the equation (5) for theamplitude of the perturbation p̃ coupled to the dynamics ofthe pressure profile p̄,∂t p̃ = –ikθ ωE xp̃ + γ 0 (∂t p̄ ) p̃ – ωt x2p̃ + χ⊥∂x2p̃

(9)∂t p̄ = –γ 0 ∂x |p̃|2 + χ⊥∂x2p̄ + S(10)In absence of a shear flow (ωE = 0), the system (9, 10) evolves
to a stationary state. However, in the presence of a static shear
flow (ωE > 0) oscillating states can be obtained (Fig. 5).

7. 1D model for oscillations with frozen
shear flow
Considering one dominant (m, n) mode, a reduced
system for the radial dynamics of this mode coupled to the
dynamics of the profiles can be obtained using therepresentation

8. Conclusions
Oscillations of turbulent transport of particles and energy
in magnetically confined plasmas can be easily observed insimulations of a variety of turbulence models. Theseoscillations typically involve a mechanism of energy exchange
438

Benkadda S. et al., Relaxation Oscillations and Transport Barrier Dynamics in Tokamak Edge Plasmas

References

between fluctuations and a poloidal shear flow. In the case of
the RBM turbulence simulations presented here, a OD model
can be constructed systematically that reproduces theseoscillations.
However, the kind of “predator-prey” mechanism
underlying these oscillations is found to be not relevant fortransport barrier relaxations. In RBM turbulence simulations
with transport barrier, relaxation oscillations of the latter are
observed even in the case of frozen poloidal shear flow. These
relaxations are due to a transitory growth of a mode localized
at the center of the barrier. A 1D model for the evolution ofsuch a mode in the presence of a shear flow describes atransitory growth of an initial perturbation. The correspondingtime scale is large enough to allow for a significant growth in
the case of low diffusivity and low poloidal wavenumber. Thisagrees well with the fact that the mode observed in the full
simulations is located at the barrier center, where thediffusivity is close to the collisional value, and that its
poloidal mode number is the lowest possible at that position.
The 1D evolution equation for the central mode does not
explain, however, the repetition of relaxation events. After the
initial growth, the amplitude decreases monotonically withtime. In order to construct a model that allows for oscillating
solutions, one has to take into account the coupling of themode to the dynamics of the pressure profile. Such a coupledsystem of equations shows periodic oscillations in the case ofa finite (steady state) shear flow. These oscillations aredifferent from the quasi-periodic relaxation events observedin the full system. A possible way to reproduce theserelaxation oscillations may be to include ambient noise in the
1D system. This work is still in progress.

[1] M. Bécoulet et al., Plasma Phys. Control. Fusion 45, A93(2003).
[2] P. Beyer, S. Benkadda and X. Garbet, Phys. Rev. E 61,
813 (2000).
[3] V. Naulin, J.J. Rasmussen and J. Nycander, Phys.
Plasmas 10, 1075 (2003).
[4] G. Manfredi and C.M. Roach, Phys. Plasmas 10, 2824
(2003).
[5] P. Beyer, S. Benkadda, X. Garbet et al., Phys. Rev. Lett.
85, 4892 (2000).
[6] S. Benkadda, P. Beyer, N. Bian et al., Nucl. Fusion 41,
997 (2001).
[7] J.L. Lumley, Atmospheric Turbulence and Radio Wave
Propagation, ed. A.M. Yaglom and V.I. Tatarski (Nauka,
Moscow, 1967), p. 166.
[8] S. Benkadda, T. Dudok de Wit et al., Phys. Rev. Lett.
73, 3403 (1994).
[9] H. Biglari, P.H. Diamond and P.W. Terry, Phys. Fluids B
2, 1 (1990).
[10] K.C. Shaing et al., Phys. Fluids B 2, 1492 (1990).
[11] A.S. Ware et al., Plasma Phys. Control. Fusion 38, 1343
(1996).
[12] C.F. Figarella, S. Benkadda, P. Beyer et al., Phys. Rev.
Lett. 90, 015002 (2003).
[13] K.H. Burrell, Phys. Plasmas 6, 4418 (1999).
[14] P.W. Terry, Rev. Mod. Phys. 72, 109 (2000).
[15] R.J. Taylor et al., Phys. Rev. Lett. 63, 2365 (1989).
[16] R.R. Weynants et al., Nucl. Fusion 32, 837 (1992).
[17] J. Cornelis et al., Nucl. Fusion 34, 171 (1994).
[18] J.F. Drake et al., Phys. Fluids B 4, 488 (1992).
[19] J.M. Finn, J.F. Drake and P.N. Guzdar, Phys. Fluids B 4,2758 (1992).
[20] H. Sugama and W. Horton, Phys. Plasmas 1, 345 (1994).

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