Broadening of electron cyclotron power deposition and driven current profiles caused by dissipative diffractive propagation

Improvements in electron cyclotron resonance heating (ECRH) and current drive (ECCD) predictions are important issues for the design and control of high-performance fusion plasmas in future devices, where these should play a more important role as actuators than in devices to date. A newly developed EC-prediction package based on the quasioptical ray tracing code PARADE revealed in JT-60SA that (i) the radial profiles of both EC power deposition and driven current are broadened and (ii) the net driven current is increased by a few kA/MW, in comparison with conventional predictions due to dissipative diffractive propagation (DDP). The mechanism of DDP is as follows: EC wave beam obliquely passing through the resonant surface is dissipated non-uniformly on its beam cross section, so that the beam trajectory shifts gradually and thus the resonant position also shifts, resulting in the broadened power deposition profile. This novel ECCD and ECRH prediction package based on PARADE is applicable not only to JT-60SA but other existing devices and even, future devices.


Introduction
Electron cyclotron resonance heating (ECRH) [1][2][3] and current drive (ECCD) [4][5][6][7] have a great advantage over other techniques, the quite local plasma heating and current drive with high power density in both real and velocity space.This phase space locality makes ECRH/CD an essential tool for fusion science.In the context of fusion plasma physics, due to the fact that ECRH/CD can selectively impart wave energy and momentum to electrons in a particular position with a particular velocity, it is expected to play the role of an actuator for highly functional control of instabilities and transport in plasmas.Neoclassical tearing mode suppression, q-profile control, and various other sensitive operations demand the precise prediction of EC power deposition and driven current profile in physical phase space [8][9][10][11].On the other hand, in the context of fusion reactor engineering, due to the fact that the port area required for the high power injection is quite small, it is even expected as the main heating and current drive source for fusion DEMO reactors [12][13][14], where large blanket area and high neutron shielding capability are required.Antenna position, steering range of injection angles, gyrotron frequency and various other designs for the ECRH/CD system also demand precise prediction of EC power deposition and driven current profile.Thus, in this paper, we report the improvement of predictions of ECRH/CD that should play a more important role as actuators in future devices than in devices to date as presented above.
It is known that ECRH/CD predictions strongly depend on the propagation and dissipation model of radio-frequency (RF) waves for ECRH/CD (or we simply call it EC waves).Hence, various quasioptical models have been developed to correctly account for the finite structure of the propagating and dissipating RF wave beam field in inhomogeneous dielectricmedia, and have been applied to ECRH/CD predictions [15][16][17][18][19][20][21][22][23][24][25][26], instead of geometrical optics (GO) [27][28][29][30].Nevertheless, these models still leave room for improvement, that is, dissipative diffractive propagation (DDP).While the quasiopticalization of ray tracing enables us to consider the evolution of RF wave intensity and phase structures in the beam cross section in the propagation direction, early models have evaluated the dissipation uniformly over the cross section [15][16][17][18][19][20][21][22].The dissipation coefficient evaluated on the central ray (or reference ray, see section 2.1) decreases the wave intensity profile of the cross section at the same rate.This conserves the structure of the field, e.g.Gaussian beams retain those Gaussian intensity and phase structures even after passing through the resonant dissipation layer.However, the actual RF wave beam, which experiences spatially inhomogeneous resonant dissipation in a realistic fusion plasma, should decay non-uniformly, and its beam structure will also be distorted in the beam cross section.If these distorted intensity and phase profiles evolve asymmetrically across the transverse direction of propagation, and then the center of these profiles gradually deviates from the conventional prediction, i.e. the central ray trajectory.This DDP is difficult to describe in early models but was first described in Balakin's quasioptical model [23][24][25][26], which is, however, a reduced model based on series expansion.
Here, the key to capture such wave beam structures more correctly is the quasioptical ray tracing code PARADE [14,[31][32][33][34][35][36], which describes wave beam propagation within an arbitrary quasioptical envelope in inhomogeneous anisotropic media, accounting for refraction, diffraction, polarization (or mode-coupling), and dissipation, simultaneously, within a reasonable computational resource.While various quasioptical codes have been proposed in the literature, PARADE has the advantage of taking into account the non-uniformity of the dissipation in the beam cross section more strictly than [26].(Although not discussed in this paper, the ability to handle mode coupling [33,36,37] is the most important feature of PARADE.)So far, we have discussed the derivation of the basic theory in [31], its application to single-mode waves in [32], its application to mode-converting waves in [33], and the more rigorous non-uniform dissipation than [26] in [34].Then, relativistic hot dispersion models were introduced in [14], and the simulation results given by PARADE were compared with the experiments in [35], where it was validated that our model qualitatively reproduces the experiments.Next, in this work, we develop new quasioptical modules to predict EC absorption power and driven current, link them to the PARADE code, and construct the ECRH/CD prediction package.Using the EC package based on PARADE, which is a new alternative to various EC models [17,22,26,38], it is expected to predict the quasioptical EC absorbed power and driven current profiles more realistically considering the DDP more strictly.Then, we apply this package to JT-60SA tokamak [39] and discuss the prediction results.
Our paper is organized as follows.In section 2, we briefly overview the theoretical models underlying PARADE and the newly developed adjoint current module.In section 3, we report on the applications of our quasioptical EC simulation package to JT-60SA and discuss the impact of DDP on ECRH/CD predictions.Finally, in section 4, we summarize our main results.

