Generalized Solovev equilibrium with sheared flow of arbitrary direction and stability consideration D.A. Kaltsas and G.N. Throumoulopoulos Department of Physics, University of Ioannina, GR 451 10 Ioannina, Greece E-mails: dkaltsas@cc.uoi.gr, gthroum@cc.uoi.gr Abstract A Solovev-like solution describing equilibria with field aligned incompressible flows [Throumoulopoulos and Tasso, Phys. Plasmas 19, 014504 (01)] is extended to non parallel flows. The solution expressed as a superposition of Bessel functions contains an arbitrary number of free parameters which are exploited to construct a variety of configurations including ITER shaped ones. For parallel flows application of a sufficient condition for linear stability shows that this condition is satisfied in an appreciable part of the plasma region on the high-field side mostly due to the variation of the magnetic field perpendicular to the magnetic surfaces. Also, the results indicate that depending on the shape of the Mach-function profile and the values of the free parameters the flow and flow shear may have either stabilizing or destabilizing effects. 1
The most known and widely employed axisymmetric magnetohydrodynamic analytic equilibrium is associated with the Solovév solution of the Grad-Sfafranov (GS) equation [1]. However, this solution being up-down symmetric and having a limited number of free parameters can not describe configurations with diverted shaping of contemporary tokamaks. This drawback was eliminated in Refs. [, 3] by introducing an arbitrary number of additional terms in the homogeneous part of the solution by means of an iterative technique. Consequently, by exploiting the respective arbitrary number of free parameters a variety of equilibria with desirable shaping and confinement figures of merit were constructed. Since sheared flows play a role in the transitions to improved confinement regimes as the L-H transition, in a previous study [4] we extended this solution to confined plasmas with incompressible flows parallel to the magnetic field. Also, we considered the linear stability of the extended solution by means of a sufficient condition [5]. Aim of the present contribution is to extend further this Solovév-like solution to plasmas with incompressible flows of arbitrary direction on the basis of a generalized GS equation [Eq. (1) below]. Unlike the usual form of the Solovév solution we employ a separable solution of the homogeneous part of the generalized GS equation expressed in terms of Bessel functions. Thus, alternative to Refs. [, 3, 4] an arbitrary number of free parameters is introduced through a linear combination of these functions. For parallel flows we reexamine the stability by applying the condition of Ref. [5] and compare the results with those of Ref. [4]. We employ the generalised GS [6, 7] u + 1 ( d X du 1 Mp ) + µ 0 R dp s du + µ R 4 0 d [ ϱ(φ ) ] = 0 (1) du in conjunction with the Bernoulli equation for the pressure [ v ( ) ] dφ P = P s ϱ ρ du () Here (R, ϕ, z) are cylindrical coordinates with z corresponding to the axis of symmetry; the function u = u(r, z) related to the poloidal magnetic flux labels the magnetic surfaces; M p (u) is the poloidal Alfvén Mach function and ϱ(u) the mass-density function being a surface quantity because of incompressibility; X(u) relates to the toroidal magnetic field, P s (u) is the static pressure and Φ(u) the electrostatic potential; ( the elliptic operator in (1) is defined as = R 1 ) R R