L-H transition dynamics in fluid turbulence simulations with neoclassical force balance
Laurent Chôné, Peter Beyer, Yanick Sarazin, Guillaume Fuhr, Clarisse Bourdelle, Sadruddin Benkadda
aa r X i v : . [ phy s i c s . p l a s m - ph ] N ov L-H transition dynamics in fluid turbulence simulations withneoclassical force balance
L. Chˆon´e , , P. Beyer , Y. Sarazin , G. Fuhr , C. Bourdelle , and S. Benkadda Aix–Marseille Universit´e, CNRS, PIIM UMR 7345, 13397 Marseille Cedex 20, France CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France.
Abstract
Spontaneous transport barrier generation at the edge of a magnetically confined plasma is inves-tigated. To this end, a model of electrostatic turbulence in three-dimensional geometry is extendedto account for the impact of friction between trapped and passing particles on the radial electricfield. Non-linear flux-driven simulations are carried out, and it is shown that considering the radialand temporal variations of the neoclassical friction coefficients allows for a transport barrier to begenerated above a threshold of the input power. E × B velocity at which fluctuations areconvected. On another hand, theory shows that the plasma gradients in the H-mode barrierare limited by pressure driven ballooning modes leading to relaxations of the barrier, knownas Edge-Localized modes (ELMs) [5]. The transition from a regime of low confinement toone of high confinement at the edge, or L-H transition occurs when externally injectingpower into the plasma and is generally followed by quasi-periodic relaxations of the barrier,which is a characteristic of the ELMs. The importance of achieving high confinement makesH-mode one of the ITER baseline scenarios, however it could be seriously hindered by theharmful nature of ELMs to the wall components. Because of this, the understanding ofthe creation, control and removal of external transport barriers is of crucial importanceto the success of magnetic fusion. Although the L-H transition has been widely observedand the conditions for triggering H-mode have been extensively studied experimentally,theoretical understanding of the underlying physical mechanisms remains unresolved [2,6]. In particular, plasma edge turbulence simulations based on first principles show self-generation of sheared flows and subsequent turbulence reduction, but no clear transition isobserved [6].In this letter we present non-linear results of flux-driven resistive ballooning simulationsof the plasma edge, taking into account the effect of neoclassical friction on the E × B flow.It is found by means of three-dimensional (3D) simulations that competition between theneoclassical friction and zonal-flows allows for the existence of two distinct regimes dependingon the imposed heat flux. These regimes correspond to a low-confinement state dominatedby turbulence, and above a certain input power, a state of improved confinement with theonset of a transport barrier. Radial and temporal variations of the friction coefficientsare found to have a strong impact on the dynamics of the system, so that taking them2nto account is necessary to obtain generation of this transport barrier. A reduced 1Dmodel which reproduces qualitatively the 3D result is derived, and 1D simulations showintermittent bursts of turbulent flux corresponding to relaxations of the established barrier.In the following simulations, the non-linear evolution of electrostatic resistive ballooningturbulence in 3D toroidal geometry is reproduced using the EMEDGE3D code [7], with thethree dimensions denoted ( r, θ, φ ) being the minor radius, the poloidal and toroidal angles,and their normalised counter-parts ( x, y, z ). This code solves the following reduced MHDmodel, in the limit of large aspect ratios and with the slab approximation: ∂ t ∇ ⊥ φ + (cid:8) φ, ∇ ⊥ φ (cid:9) = −∇ k φ − G p + ∂ x F neo + ν ⊥ ∇ ⊥ φ, (1) ∂ t p + { φ, p } = δ c G φ + χ k ∇ k p + χ ⊥ ∇ ⊥ p + S. (2)Equations (1,2) correspond respectively to the charge and energy balance, the two fields φ and p being the electrostatic potential and the total pressure. ∇ k and ∇ ⊥ are respectively theparallel and perpendicular gradients with respect the magnetic field lines and G is a toroidalcurvature operator. ν ⊥ is the classical viscosity, while χ k and χ ⊥ account for parallel andperpendicular collisional