Models
In this section, we briefly overview the several models, which are used in the main code PARADE and the sub-modules -C 3 (-cyclotron current calculation) and -H 3 (reduced module of -C 3 for heating) to predict ECCD and ECRH, respectively.In section 2.1, the most general quasioptical model underlying PARADE is overviewed.Then, the adjoint model underlying -C 3 is overviewed in section 2.2.(The specific representations of the various coefficients used in this model are summarized in appendix.)Finally, we summarize the structure of our new quasioptical ECRH/CD prediction package in section 2.3.

Brief outline of quasioptical model used in the PARADE
The most precise approach to simulate the behavior of waves in inhomogeneous anisotropic media would be the fullwave analysis.However, such analysis of EC waves with wavelengths of a few mm in three-dimensional fusion plasmas of a few tens cm to a few m class requires a considerably expensive computational resource.Thus, ray tracing, which is the main result of GO theory, has been widely used for a long time for the simulations of EC waves in fusion plasmas [3,[27][28][29][30].The GO approximation reduces the general wave equation considering the entire wave field to the Hamilton ray equation considering only the phase of a dominant plane wave, under the short-wavelength limit.This significant simplification leads to a large reduction in computational cost, while a large amount of information, e.g. the envelope structure, diffraction and polarization are lost at the same time.Therefore, various quasioptical ray tracing [15][16][17][18][19][20][21][22][23][24][25][26] has been proposed and developed to pick up the lost information again with necessary and sufficient accuracy while keeping the simplicity.Our goal here is to briefly introduce the terminology of the theoretical model used in PARADE.The full explanation of the model is described in detail in our series papers [31][32][33][34].
Let us start with the equation for the electric field E of a linear wave governed by a general dispersion operator D: We assume that the field is stationary, with constant frequency ω and has an eikonal form E = e −iωt+−i(x) ψ(x).(The time dependence is henceforth omitted for brevity.)Here, the scalar function θ is a rapidly varying 'reference phase', k .= ∇θ is the local wave vector (the symbol .= denotes definitions), and the complex vector ψ is a slowly varying envelope.Specifically, we assume where λ .= 2π/k is the wavelength, L ∥ is the characteristic scale of the beam field along the group velocity at the beam center, and L ⊥ is the minimum scale of the field in the plane transverse to the group velocity.The medium-inhomogeneity scale L media is assumed to be of the same order as L ∥ or larger.Equation ( 2) is the standard ordering used in paraxial and quasioptical codes.Under these assumptions, equation (1) can be expressed as follows: Here, L = O(ϵ ⊥ ) is a differential operator specified in [31] (also see below).Also, the matrix D is the Weyl symbol [40] of the local dispersion operator [31], which we assume to satisfy the ordering The indices H and A denote the Hermitian part and the anti-Hermitian part, respectively.For our purposes, it is sufficient to approximate D with the dispersion matrix of homogeneous plasma [31].
Since D H is the dominant part of D (see equation ( 4)), we define the modes through D H instead of D. The envelope ψ is decomposed in the basis {η s } of the orthogonal eigenvectors of D H , i.e.D H η s = Λ s η s , then is written as follows: where a o , a x , and ā are complex coefficients, η o and η x are the eigenvectors corresponding to the O-and X-modes in homogeneous plasma, and η is the third eigenvector of D H that is orthogonal to both of them.For simplicity, let us restrict the following discussions to waves excited as a singlemode, i.e.O-mode.(The X-mode case is treated similarly.The mode-converting case is then discussed in [33].)Then, the scalar envelope is a = a o = O(1) and the polarization vector is Ξ = η o .Λ o .= Ξ + D H Ξ can be used as dispersion function of the Omode and determines the GO ray trajectory of the O-wave.This trajectory is called 'reference ray' (RR) and is governed by the equations as follows: where X and K are the ray coordinates and the ray wavevector, ζ is the path along the ray trajectory, and V ⋆ .= |∂H ⋆ /∂K| is the absolute value of the group velocity.H ⋆ = Λ o is the RR Hamiltonian.Here and further, the index ⋆ denotes that the corresponding quantity evaluated on the RR.We also introduce RR-based curvilinear coordinates xµ ≡ {ζ, ρ1 , ρ2 }, where ρσ are, loosely speaking, the orthogonal coordinates in the plane transverse to the group velocity of the RR as specified in [32].(Here and further, the indices σ and σ span from 1 to 2; other Greek indices span from 1 to 3.) To simplify the field equations, we introduce rescaled complex vector amplitude ϕ = √ V ⋆ a.Then, equation ( 3) leads to the following parabolic equation: Here, ∂ σ .= ∂/∂ ρσ , summation over repeating indices is assumed, and the coefficients are expressed through D as found in [32,34].