R + z. The negative velocity term in () sets an upper limit on the flow amplitude, v, so that the relation P 0 is satisfied everywhere in the plasma. However, the positive term ϱρ (dφ/du) permits higher flow amplitudes in comparison with field aligned flows (dφ/du = 0). Derivation of (1) and () is given in [6] and [7]. For convenience we introduce the dimensionless quantities ρ = R/R 0, ζ = z/r 0, ũ = u/(b 0 R0), ϱ = ϱ/ϱ 0, P = P/(B 0 /µ 0 ), J = J/ (B 0 /µ 0 R 0 ), ṽ = v/ ( B 0 / ) µ 0 ϱ 0, where R0 is the major radius of the torus and B 0 the vacuum magnetic field at the geometric centre. Eq. (1) then remains identical in form with µ 0 set formally equal to one. In the following analysis the tildes will be dropped on the understanding that we refer to dimensionless quantities. The terms X /(1 Mp ), P s and ϱ ( ) dφ du being free surface functions can be chosen arbitrarily. We adopt a Solovev-like linearizing Ansatz: X (u) 1 M p [ dφ(u) = X0 + X 1 u, P s (u) = P 0 + P 1 u, ϱ du where X 0, X 1, P 0, P 1, G 0, G 1 are free parameters. Then Eq. (1) assumes the form ] = G 0 + G 1 u (3) u ρ 1 u ρ ρ + u ζ + X 1 + P 1 ρ + G 1 ρ 4 = 0 (4) As already mentioned in the first paragraph of this study we construct a solution of the homogeneous part of Eq. (4) with arbitrary number of free parameters. Unlike previous papers [, 3, 4], in which
such solutions were produced by using an iterative algorithm, here a homogeneous separable solution is expressed in terms of Bessel functions restricting the separation constant to integer values by exploiting the orthogonality of these functions. The homogeneous solution is: u h (ρ, ζ) = cρ j [ aj J 1 (jρ)e jζ + b j J 1 (jρ)e jζ + c j Y 1 (jρ)e jζ + d j Y 1 (jρ)e jζ] (5) where J 1 and Y 1 are the first order Bessel functions of first and second kind respectively and j is the separation constant. A particular solution of (4) can be obtained by inspection as and therefore the general solution is written u p (ρ, ζ) = P 1 8 ρ4 G 1 4 ρ6 X 1 ζ (6) u = u h + u p (7) It is noted that (7) holds for arbitrary Mach functions M p and densities ϱ. The coefficients a j, b j, c j, d j (j = 1,..., 3) can be specified in connection with the desirable boundary shaping and the parameters P 1, G 1, X 1 in connection with the desirable values of the various physical quantities, e.g, for large tokamaks the pressure, current density and electric field are on the order of 10 6 P a, 10 6 A/m and 10 4 V/m respectively. The parameter c can be fixed by the condition q a = 1.1 (so that the Kruskal-Shafranov stability criterion is satisfied) where q a is the safety factor on axis: q a = I ρ 1 Mp ( u ) 1/ u ρ ζ (8) ρ=ρa,ζ=ζa with I the poloidal current function I = (1 Mp ) X 1/ 0 + X 1u + ρ M p G 0 + G 1u (9) To completely construct the equilibrium we adopt a peaked off-axis choice for the poloidal Mach function: M p = M a ( ) m ( ) n mua nua u n (u a u) m (10) m + n m + n where u a refers to the magnetic axis and M a is the maximum value of M p. Typical values of M p for large tokamaks are of order of 10 4 according to the experimental scaling v 10 1 v s, where v s = (γp/ϱ) 1/ is the sound velocity. The parameters m and n are chosen in such a way that the flow be localised near the boundary in connection with the L-H transition phenomenology. Also we choose a linear density function: ϱ = ϱ a u/u a. Once all the arbitrary functions are specified we can construct single-null diverted equilibria, e.g., ITER like ones, by imposing