An alternative method to mimic mode conversion for ion cyclotron resonance heating

Ion cyclotron range of frequency waves in hot plasmas exhibit spatial dispersion effects and the wave equation takes the integro-differential form. Under the local plasma model assumption, the wave equation can be simplified to the differential form and adapts to the numerical scheme of the finite element method (FEM). Even though direct absorption of fast waves by ions and electrons can be described well by the local plasma model, linear mode conversion associated with non-local effects is absent. To deal with this issue, an alternative method is put forward in this paper where quasi-electrostatic fluid waves based on the multi-fluid warm plasma model are employed to take the place of ion Bernstein waves in mode conversion. On this basis, an interative fluid-kinetics (INTFLUK) code based on the FEM is developed for full-wave simulation in hot plasmas. Derivation of the wave equations as well as benchmarking of the INTFLUK code against other wave simulation codes are carried out. In both one- and two-dimensional cases, the validity of the INTFLUK code was verified by comparison of the wave field distributions and power deposition. As a useful illustration of the INTFLUK code including the scrape-off layer and a realistic antenna, the influence of the poloidal antenna phasing difference on ion cyclotron resonance heating is analyzed. Finally, it should be noted that the method in this paper has the potential to be extended to the three-dimensional case, which will be considered in the near future.

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Introduction
Ion cyclotron resonance heating (ICRH) is an efficient auxiliary heating scheme [1][2][3] which can heat target ions directly at any desired specific position in tokamaks with controllable parameter settings.This method is expected to play an important role in ion heating for future magnetic confinement fusion demonstration reactors [4,5].Full-wave simulations in the ion cyclotron range of frequency (ICRF) play an important role in the exploration of heating mechanisms [6,7] and experimental guidance [8,9].According to kinetic theory, ICRF waves in hot plasmas exhibit spatial dispersion effects and the wave equation has an integro-differential form.In order to effectively solve the full-wave equation in hot plasmas, Fourier decomposition of the wave fields is necessary.For example, the TORIC code employs spectral representation of the wave fields in the poloidal and toroidal directions while applying the finite element method (FEM) in the radial direction [10][11][12][13].The AORSA code applies the Fourier collocation method in three directions (horizontal, vertical and toroidal) [14].Such programs take the full spatial dispersion effects into account and can describe absorption by ions that occurs at the fundamental and harmonic of the cyclotron frequency, linear mode conversion between fast waves (FWs) and slow waves (SWs) and absorption by electrons through Landau and transit time magnetic damping.However, a realistic scrape-off layer (SOL) and complex geometric boundaries at the plasma edge are not easy to implement in these codes due to the numerical solution scheme used for spectral representations.The HISTORIC code represents an attempt to extend the simulation region to the plasma edge [15,16].
In order to take complex geometric structures into account, FEM-based codes [applying the FEM for both directions or at least two directions inside the two-dimensional (2D) simulation plane] represent a possible way forward.Under the local plasma model assumption, the wave equation can be simplified to the differential form and adapted to the numerical scheme of the FEM.On this basis, codes such as LION [17] and FEMIC [18] apply the FEM in radial and poloidal directions.These codes with a fast running speed can accurately simulate FW propagation and direct absorption.However, linear mode conversion associated with non-local effects (finite Larmor radius effects) would be absent.The addition of non-local effects to these FEM-based codes represents a problem.In recent years, some efforts have been made to solve this.Vallejos came up with an idea that applies wavelet decomposition to add the non-local effects iteratively to the FEM-based code [19].Despite the iterative numerical solution scheme being very time-consuming, this method has been successfully verified in the one-dimensional (1D) case.However, it is hard to extend the methodology to the 2D case.Bude converted the integrodifferential form of the ICRF wave equation into a high-order differential form by fitting a polynomial through the dielectric.This form retained the non-local effects and can be applied to 1D FEM-based codes [20,21].However, it is cannot be adapted to the 2D case because Fourier decomposition of the wave fields is still inevitable.An alternative way to consider the non-local effects in both 1D and 2D FEM-based codes is studied in this paper.
Figure 1 shows the dispersion relations obtained from the kinetic plasma model and multi-fluid warm plasma model [22] for D-(H) plasmas.As can be seen, mode conversion associated with non-local effects naturally exists for both models and the positions of the mode-conversion layers coincide.Meanwhile, the dispersion relation of kinetic FWs (the blue line) almost overlaps with FWs based on the multi-fluid plasma model (the black line), except for a slight deviation near the mode-conversion layer.Taking into account the quasielectrostatic properties of the mode-converted ion Bernstein waves (IBWs) obtained from the kinetic plasma model, quasielectrostatic fluid waves (QEFWs) based on the multi-fluid plasma model have the potential to mimic the propagation of IBWs.Nevertheless, an issue still exists because the dispersion relation curve of QEFWs (the magenta line with γ = 1) stretches from the mode-conversion layer to the low-field side rather than to the high-field side as for propagating IBWs.We therefore carefully analyzed the dispersion relation of QEFWs and found that the stretching direction can be reversed without influencing the FWs when altering the adiabatic coefficient from a positive value to a negative value.Furthermore, the dispersion relation curve of QEFWs (the magenta line with γ = −0.5)can approach that of IBWs by setting the negative adiabatic coefficient appropriately.The imaginary part of FWs and QEFWs based on the multi-fluid model is zero in the propagation region because no dissipation mechanism is included.Although an arbitrary or negative adiabatic coefficient is physically meaningless for the multi-fluid plasma model, it supplies the equivalence to some degree in the maths allowing QEFWs to take the place of IBWs in mode conversion.Therefore, in this paper we use this method to consider non-local effects under a FEM framework and develop an interative fluid-kinetics code (INTFLUK).
The paper is arranged as follows: derivation of the wave equations is given in section 2; 1D benchmarks of the INTFLUK code against SEMAL and FEMIC codes are shown in section 3; 2D benchmarks of the INTFLUK code against TORIC and FEMIC codes are shown in section 4; a useful illustration of the INTFLUK code including the SOL and a realistic antenna is given in section 5; finally, the conclusions are drawn in section 6.

