O L Krutkin

E Z Gusakov

S Heuraux

A theoretical investigation of the turbulent structures tilting measurements with radial correlation Doppler reflectometry

A new technique for measuring the tilting of the turbulent structures is studied analytically within the framework of Born approximation. Probing with O-mode, linear density profile and slab geometry are considered. Analytical expressions are derived for scattering signal and crosscorrelation function of scattering signals valid for different probing frequencies. Comparison with previously used "intuitive model" is made, discrepancies pointed out and a more general expression for narrow probing beam is derived. General conclusions about the technique are made and a possible alternative is suggested. Analytical results are validated with linear numerical simulations employing reciprocity theorem, importance of poorly localized smallangle scattering is confirmed. Now, to tie this in with the previous results one can note that in the case of l min , l max , Λ 0.5 << ρ; Δx~l cx and R 1 , R 2 ~1>> I 1 , I 2 and formula (17) coincides with Erreur ! Source du renvoi introuvable.. Another result that can be inferred from this formula is that, as expected for turbulence with Λ=0 it gives Vt max =Δxtgθ.

Introduction

One of the major physical problems arising in magnetic confinement fusion devices is the control of plasma turbulence and the anomalous transport associated with it. According to the present understanding, the turbulence, namely, drift wave turbulence is responsible for degradation of the plasma confinement. This makes it essential for confinement research to develop the diagnostic tools capable of measuring its parameters.

One of them is radial correlation Doppler backscattering, commonly referred to as radial correlation Doppler reflectometry (RCDR), proposed in [1] and used widely in experiment both in stellarators [2] and tokamaks [3,4]. The diagnostics allows, under certain conditions to directly measure radial correlation function of the electron density perturbations. The diagnostic is based on probing the plasma volume with two microwave beams at oblique incidence with respect to the magnetic surface [1]. The probing is performed in the presence of a cut-off at close frequencies one of which is kept constant while the other one is varied. For each frequency difference the correlation of the scattered signals is measured. The resulting correlation function is then related to the spatial correlation of electron density fluctuations based on the cut-off separation for the probing frequencies [START_REF] Gusakov | 38th EPS Conference on Plasma Physics[END_REF].

While diagnostic benefits from the simplicity of execution, extensive theoretical [START_REF] Krutkin | [END_REF][7][8][9][10] and numerical [11][12][13] analysis of the diagnostic have revealed that there is a number of factors that can complicate the relation between the correlation of the measured scattered signals and properties of density fluctuations. Such factors include nonlinear scattering effects [8,9,12] as well as poorly localized small-angle scattering effects in the linear case [START_REF] Gusakov | 38th EPS Conference on Plasma Physics[END_REF]7,12] or even geometrical effects [10]. These effects are well-described both in aforementioned works and in the papers dedicated to simpler diagnostic configurations [14][15][16][17]. Methods and criteria are suggested to allow for correct interpretation of the experimental data.

One of the recent developments of the RCDR diagnostic is a technique suggested for indication of tilted turbulent structures rotating in poloidal direction [18] and for determination of their tilt angle [19]. This method, however, is based on an intuitive interpretation, ignoring the complexity of wave scattering process and consequently aforementioned effects. The goal of this work, therefore, is to consider this technique in the framework of a strict linear theory and to determine applicability and limits of the intuitive approach to data interpretation.

Basic analytical equations

Within this paper, plasma probing with the wave of the ordinary polarization (O-mode) is considered in the framework of the slab geometry. This allows us to obtain straightforward expressions while retaining the qualitative results. The background density profile is assumed to be linear along the radial coordinate and uniform along the poloidal one. The coordinates, named respectively as x and y, while the z-direction is aligned with the applied magnetic field and not needed in this work. Geometry used in the consideration is illustrated on Fig. 1. In this section the goal is to outline the model used and derive general expressions for the scattered fields and a cross-correlation function measured in experiment, in much the same manner as it is done in [START_REF] Gusakov | 38th EPS Conference on Plasma Physics[END_REF]7,11]. The differences corresponding to the tilted turbulence effects and paraxial approximation utilized in this paper will be introduced in the next section.

The propagation of the O-mode in tokamak is described by the Helmholtz equation for the electric field component Ez aligned with the external magnetic field:

2 2 2 2 2 ( , ) ( , ) ( , ) 0 z z z d d E x y E x y k x y E dx dy    (1)
The wave number k for O-mode is defined as k 2 (x,y)= ω 2 /c 2 (1-n(x,y)/nc) where nc=meω 2 /4πe 2 is cut-off density. That in turn allows us to relate wave number perturbation to the density perturbation: k0 2 (x)=ω 2 /c 2 (1-n0(x)/nc)= ω 2 /c 2 (1-x/L); δk 2 (x,y)=-ω 2 /c 2 δn(x,y)/nc. In these expressions we assume that the wave is propagating through a perturbed plasma, where density can be separated into linear poloidally uniform "background" profile n0 and a perturbation δn caused by turbulence. Here, L corresponds to radial (x) coordinate of the cut-off point corresponding to the probing frequency ω and is proportional to ω 2 .

Within this paper we will restrict ourselves to considering linear scattering regime, which is adequately described within the Born approximation. Furthermore, since only the scattered signal received by antenna is of interest, reciprocity theorem [20] can be used to simplify the problem even further:

2 2 2 ( , ) ( ) ( , ) 4 s z e c n x y ie A P E x y dxdy m n              ( 2
)
where As is the amplitude of the scattered signal received by antenna at a frequency ω+Ω and P is the probing beam power. δnΩ corresponds to the spectral component Ω of the turbulence frequency spectrum and Ez is normalized to a unit of power electric field. In the considered case of drift wave turbulence Ω<<ω which is why it is going to be neglected everywhere except for the turbulence spectrum.

