Particle orbit description of cyclotron-driven current-carrying energetic electrons in the EXL-50 spherical torus

In EXL-50 plasma currents over 100 kA are non-inductively generated and maintained solely by electron cyclotron heating (ECH) power with an efficiency of ∼1 A W−1. These currents are carried by energetic electrons (EEs) in the energy range from several tens of keV up to several hundreds of keV which also account for almost all pressure in plasma. This EE component can be viewed as a large number collection of various periodic orbits of energetic particles. Based on this picture we have developed a method for particle orbit description of the EE component in a typical plasma at IP = 121 kA as analysis target. We use a fluid description as a bridge to describe successfully the EE component as a collection of various passing and trapped orbits in the approximation of monochromatic particle energy with good matching to the flux loop signals. The description has revealed characteristics of passing and trapped particles. Passing particles carry almost all toroidal current, while they account for only 20 % of total particle number of the EE component. While net current carried by trapped particles is a very small fraction, they account for a major fraction in number and carry a large positive current outside the last closed flux surface (LCFS) and a large negative current inside. As a result, trapped particles redistribute the current from inside of the LCFS to outside both radially and vertically, generating a large vertically elongated cross section in current as well as number density profiles. There is a ridge-like structure along the LCFS in the current density profile, with no such structure in the number density profile. The results suggest that forward passing particles are more advantageous in confinement than backward passing particles. This advantage increases with particle energy and contributes to the current generation observed in EXL-50 experiments.

In EXL-50 plasma currents over 100 kA are non-inductively generated and maintained solely by electron cyclotron heating (ECH) power with an efficiency of ∼1 A W −1 . These currents are carried by energetic electrons (EEs) in the energy range from several tens of keV up to several hundreds of keV which also account for almost all pressure in plasma. This EE component can be viewed as a large number collection of various periodic orbits of energetic particles. Based on this picture we have developed a method for particle orbit description of the EE component in a typical plasma at I P = 121 kA as analysis target. We use a fluid description as a bridge to describe successfully the EE component as a collection of various passing and trapped orbits in the approximation of monochromatic particle energy with good matching to the flux loop signals. The description has revealed characteristics of passing and trapped particles. Passing particles carry almost all toroidal current, while they account for only 20% of total particle number of the EE component. While net current carried by trapped particles is a very small fraction, they account for a major fraction in number and carry a large positive current outside the last closed flux surface (LCFS) and a large negative current inside. As a result, trapped particles redistribute the current from inside of the LCFS to outside both radially and vertically, generating a large vertically elongated cross section in current as well as number density profiles. There is a ridge-like structure along the LCFS in the current density profile, with no such structure in the number density profile. The results suggest that forward passing particles are more advantageous in confinement than backward passing particles. This advantage increases with particle energy and contributes to the current generation observed in EXL-50 experiments. * Authors to whom any correspondence should be addressed.
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Introduction
Recently, non-inductive current drive experiments using electron cyclotron (EC) heating power have been successful in a number of low aspect ratio spherical tokamaks (STs) [1][2][3][4][5][6][7]. To develop a solenoid-free current drive capability has been an important research endeavor for tokamaks and the STs. In particular, removing the central solenoid allows additional space to increase the toroidal field, further improving compactness and economy of ST reactors.
In the very recent EXL-50 experiments a large amount of plasma current over one hundred kilo amps is non-inductively started up and maintained solely by ECH using a 28 GHz microwave power over one hundred kilowatts with a high efficiency of I P /P ECH ∼ 1 A W −1 [7] without center solenoid. These plasmas are characterized by the presence of copious energetic electrons (EEs) that carry the plasma current as revealed by the equilibrium analysis using the four fluids model [8]. X-ray measurements suggest that this EE component is in the particle energy range from several tens of keV up to several hundreds of keV [7] while Thomson scattering measurement shows that bulk electron temperature is ≈100 eV [9]. The EE component accounts for a quite large percentage of plasma pressure, being more than 95% of the total plasma pressure, while it accounts for only a small fraction in density. The EE component also carries almost all of the plasma current. Contribution to the current from the bulk component is negligible since the loop voltage is quite low. These characteristics are similar to the plasmas in previous experiments [4,6].
While the current-carrying EEs are generated via EC resonance interaction with the wave, the detailed experimental investigations suggest that conventional EC current drive models by Fisch and Boozer [10] and also by Ohkawa [11] alone do not explain the experimental results [7]. The physics mechanism for the solenoid-free EC-driven current drive in EXL-50 is not yet well understood and quantifiable.
In this circumstance it is useful to have an overall picture of the EE component. Each confined electron of the EE component would make a periodic passing or trapped orbit in the magnetic field. The period is estimated to be in the order of microseconds, being much shorter than the EE collision time (∼1 s) and the characteristic time of plasma development during discharge. Therefore, we can replace the general image of a large number collection of various drifting electrons with the specific picture of a large number collection of various periodic orbits. In addition, the EE component is in equilibrium under the external fields in the sense that all these electrons are making periodic orbits in the field composed of the external field and the self-field from the plasma current solely generated by their periodic drift motions. Possible electrostatic potential may be 1 kV at most since the bulk temperature is ≈100 eV and the effect of the potential on the EE orbits is negligible. Hereafter this situation of collection of orbits is referred to as being in particle orbit equilibrium or simply we call such a collection an equilibrium collection.
The current associated with each orbit generates a tiny magnetic flux and contributes to the signals of flux loops installed around the plasma. Therefore, the flux loop signals would contain information about the collection of periodic orbits.
In this paper we have picked up an EC-driven EXL-50 plasma (Shot 6935, 4.5 s) at I P = 121 kA as a typical sample and have made an attempt to find an appropriate collection of periodic orbits to describe the EE component in this sample plasma in the approximation of monochromatic electron energy. We call this as particle orbit description, and 'appropriate collection' means that the collection is not only in particle orbit equilibrium but also its contribution to the flux loop signals matches well to the experimental observation. Once we find an appropriate collection it is straightforward to derive each contribution to the current profile from passing and trapped particles. This is the same for the pressure and number density profiles. These results would constitute basic information to search for the physics mechanism of the solenoid-free EC-driven current drive in EXL-50 experiments and to design next-step larger scale experiments. The idea of using particle orbits to describe the EE component in low aspect ratio plasmas has previously appeared in a primitive manner [4].
To reach this goal, we start with a fluid description based on an anisotropic pressure model. Note that significant toroidal current is flowing in the open field region outside of the last closed flux surface (LCFS), indicating that the pressure of the EE component is anisotropic. Anisotropic pressure profile can be derived from the current profile estimated using the flux loop signals. If particle orbit description is realized it also provides current and anisotropic pressure profiles. Two sets of profiles from fluid description and particle orbit description would be globally similar even if they are locally not similar. Actually the fluid description turns out to serve as a bridge to the orbit description.
In section 2, parameters and characteristics of the sample plasma are described and a model current profile that matches well to the flux loop signals is presented. In section 3, we derive the parallel and perpendicular pressure profiles of the EE component using the model current profile in section 2.
Here parallel and perpendicular denote the direction to the magnetic field. This current profile is used in section 4 as the zeroth profile for generating various equilibrium collections of orbits via iteration.
In section 4 particle orbit description in the approximation of monochromatic electron energy is attempted. In section 4.1 various characteristics of periodic orbits are presented. In 4.2 formulation of particle orbit description is given. We introduce a mid-plane velocity distribution function by which we relate the fluid quantities on the mid plane to the orbit description. Appropriate discretization of this velocity distribution in pitch angle and in radial position of the orbit crossing point on the mid plane makes it possible to calculate each contribution to current, number density and pressure in poloidal cross section from each discretized orbit. In 4.3 case 1 particle orbit equilibrium is iteratively obtained using the zeroth current profile given in section 2 and a mid-plane velocity distribution derived from the pressure profiles given in section 3 as an initial set of solution for iteration. In 4.4 cases 2 and 3 particle orbit equilibria are obtained using more peaked and more flat current profiles than that in 4.3 as initial solutions for iteration, respectively. In 4.5 characteristics and roles of passing and trapped particles are described. In 4.6 particle energy effect is presented. In section 5 discussions are given. The results are summarized in section 6.