Wave equations in warm multi-fluid plasmas
Our objective is to apply a QEFW to take the place of an IBW near the mode-conversion region.Therefore, we start from multi-fluid equations as follows: where the subscript j refers to the species number, n j is the number density, u j is the fluid velocity, q j is the species charge, m j is the particle mass, P j is the fluid pressure and ρ j = m j n j is the mass density.The gas state equation is d dt where γ j is the adiabatic coefficient.Linearizing the equations using f = f 0 + f 1 and f 1 ∼ e iωt , we obtain the linear steady state fluid equations where the subscripts 0 and 1 refer to the undisturbed equilibrium part and linear disturbed part, respectively.The undisturbed electric field E 0 is 0 as a result of the quasi-neutral condition in the plasma and therefore E = E 1 .ω is the angular frequency, T j0 is the undisturbed fluid temperature, k B is the Boltzmann constant and B 0 is the background magnetic field.
According to the motion equation ( 7), the fluid velocity can be expressed in terms of the electric field and fluid pressure as where ω cj = q j B 0 /m j is the cyclotron angular frequency and b = B 0 /B 0 is the unit vector along B 0 .
The Helmholtz equation can be obtained from equations (3) and ( 4).Meanwhile, we divide the linear electric current into the antenna current J ant and the plasma current The plasma electric current can be obtained from equations ( 8) and ( 9) directly as The first term in the bracket defines the propagation of a FW without dissipation, while the second term introduces the thermal pressure correction which describes mode conversion between a FW and slow QEFW near the hybrid resonance layer.In the coordinate system of the Stix frame [23], where the magnetic field B 0 is aligned along the z direction and the wave vector k is placed in xoz plane, the susceptibility tensor κ j of the jth plasma component is given by Λ j is related to the fluid thermal pressure correction and is given by where ω pj is the plasma angular frequency and ω cj is the cyclotron angular frequency.The linear disturbed plasma density in equation ( 11) can be obtained from the continuity equation (6), ) . (14)

