On the spatial structure of solitary radial electric field at the plasma edge in toroidal confinement devices

The solitary radial electric field in the edge of toroidal plasma is studied based on the electric field bifurcation model. Results are applied to tokamak and helical plasmas, and the dependence of the electric field structure on the plasma parameters and geometrical factors is analyzed. The order of magnitude estimate for tokamak plasma is not far from experimental observations. It is shown that, in helical plasmas, the height of electric field structure is reduced substantially owing to the ripple particle transport, while the width is influenced less. The implications of the results for the limit of achievable gradient in the H-mode pedestal are also discussed.

In these achievements of the research, study of response of edge electric field structure against external biasing has provided a unique path to understand the nonlinear mechanism in bifurcation quantitatively [34][35][36]. A solitary radial electric field was observed [35,36], and theoretical explanation has been discussed [37][38][39]. The induced bifurcation takes place under the condition where the spontaneous transition is difficult to occur. Thus this allows the study of nonlinearity in the neoclassical damping term, which is one of the origins of electric field bifurcation. In addition, even in the process of L-H transition, the magnitude of solitary electric field was found to jump without substantial change of ion pressure gradient [11]. This jump was discussed based on the electric field bifurcation model [40]. The physics of electric field bifurcation has been studied in helical plasmas as well [41][42][43]. A solitary radial electric field, which is induced by external biasing, was measured in LHD plasma [43]. Comparison of observations of tokamak and helical plasmas enriches understanding of toroidal plasma in general. Although there are many similarities in responses against biasing among tokamak and helical plasmas, a substantial difference in the magnitude of radial electric field is also noticeable. This stimulates the study of the scaling property of the solitary radial electric field, which is induced by the nonlinearity in the neoclassical damping process.
In this article, we extend the electric field bifurcation model in [38,44] to the tokamak plasmas and helical plasmas, and study the dependence of the electric field structure on the plasma parameters and geometrical factors. The order of magnitude estimate for tokamak plasma is not far from experimental observations. It is shown that the height of electric field structure is reduced substantially owing to the ripple particle transport, while the width of the solitary electric field is influenced less. This might be useful for a basis to search the H-mode plasma with high confinement-enhancement factor in helical plasmas. In addition, the implication of the results to understanding the limit of achievable gradient in the H-mode pedestal is also discussed.

Model equation
The mean radial electric field E r is governed by the charge conservation relation combined with the Poisson's relation, and is expressed as a nonlinear diffusion equation as (see chap.19 of [44]) where μ i is the ion viscosity, is the dielectric constant in toroidal plasma (in plateau regime), v A is the Alfven velocity, q is the safety factor, J r is the current in the plasma, and J ext is the component which is driven by external circuit. We are interested in the localized electric field structure, which is self-organized by the nonlinear response of the plasma. Thus the plasma parameters and their gradients (except E r ) in equation (1) are treated constant for the simplicity. Normalization is introduced and the length, time, E r and current density are normalized as where ρ p is the ion gyroradius at poloidal magnetic field, ρ ε ρ = − q p t i 1 , ε t is the inverse aspect ratio, T i is the ion temperature, and σ(0) denotes a conductivity in the linear regime The radius r 0 is the reference position, where the peak of solitary radial electric field appears, and is of the order of plasma minor radius. Then equation (1) is rewritten as In this article, we study steady state solution, which is given by the equation In studying the stataionary solution, the normalized external current I is constant in space and time.

Solitary radial electric field structure
The solution of the solitary radial electric field is known to appear, when the current J(X) has one maximum and the local equation has one stable branch and one unstable branch as is illustrated in figure 1. (In this article, the sign of electric field is chosen as positive in order to study the electric bias experiments. The extensition to the negative electric field is straightforward.) The boundary condition is chosen as for the solitary radial electric field structure. An analytic solution for the radial electric field has been derived when I is close to the peak value of the normalized current at X = X * , I * , as [44] Equation (8) denotes the solitary radial electric field (figure 2), which is localized near x = 0, where the parameter that characyerize the width of the peak is given as From equation (8a), one has an estimate, In this article, we focus upon the combination of the radial electric field and its curvature, XX″, as a key parameter for the suppression of turbulence [33]. A brief explanation is made in the appeandix. It is evaluated as

Bifurcation of electric field at tokamak edge
In tokamak configurations, the radial current by the neoclassical process has been derived as [37] ρ π ν ω is the ion transit angular frequency, E r,a represents the radial electric field which is induced by the neoclassical bulk viscoity, and is normalized as, γ i is the specific heat ratio of ions, and V i,// is the mean ion velocity along the magnetic field. From equation (12a), one has the conductivity (in the limit of weak electric field) as (13) and the normalized current function as We are interested in the case that the bifurcation to the solitary radial electric field structure occurs away from the spontaneous L-H condition, so that the simplification is employed. This condition (15) may also be used in the circmustance where the jump of radial electric field is large enough (in comparison with the ion-pressure-gradient driven radial electric field) in the H-mode plasma, as was reported in [11]. From equations (14) and (15), we have an order of magnitude estimate and See, e.g. [37] for more accurate evaluation of the second derivative of J(X). Thus, one has (apart from a numerical factor of the order unity) in the normalized unit. An order of magnitude estimate of the radial electric field is given in the experimental units, by rewiting equation (16) by use of equation (3c), as Equation (13) for the conductivity gives the ratio of σ(0) to dielectric constant as Substituting this relation (20) into the formula of normalizing length, equation (3a), one has This length l denotes the characteristic value for the width of the peaked profile. Following the similar process, equation (18) is expressed in the dimensional form as Equations (21) and (22) show that the solitary structure becomes steeper when the turbulence transport is suppressed. If one uses a neoclassical value for the ion shear viscosity, for the analytic insight of the problem, one has the characteristic radial scale length of the solitary electric field from equations (21) and (23) as This is the estimate of the upper bound of steepness of the solution. The peak curvature of electric field, combined with the electric field strength, is estimated from equations (22) and (23) as It is characterized by the ion temperature and poloidal gyroradius.
One might be interested in how the result behaves in the banana regime. In comparison with the plateau regime, the perpendicular dielectric constant and shear viscosity are modified as