appropriate boundary conditions. For the boundary shaping we follow the method of Refs. [], [4]. We impose the shaping conditions at four prescribed characteristic points of the boundary (inner, outer, upper and lower): (1 ε, 0), (1 + ε, 0), (1 δε, κε), (1 δε, κε), where a is the minor radius of the torus, δ is the triangularity, κ the elongation and ε = a/r 0 the inverse aspect ratio. For the ITER machine the respective planned values are a = m, δ = 0.33, κ = 1.33 and the major radius of the torus is R 0 = 6. m. The conditions are: u(1 ε, 0) = u(1 + ε, 0) = u(1 δε, κε) = u(1 δε, κε) = 0 (11) u ρ (1 δε, κε) = 0 (1) u ζ (1 ε, 0) = u ζ (1 + ε, 0) = 0 (13)
u ζ (1 δε, κε) = u ρ (1 δε, κε) = 0 (14) (1 α) u ζζ (1 ε, 0) + εκ u ρ (1 ε, 0) = 0 (15) (1 + α) u ζζ (1 + ε, 0) εκ u ρ (1 + ε, 0) = 0 (16) κ u ρρ (1 δε, κε) ε cos α u ζ(1 δε, κε) = 0 (17) where α = arcsin δ. Conditions (11) impose u to vanish at the four characteristic points, (1) ensures the local symmetry of the boundary curve at the upper point, while (13) relates to local up-down symmetry near the plane z = 0. Conditions (14) produce the x-point topology of the lower point and relations (15), (16), (17) (which are proved in Ref. [4]) are related to the curvature of the boundary curve at the inner, outer and upper point respectively. On account of the above system of 1 equations we truncate the summation in expression (5) so as to include the first 1 terms. The constants a i, b i, c i, d i (i = 1,..., 3) are determined by numerically solving the set of Eqs. (11-17). As an example, an ITER-like equilibrium is given in Fig. 1. Also we constructed D-shaped and Double-null diverted configurations by making the Ζ 0. 0. 0.6 0.8 1.0 1. 1.4 Fig. 1: An ITER-like equilibrium with flow of arbitrary direction in connection with solution (7). The basic geometrical parameters are a = m, δ = 0.33, κ = 1.33, R 0 = 6. m. proper changes in the shaping conditions regarding the upper and lower points. The equilibria have safety factor profiles increasing monotonically from the magnetic axis to the plasma boundary, current density profiles increasing throughout the plasma from the inner to the outer point in connection with the ansatz (3) (Fig. ), electric fields localized in the flow region and possessing an extremum, and peaked pressure profiles vanishing on the boundary by adjusting the free parameter P 0 (Fig. 3). An interesting result is that the flow creates local pressure extrema (Fig. 3) associated with the formation of pressure islands shown in Fig 4. Also, sufficiently large flows can create paramagnetic current layers. Such an example is shown in Fig. 5. We now consider the linear stability of the equilibria constructed for field aligned flows by applying the sufficient condition of Ref. [5]. This condition states that a stationary equilibrium with field aligned incompressible flow and constant mass density is stable to small three dimensional perturbations if M p < 1 and A 0 where A = A 1 + A + A 3 + A 4 (18) A 1 = (J u) (19)
J φ,0 A m 1.5 10 6 1.0 10 6 500 000 0.8 0.9 1.0 1.1 1. 1.3 Fig. : Toroidal current density profile on the plane ζ = 0 increasing from the inner to the outer point. The local extrema are formed in the toroidal layer where the flow takes place..5 10 6 P,0 Pa.0 10 6 1.5 10 6 1.0 10 6 500 000 0.8 0.9 1.0 1.1 1. 1.3 Fig. 3: Pressure profile on the plane ζ = 0. In the regions where the flow is located the profile has local extrema in accordance with Bernoulli s law.