heat diffusivities. S ( x ) is a heat source term (all numerical resultspresented here are from flux-driven simulations). A term for poloidal flow damping whichaccounts for friction between trapped and circulating particles ∂ x F neo (cid:0) ¯ φ, ¯ p (cid:1) is added inEq. (1) ( ¯ f denotes the flux-surface average of quantity f ).The system (1,2) is dimensionless: time is normalised to the interchange time, τ int = √ R L p √ c S , with ˜ c S the acoustic speed and L p the characteristic length of pressure gradient.The perpendicular length scale is the resistive ballooning length, ξ bal = q ρη k τ int L s B , with themagnetic shear length L s being the parallel length scale. The fields φ and p are normalisedrespectively to B ξ bal τ int and ξ bal p L p . Because the MHD model doesn’t separate density andtemperature, an assumption is made that the former is constant n = n , therefore p = n T .Furthermore, a fixed ratio between the temperatures T i = ǫ T T e is also assumed. This isnecessary to carry out the derivation of the neoclassical friction term.The starting point of this reasoning is the radial force balance equation which, if weconsider toroidal rotation to be negligible (generally true in the absence of torque injection),3an be written thus: ∂ x ¯ φ + ǫ T ǫ T + 1 τ int p ξ bal L p en B ∂ x ¯ p = ¯ u y . (3)In the fluid model, the poloidal velocity is not normally constrained, however an expressionemerges from the neoclassical theory: ¯ u neoy = ǫ T ǫ T +1 τ int p ξ bal L p en B K ( ν i, ∗ ) ∂ x ¯ p [8, 9]. The colli-sionality is expressed as a function of ¯ p (since ν i, ∗ ∼ nT − i ∼ ¯ p − , under the assumptionsmentioned above), and a heuristic closure [10] allows for this constraint to be taken intoaccount in the fluid model through a friction term which enforces relaxation towards thisequilibrium: F neo = − µ neo (¯ p ) (cid:2) ∂ x ¯ φ − K neo (¯ p ) ∂ x ¯ p (cid:3) , (4)where K neo = ǫ T ǫ T +1 τ int p ξ bal L p en B [ K ( ν i, ∗ ) − µ neo = µ i h q ( x ) ε ( x ) i , with q ( x ) the safety factorand ε ( x ) the inverse aspect ratio. The Hinton and Hazeltine formula is used to determine K ( ν i, ∗ ) for all neoclassical regimes [8], and an approximate fit for µ i is found in [10]. Asimple form of this friction term is obtained by considering that collisionality is constantand in the range of the plateau regime, so that K ( ν i, ∗ ) ≃
0. One can notice that this termallows for a coupling between the flow and the pressure gradient, therefore a possible positivefeedback between the two. Moreover the value toward which the rotation relaxes dependson the value of K neo , so that radial variations of K neo will introduce a shear.The simulations are carried out in the range of minor radius between 0 . < r/a < χ ⊥ and ν ⊥ . All simulations are flux-driven by a source S ( x ) locatedin the x < x in buffer zone, imposing the heat flux Q = R S ( x ) dx . Here x denotes thenormalised minor radius (to ξ bal ), x in and x out are the positions of the main simulationdomain’s boundaries, with x out corresponding to r = a . The safety factor is hyperbolic,between q ( x in ) = 2 . q ( x out ) = 3 . − <ν i, ∗ < (near banana to collisional regime), which is in agreement with what is observedin L-mode at the edge [11], and happens to be where K ( ν i, ∗ ) varies the most rapidly. Thisis with the exception of ν ⊥ and χ ⊥ , chosen large enough to ensure damping at sub-Larmorscales. The focus of the study being the L-H transition, competition between the mean flow(here in particular due to the F neo term) and zonal-flows is expected. While zonal-flows4enerated by turbulence are included in the model, it doesn’t account for certain zonal-flowsaturation phenomena. In particular, it turns out that to maintain competition betweenboth contributions to the flow, it is necessary to increase the influence of F neo , which is doneby multiplying µ neo by a factor 6 (sufficient in this set of parameters). Several simulations aredone in the range of 5 ≤ Q ≤
30 to study the impact of this friction on confinement. The τ E ( − s ) (a) 050100150200250 ¯ p (c)0 5 10 20 30 Q − (cid:0) ∂ x ¯ p (cid:2) x (b) 0.90 0.95 1.00 r/a ∂ x ¯ φ (d) Q = 5Q = 10Q = 20Q = 30 FIG. 1. (Color online). Evolution of the confinement efficiency as a function of the heat sourceamplitude in the 3D case. Left panels shows volume-averaged quantities, with error bars being thestandard deviation in time. Right panels are flux-surface averaged quantities. results on figure 1 show the confinement deterioration expected in L-mode with increasingheat flux for Q <