Sample plasma as analysis target
Figure 1(a) shows a visible image of the sample plasma, where line density along the horizontal chord on the mid plane with the tangential radius R = 0.49 m is 1.02 × 10 18 m −2 , indicating a line average density of ≈6 × 10 17 m −3 . Averaged bulk electron and ion temperatures estimated from the four fluid model [8] are 200 eV and 35 eV, respectively. These estimations are consistent with a Thomson scattering measurement to a similar plasma [9] and indicate that the pressure of the bulk components is ≈20 Pa.
Solid square points in figure 1(d) show signals of 39 flux loops surrounding the plasma depicted in figure 1(b). Note that contributions from the external coil currents are subtracted and these signal values are solely from the plasma current. Both the plasma image and the flux loop signals indicate that the plasma has upper and lower symmetry around the mid plane at Z = 0 under the symmetric external field shown in figure 1(e).
In the case of low beta plasmas as the present sample plasma, we can obtain the pressure profile from the profiles of the toroidal plasma current and the poloidal field from the external coil currents. Therefore, we first search for a symmetric current profile from which flux intensities match well to the flux loop signals, using a model formula for the current profile (see appendix in [12]). Figure 1(b) shows an optimal current profile which is obtained by adjusting seven parameters of the formula including major and minor radii R 0 and a, peaking factor α, triangularity δ, vertical elongation κ, radial shift of current peak σ and total toroidal current I p . Open squares in figure 1(d) are flux intensities at the flux loops from this model profile and match well to the experimental ones (solid squares). From this current profile with its self-field and external field we obtain the corresponding pressure profile as described in the next section.