Brief outline of adjoint model used in the -C 3
The direct calculations of the Fokker-Planck equation [41,42] are comprehensive for precise predictions of currents driven by the RF waves but are time consuming and not convenient for ECCD modeling in tokamaks with some purposes, e.g. a brief analysis between discharges or scanning surveys to optimize the launching conditions.We therefore develop the -C 3 module, which calculates driven currents accounting for the quasioptical structure of the envelope given by PARADE, based on the adjoint technique [43][44][45][46][47][48][49][50][51][52].The adjoint technique, which exploits the self-adjoint property of the linearized collision operator in the linear Fokker-Planck equation, is a convenient method to estimate currents through Green's function.It meets the concept of a quasioptical approach, that is convenient but still gives reasonable predictions and has been used in several codes to predict ECCD [17,22,29].Our object here is to briefly review the adjoint model used in the -C 3 module, mainly along the context of [50,51].Careful and detailed explanations are found in the many literatures [41,[43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58].
Let us start our review from the Ohm's law.Parallel components to the magnetic field lines of the currents driven by the RF waves will be presented as, where, u = p/m e = vγ is the normalized relativistic momentum, γ .= 1 + u 2 /c 2 is the Lorentz factor, and δf e = f e − f eM is the first order difference of the distribution function f e from its bulk component, with µ .
= m e c 2 /T e .= 2(c/v th ) 2 and thermal velocity v th .This is a well known relativistic Maxwell distribution function.
As shown in equation (8), δf e should be found to determine j ∥ .Then, let us consider linear Fokker-Planck equation that δf e obeys as, where Q rf on the right side is the quasilinear diffusion term [41,[53][54][55] representing the RF source.(Specific representation of Q rf is summarized in appendix A.1.)The left side represents the linear response of the electrons due to Q rf .C lin represents linearized collisions of electrons [56] and V = v ∥ ∇ ∥ corresponds to the Vlasov behavior, where ∇ ∥ .= ∂ ∥ is a derivative along the magnetic field lines.To solve linear Fokker-Planck equation (10) in reasonable computational costs, we introduce the adjoint equation, where ν e0 is a collision frequency of electrons, b .= |B|/|B max | is a normalized magnetic field strength with |B max | the maximum strength of B along a given field line, and g = g(w = u/v th ) is Green's function, which is the key of the adjoint technique.Formally integrating equations (10) and (11) with u, and flux surface averaging them, we can obtain the following relations with a line element along the field line dl denotes a flux surface average.)Here, Vlasov V and linearized collision C lin operators have a self-adjoint property as, .
(14) By exploiting this property (14), the left sides of both relations ( 12) and ( 13) can be cancelled identically, then Combining equation (15) with flux averaged equation ( 8), we obtain the resulting representation of driven currents depending on Green's function g and the quasilinear diffusion function Q rf as follows, As a consequence of the above procedure, the problem to consider δf e obeying equation (10) to determine j ∥ in equation ( 8) is translated into the problem to consider g obeying equation (11) to determine ⟨j ∥ ⟩ in equation ( 16).
Next, we introduce an approximation to solve equation ( 11) analytically.Assuming a sufficiently long mean free path of the electron, V can be removed from equation (11) by taking the flux surface averaging as, where σ .= ξ /|ξ|, and ξ .= u ∥ /u is the pitch of the electrons.This is analytically solvable for g as, where A is summarized in appendix A.2 (or see [49]) and φ .= (1 − ξ 2 )/b is a normalized magnetic moment.Equation (18) leads to the problem of finding g(w) which boils down to the problem of finding χ(w), which is the coefficient of the generalized Spitzer function K = χ f eM for a toroidal magnetic field condition [58].K corresponds to the distribution function in the weak parallel electric field to magnetic field lines.χ(w) is governed by following equation as, where F p is the fraction of the passing electrons [59] and υ e (w) represents a pitch angle scattering [57].Then, it is convenient to assume the polynomial expansion for the specific representation of χ with coefficients d i [60] as, Exploiting the variational technique [50,57] to equation (19), we can obtain the simultaneous equations approximately equivalent to equation (19) as follows, where the coefficients M ij , W ij , and G j are specified in appendix A.3.Solving equation ( 21) for d 0 and d i , and substituting them into equation ( 20) yield an approximate solution for χ.An overview of this variational technique to determine χ are briefly summarized in appendix A.3 and explained in detail in [50,57].
Finally, because the absorbed power from the RF waves can be presented as, the current drive efficiency η = ⟨j ∥ ⟩/⟨p abs ⟩ is specified as where we perform the integrations by parts and simple algebra for convenience in actual simulations.∇ u .= ∂/∂u and D rf is a quasilinear diffusion coefficient as in appendix A.1.