Wave equations in the INTFLUK code
Solving the equation set ( 10)-( 14) can give the wave solutions containing mode conversion between FWs and slow QEFWs.By adjusting the adiabatic coefficient γ j , a slow QEFW can mimic the propagation of an IBW, as shown in figure 1.However, other kinetic effects relevant to FW absorption are not considered.The susceptibility tensor in equation (12) based on the fluid model can only define the propagation of a FW without dissipation.Hence, we modify the equation set as follows.Firstly, the fluid susceptibility tensor κ j in equation ( 11) is replaced with the kinetic susceptibility tensor χ j to obtain equation ( 16).Secondly, the fluid susceptibility tensor κ j in equation ( 14) is replaced with the Hermitian part of the kinetic susceptibility tensor χ j H to obtain equation ( 17).As a result, the equation set governing both direct FW absorption and linear mode conversion is introduced.A problem that may arise in the simulation is the numerical pollution of n j owing to the drastic variation of χ j H near the fundamental cyclotron resonance layer for species j.The way to cope with this is simply to force the linear disturbed plasma density n j to approach zero near the fundamental cyclotron resonance layer for species j.It is also worth noting that the wave equation set here will degrade to the FEMIC circumstance [18,24] if the second term in the bracket of equation ( 16) is neglected ) ) . ( Similar to the FEMIC code [18], we use a quasihomogeneous approximation of the susceptibility tensor.It should be pointed out that equations ( 14) and ( 17) are not explicit expressions of the density perturbations.Therefore, we apply the multi-physics coupling modules of COMSOL software to solve the equation set numerically and obtain self-consistent solutions.The susceptibility tensor for a hot Maxwellian plasma in a Cartesian coordinate system of the Stix frame [23] with the assumption k y = 0 is given by The elements of the susceptibility tensor χ j are given by where n is the harmonic number, ε j is the charge sign, q j is the species charge, λ j = k x 2 ρ L, 2 j /2 is the FLR parameter, ρ L,j is the Larmor radius, v j is the thermal velocity, I n = I n (λ j ) is the modified Bessel function of order n and Z(ζ n,j ) is the plasma dispersion function evaluated at ζ n,j = (ω + nω cj )/(k z v j ).The prime denotes a derivative.The perpendicular wave number of the FW can be calculated in advance by where n || is the main parallel component of the refractive index and K xx and K xy are the elements of the dielectric tensor K with The local absorption from the individual species is calculated approximately by where χ j A is the anti-Hermitian part of the susceptibility tensor for species j.

Physical model descriptions
In this section we compare the simulation results of the 1D INTFLUK code, the SEMAL code [25,26] and the 1D FEMIC code.
The SEMAL code solves the electromagnetic wave equations to all orders of the Larmor radius and takes the non-local effect into account for hot plasmas with Maxwellian equilibrium distribution functions [26].One assumes a slab geometry with inhomogeneity along the x direction and equilibrium magnetic field B 0 along the z direction.A single parallel wave number k z and poloidal wave number k y need to be chosen in the SEMAL code for 1D simulations.
In order to match the simulation conditions of the SEMAL code as much as possible, a slab geometry without curvature and identical coordinate system are adopted in both INTFLUK and FEMIC codes.Referring to the EAST tokamak for parameter settings, the major radius is 1.85 m, the minor radius is 0.45 m and the central magnetic field strength is 2.5 T. The plasma is composed of electrons, majority D ions and minority H ions.The plasma parameter profiles are shown in figure 2. The ICRF antenna strap is fixed at x = 2.35 m on the low-field side.In order to avoid numerical pollution of the INTFLUK code (as pointed out in section 2.2), the linear disturbed plasma density n j is forced to approach zero near the fundamental cyclotron resonance layer for species j and near the plasma boundaries.The mesh size in the x direction is less than 5 mm.