Fluid description as a bridge to particle orbit description
The current profile in figure 1(b) shows that the current at outside of the LCFS is significant, indicating the presence of pressure gradients along the open field lines connected with the vessel wall. The isotropic pressure does not support pressure gradients along field lines. Therefore, we study details of the local equilibrium by using the equation for anisotropic pressure [4] , Here P = P ⊥ I + (P ∥ − P ⊥ )bb is the pressure tensor with I being the unit tensor and b = B/B the unit vector along the field line, and P ∥ = n e m e < γv 2 ∥ > v and P ⊥ = n e m e < γv 2 ⊥ /2 > v where γ is the relativistic factor, ∥ and ⊥ denote the parallel and perpendicular components to the magnetic field, respectively, and < > v the average over the velocity space.
In the cylindrical coordinate system (R, ϕ, Z) with Z axis being the axisymmetric axis of the torus, components of the tensor read P RR = P ⊥ + (P ∥ − P ⊥ ) sin 2 α sin 2 β P Rϕ = P ϕ R = (P ∥ − P ⊥ ) sinα cosα sinβ P RZ = P ZR = (P ∥ − P ⊥ ) sin 2 α sinβ cosβ Here α is the angle between the field line and the toroidal direction, and sinβ = B R /B P and cosβ = B Z /B P with In the present axisymmetric case the radial and vertical components of the equation (1) are approximately given by  Root mean square deviation, rmsdev, versus fitting parameters of α, R 0 , δ, σ and Ip are plotted in (a)-(e), respectively, using gray squares. Black squares denote an optimal set of these fitting parameters used in the profile in figure 1(b). Elongation κ and minor radius a are adjusted so as that the current boundary is in touch with the inboard and top and bottom limiters when R 0 is changed. Red rectangles in (a) denote rmsdev of the current profiles of the particle orbit equilibrium which is generated based on the model current profile having various α with keeping other fitting parameters unchanged.
respectively, by neglecting small correction terms proportional to sin 2 α. By combining these equations with the constraint on currents, we have This relationship reduces to the following equation with and resolves the sum pressure profile into the parallel and perpendicular pressure profiles as shown in figures 3(b) and (c), respectively.

Orbit characteristics
There are three constants of motion in the present axisymmetric case, which are the angular momentum, magnitude of the velocity and the magnetic moment. By taking advantage of this merit, the guiding center orbits are numerically obtained using the same method as that in [12] (see section 3). Various guiding center orbits starting from the mid plane in the field composed of the self field from the model current  Figure 4(a) shows a set of orbits starting with pitch angles from 4 to 176 • on the mid plane from the high field side of the magnetic axis. These are passing orbits and their other crossing points on the mid plane are located in the low field side of the magnetic axis. Figure 4(b) shows the ratio of toroidal current to electron energy (i/w) of each orbit upon one cycle of periodic motion versus pitch angles at the starting point on the mid plane. The ratio i/w is normalized to I P /W ( = 18.74 [A J −1 ]), where I P and W are total current and total kinetic energy of the model plasma shown in figures 1 and 3 for the case of v = 1.65 × 10 8 m s −1 . I P /W is a useful measure to resolve trapped and passing orbits. The orbit with θ = 92 • is mostly forward passing one but drift direction in small part of the orbit is backward during one cycle of periodic motion. While this is a trapped orbit in usual definition of passing and trapped orbits, i/w is much larger than I P /W and horizontal width of the orbit is in the same order as its vertical height, and this orbit is categorized into passing orbits in the present paper. Figure 5(a) shows a set of orbits starting with pitch angles from 4 to 176 • from the low field side of the magnetic axis inside of the LCFS. Orbits with pitch angles from 52 to 140 • are trapped orbits and others are passing orbits. The ratio i/w of trapped orbits is much smaller than I P /W, while i/w of passing orbits is much larger than I P /W. Other crossing points of passing orbits on the mid plane are located in the high field side of the magnetic axis. Figure 6(a) shows a set of orbits starting with pitch angles from 4 to 176 • at outboard side of the LCFS. Orbits with pitch angles from 28 to 136 • are trapped orbits and others are lost orbits. While i/w of trapped orbits is much smaller than I P /W, magnitude of currents from both the forward and backward portions of trapped orbits that extend to the top and bottom of the LCFS is large as shown in figure 6(b).    Figure 8 shows orbits starting from the point slightly higher field side of the magnetic axis and their other crossing points on the mid plane are all distributed on the low field sides of the starting point, which is the same situation as the case in figure 4. Thus there is a watershed radial position on the mid plane at R ≃ 0.61 m, slightly higher field side of the magnetic axis. Orbits starting from higher field side of this position have other crossing points always on the low field sides of their starting points, while orbits starting from lower  field side of this position have other crossing points on both high and low field sides of their starting points. Hereafter this radius is referred to as the change-count radius and denoted by R chng-cnt .
Orbits starting from the radial position at higher field side of R chng-cnt are all passing orbits and referred to as off-axis passing orbits, while passing orbits of which two crossing points are both located in the lower field side of R chng-cnt are referred to as on-axis passing orbits. Two crossing points of any trapped orbit are both located in the lower field side of R chng-cnt .