Structure of the EC physics calculation package based on PARADE
Based on the model overviewed in section 2.2, we newly developed the prediction modules -C 3 for ECRH/CD and -H 3 only for ECRH, which evaluate driven current and absorbed power from the quasioptical wave beam structure computed by PARADE.Specifically, -C 3 computes the local current density using equation ( 16) and other necessary equations in section 2.2, for all spatial mesh points where the wave power given by PARADE is sufficiently finite.Calculation time of the integration of resonant electrons in momentum space is substantially reduced by using the 1D integral technique along the resonance curve (described in detail in [14]).Quasioptical structures of wave beams are introduced into the current and absorbed power density via spatial profiles of the amplitude |a(x)| and wave vector k(x) generated from the quasioptical complex amplitude a(x) of PARADE as follows, Substituting them, we can obtain the map of quasilinear diffusion coefficient D rf (x) = D rf [k(x), |a(x)|] in quasioptical limit, which is used in the equations ( 16) and ( 22) through equation (A.1).-H 3 is the reduced module of -C 3 only for evaluating the plasma heating.It can compute only power deposition faster than -C 3 .
As shown in figure 1, both -H 3 and -C 3 are incorporated as the components in the EC-prediction package based on the main code PARADE.A typical calculation of flow is as follows: Read the plasma equilibrium and the radial profiles of electron plasma parameters from the data set directory named FLX (abbreviated from magnetic FLuX).We compute the propagation and dissipation of the EC wave power profile injected from an antenna using PARADE.Calculate the radial 1D profiles of the absorbed power density and the driven current density using -C 3 (or only the absorbed power using -H 3 ) with the 3D EC wave power profile given by PARADE.The wave propagations projected on the arbitrary cross section of the plasmas are visualized using the CROSS code.[The detail of -E 3 (Electron Emission Evaluation) is described in [61]] (To ensure there are no programming bugs in absorbed power computation, the deposition profiles calculated by -H 3 and the same profiles computed from the dissipated wave power have been compared and are checked both are in agreement.)