To obtain the signal corresponding to linear scattering it is enough to substitute expression for the unperturbed electric field (given by the solution of equation ( 1) when perturbation δk is neglected) into the expression (2). Considering the background density profile to be linear, the solution of equation ( 1) without the density fluctuations can be expressed in terms of an integral over Airy functions corresponding to poloidal harmonics of the probing beam:

0 2 3/2 (k ) 2 2 0 0 ( ) 8 ( , ) ( ) ( ) exp( ( ) ); 2 4
( ) 2 ( ) (x) ; 3 y y y z y y L y y y dk x L k i E x y f k Ai i k c Lk k k k dx c                         (3)
where L(ky)=L×(1-ky 2 c 2 /ω 2 ) is the radial coordinate at the turning point, α=(c 2 L/ω 2 ) 1/3 is Airy scale and f(ky)is the probing antenna pattern. We will consider a Gaussian antenna pattern given by the following expression:

  2 2 ( ) 2 exp( / 2); y y f k k K       (4)
where K= ω sinθ /c is the probing beam poloidal wavenumber and θ is the antenna tilt angle with respect to normal probing and ρ is the beam waist. Substituting the solution (3) into expression (2), we obtain an expression for the scattering signal in the linear approximation:

f k f q k C q k i k i q k                              (5)
Here, κ and q represent turbulence radial and poloidal wavenumbers respectively and C(κ, q, ky) corresponds to the scattering efficiency, definition of which can be found in [14]. Following the approach used in [7,11] we will use the following asymptotic expression derived in [14] and corresponding to the localized Bragg backscattering:

  3 2 2 2 2 2 2 ( ) 2 / (2 ) ( , , ) exp ; 12 2 4 y y y y q k k L q k q i C q k i                                   (6)
Substituting it into the formula (5) and performing integration over ky we arrive to the following expression for the scattering signal:

      2 ) s e qK q Lc i iL i i q e i d dq A P n q mc Lc qc Lc i q qc                                                    (7)
Before proceeding from this expression to the correlation function a few things should be mentioned. First of all, L has a quadratic dependence on the probing frequency, which should be taken into account when we consider correlation of two scattering signals corresponding to different probing frequencies. Consequently, the Airy scale α does not depend on ω.

Normalized cross-correlation function (CCF) of scattering signals separated in frequency by Δω and in time by Δt can be expressed as follows in the case of statistically stationary turbulence:

( ) * ( ) ( , ) ; ( ) * ( ) ( ) * ( )
it ss s s s s A A e d CCF t A A d A A d                                            (8)
where brackets designate averaging over the statistical ensemble of the turbulence. Which means they are applied to δn. In this work density perturbations are assumed to be statistically homogeneous, stationary and independent, which means their two-point cross-correlation function can be determined from Wiener-Khinchin theorem:

    2 3 ( , , ) ( , , ) * ( , ) exp ( ')
( ') ( ') ; 2 d dqd n x y t n x y t n q i x x iq y y i t t                      (9)
Keeping in mind that in experimental conditions Δω/ω<<1, we can now express the correlation function in terms of an integral over turbulence wavenumbers and frequency while expanding the argument of exponent in (7) into Taylor series over Δω/ω:

      2 2 2 2 2 2 22 2 2 2 2 2 2 2 ( , ) ( , ) exp ; (0, 0) 2 ( , ) ( , ) exp 2 4 exp 2 2 ( ) 2 ; 2 ( / ) 1 ( / 2 ) Jt CCF t K J qK J t d dqd n q q K K dq i L i i t d Lc qc Lc q qc                                                                                     (10)
This expression does not yet take into account specifics of the considered problem, and is very close to the ones obtained in [START_REF] Gusakov | 38th EPS Conference on Plasma Physics[END_REF]7,11]. The particularities of the tilted turbulent structure come from the shape of turbulent spectrum and its temporal behavior, which did not play a role in the stationary case considered in previous works.

CCF in case of poloidally rotating tilted turbulence.

To model the tilting angle measurement using RCDR, much like in [18,19] we will consider turbulence with a tilted Gaussian spectrum over wavenumbers rotating uniformly in poloidal direction with velocity V:

  2 2 min max 0 2 2 2 2 min max ( , ) 2
exp( ( cos sin ) / 4 ( sin cos ) / 4);

n q qV l l n q l q l                    (11)
Here, φ corresponds to the turbulence tilt angle between the poloidal direction (y-direction) and the stretching direction of the turbulent structure (see Fig. 1), so φ=0 corresponds to the turbulence being aligned to the poloidal direction. This in turns means that for φ=0 lmin and lmax are radial and poloidal correlation lengths of the turbulence respectively.

To further simplify the task, we will exploit the paraxial approximation, assuming |q-2K| << ω/c. This assumption will allow us to expand phase Φ into Taylor series over q-2K:

    3/ 2 2 2
L qc L Lc L q K q K c c L Lc L q q c d L Lc Lq dc                                                                        2 ; cos q      (12)
Substituting ( 11) and ( 12) into (10) performing integration over fluctuation natural frequency Ω and carrying constant part of Φ out of the integral:

    2 3 ( ) 2
l l qtg qctg q K K                                                                            2 2 2 2 2 qK             (13)
where

  22 2 2 sin sin 2 / 2 cos L c        
is the phase that can be excluded by calibration of the experimental scheme and for that reason will be omitted as well as constant numerical coefficients that do not influence the value of CCF in the subsequent formulae.

Integral over poloidal wavenumber q can be estimated using saddle-point method:

    2 2 2
l l i L K K Jt Lc q c Lc q qc ll K iVt i                                                                                         2 3 sin ; cos Ld                    (14)
Here, q* and 1/ρ* are defined as follows:

 

ll q K iVt i L xc l l i                                       (15)
q* corresponds to the saddle point and 1/ρ* is the characteristic width of the region of the main contribution into the integral over poloidal wavenumbers and for realistic parameter values lmin, lmax<< ρ corresponds to one over probing beam waist.

For a wide enough beam the expression for saddle point simplifies becoming q*=-2K which corresponds to the Bragg condition. Here Δx is equal to the separation of cutoffs 2LΔω/ω and therefore the CCF appears to be a function of cutoff separation rather than of turning points separation given by 2Lcosθ 2 Δω/ω. The same result was noted in previous studies [7,10,11].

Next, an integral over radial wave number should be determined. Integrand in this case has a saddle point as well as two branching points in the denominator. We will start our consideration with only taking into account the saddle point, which was shown [7] to correspond to a situation when small-angle scattering off long-scale fluctuation is suppressed and the CCF reproduces turbulence correlation function. Other situations will be considered in the later sections.

CCF at suppressed small-angle scattering.

The saddle point κ* and the second derivative of the integrand phase in its vicinity are determined by the following expressions:

    3
i x iVt i x K l l ll ll d l l l d                                        % (16)
The important thing to notice here is that anisotropy of turbulence spectrum introduces real part into the saddle pointdue to the tilt of the turbulent structure the nonzero radial wavenumber is prevalent in the spectral component with nonzero poloidal wavenumber. This effect can play a role for high enough values of probing angle and stronger anisotropy.