Formulation of particle orbit description
Electron orbits are labeled by the radial coordinate of the crossing point on the mid plane, magnitude of the velocity v and the pitch angle θ at the crossing point as shown in figures 4-8. We introduce a mid-plane velocity distribution function in order to reflect the current and pressure profiles on the mid plane. The function is characterized by three parameters and reproduces these current and pressure profiles by adjusting these parameters. Thus orbits are distributed in pitch angle and crossing point on the mid plane in the present monochromatic velocity case and after discretization of distribution we trace out each orbit and sum up its contributions to the pressure and current profiles on the poloidal cross section.

Velocity distribution on the mid plane.
We employ spherical coordinate system for the velocity distribution function on the mid plane with v being magnitude of velocity and θ pitch angle, and introduce a simple model of the velocity distribution which has θ distribution in the form of exponentials of sinθ and monochromatic v distribution as follows, for backward range of π /2 < θ < π.
Here A, f and b are functions of R.
Then particle number density is given by where dVol = 2π v 2 sinθdθdv is the differential volume element in velocity space with integrations from θ = 0 to π and v = 0 to ∞. N f and N b are densities of forward and backward particles, respectively, and given by Toroidal current density is given by where e is the elementary charge and Here we define positive direction of toroidal current as the current flowing direction in EXL-50 experiments (see figure 1(b)).
Parallel pressure is given by where Perpendicular pressure is given by where After some algebra we have following two relationships and with the following notation We have current and pressure (J, P ∥ and P ⊥ ) profiles on the mid plane from the fluid description. First we find out numerically the exponents f and b of velocity distribution function from the relationships (13) and (14). Then we can determine the coefficient A of the velocity distribution function (8) by substituting above f and b into any equation for J, P ∥ or P ⊥ (equations (10)- (12)). Thus we obtain the set of three parameters (A, f and b) for velocity distribution at each location on the mid plane that gives the same current and pressure profiles as the fluid results.
Hereafter we drop the subscript zero from v 0 and γ 0 , and write simply as v and γ, respectively. Figure 9(a) shows the current and pressure profiles on the mid plane in the case of fluid description shown in figures 1(b), 3 and 9(b) shows the corresponding parameters A, f and b of the velocity distribution (equation (8)) in the case of v = 1.65 × 10 8 m s −1 .

Discretization of distribution on the mid plane.
For numerical calculation we setup starting cells distributed on the mid plane as shown in figure 10. Each cell is an axisymmetric toroidal ring with a very small square cross section and has one starting point in the center of the cross section. Thus the starting points are discretely distributed along the radial coordinate on the mid plane. In addition we discretize integration over pitch angle as follows.
The number of particles with pitch angles from θ i − ∆θ/2 to θ i + ∆θ/2 in the starting cell ring located at R sc is Here A and f and b take values at R = R sc and V sc = 2π R sc ℓ 2 sc is the volume of the starting cell ring. All these n sci particles drift along the vertical line of orbit in the starting cell and have the pitch angle θ i when they pass through the starting point. Note that the term 'starting' here is only for orbit calculations. All confined orbits are closed as shown in figures 4-8 and there is a steady flow of particles along each closed orbit and at any moment there are n sci particles evenly distributed along the vertical line of orbit in the starting cell.
At this moment it is useful to see figure 11, where grey line denotes a trapped particle orbit and hundred black dots on the orbit denote the particle transient locations of a single particle with the time step of hundredth of the period. The particle proceeds from 0 point to 1, 2, 3, . . . as the time proceeds from 0 to T/100, 2T/100, 3T/100, . . . where T is the period of drift motion of the particles along the closed orbit. We can view this figure another way, that is, we view the hundred black dots as hundred particles on the same orbit. Hundred particles make the same orbit with the time delay of hundredth of the period between the neighboring particles. Distribution of these hundred particles on the orbit looks exactly the same on the orbit when time = 0, T/100, 2T/100, 3T/100, . . .. Suppose thousand particles distributed with the time delay of thousandth of the period between the neighboring particles on the orbit. They would look nearly the same at any moment, representing a steady flow of particles. The number of particles between any two points on the orbit is proportional to the time delay of the drift motion between the two points.

Contribution to current and pressure in poloidal cross
section. Every confined electron makes a periodic orbit on the poloidal cross section. We divide the poloidal cross section into fine square cells numbered as k = 1, 2, 3,. . . which constitute a vessel cell section, as shown in figure 10. Three dimensionally each vessel cell is an axisymmetric cell ring along the toroidal field. We trace out one cycle of the orbit for a particle that has pitch angle θ i on the mid plane, and obtain the period T i , the spans of the time t ik and the toroidal angle ϕ ik upon the traverse through the kth cell ring along the three dimensional orbit. The number of particles and the current in the kth vessel cell ring contributed from the n sci particles in the starting cell are given by Here t sci is the time to traverse vertically across the starting cell and given by is the vertical drift velocity at the starting point. Using the conservation relationship of magnetic moment along the orbit, the perpendicular and parallel pressures in the kth cell ring contributed from the n sci particles in the starting cell are given by respectively, where V k = 2π R k ℓ 2 vc is the volume of the kth vessel cell ring.
In numerical calculations described in the following sections side lengths of square cross sections of starting and vessel cells are ℓ sc = 0.0005 m and ℓ vc = 0.01 m, respectively.