Simulation results in JT-60SA tokamak
In this section, the newly developed EC-prediction package is applied to the typical operation scenarios expected in JT-60SA and is compared with the conventional models.We first compare with the single ray prediction and show that the finite beam width broadens the power deposition, as is well known.We then show that the DDP, which is a unique advantage of our quasioptical EC-prediction package, must be taken into account for ECRH/CD predictions, by comparing with conventional quasioptical calculation.
Figure 2 shows the radial profiles of the electron density n e and temperature T e for typical scenarios of JT-60SA [62,63] used in this section.Scenario 2 is a full plasma-current scenario, in which RF waves with 138 GHz is injected for the 2nd harmonic X mode ECRH/CD.On the other hand, scenario 5 is a high-beta scenario and is operated with a lower magnetic field than scenario 2, hence the RF frequency is switched to 110 GHz.Such operation switching at different frequencies is possible by employing multi-frequency gyrotrons [64].In the following calculations, we assume a de-focused Gaussian beam with the waist w 0 = 17.97 mm and the center of the antenna (R, z) = (4.13,0.8) m as the initial position of the RR.The orientation of the RR is also defined via the angles for toroidal and poloidal direction (θ tor , θ pol ), where θ tor = 0.0 • on poloidal plane and θ pol = 0.0 • on horizontal plane.(The final parameters of the RF antenna to be installed in JT-60SA are being designed, but the above are sufficiently reasonable settings to validate and demonstrate our newly developed ECprediction package.)

Power deposition profile broadened by finite width of the beam
Here, we show the broadening of the wave power deposition profile by considering the envelope profile over the beam The radial profiles of the electron density ne and temperature Te for typical operation scenarios of JT-60SA tokamak [62,63], where scenario 2 is a full plasma-current scenario, and scenario 5 is a high-beta scenario with a lower magnetic field than scenario 2. cross section.The quasioptical EC-prediction package based on PARADE is applied to the typical scenario 2 of JT-60SA (see, figure 2), and its simulation results are compared with the single GO ray prediction as shown in figure 3.In this simulation, we chose a weakly relativistic model [65] for the Hermitian part of the plasma dielectric tensor and a full relativistic model [14] for the anti-Hermitian part.(Farina [66] has also been proposed as a dispersion model for high temperature plasmas.It is also important for future work to implement such other models in our EC package as options and to discuss the applicable range for more precise simulations.)We assume (θ tor , θ pol ) = (0.0 • , 4.0 • ) for the injection angles and 138 GHz for RF wave frequencies.
Figure 3(a) shows the propagation of wave beams in plasma calculated with the single GO ray model (black line) and PARADE (black contour).Figure 3(c) shows that the radial profile of deposited power density by PARADE (orange line) is broadened for ρ > 0.48 and has a reduced peak, compared to the result by a single GO ray (blue line).This result is due to the fact that the wave intensity of PARADE has a finite profile over the beam cross section, unlike the single GO ray model, where the wave intensity is completely localized on the ray trajectory.Figure 3(b), an enlarged view of the area near the absorption position in figure 3(a), shows that the intensity profile of PARADE considering the envelope (black contour) passes ρ > 0.48 near the resonance layer, while a single GO ray (black line) does not pass ρ > 0.48.Although the validity of the standard GO approximation (λ/L media ≪ 1) improves with larger plasmas, it is shown here that single GO ray tracing is insufficient and quasioptical envelopes are needed even for the JT-60SA plasma, which is the world's largest tokamak until the completion of ITER [67].This simulation might be just a confirmation of the conventional physics shown in many past papers, but works as an introduction to correctly understand the impact of DDP shown in the next section.

Impact of dissipative diffractive propagation for EC predictions
Our main goal in this section is to show that even the conventional quasioptical model is insufficient to predict detailed ECRH/CD, and the upgrade model accounting for DDP is needed.Specifically, we show that the DDP allows further broadening of power deposition and driven current profiles in section 3.2.1 and elaboration of net driven current in section 3.2.2.To accomplish this, let us compare predictions using the full version of our quasioptical EC package based on PARADE with DDP and a reduced version of the package.The reduced package is a same quasioptical package but with 'uniform' dissipation in the beam cross section so corresponds to the earlier model of quasioptical ray tracing.