The contribution of the saddle point to the integral takes the form:

  2 2 3 2 2 2 2 1/
K K K iVt i x Jt Lc Lc l q i x iVt i x K l l                                                                                                     % 2 2 2 2 / 4 * ; cx l                      % (17)
Since tilt angle measurement technique [18,19]

Vt i K i x l                                                                        % % % (18)
Next, we will consider situation when lmin, lmax << ρ which will allow to significantly simplify the resulting formulae. A complete formula for tmax without this assumption is derived in Appendix 1. The simplified version of formula (18) takes the following form:

    2
                                    (19)
Substituting expression for ρ* into it and selecting real component of the exponent argument after neglecting smaller terms one can arrive to the following expression:

    2 22
                             (20)
Using it, the final result for tmax as well as the temporal Gaussian width Δt of the CCF can be obtained:

  22 max min max 2 2 2 2 sin 2 1 ; 2 cos sin 2; t x l l Vt xtg l l V             P (21)
The last term in the first line is a result of combining the last two terms of (20). Some remarks should be made about this result. Primary one is that while expression for tmax looks similar to the one obtained in the intuitive approach [19] the definition of Δx there is different, as it is defined as the turning point separation, which in this case takes the form Δx=2Lcos 2 θΔω/ω. This in turn means that for higher probing angles formula (21) gives drastically different values compared to the one introduced in [19] due to a difference of a factor cos 2 θ. At the same time for low probing angles, as described in [7], poorly localized scattering considered in the next section starts playing a role and provides qualitatively different results.

Another thing to note is that from the expressions ( 17) -( 19) radial correlation length of the CCF for the case of tilted turbulence can also be estimated as lcx 2 = lmin 2 cosφ 2 + lmax 2 sinφ 2 , which corresponds to a projection of the tilted turbulence spectrum on the radial coordinate.

Influence of small-angle scattering on CCF.

Aside from the saddle point domination described in the previous section, another situation can be considered. When the saddle point provides exponentially small input into the integral ( 14) compared to the input of branching point vicinity, its contribution can be neglected. We will try to estimate the CCF in this case.

To determine branching points, we will simplify denominator in expression ( 14) by substituting q*=-2K into it. The reason for such treatment is the assumption of correlation lengths being much smaller than ρ coupled with the fact that Δx and t can be estimated as having the same order of magnitude as lcx and Δt In this case, the expression for the branching points in ( 14) takes form, which, under the same assumptions, naturally coincides with the one obtained in [7]:

2 2 2 4 2 2 sin 2 br Lc i L Lc             (22)
Similarly, a criterion for the critical angle above which the small-angle scattering contribution should be negligible can be obtained. The only difference to the one introduced in [7] is due to the turbulence tilt, which causes dominance of nonzero κ in the turbulence spectrum, as mentioned before and also complicates the expression for lcx. The final criterion of small-angle scattering suppression in this case takes a more restricting form:

                ( 23
)
The case when the branching provides a dominant contribution into the integral corresponds to the situation when the value of the integrand in κbr is exponentially bigger than the value at κ* (which corresponds to large value of Δx, when the two points are moved far apart). In this case the main temporal dependence is determined by the value of the integrand at the branching point:

  2 2 2 3 max min 2 2 sin 2 1 sin ( , ) (t) exp 2 4 ; 2 * 4 cos br ll J t Const iVt K i L                                           (24)
This once again allows determining of the values of tmax and Δt:

  22 2 max min max 2 2 2 sin 2 sin ; 4 2 1 2; t ll L Vt xtg Lc V                (25)
Same as in the transition from (19) to (20), the tangent function in (25) arises from combining the last two terms of (24) when extracting the real part. This formula illustrates that in the case when the scattering signal is dominated by the poorly localized small-angle scattering, the dependence of tmax on Δx might not provide information on the turbulence reproducing antenna tilt angle instead. This effect will be observed in numerical calculations in further chapters and it seriously limits applicability of this method for the measurement of the turbulence tilt angle. This result is certainly not strict, as the temporal dependence only considered within exponent phase. Moreover, we cannot offer the limits of validity for this result without a much deeper consideration of the integration contour and the saddle point method. But we felt it was important to underline the qualitative possibility on the tmax not being dependent on turbulence parameters (as the first term in (25) is often negligible) and moreover, such situation will be demonstrated in calculations in the further sections.

Finally, aside from two considered cases there is an intermediate situation when the CCF is not dominated by small-angle scattering, but it still plays a significant role in determining the maximum of the CCF absolute value. In this case the interplay between small-angle scattering and back-scattering needs to be taken into account, since it is impossible to neglect either. Obtaining precise results for such a situation is problematic and its existence further complicates analysis of calculation results.

Analytical results discussion

3.1.The model limitations

One improtant thing to notice is that while obtained results are derived rather strictly, there are still limitations to their applicability due to the model assumptions. Such limitations are listed in this section to give the reader a better idea on the applicability of the obtained results. First and foremost, the results of this paper are derived in Born approximation, which means they are not applicable for nonlinear regime of scattering, which corresponds to higher values of density perturbation amplitude [8,12]. Another assumption made in the paper is the linear dependence of the density profile on the radial coordinate, which allowed performing explicit integration both in the scattered field [START_REF] Gusakov | 38th EPS Conference on Plasma Physics[END_REF] and in the phase expressions (3). In experimental situation, density profile is often nonlinear and moreover, X-mode probing is often used, with k 2 having dependence on both density and magnetic field resulting in a nonlinear radial profile. However, the most efficient scattering takes place in the vicinity of the cut-off and k 2 there can be approximated by a linear dependence. In such a case, the corresponding scale length should replace L in the theoretical results presented here (for X-mode the expression for it is more complicated, but can be inferred from [9]). Such a method could for example be relevant for the H-mode plasma where pressure gradient in the edge area is much steeper than in the core plasma. Due to the main contribution coming from the vicinity of the cut-off deeper areas of the plasma pedestal are well described by such a substitution of gradient scale length.

The simplification of the beam waist being much larger then turbulence correlation lengths used to obtain results ( 21) and (25) generally holds true in RCDR experiments. Similarly, paraxial approximation used in the section 2 is usually applicable. One more simplification was implicitly made by using the specific form (6) for scattering efficiency. While generally this form is appropriate, in some cases, for example in situations when plasma edge plays a significant role (such a case will be presented in chapter 4), a different scattering efficiency should be used. Further details on scattering efficiency and the conditions under which ( 6) is applicable can be found in [14].

Another simplification, which was not explicitly stated before is the infinite limits on the integral in (14). The actual limits correspond to κ having to be smaller than 2ω/c for the possibility of scattering to happen. Mathematically, this corresponds to the condition of existence of stationary point in the integral over radial coordinate, when calculating As (see, for example [14]). While this limitation is not relevant for isotropic turbulence, in the anisotropic case, according to formula (16), the saddle point can move out close to this limit and make it meaningful. Finally, the slab geometry considered in the paper is applicable for medium and large experimental devices. The effects of plasma curvature on RCDR results including some qualitative differences were analyzed in [10].