Particle orbit equilibrium from α = 0.45 profile (case 1)
As stated previously we handle the case of mono-EEs and investigate the case of v = 1.65 × 10 8 m s −1 . (w = 100 keV) in this and next subsections. In this subsection we generate an equilibrium collection using the mid plane pressure and current profiles cut out from the profiles in figures 1(b) and 3(a)-(c) as a starting point of iteration. Figure 9(a) shows these profiles and figure 9(b) shows the profiles of parameters A, f and b with which on-mid-plane velocity distribution function (equation (8)

2).
Most unknown factor is the internal current profile, which is represented by the current peaking factor α in the model current formula. The equilibrium collection generated in this subsection as case 1 is from the optimal model profile with α = 0.45. In next subsection equilibrium collections are generated from α = 0.90 and 0.15 profiles as cases 2 and 3, respectively.
Every confined orbit has two crossing points on the mid plane and is categorized into off-axis passing orbits or onaxis passing orbits or trapped orbits as described at the last paragraph in section 4.1. Two crossing points of any on-axispassing orbit and any trapped orbit are close to each other and locate at R > R chng-cnt . High field side crossing point of any off-axis-passing orbit locates at R < R chng-cnt while its low field side crossing point locates at R > R chng-cnt as seen in figures 4-8.
At R < R chng-cnt on the mid plane both the pressure and the current are contributed solely from off-axis passing particles at high field side crossing points. Then the contributions to the number of particles, current and perpendicular and parallel pressures to kth vessel cell from the off-axis passing particles are obtained by summing up the elemental contributions given by equations (17)-(20), respectively, over all discretized elements in pitch angle and radial position located at 0.19 m < R < R chng-cnt ≃ 0.61 m. The contributions to the vessel cells on the mid plane are shown in figure 12(b), which reproduce well the fluid profiles in figure 9(a) at R < R chng-cnt . In the same way the contributions to the number of particles, current and pressure to kth vessel cell from the on-axis passing particles and trapped particles are obtained by summing up the elemental contributions given by equations (17)-(20) over all discretized elements in pitch angle and radial position at R chng-cnt < R < 1.23 m. In this radial range the elemental contributions given by equations (17)-(20) are all multiplied by 1/2 since every orbit has two crossing points and is counted twice in this range in contrast to single count of off-axis passing orbits at the range of R < R chng-cnt . In addition in summing up the elemental contributions those from the off-axis passing particles are omitted since they have been already counted as stated in the previous paragraph. The contributions to the vessel cells on the mid plane from the trapped particles and the on-axis passing particles are shown in figures 12(c) and (d), respectively. Figure 12(a) shows the sum profiles of those in figures 12(b)-(d). In addition to the single count range (R < R chng-cnt ), particle orbit results reproduce well the fluid profiles at outside of the LCFS (R > R lcfs =0.945 m) where only trapped orbits contribute and two crossing points of each trapped orbit is quite close. In the intermediate range (R chng-cnt < R < R lcfs ), the pressure profiles still reproduce the fluid results while the current profile significantly differ from the fluid result.
The information of the current at every vessel cell contributed from all particles brings about the current profile on the poloidal cross section. The total current summed up over all vessel sells is turned out to be I P = 124.55 kA, being somewhat larger than I P = 121.0 kA, the total current of the zeroth profile in figure 1(b). The current density on poloidal cross section is multiplied by 0.9715 so as that total current becomes I P = 121.0 kA and the resulted current density profile is termed as 1st current profile and plotted in figure 13(a), which is quite different from the zeroth profile in figure 1(b) as shown in figure 13(b). This multiplication is equivalent to multiply A of mid-plane velocity distribution function with the same multiplication factor.
The difference of 1st current profile from 0th profile is large and remarkable as shown in figure 13(b). We calculate root mean square of current differences from 0th profile over vessel cells using the following formula and have a large value of rmsjd = 0.2424, Here summation is done over vessel cells where |J k /J peak45 | > 10 −5 for the 0th or 1st profiles. We iterate the same procedure using the field from the 1st current profile for orbit calculations instead of the zeroth profile to obtain 2nd current profile, where multiplication factor used to adjust the total current to be 121.0 kA is 0.9960. When we continue iteration of the same procedure, multiplication factor to adjust total current to be I P = 121.0 kA decreases and finally converges to ≃0.958 after 15 iterations as shown in figure 14(b). In accordance with this convergence RMS of current profile difference between the adjacent rounds of iteration converges to a small value of ≃0.01 as shown in figure 14(a). The 15th current density profile and the difference from 14th profile are plotted in figures 15(a) and (b), respectively. The difference is quite small all over the profile, indicating that the collection of orbits is essentially in particle orbit equilibrium under the external field shown in figure 1(e). In addition RMS deviation of the flux loop signal intensities from this 15th current profile to the observed ones is 0.0540, being as good as that of 0th current profile (0.0507). Figure 16(a) shows kinetic pressure profiles obtained using equations (19) and (20) for this particle orbit equilibrium while figure 16(b) shows fluid pressure profile obtained by applying equations (5) and (7) to the particle orbit current profile shown in figure 15(a). Indeed both results are coincident and surprisingly similar to the fluid pressure shown in figure 3, while the detail of the current profiles are quite different between 0th ( figure 1(b)) and 15th current profiles ( figure 15(a)).