Broadening of power deposition and driven current pro-
files. Figure 4 shows comparisons of power deposition and driven current profiles between both models.These simulations are performed for JT-60SA scenario 5 plasma.(As a reminder, its radial profiles of electron density and temperature are shown in figure 2.) In scenario 5, which is planned as a high-beta scenario, the confinement magnetic field is set to be lower than in scenario 2, which was used in the demonstration in the previous section 3.1, and the injecting RF wave frequency for 2nd harmonic X-mode heating is set to 110 GHz, which is also lower than in scenario 2. Nevertheless, the central  The RR is at (ρ 1 , ρ2 ) = (0.0, 0.0) m for all cross section.Due to the transversely non-uniform resonant-dissipation, the intensity profile transversely shifts toward ρ2 direction, that almost corresponds to the minor-radial direction of the flux surface of figure 4(a).This is the DDP.
density n e (ρ ∼ 0) does not change substantially, so the injected wave beam of scenario 5 is more prone to be affected by refraction and diffraction than scenario 2. This leads to the importance of the fact that PARADE can describe DDP.The toroidal injection angle is fixed to be θ tor = 10.0 • , and the poloidal injection angles are scanned for θ pol = −15.0• , −5.0 • , and 5.0 • (figure 4(a)).
Comparison results are shown in figures 4(b) and (c).It is clearly shown that both radial profiles of power deposition and driven currents by the full EC package (orange lines) are broadened to the outward side rather than those by the reduced EC package (blue lines).Furthermore, the more outwardly absorbed radial profiles are, the more significantly broadened they are to the outward side.Such difference between orange and blue lines and the correlation between outward absorption and broadening cannot be explained only by diffraction since both quasioptical models take diffraction into account.
To explain this, we need to understand the DDP process, which is the wave beam evolution with both diffraction and nonuniform dissipation, simultaneously.
Let us look at figure 5, which is the evolution of the crosssectional profile of beam intensity along the RR trajectory using the full package.The simulation is performed for the same situation with θ pol = −15 • of figure 4. It will help us to understand what the DDP process is.As defined in section 2.1, ρσ are the vertical and horizontal two directions orthogonal to ζ, and near the resonance, the ρ2 direction almost corresponds to the minor-radial direction of the flux surface of figure 4(a).When an EC wave beam obliquely passes through the resonant surface [red line in figure 4(a)], if the condition, L ⊥ /L dissip.≪ 1, where beam width scale L ⊥ is defined in section 2.1 and L dissip. is the spatial variation scale of resonant dissipation, is violated, the beam power should be nonuniformly dissipated in its beam cross section.The power profile of the non-uniformly dissipated beam is then distorted from the original axisymmetric profile, which causes a shift of the beam center position in the cross section.(i.e. at figure 5(c), beam profile shifts to ρ2 > 0 direction, which correspods to ρ > 0 direction of figure 4(a)) Such propagation shift also makes the shift of resonant position and this makes further propagation shift.(i.e. in figure 5(d), beam profile further shifts from figure 5(c)) Integrating this DDP process along the beam trajectory, the resonant position is gradually shifted, and as a consequence, the deposition and current profiles are broadened toward an outside radial direction and those peaks are decreased as shown in figures 4(b) and (c).As shown in figure 4(a), the more outwardly propagating RR for a minor-radius is the more obliquely injection to the resonance layer, and as shown in figures 4(b) and (c), the more outwardly propagating RR for a minor-radius is the more broad power deposition and current profiles.These facts support our claim that these broadenings are due to DDP.