Qualitative conclusions

Aside from the aforementioned differences with the results obtained in [19] and limitation caused by small angle-scattering effects, a few other conclusions can be drawn.The main one is that, considering equation ( 14), one can see that aside from negligible addition to ρ* parameters of turbulent spectrum only participate in the formula in two combinationsone lmin 2 cosφ 2 + lmax 2 sinφ 2 corresponding to lcx and the other (lmax 2 -lmin 2 )sin2φ. These are exactly the combinations that can be determined from tmax dependence on Δx and from the CCF decay with Δx at tmax =0. Thus, formula ( 14) shows, it is not possible to resolve individual parameters lmax, lmin, φ and obtain more information on turbulence tilt from CCF, even if a different technique is used.

Another method however, seems to be applicableformula (17) indicates that dependence of imaginary part of CCF on Δx also provides information on parameter (lmax 2 -lmin 2 )sin2φ, which can be extracted. Advantage of this method is that it does not require temporal analysis of CCF. We also expect this method to be more suited for the reconstruction procedure proposed in [7] and validated in [11] to counteract the influence of small-angle scattering, as the tmax approach did not work well with reconstruction procedure in preliminary calculations due to strong influence on numerical errors. Verification of this speculation as well as more detailed study of this alternative method will be the subject of future work. In the opposite case, when ρ is comparable with the correlation lengths, additional combination of parameters is present in ρ* and one could theoretically obtain parameters lmax, lmin, φ by fitting them into (27) and into dependence of CCF on Δx given by (17). This approach, however, was not further explored in this paper due to improbability of this situation.

Numerical validation

To validate the results obtained in the previous chapters, numerical modeling was performed using wave propagation solvers. Since full-wave calculations are demanding and results of the paper were obtained within the Born approximation, a faster approach, similar to the one described in [21] was used. Namely, the solution (3) of the unperturbed Helmholtz equation combined with turbulence spectrum (11) was used to calculate the CCF according to (2) and (8) without the need to average over random fluctuations or to calculate perturbed electric field.

While simple, such an approach allowed for recovery of all the relevant physical effects in linear scattering regime.

After calculating the CCF this way, the maximum of its absolute value for each frequency separation Δω was determined and corresponding tmax was obtained. The dependence of tmax on Δx was compared with the ones described by formula (21).

A result of applying such a procedure can be seen on the figure 2. On the figure ymax=Vtmax and Δx=2LΔω/ω. Parameters used for presented calculation were chosen to be relevant to experimental condition and are as follows: L=10 cm, lmin=1 cm, lmax=1.5 cm, ρ=3 cm, φ=45°,f=30 GHz. It can be seen that for higher probing angle 30° the agreement between theory and calculation is rather good (although not for the angle 60°, which we attribute to edge effects rendering scattering efficiency (6) inapplicable), whereas for low angle 10° the agreement deteriorates due to small angle scattering. This is in agreement with the theoretical results, predicting that higher probing angles benefit from suppressed small-angle scattering.

A more favorable situation is shown at figure 3. In this case the input parameters are as follows: L=4 cm, lmin=0.2 cm, lmax=1.4 cm, ρ=1.6 cm, φ=45°, f=70 GHz. As can be seen from the figure significant disagreement can only be observed for the probing angle 10°. Improvement of the situation for these parameters goes in line with the small-angle scattering suppression criterion (23). To illustrate the situation when small-angle scattering dominates the scattered signal and dependence on the turbulence parameters becomes negligible, the numerical results for this case are shown on the figure 4. In agreement with the explanation given in section 2.2, for small values of Δx the branching point does not influence integral ( 14) and computation results are adequately described by formula (21). As the radial separation grows, the branching and saddle points move further apart and the dependence on the turbulence parameters disappears in agreement with (25). The red line seemingly crossing origin corresponds to the fact that constant offset in formula (25) for these parameters has the value of 0.01 cm. This corresponds to the fact that small-angle scattering contribution into scattering signal decays much slower with radial separation than that of the backscattering. As it can be seen from the figure 4, the influence of small angle scattering can potentially even change the sign of the linear dependence coefficient, which can be used for experimental confirmation of the theoretical predictions.

Finally, to illustrate significance of the difference in Δx definition in this paper and in [19] figure 5 demonstrates calculated value of ymax together with theoretical predictions given by both formula (21) and intuitive interpretation given in [19] as a function of frequency separation. This difference being a factor cos 2 θ is quite significant for larger probing angles, which seem to be more suitable as they allow suppressing small-angle scattering effects. The calculation used is the same at the one presented at figure 3, while probing angle is 30°. To see the impact of difference between analytical formulae we have calculated the turbulence tilt angle from calculation results according to formula (21) and to [19] neglecting l⊥ in both cases. The results obtained were 45° and 41° respectively, which mean that while rigorous analytical formula provides a better estimate, the difference in the considered case is not that large.

Overall, the modeling results confirm analytical conclusions and highlight the importance of the effects caused by small angle scattering. We would also like to empathize that various effects mentioned in section 3.2, such as simplified scattering efficiency effects at higher probing angles, finite probing beam waist and saddle point κ* moving close to integration limit 2ω/c for small turbulence tilt angle manifested in a number of calculations not presented in this paper.

Together with small-angle scattering effects, they made the range of parameters where the calculations were in agreement with analytical prediction (21) rather limited. This observation casts doubt on the ability of the diagnostic to produce reliable quantitative experimental results and suggests that the measurements should be interpreted very carefully.

Conclusions

In this paper we have investigated the turbulence tilt angle measurements technique within the linear approximation of analytical theory. Analytical expression for CCF was derived and key features, such as the influence of small angle scattering as well as dependence on cut-off separation rather than turning point separation of "intuitive interpretation" were pointed out. New formulae for the maximum of the CCF were derived for both a case with no small-angle scattering influence and for the one in which small-angle scattering dominates the scattered signal. A general expression, taking into account finite beam width was also obtained, demonstrating that for smaller beam waists direct interpretation is impossible.

Analytical results were validated with the use of numerical simulations employing reciprocity theorem and linear approximation. The contribution and the role of the small angle scattering that have been identified in the new formulae and the difference (which one, precise it) with "intuitive" interpretation [21] was confirmed.

Fig. 1 .

 1 Fig.1. Simplified scheme of the theoretical model geometry defining part of variables.

Figure 2 .

 2 Figure 2. CCF maximum dependence on the radial separation. L=10 cm, lmin=1 cm, lmax=1.5 cm, ρ=3 cm, φ=45°,f=30 GHz. Lines correspond to formula (21), points correspond to calculation results, while different colors, line and symbol types designate different probing angles θ.

Figure 3 .