Results and discussions
The electric field distributions calculated by the SEMAL (top row), INTFLUK (second row) and FEMIC codes (bottom row) are all shown in figure 3. The first, second and third columns show E x , E y and E z , respectively.The wave frequency is 37 MHz, the parallel wave number k z is 13 rad m −1 and the poloidal wave number k y is 0 rad m −1 .In order to observe a significant mode-conversion effect, the minority concentration η is chosen to be 20%.The adiabatic coefficient in the INTFLUK code is set to −0.5 (γ e = γ H = γ D = −0.5).Compared with IBWs (shortwavelength signals near x = −0.19m) obtained using the SEMAL code, the INTFLUK code gives similar field patterns for the mode-converted waves.Coincidence of the modeconversion region between the SEMAL and INTFLUK codes can be observed.These results are consistent with the dispersion relation in figure 1.Furthermore, short-wavelength signals only occur in E x and E z because IBWs in the SEMAL code and slow QEFWs in the INTFLUK code are both quasielectrostatic modes.This is an important reason why QEFWs in the INTFLUK code can mimic the behavior of IBWs.The different field patterns in the INTFLUK and SEMAL codes can be attributed to the phase difference originating from the different physical models.This has no substantial influence on the power deposition.On the other hand, the FEMIC code can simulate FW propagation and absorption based on the local plasma model.The non-local effect and mode conversion are excluded.Hence, there is a singular point in E x and E z near the mode-conversion layer for the FEMIC code.However, the INTFLUK code removes the singular point by the fluid thermal correction.This illustrates that the INTFLUK code makes an effective correction near the mode-conversion layer without changing FW propagation compared with the FEMIC code. Figure 4 shows the power deposition profiles obtained from the three codes.H ion and D ion absorption obtained from the three codes are similar.The discrepancy occurs in the electron absorption profiles.The electron absorption profile given by the FEMIC code is thin and sharp near the mode-conversion layer due to the existence of the singular point.Taking into account the mode-conversion effect, the electron absorption calculated by the INTFLUK code is broadened near the modeconversion layer and more similar to that using the SEMAL code.

Physical model descriptions
The formalism of equations ( 15)-( 17) also applies to the 2D situation.In this section we compare the simulation results between the 2D INTFLUK code, the TORIC code [12] and the 2D FEMIC code.
The TORIC code solves the finite Larmor radius ICRF wave equations in axisymmetric toroidal geometry.Based on the finite Larmor radius approximation to second order,  TORIC describes linear mode conversion between FWs and IBWs.Absorption by ions occurs at the fundamental and first harmonic of the cyclotron frequency, and by electrons through Landau and transit time magnetic damping.The TORIC code employs a spectral representation of the wave fields in the poloidal and toroidal directions while applying the FEM in the radial direction.With this in mind, the TORIC code can only simulate wave propagation and absorption inside the last closed flux surface (LCFS) with simple geometric boundaries.The extension of TORIC to take into consideration a realistic SOL and complicated antenna structures is a subject worth further attention.
In view of the fact that the kinetic susceptibility tensor in equation ( 18) has a strong dependence on the magnitude and a weak dependence on the phase of the perpendicular wave vector k ⊥ , the phase effect of k ⊥ , which causes rotational transformation of the susceptibility tensor around an equilibrium magnetic field B 0 for different poloidal modes, has been neglected in the INTFLUK and FEMIC codes.The benefit of this treatment is that the spectral representation of the wave fields in the poloidal direction can be omitted.Both the radial and poloidal directions apply the FEM directly in the INTFLUK and FEMIC codes.Hence, the superiority of the INTFLUK and FEMIC codes is their ability to perform simulation with a realistic SOL and complicated geometric constructions.Nevertheless, the validity of the INTFLUK code is our main objective, and the simulations are only carried out inside the LCFS in this section.In order to match the simulation conditions of the TORIC code, a cylindrical coordinate system (R, φ, Z) is adopted in the INFLUK and FEMIC codes.Therefore, the solution of the wave equation has the form where E is the complex amplitude and n φ is the toroidal mode number.The transformation of the tensor in equation ( 16) from the Cartesian coordinate system of the Stix frame to the cylindrical coordinate system is given as For simplicity, the poloidal magnetic field is neglected and the magnetic field is in the negative φ direction.Assuming that the perpendicular wave vector is in the radial direction, the transformation matrix R can be given as follows: Neglecting the up-and downshift effects, the parallel wave number in toroidal geometry can be calculated by The analytical poloidal magnetic flux surface function ψ = ψ(R, Z), referring to the EAST tokamak for parameter settings, is adopted in all three codes.The major radius is 1.85 m, the minor radius is 0.45 m, the ellipticity of the LCFS is 1.75, the triangularity of the LCFS is 0.45 and the Grad-Shafranov shift is 0.05 m.The center of the antenna strap is positioned at 2.35 m and the strap length is 0.754 m.The plasma is composed of electrons, majority D ions and minority H ions.The plasma parameter profiles are shown in figure 5.The central magnetic field strength is 2.54 T and the antenna frequency is 37 MHz to achieve on-axis minority H ion heating.The toroidal mode number is 25.The adiabatic coefficient in the INTFLUK code is −0.5 (γ e = γ H = γ D = −0.5).The linear disturbed plasma density n j is forced to approach zero near the fundamental cyclotron resonance layer for species j and near the LCFS in the INTFLUK code.The mesh size in x and y directions is less than 5 mm.The total mesh number is about 200 000 in 2D simulations.The model applied here has 2 million degrees of freedom and requires approximately 40 GB of RAM for the direct solver.The time spent on each run is less than 3 min.