Elaboration of net driven current.
Here, we discuss how the DDP improves the net current magnitude.While a detailed discussion requires direct Fokker-Planck computations, a reasonable interpretation can be obtained from the results of the adjoint method as follows.Figure 6(a) shows the net driven currents I = ´⟨j ∥ ⟩dA for each of the six EC driven current densities ⟨j ∥ ⟩ in figure 4(c).We can see that all the predictions of net currents with DDP (orange dots) can be elaborated to be increased from conventional blue dots.This is because the 3D spatial structures of the parallel wave vector k ∥ (x) and the absorbed power p abs (x) are modified due to DDP (and the modifications are reflected into the current via D rf ).To show this, let us introduce the effective parallel wave vector, which is weighting averaged k ∥ (x) with absorbed power p abs (x).This representation reflects the fact that the parallel wave vector k ∥ at the position x ′ with greater absorbed power p abs (x ′ ) should have a larger contribution to the net driven current I.We can see that all the K ∥ elaborated with DDP (orange dots) are larger than conventional blue dots.
The wave-electron resonant-interaction conditions in momentum space {u ∥ , u ⊥ }, for different two K ∥ with and without considering DDP, are shown as orange and blue elliptical curves in figure 7.
(Other variables in equation ( 26) are detailed in appendix A.1.)Figures 7(a)-(c) correspond to the case of θ pol = 5 • , −5 • , and −15 • , respectively.Let us look at the resonance curves (RCs) without considering DDP (blue curves).These conventional RCs tell us that the lower driven current for the more radially outward power deposition is due to the increase of trapped particle area in momentum space.In figure 7(a), since the RC is located below the trapped/passing boundary (straight dashed lines), an injected wave resonates with the passing particles and drives the Fisch-Boozer current in the positive direction.However, because it is close to the boundary, the driven current magnitude can be reduced by the Ohkawa effect.(The direct Fokker-Planck computation should be required for quantitative evaluation of it.)In figure 7(b), a part of the RC is located above the boundary because the boundary pitch goes down toward the radially outward direction.Since a part of the injected wave power is used to accelerate trapped particles, the effective power to accelerate passing particles carrying the current is reduced, which causes a decrease in the net driven current.In figure 7(c), the wave beam would not be able to accelerate a sufficient number of passing particles, since most of the RC is located above the boundary in the bulk-region.Because of this, the current is barely driven at the θ pol = −15 • case, as shown in figure 6(a).Let us look at the orange RCs considering the DDP.In all figures 7(a)-(c), the RCs are shifted toward a larger u ∥ direction and the radius of the RCs also becomes smaller.This is due to the fact that the DDP makes the larger K ∥ and the wave-electron resonant-interaction in lower magnetic field, as explained later.Because orange RCs in all figures 7(a)-(c) move under of the boundary and are entirely interacting with the passing particles carrying the current, the net driven currents are improved.Especially, driven current at θ pol = −15 • is greatly improved, since the conventional blue RC, which interacted almost exclusively with trapped particles, is modified to an orange RC, which interacts entirely with passing particles.Finally, we shall illustrate the correcting mechanism of K ∥ prediction by DDP, using a schematic image figure 8. First, the DDP shifts the beam trajectory to the weaker-dissipation side.(See, figure 2 of [34].)In tokamaks, this shift direction corresponds to the toroidal direction.(See, bending orange arrow in figure 8.) Which means beam propagation is toward the field line direction.(See, red arrow in figure 8.) This process eventually increases K ∥ estimated as the inner product of the wave vector and the unit field vector.Also, especially for lowfield side launch, the wave beams with DDP (orange arrow in figure 8) stay longer on the low-field side and will be absorbed in the low-field side, because of the up-shifted absorption due to the Doppler effect with the increased K ∥ by DDP.In accordance with this mechanism, the variation of K ∥ due to DDP can be estimated to be at most first order O(ϵ ⊥ ) in quasioptical limit, and this estimation is consistent with the variation level of the net current (at most few kA).We show here that the DDP elaborates an effective parallel wave vector K ∥ , and the improved K ∥ makes the increased net current I.

Conclusions
Improvement of ECRH/CD predictions is an important issue in the design and control of high-performance fusion plasmas in future devices, where these should play a more important role as actuators than in devices to date.ECRH/CD predictions strongly depend on the propagation and dissipation model of EC waves.Quasioptical models have therefore been developed to describe the evolution of the phase and intensity structure of the EC wave beam field in the cross section with respect to the propagation direction.A quasioptical EC wave beam obliquely passing through the resonant surface is dissipated non-uniformly in its beam cross section so that the beam profile shifts gradually and thus the resonant position also shifts.Integrating this process step by step along the propagation, beam trajectory and deposition profile of its dissipated power is altered from conventional predictions.Quasioptical ray tracing code PARADE has an advantage in the treatment of this 'dissipative diffractive propagation' [34] in reasonable computational resources, and was experimentally validated [35].
Recently, -C 3 , which is the module for PARADE to predict both EC power deposition and driven current profiles within the quasioptical assumption, and -H 3 , which is the reduced module of -C 3 only for evaluating power deposition, have been developed.These are successfully linked to the PARADE code, and the quasioptical ECRH/CD prediction package is constructed.In section 2, we briefly overview the theoretical models underlying PARADE code, and -C 3 and -H 3 modules.Then, in section 3, the EC-prediction package is applied to the typical operation scenarios of JT-60SA and is compared with other conventional models.In section 3.1, we first compare with the single GO ray prediction and show that the quasioptical beam envelope broadens the wave power deposition profile, as is well known.We then performed in section 3.2 that the comparison between our quasioptical calculation considering DDP and conventional quasioptical calculation NOT considering DDP.We found that DDP further broadens both the power deposition and driven current profiles in section 3.2.1, and increases the magnitude of the net driven current in section 3.2.2,from the conventional quasioptical calculations.This novel ECCD and ECRH prediction package based on PARADE is applicable not only to JT-60SA but other existing devices and even, future devices, where ECCD and ECRH should play a more important role as an actuator than devices to date.Numerical and experimental validations of PARADE simulations including DDP are the next important works.A careful and comprehensive comparison works with various EC codes and experiments is in progress and will be reported in the near future.