Particle orbit equilibria from α = 0.9 and 0.15 profiles (cases 2 and 3)
Other particle orbit equilibria can be generated starting from other model current profiles using as 0th current profile and using the method described in previous sections. Red rectangles in figure 2(a) denote rmsdev of the current profiles of the particle orbit equilibria generated from the 0th current profiles having various α with keeping other fitting parameters unchanged. While rmsdev of 0th profiles only gradually increases as α is increased from the optimal value of α = 0.45, rmsdev of generated particle orbit current profile strongly increases with α. On the other hand, rmsdev of generated particle orbit current profile keeps low level when α is decreased down to 0.15 at which rmsdev of particle orbit current profile is slightly lower than that of the 0th current profile. Figures 17 and 18 show detailed results of the cases with α = 0.90 and 0.15, respectively. Figure 17(b) shows that vertical elongation of generated particle orbit current profile is significantly smaller than that of 0th current profile in figure 17(a) and the particle orbit current profile lacks of current at the top and bottom regions at outside of the LCFS. As a result matching to the flux loop signals from particle orbit current profile is poor compared with that of the 0th current profile as shown in figures 17(d) and (e). This tendency enhances as α is increased further and the matching to the flux loop signals deteriorates. On the other hand, vertical elongation of generated particle orbit current profile is nearly the same as that of 0th current profile in the case of α = 0.15 as shown in figures 18(a) and (b) and matching to the flux loop signals from particle orbit current profile is slightly better than that of 0th current profile as shown in figures 18(d) and (e).

Passing and trapped particles as current carrier
Let us investigate the detail of the particle orbit equilibrium in case 3 since its particle orbit current profile matches well to the flux loop signals as shown in figure 18.
Profiles of total particle number density, passing particle number density and trapped particle number density in case 3 are plotted in figures 19(a)-(c), respectively. Profiles of total current density, passing particle current density and trapped particle current density in case 3 are plotted in figures 19(d)-(f ), respectively.
Total particle number of the EE component is 4.04 × 10 17 of which 80% are trapped particles. While passing particles account for the remaining 20% of particles, they carry 101.5% of the total current indicating that net current carried by trapped particle is slightly negative in this case as shown in table 1. While net current carried by trapped particles is a very small portion of the total current, they carry a large positive current at outside of the LCFS and a large negative current at inside of LCFS as seen in figure 19(f ) and table 1. As a result, trapped particles redistribute the current from inside of the LCFS to outside both radially and vertically, generating a large vertically elongated cross section both in particle number density profile and current density profile as seen in figures 19(a) and (d), respectively. It is remarkable that there is a ridge-like structure along the LCFS in the current density profile (figures 18(b) and 19(d)), while there is no such structure in the number density profile ( figure 19(a)).
In the sense of net current passing particles carry almost all of the plasma current as shown in table 1. This current is composed of a forward current carried by forward passing particles and a backward current carried by backward passing particles as shown in figure 20. Forward passing particles and current occupy larger area than the LCFS area while backward passing

Particle energy effect (cases 4 and 5)
In cases 4 and 5, we replace the particle energy from 100 keV to 30 keV and to 300 keV, respectively, and corresponding equilibrium collections are obtained by starting from the same α = 0.15 model profile as shown in figures 21(a) and (b) after 14 and 9 iterations, respectively. Let us see particle energy effect on the outside current carried by trapped particles by comparing the results from case 4 (30 keV), case 3 (100 keV) and case 5 (300 keV).
Deviation to flux loop signals are plotted in figures 21(c), 18(e) and 21(d), respectively and their RMS deviations to flux loop signals are 0.0527, 0.0533 and 0.0584, respectively. In the 30 and 100 keV cases the deviations are nearly the same while it somewhat deteriorates in the 300 keV case. This change comes from change of current density at the top and bottom regions at outside of the LCFS and is correlated with the change of percentage of trapped particles in total particle number and also with the change of positive portion of trapped particle current. The former is 80.5%, 79.7% and 76.3% and the latter is 39.7, 38.1 and 32.1 kA, being 32.8%, 31.5% and 26.5% of the total current, in cases 4 (30 keV), 3 (100 keV) and 5 (300 keV), respectively, as listed in table 1. Such a correlation between the matching figure (rmsdev) and magnitude of the positive portion of trapped particle current holds over all cases in table 1.
While the relative number of trapped particles and amount of current at the top and bottom outside regions of the LCFS gradually decrease with the increase of particle energy, trapped particles still redistribute the current from inside of the LCFS to outside even in the 300 keV case. Thus trapped particles work as outside current carrier to increase plasma cross section over a wide range of particle energy from 30 keV to 300 keV.
Contribution of trapped particles to the total current of I P = 121 kA is negligible. The total current is generated from the difference between forward and backward passing particles. The ratios of particle number and absolute current of forward passing particles to backward passing particles increase as the particle energy increases as shown in cases 3-5 in table 1. The ratio in particle number are, 1.5, 2.0 and 3.5 and that in absolute current are 1.4, 1.9 and 3.1 in cases 4 (30 keV), 3 (100 keV) and 5 (300 keV), respectively. Both ratios are nearly the same and increase with energy.