 3 Figure 3. CCF maximum dependence on the radial separation. L=4 cm, lmin=0.2 cm, lmax=1.4 cm, ρ=1.6 cm, φ=45°,f=70 GHz. Lines correspond to formula (21), points correspond to calculation results, while different colors, line and symbol types designate different probing angles θ.

Figure 4 .

 4 Figure 4. CCF maximum dependence on the radial separation for L=8 cm, lmin=0.5 cm, lmax=1 cm, ρ=2 cm, φ=45°,f=70 GHz. Black line with points corresponds to computation results, solid red line corresponds to theoretical formula (21) while dashed blue line corresponds to formula (25)

Figure 5 .

 5 Figure5. Dependence of ymax on the probing frequency. L=4 cm, lmin=0.2 cm, lmax=1.4 cm, ρ=1.6 cm, φ=45°,f=70 GHz. Black symbols correspond to calculation, solid red and dashed blue lines are given by formula(21) and intuitive interpretation given in[19] respectively.

  is based on determining the tmax corresponding to maximum of absolute value of CCF(Δω,t), we can only consider term of exponential dependence on t:

		saddle		() Vt	2				l	2 max	l	2 min		2	sin	2	2	xVt		l	2 max	l	2 min		sin	2
	( Jt	)	exp			1
					2									2		2						2	2
				2*								8	*		l						2	*	l
																	cx						cx
				2 max ll	2 min		2	sin	2	2							2			sin	3
	exp	1														2			2		;
			2			22
			*		8*																	cos
									cx

Acknowledgements

The analytical investigation was performed with support of the Russian Science Foundation grant 17-12-01110, whereas numerical analysis was supported by the Ioffe Institute. This work has been carried out partially within the framework of the EUROfusion Consortium and of the French Federation for Magnetic Fusion Studies (FR-FCM) and has received funding from the Euratom research and training program 2014-2018 and 2019-2020 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

Appendix 1:

If the formula (17) was to be considered for arbitrary relation between l min , l max and ρ, one could still obtain the expression for t max by finding real part of the exponent argument. The key point is to keep in mind imaginary parts of both ρ* and  l cx (17) using them:

And as a result:

;


O L Krutkin
email: krutkinoleg@rambler.ru

E Z Gusakov

S Heuraux

C Lechte

Nonlinear Doppler reflectometry power response. Analytical predictions and full--wave modelling

Keywords: turbulence diagnostics, Doppler reflectometry, plasma

Nonlinear Doppler reflectometry power response.

Analytical predictions and full-wave modelling

Introduction

Turbulent transport plays a key role in limiting confinement in magnetic fusion devices. Investigation of underlying physical processes behind it requires suitable tools to study plasma turbulence.

One of such tools is Doppler reflectometry or Doppler Backscattering based on the analysis of time series obtained at fixed frequencies, which allows to study poloidal spectrum of electron density fluctuations [1] as well as poloidal rotation velocity [2]. The diagnostic probes the plasma with microwave beam in the presence of a turning point. This probing beam is tilted with respect to magnetic surface and only the backscattering signal is measured, not the direct reflection. The difference in frequency between scattering and probing signals is interpreted as a Doppler effect allowing to define the poloidal velocity of density fluctuations. This velocity is further decomposed into plasma rotation velocity and phase velocity of plasma density fluctuations. The tilting angle of antenna in these measurements corresponds to fluctuation poloidal wavenumber defined by Bragg rule, which makes the measurement of fluctuation poloidal wavenumber spectrum possible by varying the probing angle and the probing frequency assuming a poloidal invariance of the turbulence properties.

However, such a straightforward interpretation is only possible in linear regime of scattering, where Born approximation over the amplitude of density fluctuations is applicable. This case was extensively studied both analytically [3] and numerically [4], [5].

Analytical predictions have also been made for strongly nonlinear (or saturated) case [6] and nonlinear effects were observed in full-wave numerical modeling [7]- [START_REF] Lechte | [END_REF]. In the analytical theory a nonlinearity criterion was derived, that would allow to distinguish strongly nonlinear regime of Doppler reflectometry.

Finally, a study of intermediate nonlinear regime, when the amplitude of scattered signal scales nonlinearly with density fluctuations r.m.s. was performed [10] and criterion of transition from linear regime to nonlinear regime was derived. This analysis, however, was done in the physical optics approximation, which has limited domain of validity due to the fact that only plasma at turning point region is taken into consideration.

In the present paper, intermediate nonlinear regime of Doppler reflectometry will be considered in the framework of perturbation theory applied to Helmholtz equation, to overcome limitations of physical optics.

Analytical treatment

In this work, plasma probing with the microwave beam of O-polarization will be considered. Analysis is perfomed in a slab geometry, background density profile is assumed to be linear over radial coordinate and uniform over poloidal one. Model geometry of this treatment is shown on the figure 1.

Like in [10], the transition to nonlinear scattering regime is assumed to happen when amplitude of scattered signal corresponding to linear scattering becomes smaller than that of the signal corresponding to multiple scattering. To determine these signals perturbation theory is used.

Basic equations and perturbation theory

Microwave propagating through plasma at fixed frequency is described by Helmholtz equation:

2 2 2 2 2 ( , ) ( , ) ( , ) ( , ) 0; d d E x y E x y k x y E x y dx dy + + = (1)
which in the case of perturbed plasma can be written as:

2 2 ( , ) ( ) ( , ) ( , ) ( , ) 0; E x y k x E x y k x y E x y δ Δ + - = (2)
The wave number for O-mode is defined as k 2 (x,y)=1n(x,y)/n c where n c =m e ω 2 /4πe 2 is cut-off density. That in turn allows us to relate wave number perturbation to density perturbation: k 2 (x)=1-n 0 (x)/n c ; δk 2 (x,y)=δn(x,y)/n c While for the linear density profile exact solutions can be obtained, we will use WKB approximation. This allows us to obtain all the relevant physical results without dealing with complicated mathematical expressions. Assuming δk=0 in equation to (2) and separating variables, WKB approximation, solution of unperturbed Helmholtz equation for a poloidal harmonic k y is:

0 ( ) ( ) 4 (0, ) ( , ) ( ) ( , ) e cos ( , ) ; 4 y y y y y y L k i k i y x k k E x k f k k x k k x k dx π π Φ + = × ⎛ ⎞ × + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∫ (3)
where

( ) ( ) ( ) 0 Φ , y L k y y k k xk d x = ∫
, f(k y ) is the probing beam amplitude distribution over poloidal wavenumbers. L(k y ) is radial turning point position for a given poloidal wave number and k(x,k y ) 2 =k 2 (x)-k y 2 . Equation describing the next order of perturbation theory for a single poloidal mode can be written as:

2 2 2 1 2 1 2 2 2 ( ) ( , ) ( , ) ( , )
y x d c c E x k k x k E x k dx k k n x q i k q i k x k n k x k q dx f k q dq ω ω δ π π ∞ -∞ - + = ⎛ ⎞ = Φ -+ × ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ × - + - ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∫ ∫ (4)
Where q corresponds to the poloidal wave number of density perturbation over which integration is carried out. For further analysis, we will assume that density fluctuations are uniform over x, which corresponds to large (l cx ~L(0)) radial correlation length and is in line with physical optics description. In this case, the solution to this equation with boundary conditions defined by perturbation theory can be written as:

2 (t 1) 4 2 1 2 2 1 ( ) 2 (s 1) 4 2 2 ( ) ( ) (0, ) ( , ) ( , ) ( , ) exp ( ) ( ', ) ' ( ) ( ) (0, ) 1 1 2 ( 0 , ) ( 0 , ) ( 0 ,
n q f k q e k k q c n E k x k k x k q k x k q i k q it k x k q dx n q f k q e k k q c n t k k k k k k π π ω δ ω δ + =± - + - - = × - - - ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ × Φ -+ - - ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ - ⎛ ⎞ - - - ⎜ ⎟ ⎜ ⎟ - - ⎝ ⎠ ∑ ∫ 2 1 1 ( ) ) (0, ) exp ( ) ( 1) ( ) ( ', ) ' ; ( , ) y s t L k y y y y y x q k k i k t k is k x k dx k x k =± =± × ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ × Φ + + Φ + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∑ ∑ ∫ (5)
The solution (5) includes two terms -one of them represents a solution of the differential equation ( 4) when it is homogeneous, whereas the other does not. The latter term can be interpreted as a "forced" oscillation of system, while the former is the propagating one. For this reason, we will designate this terms as A F and A P .

( ) 1 1 1 ( ) (0, ) exp ( ) ( ', ) ' ; ( , ) 4
1 ( 0 , ) ( 0 , ) 1 exp 2 (0, ) ( , ) 4 exp
( ) ( 1) ( ) ( ', ) ' ; L K F p x P p t L K q x k K A i K ip k x K dx k x K tk K k K q A i p k K q k x K q i K q i t K ip k x K q dx π π =± =± =± + ⎞ ⎛ ⎞ ⎛ = Φ + + ⎟
δ ω ω ∞ ∞ -∞ -∞ = ∫ ∫ (6)
Here A s is the amplitude of scattered signal received by antenna and P is probing beam power. This formula allows us to derive the signals corresponding to linear and quadratic scattering without calculating corresponding electric field in the whole space.

Linear response

To calculate the scattering signal corresponding to a single scattering, electric field given by formula (3) should be substituted into (6).

( )

k k k k q f k q i k q i k k x k k x k q f k it k x k dx is k x k q dx π δ ω + ∞ ∞ + =± -∞ -∞ + = - × + -- Φ + + Φ × + ⎛ ⎞ × + + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∑ ∫ ∫ ∫ ∫ ∫ (7)
We will consider the version of diagnostic where the same antenna used as receiver and emitter. Also the diagram of antenna will be assumed Gaussian so that:

( ) ( ) ( ) 1/2 2 2
2 e x p /2 ;

y y f k k K π ρ ρ ⎡ ⎤ = - - ⎢ ⎥ ⎣ ⎦
where K=ωsinϑ/c, ϑ is a probing angle with respect to normal to magnetic surface (which in our case is directed along x) and 2ρ is probing beam waist. The integrals over q and k y in (7) can be calculated using saddle-point method, however to simplify further derivation we will assume wide enough beam corresponding to the cutoff situated in the antenna near field.

( ) ( ) 0 2 ; , L K dx k x K ρ >> ∫ (8)
In this case antenna diagram determines the main contribution to integrals over poloidal wave numbers in (7) and the result of integration is

( ) 2 3 / 2 2 1 1 0 ( ) (0, ) 4 ( 2) 4 ( , ) exp 2 ( ) ( ) ( , ) ; 2 4 y L K y s t s e c L k x k k ie n K A P d x m n k x K i K i i s t k x K dx π δ ω ω ρ π π =± =± - = × ⎛ ⎞ ⎡ ⎤ ⎜ ⎟ × Φ + + + + ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ ∑ ∑ ∫ ∫ (9)
In the formula [START_REF] Lechte | [END_REF] two contributions of different origin can be seen. Contributions possessing same indexes s=t, which correspond to backscattering in radial direction and contributions possessing s=-t which correspond to the forward-scattering.

Up to this point, we didn't make any assumptions about density profile shape. To calculate integral over x we will assume that density profile is linear, which means that k(x) can be written as

( ) x ( , ) (0)
y y L k k x k c L ω - =
Which allows us to calculate the scattering signal, which for large radial correlation length (l cx >>(Lc 2 /ω 2 ) 1/3 ) is dominated by forward-scattering.

3/2 2 3/2 1/2 ( 2 ) 4 () 4 ( ) exp ; 3 (0)
lin s e c e P n K A L K m n L K i cL π δ ωρ ω - = - × ⎛ ⎞ × ⎜ ⎟ ⎝ ⎠ (10)

Quadratic response

Next, we calculate the quadratic response to the backscattering signal by substituting formulae (3) and ( 5) into (6). In the case of wide enough antenna beam (8) that we are considering, this result in an expression:

( ) min 2 3/2 2 2 1 0 ( ) 4 ( )( 2 ) 4 (0, ) exp ( ) ( , ) ; 4 ( , ) ( 2 )
L s s e c c F N L K x e n q n qK A P d q d x m c n n A A k K i K is k x K dx is k x K q q K π ω δ δ ω ρ π ∞ =± -∞ -- = × - ⎞ ⎛ Φ + + ⎟ ⎜ ⎜ ⎟ + ⎝ ⎠ ∑ ∫ ∫ ∫ (11)
Here L min is the smallest of L(q) and L(q+K). Firstly, we will consider A F together with the t=-1 term of A P since they can cancel each other out for specific values of q. Integrals over x can be written as:

( ) min ( ) ( ) , 1 0 , 1 0
( ) ( ) (0, ) exp ( ) ( ) ( ', ) ' ; ( , ) 4
(0, ) (0, ) exp ( ) ( , ) ( , ) exp ( ', ) ' ( ', ) ' 4 4 L K L K F s p x L P s p L K L K q x x k K I i K i s p k x K dx dx k x K k K k K q I i K q k x K k x K q is k x K dx ip k x K q dx π π π =± =± + ⎛ ⎞ ⎡ ⎤ = Φ + + + ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ + = Φ +× + ⎛ ⎡ ⎤ ⎡ ⎤ × + + + + ⎜ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎝ ∑ ∫ ∫ ∑ ∫ ∫ ∫ ; dx ⎞ ⎟ ⎜ ⎟ ⎠ (12)
Firstly, the main contribution to the signal is once again provided by forward-scattering (s=p) due to faster variation of phase in the backscattering term integrand. I P coincides with I F only for q=0,-2K, and is smaller for other values of q, also due to variation of phase of the integrand. For that reason we will consider the vicinity of q=0 where phase of integrand in I P can be expanded into Taylor series. Since the integrand in ( 12) is symmetrical around q=-K, to calculate the integral over q it is enough to consider q>-K. That area is defined by condition 1 (0) ( )

q q L L K sinϑ << Δ =
, which corresponds to slow variation of integrand phase over integration domain. This consideration however is only valid, when q=0 and q=-2K can be considered separately, which corresponds to K >Δq.