Results and discussion
The dependence of the power partitions among different species on the minority concentration η is shown in figure 6.The fraction of power absorbed by each plasma species obtained from the INTFLUK code agrees with the TORIC code to within almost 5%.The larger simulation disparity between the two codes occurs near η = 20% and η = 45%, where the calculation error is as large as 7%.Taking into account the    correction.The poloidal magnetic field as well as the up-and downshift effects are excluded in both codes for simplicity.Therefore, the short-wavelength signals represent IBWs rather than ion cyclotron waves (ICWs) for the TORIC code.The difference in the field patterns between INTFLUK and TORIC codes exists in the mode-converted waves near the LCFS where the field strength given by INTFLUK is somewhat higher than that using TORIC.This may be attributed to the significant fringe effect of the antenna model in the INTFLUK code.
The flux surface-averaged power absorption profiles with different minority concentrations η are shown in figure 10.As can be observed, the absorption profiles of each plasma species calculated with the INTFLUK code (first line) agree well with the results for the TORIC code (second line).The modeconversion effect becomes significant and electron absorption (red lines) increases with increasing η for both codes.It should be noted that electron absorption consists of FW dissipation and the self-loss of IBWs with the TORIC code.The INTFLUK code only considers the former and neglects the self-loss of mode-converted waves because the tensor in equation ( 13) related to the fluid thermal correction is a Hermitian matrix.Nevertheless, neglecting the self-loss of QEFWs does not lead to too much calculation error, as shown in the results.
The 2D distributions of electron absorption with η = 20% obtained from the three codes are shown in figure 11.There is a sharp absorption band coinciding with the mode-conversion     In order to validate the universality of the INTFLUK code, we also carried out benchmarking for H-(He-3) plasmas and D-H-(He-3) three-ion plasmas against the TORIC code as shown in figure 13.The central magnetic field strength is 2.8 T and the antenna frequency is 27 MHz to achieve onaxis minority He-3 ion heating.The toroidal mode number is 25.The adiabatic coefficient in the INTFLUK code is set to be 0.1 and −0.1 for the H-(He-3) plasma case and D-H-(He-3) plasma case, respectively.Figure 13(a) shows the power partition among different species as a function of He-3 ion concentration η.The inverse minority heating scenario gradually switches to the mode-conversion heating scenario when η changes from 1% to 10%.The fraction of power absorption by each plasma species calculated by the INTFLUK code agrees with TORIC code within 5%. Figure 13(b) gives the results for the D-H-(He-3) three-ion plasma case.The concentration of He-3 ions is fixed to 0.1% and the H ion concentration η ranges from 50% to 90%.The maximum fraction of He-3 ion absorption occurs at η = 70% where the ionion hybrid resonance layer of D and H ions coincides with  the fundamental cyclotron resonance layer of He-3 ions [27].The calculation error between INTFLUK and TORIC codes is less than 10% in this case.Even though the self-loss of modeconverted QEFWs is not considered in the INTFLUK code, the calculation error can still be controlled within a certain range by choosing appropriate adiabatic coefficients for the different plasma compositions.
Several principles for choosing the adiabatic coefficient in practical application of the INTFLUK code can be summarized as follows after a lot of program tests.The absolute value of the adiabatic coefficient |γ| often ranges from 0.1 to 0.5 and need not be too precise.The uncertainty of |γ| in this range affects the power partition among different species within a few per cent and has almost no influence on the power deposition profiles.However, the sign of γ is important.For a normal minority heating scenario, such as D(H) plasmas, γ is chosen as a negative value.For an inverse minority heating scenario, such as H(He-3) plasmas, γ needs to be a positive value.The three-ion heating mechanism can be considered as a special case of the normal minority heating scenario and γ is naturally set to be a negative value.