Current profile
The strongly elongated D-shape of plasma image in very low aspect ratio ( figure 1(a)) suggests that the current profile would also have the same outer shape. The employed model formula of current profile matches well to such outer shape and in addition uncovers the internal current distribution in terms of the current peaking factor and radial shift of current peak (figures 2(a) and (d)). Note that while the outer field hardly depends on internal current distribution in the case of circular current cross section in large aspect ratio, the outer field depends on internal current distribution in the case of strongly elongated cross section in low aspect ratio. Thus the current profile turns out to be a very broad one such as shown in figure 1(a). In the present case, however, almost all the current is carried by the EE component in periodic motion. The particle orbit description of these periodic EE component further corrects the current profiles in model formula locally and reveals likely characteristics of the real current profile as described below.
The particle orbit current profiles are locally different from the 0th profiles as shown in figure 18. Both profiles match well to the flux loop signals to the same degree. The particle orbit current profile has a ridge-like structure along the LCFS which may compensate for the lack of current outside the LCFS and gives a good matching to the flux loop signals. The question is which profiles represent well the likely real profile. Present particle orbit analysis shows that the current at outside of the LCFS is carried mainly by trapped particles. Figures 18(a)-(c) indicates that the current density of the trapped particles is much lower than the 0th current profile in at the top and bottom regions of this area. On the other hand the 0th current profile is almost flat from the center to the periphery beyond the LCFS. Therefore not only peaked but also flat current profiles inside the LCFS is not acceptable since 'rmsdev' of such current profiles would significantly deteriorate. Thus, particle orbit analysis suggests that the hollow profile with a ridge-like structure along the LCFS and the lower current density at the top and bottom area outside the LCFS represents a more proper current profile than the 0th profile. In other words, while the 0th profiles in figures 1(b) and 18(a), and the particle orbit profiles in figures 15(a), 18(b), 21(a) and (b) are all appropriate in the sense that they are well-matched to the flux loop signals, the 0th profiles are not proper in the sense that they do not represent characteristics of passing and trapped particle currents shown in figures 19(d)-(f ).

Difference in confinement area of forward and backward passing particles
In the relativistic energy range the confinement area for forward passing electrons is significantly larger than that for backward passing electrons, as shown in figure 20. The difference comes from the difference between the outward and inward shifts of forward and backward passing electron orbits from the lower field side LCFS on the mid plane. The difference in both shifts is derived from conservation of three constants of particle motion and given by depending on the pitch angle at the higher field side LCFS on the mid plane. The former formula is for the case when the pitch angles are close to 0 and 180 • for forward and backward passing electrons, respectively. The latter formula is for the case when they are both close to 90 • . Both formula indicates that the difference increases when aspect ratio decreases and they give nearly the same difference in the present case of very low aspect ratio as seen in figure 4. Here is the poloidal Lamor radius at the lower field side LCFS on the mid plane, and a and R 0 are the minor and major radius of the LCFS, respectively. Thus the difference in confinement area between forward and backward particles increases with the energy. This increase in difference may result in the increase of fractional difference in population and current between the two kinds of particles with energy, as appeared in cases 3-5 in table 1, and may contribute to efficient current generation observed in EXL-50 experiments [7].