14, followed by a sharp increase and again a deterioration if the sourceis increased further. This corresponds to strong changes in the profiles of the flux-surfaceaveraged pressure and poloidal velocity, the latter being defined as ¯ u y = ∂ x ¯ φ . Indeed,when the heat flux is below Q = 14, the pressure profile is roughly a straight line andthe poloidal velocity is low with some radial variations. Above the threshold, the poloidalvelocity profile is strongly modified: in the main part of the simulation domain it stays atlow amplitude and changes sign, but between 0 . < r/a < u F By = K neo ∂ x ¯ p , while it departs from it for the lower sources. Furthermore, the shape of thevelocity profile shows good qualitative agreement with measurements of the radial electricfield in H-mode [18–20], even though the radial electric field at the LCFS is not constrainedby SOL physics. We also show that the mean value of the poloidal velocity at the peak,and consequently the associated shear, is significantly increased (here about 3 times larger)when K neo and its radial variations are taken into account (see Fig. 2). Correspondingly,the friction coefficients, as calculated from the equilibrium pressure in the code, show largechanges before and after the transition.In particular, as shown on Fig. 3 (left panel) the maximum value of K goes from -1 for Q <
14 (transition from collisional to plateau neoclassical regime) to 0 (plateau regime)after the transition. Moreover, after the transition the profile of K is ranging from -2.1to 0 with a strong gradient at the position of the barrier. Correspondingly, the value of µ neo doubles at the position of the barrier after the transition (as illustrated in Fig. 3, rightpanel), and shows a very sharp gradient outward from this position. This supports the factthat the radial and temporal variations of both coefficients should be taken into account, K in order to allow for strong enough shear flows, and µ neo in order to allow for competitionbetween neoclassical friction and zonal-flows. r/a ∂ x ¯ φ Q = 10 r/a ∂ x ¯ φ Q = 20 K(ν i,∗ ) = K(¯p)K(ν i,∗ ) = 0
FIG. 2. (Color online). Comparison of the calculated mean poloidal velocity ¯ u F By with K ( ν i, ∗ ) = 0and K ( ν i, ∗ ) = K (¯ p ), in the lower and higher range of heat flux. Smooth approach of the threshold has shown dithering of the poloidal velocity, reminis-cent of the I-phase in slow L-H transitions [21, 22]. This is clearly seen when looking at the6 .90 0.95 1.00 r/a −2.0−1.5−1.0−0.50.0 K ( ν i , ∗ ) r/a µ n e o Q = 10Q = 20 FIG. 3. (Color online). Profiles of K ( ν i, ∗ ) and µ neo shown for two values of input power, beforeand after the transition. time evolution of the poloidal velocity and the associated shearing rate, as shown on Fig. 4.Before the formation of the transport barrier the velocity shear fluctuates around 1 to 2 (cid:1) | ∂ x ¯ φ | (cid:0) . − time (10 −3 s) r / a ∂ x ¯ φ FIG. 4. (Color online). Time evolution of the the poloidal velocity when crossing slowly thethreshold. Upper panel shows evolution of the average shearing rate in the range 0 . ≤ r/a ≤ r/a =0 .
95. Here the statistically stationary phase is not shown. (normalised unit). An increase to twice this value is then observed shortly after 1 ms, soonfollowed by a sharp fall back to its original level. This is repeated twice, each time towardshigher velocities, before a new state is reached at t > k k = 0 to overlook the toroidal direction. If we retain only one poloidal wave-number k , the two fields p and φ are decomposed in terms of equilibrium and fluctuatingquantities thus: f = ¯ f + ˜ f e ıky + c . c . , and the following four-fields 1D system is obtained [23]: ∂ t ¯ p = − ık∂ x (cid:16) ˜ p ˜ φ ∗ − ˜ p ∗ ˜ φ (cid:17) + χ ⊥ ∂ x ¯ p + S ( x ) , (5) ∂ t ¯ V = ık∂ x (cid:16) ˜ φ∂ x ˜ φ ∗ − ˜ φ ∗ ∂ x ˜ φ (cid:17) − µ neo (cid:0) ¯ V − K neo ∂ x ¯ p (cid:1) + ν ⊥ ∂ x ¯ V , (6) ∂ t ˜ p = ık h ˜ φ ( ∂ x ¯ p − κ ) − ¯ V ˜ p i − α p | ˜ p | ˜ p + χ ⊥ ∂ x ˜ p, (7) ∂ t ˜ φ = ı (cid:18) gk ˜ p ¯ p − k ¯ V ˜ φ (cid:19) − α φ (cid:12)(cid:12)(cid:12) ˜ φ (cid:12)(cid:12)(cid:12) ˜ φ + ν ⊥ ∂ x ˜ φ, (8)with the equilibrium poloidal velocity ¯ V = ∂ x ¯ φ . The α f (cid:12)(cid:12)(cid:12) ˜ f (cid:12)(cid:12)(cid:12) ˜ f terms account for satura-tion via mode coupling. Here t is normalised to ω S = m i eB , x to ρ S = √ m i k B T e eB .In this case, partial stabilisation of the turbulence is achieved above a certain thresholdof the injected power, as illustrated on Fig. 5, showing that this reduced model still containsthe minimal elements to reproduce this behaviour. In the parameter range considered so far,it turns out that for low fluxes, the collisional and turbulent fluxes are of the same order ofmagnitude (Fig. 5, right panel). These results show particularly interesting dynamics of thesystem once the turbulence level is strongly reduced: here turbulence is not steadily sup-pressed but shows instead quasi-periodic bursts. Interestingly the pseudo-period increaseswith the injected power (see Fig. 6). This behaviour bears similarities with type-III ELMs,which were already suggested to be governed by the resistive ballooning instability [7, 14, 15].In conclusion, 1D and 3D fluid, non-linear flux-driven simulations of edge turbulenceincluding self-consistent friction between passing and trapped particles have shown the ex-istence of two distinct regimes depending on the imposed heat flux. At low heat flux, thepoloidal velocity is dominated by zonal-flows, and only poor confinement is achieved. Whenthe input power exceeds a certain threshold, the effect of friction takes over and the poloidalvelocity rises sharply near the LCFS. This is associated with strong velocity shear, which8 .0 0.5 1.0 1.5 2.0 2.5 3.0 Q × 10 τ E ( − s ) τ E Q × 10 Q c o ll , Q t u r b × Q turb Q coll FIG. 5. (Color online). Evolution of the confinement efficiency as a function of the heat sourceamplitude in the 1D case. Q t u r b time (10 −3 s) Q = . · − , . · − , · − FIG. 6. (Color online). Time evolution of the turbulent flux in the presence of a barrier for differentinput powers, in the 1D case. The panels are in order of increasing power. governs the reduction of the turbulent transport, and the subsequent better confinement.In addition, oscillations of the poloidal velocity are observed in 3D simulations when ap-proaching the threshold slowly, which is reminiscent of the limit-cycle oscillations in L-I-Htransition experiments [21, 22, 24]. Long 1D simulations have shown quasi-periodic relax-ations of the transport barrier with a period increasing with the input power, as observedfor type-III ELMs.The authors acknowledge fruitful discussions with X. Garbet and Y. Camenen. Thiswork is supported by the French National Research Agency, project ANR-2010-BLAN-940-01. Computations have been performed at the M´esocentre d’Aix-Marseille Universit´e.9
1] G. M. Staebler, Plasma Phys. Control. Fusion (1998) 569.[2] J. W. Connor et al. , Plasma Phys. Control. Fusion (2000) R1–R74.[3] J. G. Esler, Phys. Fluids (2008) 116602.[4] F. Wagner et al. , Phys. Rev. Lett. , 1048 (1982).[5] H. R. Wilson et al. , Plasma Phys. Control. Fusion , (2006) A71–A84.[6] J. W. Connor, “Recent Progress in the Theoretical Modelling of the L-H Transition”, 17thJoint EU-US Transport Task Force Meeting, Padua, Septembre 3-6, 2012.[7] G. Fuhr et al. , Phys. Rev. Lett. , 195001 (2008).[8] F. L. Hinton et al. , Rev. Mod. Phys. , 239 (1976).[9] P. Helander, D. J. Sigmar, Collisional Transport in Magnetized Plasmas, Cambridge UniversityPress (2005).[10] T. A. Gianakon et al. , Phys. Plasmas , 536 (2002).[11] C. Bourdelle et al. , Plasma Phys. Control. Fusion (2012) 115003.[12] H. Biglari et al. , Phys. Fluids B , 1 (1990).[13] F. L. Hinton et al. , Phys. Fluids B , 1291 (1993).[14] P. Beyer et al. , Phys. Rev. Lett. , 105001 (2005).[15] P. Beyer et al. , Plasma Phys. Control. Fusion (2007) 507–523.[16] M. A. Malkov et al. , Phys. Plasmas , 122301 (2008).[17] A. Strugarek et al. , Plasma Phys. Control. Fusion (2013) 074013.[18] P. Sauter et al. , Nucl. Fusion (2012) 012001.[19] E. Wolfrum et al. , Plasma Phys. Control. Fusion (2012) 124002.[20] E. Viezzer et al. , Nucl. Fusion (2013) 053005.[21] H. Zohm et al. , Phys. Rev. Lett. , 222 (1994).[22] R. J. Colchin et al. , Phys. Rev. Lett. , 25502 (2002).[23] S. Benkadda et al. , Nucl. Fusion (2001) 995.[24] L. Schmitz et al. , Phys. Rev. Lett. , 155002 (2012)., 155002 (2012).