Influence of the poloidal antenna phasing difference
The advantage of the INTFLUK code is its ability to make a simulation with a realistic SOL and a complicated geometric construction.Inclusion of the SOL is important for analyzing effects such as wave coupling, surface waves around the LCFS and mode conversion in the SOL [24].The 2D full-wave simulation model for EAST is shown in figure 14.Previously, this model has been applied to study the influence of plasma density perturbations on ICRH [28].The complex structures of the first wall and antenna straps are retained.The realistic magnetic surfaces obtained from the EFIT code are represented as red lines.The first wall, divertor and vessel wall of the antenna, which are represented by black lines, are assumed to be perfect electrical conductors.The current straps are positioned within the vacuum antenna box and are represented by green lines.The antenna straps are composed of an upper loop and a lower loop.The phase difference ∆Φ pol between them can be  adjusted on EAST.As a useful illustration of the INTFLUK code, the influence of the poloidal antenna phasing difference on the ICRH scenario, in which mode conversion is important, will be studied in this section.The simulation was carried out in D-(H) plasmas.The central magnetic field strength is 2.54 T and the antenna frequency is 37 MHz.In this case, the minority concentration is 30% and mode conversion plays an important role.The toroidal mode number is chosen to be 25 so that the up-and downshift effects can be neglected.
The dependence of the power partitions of different species on the poloidal antenna phasing difference ∆Φ pol is shown in figure 15.The fraction of electrons absorbed ranges from 50% to 80% when ∆Φ pol is adjusted between −180 • and 180 • .The maximum fraction of electrons absorbed is achieved at ∆Φ pol = 140 • .The asymmetry of the curve with respect to ∆Φ pol = 0 • is mainly caused by the anisotropy of the magnetized plasma.To investigate the reason for the dramatic variation of electron absorption with ∆Φ pol , field distributions of the left-handed electric field component are displayed in figure 16.As can be seen, the left-handed electric field component of FWs changes significantly with ∆Φ pol .Taking into account the lower single-pass absorption of FWs in this case, the phenomenon can be explained by the fact that the field distributions formed by multiple reflections are more sensitive to ∆Φ pol .Furthermore, the field intensity of mode-converted waves also varies with ∆Φ pol .Finally, as shown in figures 17 and 18, different electron and ion absorption with different ∆Φ pol can be achieved.

Conclusions
The similarities between IBWs and QEFWs can be summarized as: (1) the mode-conversion layers coincide with each other; (2) the dispersion relation of QEFWs can approach that of IBWs by setting an appropriate adiabatic coefficient; (3) a quasi-electrostatic property is fulfilled.Hence, in this paper we put forward an alternative method to employ QEFWs to take the place of IBWs in mode conversion.On this basis, the INTFLUK code based on the FEM is developed for full-wave simulations.Derivation of the wave equations and benchmarking the INTFLUK code against other wave simulation codes are carried out.
In the case of a 1D slab D-(H) plasma, we compare the calculation results between the INTFLUK, SEMAL and FEMIC codes.Compared with the FEMIC code based on the local plasma model, the INTFLUK code can describe mode conversion effectively and gives more similar field patterns and power deposition to the SEMAL code in the mode-conversion region.In 2D axisymmetric D-(H) plasmas, benchmarking of the INTFLUK code against the TORIC code was carried out.The fraction of power absorbed by each plasma species obtained by the INTFLUK code agrees with the TORIC code to within almost 5%.The dependence of field patterns on the minority concentration is rational and agrees well with the TORIC code.In contrast to the FEMIC code, INTFLUK effectively removed the singularities near the mode-conversion layer by fluid thermal correction.In order to validate the universality of the INTFLUK code, benchmarking for H-(He-3) plasma and D-H-(He-3) three-ion plasmas against the TORIC code was carried out.Even though the selfloss of mode-converted QEFWs is neglected in the INTFLUK code, the calculation error can still be controlled within a certain range by choosing an appropriate adiabatic coefficient for the different plasma compositions.As a useful illustration of the INTFLUK code including the SOL and a realistic antenna, the influence of the poloidal antenna phasing difference on the ICRH scenario was analyzed.Electron absorption varies dramatically with the poloidal antenna phasing difference for the case of low single-pass absorption.This implies that adjusting the phasing difference of the poloidal antenna can help to achieve a mode-conversion heating scenario.The poloidal magnetic field is neglected in 2D models, therefore the mode-converted waves are limited to IBWs, rather than ICWs.Finally, it should be noted that the method in this paper has the potential to be extended to the 3D case, and this will be considered in the near future.