Absorption of X wave via harmonic EC resonances
Experimental results suggest that both the bulk and EE components are maintained by EC resonance absorption of 28 GHz power. While the bulk component could be maintained via absorption of X waves at the fundamental [13] and 2nd resonances, it is not obvious that the trapped EE component which locates from the 3rd to 5th harmonic resonance layers   figure 19(c)) can absorb the wave power via such high harmonic resonances. In order to clarify this question we estimate absorption of X-waves that propagate on the mid plane in a model plasma composed of the bulk and energetic components as shown in figure 22. For simplicity we employ relativistic Maxwellian momentum distributions f M (p) to model the energetic component. As shown in figure 22(b) the electron density is set to be a broad profile on the mid plane, n e /n e0 = n b /n b0 = n h /n h0 = (1 − x 6 ) 2 with n e = n b + n h and x = (R − R p )/a p , where R p = 0.71 m and a p = 0.52 m and subscripts b and h denote bulk and high temperature components, respectively and 0 does the flat top value with n e0 = 0.6 × 10 18 m −3 .
While bulk electron temperature is set to be T b = 200 eV, three temperatures of T h = 30, 100 and 300 keV are considered for the energetic component. For simplicity both temperature profiles are set to be uniform. The density fraction of energetic component n h0 /n e0 is adjusted so that total pressure at the flat top P 0 = n b0 T b + n h0 T h is 800 Pa to simulate the sample plasma. The fractions are 0.27, 0.081 and 0.027 for the cases of T h = 30, 100 and 300 keV, respectively. We neglect the ion pressure.
The flat top density is 6% of the plasma cutoff density for 28 GHz waves and X-waves propagate straight except for the vicinity of the cyclotron cutoff layer as shown in figure 22(a). The cutoff layer is very close to the fundamental EC resonance layer to the lower field side and X-waves are reflected at the cutoff layer.
Damping characteristics of wave power flux along the ray for virtual injections with θ inj = ±5.7 • , ±17.4 • and ±27.0 • are plotted in figures 22(c)-(e), respectively. Absorption rate is estimated from the energy gain of resonance electrons via the quasilinear diffusion along the relativistic EC resonance characteristics (see appendix A in [14]) up to the eighth harmonic resonance in the electron momentum space as follows.
where P(ℓ) = Real(E × B * )/2µ 0 is the Poynting vector, that is power flux, and ℓ is the length along the ray, and Γ w = −D · ∇ p f M (p) is the flux in the electron momentum space due to the quasilinear diffusion with the diffusion coefficient Here, and E R and E L are the right handed and left handed circular components of wave electric field, respectively and E ∥ is the parallel component to the magnetic field. Dominant polarization of the oblique X wave is E R . In addition, becomes larger than 1 for energetic resonance electrons at the high harmonic region of ω/Ω ce > 3. Thus the term E R J n−1 (δ) becomes significant at high harmonic region for energetic resonance electrons and oblique X waves are absorbed as shown in figures 22(d) and (e). Here it is noted that the EE component is widely spread in the momentum space and the EE component interact with the wave simultaneously at multi resonance characteristics in momentum space over a wide range around the LCFS and the outside of the LCFS (see, for example, figure 8 in [15]). On the other hand the E R component of O waves is quite small and absorption of O waves is negligible. In this particular shot, two 28 GHz waves were injected from two launchers. The first launcher delivered O waves horizontally on the mid plane at an estimated power of 20 kW from Port 3 with a slightly oblique angle of 10 • to the toroidal direction and opposite to the direction of I p . The second launcher delivered a mixed O and X waves at an estimated power of 120 kW from an elevated position of Port 1 directed at 13.5 • downward in the same toroidal direction as the first launcher (see figure 1 in [7]). The injected waves spread themselves during propagation and figure 22 shows that even in the case of X waves single path absorption is small, suggesting that the wave power would be absorbed via multi reflections with mode-changes between O and X waves on vessel wall and center post. Thus a fractional power may be consumed to maintain the energetic trapped electrons over a region across the high harmonic resonance layers.

Summary
We have introduced particle orbit description for the EE component in a sample plasma at I P = 121 kA in EXL-50. Almost all of the pressure and the toroidal plasma current in this sample plasma is generated and carried by the EE component having the particle energy range from several tens of keV to several hundreds of keV and the contributions from the bulk components is negligible. This collisionless energetic component can be viewed as a large number collection of various periodic orbits. Based on this picture we have developed an iterative method to generate particle orbit equilibria of the EE component in the approximation of monochromatic kinetic energy.
Initially we set a velocity distribution on the mid plane and a 0th toroidal current profile. Each particle starting from the mid plane makes a periodic orbit in the field composed of the self field from the 0th current and the external field from the external coil current. We trace out every orbit, which generates the 1st toroidal current profile. In the next iteration we use the 1st current profile for orbit calculation with the adjustment of velocity distribution on the mid plane to keep the total current to be I P = 121 kA and obtain the 2nd current profile. We continue the iteration until the adjustment and the current profile converge. We use model current profiles adjusted with seven fitting parameters as the 0th current profile. Fluid theory based on anisotropic pressure model generates anisotropic pressure profiles from the self and external fields which give corresponding mid-plane velocity distribution.
Model current profile best matches to the flux loop signals when the current peaking factor is α = 0.45. Various equilibrium collections are generated by starting from model current profiles with various α. Matching of generated equilibrium collections to the flux loop signals strongly deteriorates as α increases from the above optimal value. On the other hand matching of generated equilibrium collections to the flux loop signals is as good as that of the above optimal model current profile when α decreases from 0.45 to 0.15, suggesting that these latter equilibrium collections reflect the characteristics of EEs in the sample plasma as follows.
Passing particles carry almost all toroidal current while they account for only 20% of total particle number of the EE component. While net current carried by trapped particles is a very small portion of the total current, they account for a major fraction in number and carry a large positive current at outside of the LCFS and a large negative current inside the LCFS. As a result, trapped particles redistribute the current inside the LCFS to outside both radially and vertically, generating a large vertically elongated cross section both in number density profile and current profile. There is a ridge-like structure along the LCFS in the current density profile, with no such structure in the number density profile.
Above-mentioned characteristics of trapped and passing particles essentially stay unchanged for the wide range of electron energy from 30 keV to 300 keV, suggesting that trapped and passing particles most likely play the same roles in the actual plasma in which EE energy is distributed according to proper diffusion mechanisms in the velocity-pitch angle space.
Forward passing particles occupy a larger area than the LCFS area in the poloidal cross section, while backward passing particles do a smaller area than the LCFS area. In addition, the number of forward passing particles is larger than the number of backward passing particles. These differences suggest more favorable confinement of forward passing particles over backward passing particles. This difference in confinement increases with particle energy and may contribute to the current generation observed in EXL-50 experiments [7].