Large probing angle K >Δq.

After performing the integration over x under the same assumptions as in 1.2 we obtain the following:

A similar result can be obtained for t=1 term in A P . It is however, smaller than a combination of terms in (13) and can be neglected. Due to symmetry of integrand, same results can be obtained in vicinity of q=-2K.

Substituting ( 13) into ( 11) and neglecting A P where it is appropriate we get

2 3 / 2 2 3 / 2 2 2 1 / 2 2 2 2 4 4 () 2 4 ( ) exp 4 3 (0) ( ) ( 2 ) ( ) ( 2 ) ( 2 ) ( 2 ) ( ) ( 2 ) (0) ( ) ; s e q c c K q q c q e L K A P L K i m c c L n q n q K n q n q K dq dq q q K n q q K n с n q n q K i L L K dq n π ω ω ω ω ρ δ δ δ δ δ δ ω -Δ ∞ - Δ Δ -Δ ⎛ ⎞ = × ⎜ ⎟ ⎝ ⎠ ⎡ -- -- × + + ⎢ + + ⎢ ⎣ ⎤ -- + ⎥ ⎥ ⎦ ∫ ∫ ∫ (14)
To continue analysis, we will assume the Gaussian spectrum of the density fluctuations

2 2 0 ( ) exp / 8 cy cy n q n l q l δ δ ⎡ ⎤ = - ⎣
⎦ . In this case, estimating the main contribution to A s one can obtain:

2 2 2 0 2 2 2 2 3/2 1/
L K l n e P A m c K n Kl l K L K i cL L K l n e P A m c K n Kl L K i cL δ ω ωρ ω δ ω ωρ ω ⎧ ∝ × ⎪ ⎪ >> ⎪ ⎛ ⎞ ⎪ × - ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎨ ⎪ ∝ × ⎪ ⎪ < ⎪ ⎛ ⎞ × ⎪ ⎜ ⎟ ⎪ ⎝ ⎠ ⎩ (15)
In the intermediate case Kl cy ~1 estimating integral is difficult, but these two parametric dependencies coincide with each other in this case which points to the conclusion that parametric dependencies properly describe the whole parameter interval.

Small probing angle K <Δq.

As in the previous case, we expand the integrand phase in (12) in the vicinity of q=0 in Taylor series. This time the area of validity of such expansion is ( )

2 / 0 cos q q cL ω ϑ << Δ = .
After performing integration over x under the same assumptions as before we get

3/2 1/2 3/2 1/2 2 ( ) 4 ( ) exp ; 3 (0) 2 ( ) 4 ( ) exp ( ) (0) ( ) 3 (0) sin ( ) (0) ( ) ; ( ) (0) ( ) FS F FS P L K I LK i cL L K с I LK i i LK L qq K cL с L K L q q K с L K L q q K ω ω ω ω ω ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ≈ - + × ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ + ⎜ ⎟ ⎝ ⎠ × + (16)
Once again, taking the contribution of A P into account only in vicinity of q=0 and q=-2K, we can divide A s into two integrals over different domains:

2 2 2 3 / 2 2 3 / 2 2 1 / 2 2 2 2 ( ) 8 (K) exp 3 (0) ( ) ( 2 ) ( 2 ) ( ) ( 2 ) ( ) (0) ; 2 s e c q q c K e L K A P L i m c c L n q n q K dq q q K n с q K n q n q K i L K L dq q K n π ω ω ω ρ δ δ δ δ ω ∞ Δ Δ - ⎛ ⎞ = × ⎜ ⎟ ⎝ ⎠ ⎡ -- × + ⎢ + ⎢ ⎣ ⎤ + -- + ⎥ + ⎥ ⎦ ∫ ∫ (17)
Estimating the main contributions into integral over q for different values of cy l we obtain

2 2 3 0 2 3/2 2 2 1/2 2 2 2 2 0 2 2 3/2 1/2 ( ) (0) 1; 4 ( ) exp / 4 ; 3 (0) ( ) 1; ( ) (0) 4 ( ) exp ; 3 (0) s c y e c cy cy s c y e c cy n e P A l LK L m n ql L K i Kl c c L n e P A lLK m n c ql L K L c L K i cL δ ωρ ω ω δ ω ωρ ω ω ⎧ ∝ × ⎪ ⎪ Δ >> ⎪ ⎛ ⎞ ⎪ × - ⎜ ⎟ ⎪ ⎝ ⎠ ⎪ ⎨ ⎪ ∝ × ⎪ ⎪ Δ << ⎪ ⎛ ⎞ × ⎪ ⎜ ⎟ ⎪ ⎝ ⎠ ⎩ (18)
These parametric dependencies also coincide in the intermediate parameter range at Δql cy ~1. Moreover at K~Δq formulae [START_REF] Gusakov | [END_REF] and (18) are in agreement with each other.

Criterion for transition to nonlinear backscattering regime

3/2 1/2 3/2 1/2 2 ( ) 4 ( ) exp ; 3 (0) 2 ( ) 4 ( ) exp 2 (0) ( ) 3 (0) sin 2 (0) ( ) ; 2 (0) ( ) FS F FS P L K I LK i cL L K с I LK i i L LK K q cL с L L K Kq с L L K Kq ω ω ω ω ω ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ = - × ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ × (13)
We have obtained both linear and quadratic contributions to the scattering signal, which correspond to single and double scattering of the probing wave. Comparing these two contributions allows us to derive the threshold amplitude for δn exceeding which lead to nonlinear effects dominating in the scattering signal.