Figure 1 .
Figure 1.Dispersion relations in D-(H) plasmas obtained from the multi-fluid plasma model and the kinetic plasma model under identical plasma parameter conditions.The minority concentration is 15%.γ (γe = γ H = γ D = γ) represents the adiabatic coefficient of a multi-fluid warm plasma.

Figure 2 .
Figure 2. 1D profiles of the electron density ne and temperature Te.The ion temperature T i = Te.

Figure 3 .
Figure 3. Electric field calculated by the SEMAL (top row), INTFLUK (second row) and FEMIC codes (bottom row).Real parts are drawn in blue and imaginary parts in red.The dashed lines mark the mode-conversion layer.The parallel wave number kz = 13 rad m −1 .

Figure 5 .
Figure 5. 1D profiles of the electron density ne, electron temperature Te and ion temperature T i .

Figure 6 .
Figure 6.Power partition among different species as a function of H ion concentration η.Solid lines represent the results calculated with the INTFLUK code and dashed lines represent the results calculated with the TORIC code.

Figure 7 .
Figure 7. Dependence of the left-handed electric field component on H ion concentration η obtained from the INTFLUK code.The toroidal mode number is 25.

Figure 8 .
Figure 8. Dependence of the left-handed electric field component on H ion concentration η obtained from the TORIC code.The toroidal mode number is 25.

Figure 9 .
Figure 9. Left-handed electric field component calculated by the TORIC (a), INTFLUK (b) and FEMIC codes (c).The toroidal mode number is 25 and the H ion concentration η is 20%.

Figure 10 .
Figure 10.Flux surface-averaged power absorption profiles calculated with the INTFLUK (first line) and TORIC codes (second line).The toroidal mode number is 25 and the total absorbed power is normalized to 1 W.

Figure 11 .
Figure 11.Electrons absorption calculated by the TORIC (a), INTFLUK (b) and FEMIC codes (c).The toroidal mode number is 25 and the H ion concentration η is 20%.The total absorbed power is normalized to 1 W.
Figure 12  shows the corresponding flux surfaceaveraged power absorption profiles.Combined with figure11, electron absorption given by the INTFLUK code is more similar to the TORIC code results.Even so, a discrepancy exists in ion absorption between INTFLUK and TORIC codes in this case.The reason may be the difference in the kinetic physical models.

Figure 12 .
Figure 12.Flux surface-averaged power absorption profiles calculated with the TORIC (a), INTFLUK (b) and FEMIC codes (c).toroidal mode number is 25 and the H ion concentration η is 20%.The total absorbed power is normalized to 1 W.

Figure 13 .
Figure 13.Power partition among different species as a function of He-3 ion concentration η in H-(He-3) plasmas (a), and as a function of H ion concentration η in D-H-(He-3) plasmas (b).Solid lines represent the results calculated using the INTFLUK code and dashed lines represent the results calculated using the TORIC code.

Figure 14 .
Figure 14.(a).Geometry of the EAST cross section.The first wall, divertor and vessel wall of the antenna box are represented as black lines (perfect electrical conductors), magnetic surfaces are represented as red lines, the current straps are represented as green lines (source boundary) and the Faraday screen is represented as a black dotted line in front of the current straps.(b).Plasma parameter profiles including the SOL.

Figure 15 .
Figure 15.Power partition among different species as a function of the poloidal antenna phasing difference ∆Φ pol .The toroidal mode number is 25 and the H ion concentration η is 30%.

Figure 16 .
Figure 16.Dependence of the left-handed electric field component on the poloidal antenna phasing difference ∆Φ pol .The toroidal mode number is 25 and the H ion concentration η is 30%.

Figure 17 .
Figure 17.Dependence of the electron absorption on the poloidal antenna phasing difference ∆Φ pol .The toroidal mode number is 25 and the H ion concentration η is 30%.The total absorbed power is normalized to 1 W.

Figure 18 .
Figure18.Dependence of the flux surface-averaged power absorption profiles on the poloidal antenna phasing difference ∆Φ pol .The toroidal mode number is 25 and the H ion concentration η is 30%.The total absorbed power is normalized to 1 W.