0. 0. 0.6 0.8 1.0 1. Fig. 4: Isobaric surfaces with flow created pressure islands around the local extrema of Fig. 3 1 10 6 J Ζ,0 A m 500 000 0.8 0.9 1.0 1.1 1. 1.3 500 000 Fig.5 The ζ component of the current density. Owing to the flow a paramagnetic current layer of J ζ is formed. A = (J u) ( u )B (0) A 3 = 1 4 (1 M p ) 1 dm p du u u B (1) A 4 = 1 (1 M p ) 3/ dm p du u 4 g () ( 1 dp g = s 1 Mp du dm ) p B (3) du The quantity A 1 being always negative consists a destabilizing contribution potentially related to current driven modes. The other terms can be either stabilizing or destabilizing. Specifically, the term A relates to the current density and the variation of the magnetic field perpendicular to the magnetic surfaces. The term A 3 involves the shear and magnitude of the flow in conjunction with the variation of the magnitude of the magnetic field perpendicular to the magnetic surfaces. A 4 is mostly a flow term depending on the magnitude and the shear of the flow. The flows satisfying (1) are inherently sub-alfvénic (M p < 1) because of an integral transformation involved [7]. For parallel flows the equilibrium is obtained from the solution (5)-(7) with G 1 = 0. Also, M p is the Alfvén Mach function of the parallel velocity because in this case it can be shown that the poloidal Mach function, the respective toroidal Mach function and
Ζ Ζ 0.6 0.6 0. 0. 0. 0. 0.6 0.8 1.0 1. 1.4 0.6 0.8 1.0 1. 1.4 Fig. 6: Stability diagrams for the quantity A for the Mach function (10) and two different values of M a: M a = 10 (left) and M a = 4 10 (right). In the red (gray) coloured regions the stability condition is satisfied. The stability area slightly shrinks as M a takes larger values. the (total) parallel Mach function are equal one another. In this paragraph in addition to the localized peaked-off-axis Mach function (10) we adopted the extended peaked-on-axis Mach function: ( ) l u (4) M p = M a u a where the parameter l is associated with the flow shear. The quantity A was calculated analytically. After a broad variation of the values of the free parameters (in particular we permitted M p to vary up to the order of 10 1, close to the non-negative-pressure limit) we came to the following conclusions: 1. The condition A 0 is satisfied in a part of the plasma region on the high field side (Figs. 6-7) where the current density, J, is sufficiently small so that the term A can overcome A 1. Unlike, such a stable region was not found for the equilibria of [4] corresponding to the same ansatz (3) for the free functions (with vanishing non-parallel-flow term). The reason should be the different homogeneous solution constructed here. Since J is an increasing function of ρ the non positive term A 1 dominates over the other terms for sufficiently large values of ρ. Therefore the current density J plays a destabilizing role in connection with the negative term A 1.. In the majority of the cases considered the term A related to the magnetic shear is stabilizing. 3. Depending on the shape of the Mach function and the parametric values, the flow and flow shear can have either destabilizing or stabilizing effects. An example in the former case is given in Fig. 6 and another in the latter case in Fig. 7. We have constructed analytically a Solovev-like equilibrium solution with arbitrary number of free parameters by means of a superposition of Bessel functions which can be assigned in connection with desirable boundary shaping. Hence using appropriate sets of boundary conditions we derived a variety of configurations including single null diverted, ITER-like ones. For parallel flows application of a sufficient condition for linear stability implies that this condition is satisfied in an appreciable part of the plasma on the high field side restricted however for large distances from the axis of symmetry by respective large current densities. The magnetic shear in general plays a stabilizing role while the flow and flow shear can be either stabilizing or destabilizing.
Ζ Ζ 0.6 0.6 0. 0. 0. 0. 0.6 0.8 1.0 1. 1.4 0.6 0.8 1.0 1. 1.4 Fig. 7: Stability diagrams for the quantity A for the Mach function (4) with l = and M a = 4 (left), M a = 0.5 (right). The red (gray) coloured region in which the stability condition is satisfied broadens as M a increases. One of the authors (GNT) would like to thank Henri Tasso, George Poulipoulis and Apostolos Kuiroukidis for very useful discussions. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the by (a) the National Programme for the Controlled Thermonuclear Fusion, Hellenic Republic, (b) Euratom research and training programme 014-018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. References [1] L. S. Solovév, Sov. Phys.-JETP 6, 400 (1968). [] A. J. Cerfon and J. P. Freidberg, Phys. Plasmas 17, 0350 (010). [3] R. Srinivarsan, L. L. Lao and M. S. Chu, Plasma Phys. Controlled Fusion 5, 035007 (010). [4] G. N. Throumoulopoulos and H. Tasso, Phys. Plasmas 19, 014504 (01). [5] G. N. Throumoulopoulos and H. Tasso, Phys. Plasmas 14, 1104 (007). [6] H. Tasso and G. N. Throumoulopoulos, Phys. Plasmas 5, 378 (1998). [7] C. Simintzis, G. N. Throumoulopoulos, G. Pantis, and H. Tasso, Phys. Plasmas 8, 641 (001).