Figure 1 .
Figure 1.Codes and modules contained in prepared EC-prediction package RAPLUS (RAy '+' other functions).PARADE code, -C 3 module, and -H 3 module targeting in this paper are marked as red.

Figure 2 .
Figure 2.The radial profiles of the electron density ne and temperature Te for typical operation scenarios of JT-60SA tokamak[62,63], where scenario 2 is a full plasma-current scenario, and scenario 5 is a high-beta scenario with a lower magnetic field than scenario 2.

Figure 3 .
Figure 3. Comparisons of ECRH predictions for the typical scenario 2 of JT-60SA plasma using quasioptical EC-prediction package based on PARADE and the conventional single GO ray model.(a) shows the poloidal cross section of normalized flux surface (blue lines), cold electron cyclotron resonance layer for the 2nd harmonic (red line) as a reference, single ray trajectory of GO (which also corresponds to the RR, black line), and the intensity profile of quasioptical envelope (black contour).(b) is an enlarged view of the area near the power absorption in (a).We can see that the quasioptical wave beam is transversely extended across the flux surfaces around the cold resonance layer.(c) shows that the power deposition profile by the PARADE-based package (orange line) is broadened from that by GO (blue line).

Figure 4 .
Figure 4. Comparisons of ECRH and ECCD predictions for the typical scenario 5 of JT-60SA plasma with RF wave frequency 110 GHz, using two EC-prediction models.(a) shows the poloidal cross section of normalized flux surface (blue lines), cold electron cyclotron resonance layer for the 2nd and 3rd harmonics (thick and thin red lines) as a reference, and the RR trajectories (black lines) for different poloidal injection angles θ pol = −15.0• , −5.0 • , and 5.0 • .(toroidal injection angle θtor = 10.0 • is fixed.)(b) and (c) show radial profiles of power deposition and driven current for each θ pol , respectively.Orange and blue lines correspond to full quasioptical EC-prediction based on PARADE with DDP and reduced quasioptical EC-prediction without DDP (thus corresponds to conventional qusioptical model), respectively.

Figure 5 .
Figure 5.The cross-sectional intensity profiles of EC wave beam along the RR trajectory at various propagation distances: ζ = 1.0, 1.4, 1.5, and 1.6 m.This simulation is performed for the same case with θ pol = −15 • of figure 4. The RR is at (ρ 1 , ρ2 ) = (0.0, 0.0) m for all cross section.Due to the transversely non-uniform resonant-dissipation, the intensity profile transversely shifts toward ρ2 direction, that almost corresponds to the minor-radial direction of the flux surface of figure 4(a).This is the DDP.

Figure 6 (
b) shows K ∥ for each simulation of figure 4(c).

Figure 6 .
Figure 6.(a) shows the net driven currents I = ´⟨j ∥ (ρ)⟩dA(ρ) spatially integrating each current densities ⟨j ∥ (ρ)⟩ of figure 4(c).(b)shows the effective parallel wave vector K ∥ for each simulations of figure4(c).Orange dots are for full predictions with DDP and blue ones are for conventional predictions.The leftest two dots correspond to the most radially inside injection case and the rightest are for the most outside case.

Figure 7 .
Figure 7. Orange and blue curves are the EC resonance conditions (equation (26)) for different two K ∥ with and without considering DDP in momentum space, respectively.Orange and blue straight dashed lines are the boundaries of trapped and passing particles for each resonance condition.(a)-(c) correspond to the case of θ pol = 5 • , −5 • , and −15 • , respectively.The cyclotron frequency is computed using a weighting averaged magnetic field strength by absorbed power profile, ωc = (e/me) ´dx p abs (x) |B(x)|/ ´dx p abs (x).Black dashed contours represent the corresponding thermal velocities, as references.

Figure 8 .
Figure 8.A schematic of tokamak top view with RF wave beam obliquely injected from the outside port antenna.The red arrow represents the toroidal field line dominating most of the confinement field.The blue dashed and orange solid arrows represent RF wave beams predicted by conventional model and our model with DDP, respectively.The orange solid arrow is shifted from the blue dashed arrow to the weaker-dissipation side due to DDP.Inner product of orange and red arrow (∝ K ∥ ) is increased from conventional blue arrow case.