Case of helical plasmas
This model is applied to helical plasmas, in order to examine the observation in [43]. We take a simple model of magnetic field for the helical plasma, the multi polarity of which is 2, as where ε h is the helical ripple amplitude, θ and ζ are the poloidal and toroidal angles, respectively, and M is the number of field period. The regime of helical ripple transport, is chosen. The ratio between the effective collision frequency (LHS) and the transit frequency in the helical ripple (RHS) is sometimes referred to as In addition, we study the case of so-called 'ion-root', i.e. the helical ripple transport of ions in the absence of radial electric field is much larger than that of electrons, The modification of the ion-root branch by external current is studied here. The radial current is approximately given by This is for the clarification of the problem, and the study of general cases, in which change between the electron-root and ion-root can take place, is left for future work. The radial particle flux of ions has been derived in [45] as where c n and c T indicate the neoclassical coefficients, and the fitting formula of the ripple transport coefficient has been derived as Comparing the condition for the ripple transport regime, equation (28), one sees that the condition holds in the regime of helical ripple transport. Equations (31a) and (32b) indicate that the ripple particle transport is more strongly reduced by the radial electric field, in comparison with the case of tokamak plasmas. By employing the normalization of equation (3), one has and the normalized current function where is the contribution of neoclassical bulk viscosity effect, and coefficients are given as = ( + ) c c S X 1 n n 3/2 3/2 and = ( + ) c c S X 1 T T 3/2 3/2 . As is the case in the preceding section, we neglect the contribution of X a, h , and use a simplified form Model equation (35) tells that the peak of the current function appears at and it gives Substituting equations (36) and (37) into equation (10), one has (apart from a numerical factor of the order unity) With the help of equations (36) and (38a), the peak curvature of electric field, combined with the electric field strength, is estimated as, in the normalized unit. Equations (3a) and (33) give the normalization length as Considering the case that the helical ripple transport is dominant in the transport process, one takes an approximation of Substituting equation (40) into equation (39), one has ε ρ l~t p (41) in the experimental unit. Then, equation (38b) is rewritten in the dimesional unit as  (42) Comparing results for tokamaks and helical systems, equations (24b) and (42), respectively, one sees that the electric field curvature effect is reduced by the factor S −2 if helical ripples affect the transport. The difference between helical systems and tokamaks appears in the magnitude of the localized electric field, and the radial scale length obeys similar dependence.

Summary and discusssions
In this article, the solitary radial electric field structure in the edge of toroidal plasmas are investigated. Based on the electric field bifurcation model, order of magnetidue estimates of the electric field, curvature and radial scale length are given. The result is summarized in the table 1. It is noted that the scale length and curvature depend on the model of the shear viscosity of ions, but the magnetide of peak electric field is independent of the model of shear viscosity.
Although the results are obtained as scaling relations (apart from numerical coefficients of the order of unity), it might be worth examining the absolute values which the model provides. For the parameters of the tokamak edge (see, [13] for detailed measurements), equation (24b) or (26) (table 1) is not bad in understanding the experimentally observed solitary radial electric field structure in tokamaks. For the case of helical plasmas, the value (for the peak curvature of electric field multiplied by the electric field strength) is suppressed by the factor of S −2 owing to the helical-ripple trapped particles, while the width of the peak is independent of S. Considering the fact that the parameter S is in the range of a few 10 [46], the reduction factor is about ~10 −3 times smaller compared to tokamaks. Experiments have reported the value of ~10 9 [V 2 m −4 ] [43]. The width of the solitary peak is not far from those in tokamaks. Thus, the scaling relations in this article give qualitative understanding for the differences in the solitary radial electric field structures between tokamaks and helical plasmas.
When the electric field inhomogeneity is weak and turbulence is not suppressed, the viscosity is larger than equation (23). If one substituites the gyro-Bohm dependence, The curvature of the electric field (multiplied by electric field) is modified accordingly as, electric field is of the order of the decorrelation rate of microscopic drift wave turbulence. The nonlinear decorrelation rate of drift wave fluctuations is evaluated as for ρ ⊥ k~1 i , where ⊥ k is the wavenumber of drift wave fluctuation and L n is the density gradient scale length. The condition that the rate of modulation by the inhomogeneous radial electric field is larger than the nonlinear decorrelation rate of fluctuations,  (45) for tokamaks. Here, equation (24b) is used for plateau regime. For given geometrical parameters, the suppression of turbulence (by the bifurcation to the state with solitary radial electric field structure) stops when the gradiel becomes steep and the gradient scale length becomes of the order of qρ p . This describes a scaling property for one of the achievable limits of density gradient in the H-mode pedestal. For helical systems, combination of equations (42) and (44) (46) in the helical ripple particle regime. This condition (46) is more stringent than equation (45). The achievable gradient is weaker by the factor S −1 than equation (45). Thus, in helical plasmas, the effective suppression of turbulence by the solitary radial electric field occurs more easily in the plateau regime than in helical ripple particle regime, if other parameters are common.
The impacts of magnetic field ripples on the edge barrier in tokamaks have attracted attentions, particularly in conjunction with symmetry-breaking-magnetic perturbation for ELM control [47]. As has been discussed in [40,48], the influence of small ripples is complicated, and may introduce a new bifurcation in addition to the one, which is discussed in this article. The issue requires future intensive studies.