Combining (10) with ( 15) and ( 18) we obtain following criteria

n c q l a n L q K n c c q l b n l L n K l c n q K n c K l d n l δ ω ϑ δ ω ω ϑ δ ϑ π δ ϑ ω - - ⎧ ⎫ >> Δ >> ⎪ ⎪ ⎪ ⎪ Δ >> ⎬ ⎪ ⎪ >> Δ << ⎪ ⎪ ⎪ ⎭ ⎨ ⎫ ⎪ >> >> ⎪ ⎪ ⎪ ⎪ Δ << ⎬ ⎪ ⎪ >> << ⎪
⎪ ⎭ ⎩ (19) While this system is consistent in itself, interpreting different parameter ranges using it is difficult. This system of inequalities can be rewritten with respect to the values of L, l cy , c/ω and ϑ. Doing that, we obtain two cases corresponding to different value of poloidal correlation length of turbulence.

For l cy 2 >>Lc/ω:

K l n c c a n L L K l n c b n L δ ϑ ω ϑ ω ϑ δ ϑ ϑ ω ϑ ⎛ ⎞ - >> << ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ - >> << ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ (20)
In the opposite case of l cy 2 <<Lc/ω:

n c c c a n l L L c n c L b c n l l K l n c c n l δ ϑ ω ω ω ϑ ϑ δ ϑ ω ϑ ω ϑ ω δ ϑ ϑ ω >> << << >> << ⎛ ⎞ - >> << ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ (21)
The latter case (21) seems to be relevant to the Doppler backscattering experimental measurements, however for these results to be applicable the density perturbation should be smaller than that prescribed by the criteria of transition to fully nonlinear regime. This regime is characterized by the strong probing wave phase modulation and by saturation of the scattering signal power growth with the probing microwave power. Using the approach described in [12] together with assumptions of linear profile and large radial correlation length the condition of strong phase perturbation of the probing beam can be written in accordance with appendix A as:.

Comparing ( 22) to ( 20) and ( 21) we can see that even for a small nonzero probing angle large enough value of l cy can lead to an existence of nonlinear regime that is not saturated. It can also be seen that in case of l cy <c/ω the strong phase modulation threshold is never higher than the threshold of nonlinear regime we are considering in this paper.

To draw comparison between obtained criteria and predictions of the physical optics model [10] we first need to change their normalization based on [12] [13] allowing to take the plasma volume phenomenon into account. In the case of 1D turbulence considered in the present paper the condition of strong phase modulation utilized for normalisation in [10] instead of ( )

ln / c cx cx n c n l L L l δ ω >> takes a form (22).
With that the condition of nonlinear regime onset derived in [10] coincides with (20). According to [10], this criterion should be valid for the whole parameter range and other criteria corresponding to (20) are not obtained. The reason for that might be Kirchhoff approximation [START_REF] Beckmann | The Scattering of Electromagnetic Waves from Rough Surfaces[END_REF] used in [10]. Under our assumption, the Kirchhoff aproximation imposes condition δn/n c <<c/ωL which contradicts most of parameter ranges in (20) and (21) Finally, in order to generalise criteria (20) and (21) to the case of real 2D turbulence we will employ the same approach as described in [10]. Firstly, we will note that in the case of 1D density fluctuations and normal probing of plasma the criterion (20) coincides with the saturation criterion (22). While we are unable to derive nonlinearity criterion that would account for radial inhomogenity of the turbulence and include redial correlation length, a saturation criterion at normal incidence of the probing wave was obtained in [ [13]. Assuming that the two criteria coincide in general case of the 2D turbulence and using the strong phase modulation criterion ( )

ln / c cx cx n c n l L L l δ ω
>> obtained for the 2D turbulence in [12] and [13] one can obtain from (20) the following criteria: [ [13]. Therefore the dependence of criteria on radial correlation langth is inherited from [[13] and explanation of physics behind this dependence is unclear. While this procedure is based on assumption of the concidence of nonlinearity and saturation criteria in the case ϑ=0, direct confirmation of which seems challenging, the numerical validation is possible. The result of full-wave modelling presented both in [10] and in latter sections of this paper demonstrate agreement with the generalized formulae (23) and (24).

The generalized expression (23) explicitly coincides with the one obtained in [10]. While the parameters determining transition to nonlinear regime include turbulence correlation length, that is not nesecessarily known in experiment, formulae (23) and (24) still provide some information about possibility of obtaining linear scattering regime in experiment. .

Analytical treatment discussion

Having obtained criteria for transition to nonlinear regime, we would like to consider their relevance to experiment. But before doing that all the limitation of the model should be accounted for.

Model limitations

First and the most experimentally limiting approximation of the model is that of the radially uniform density fluctuations in the turning point area. It was used to obtain the solution of equation (4). To overcome this limitation, generalized formulae (23) and ( 24) are proposed, though they are not rigorously derived from basic equations.

Another simplification used in this model is linear dependency of k 2 on x, which allowed to compute explicitly integrals over x in [START_REF] Lechte | [END_REF] and (12). In experiment, the k 2 profile in the turning point vicinity, where Doppler backscattering takes place, can be approximated by a linear behavior. The corresponding k 2 profile scale length should replace parameter L in the obtained criterea. In this form criterea (21), ( 22) are also applicable to the Xmode probing.

This treatment also makes the description of the probing of an important case of H-mode plasma possible. Such plasma features a strong density gradient at the edge followed by a flat area deeper in the plasma. While the gradient scale length inside the pedestal is different than at the edge, the edge area does not contribute strongly into scattered signal when cut-off is located in the flat area due to the reverse dependence on k(x,K) of integrals over x in formulae ( 9) and (12). Therefore analytical results are still relevant inside the pedestal when a gradient scale length corresponding to a flat area is used as L.

Another assumption that was used is [START_REF] Krutkin | Proc. 44th EPS Conf. on Contr. Fusion and Plasma Physics (Belfast) ECA[END_REF], which generally holds true in the Doppler backscattering experiment.

The slab plasma geometry considered in the paper is consistent with experiments at the medium and large scale devices. The geometrical effects introduced by magnetic surface curvature are not leading to qualitative effects even on small device, as it was shown recently in [START_REF] Gusakov | [END_REF].

Finally, we used WKB approximation, but the same results can be obtained by using exact Airy solutions, though derivation includes more complicated expressions. This was demonstrated in [START_REF] Gusakov | [END_REF] for the Born approximation.

Experimental relevance

While formulae (23) and (24) provide criteria of nonlinear transition for a wide parameter range we would like to note the ones relevant to experimental parameters.

In FT-2 Doppler reflectometry experiment [17] the probing parameters for O-mode lead us to L=1.5 cm, f= 30 GHz. Radial and poloidal correlation lengthes determined from experimental data are 0.2-0.4 cm and are in agreement with gyrokinetic simulation results. In such a scenario, formula (24) is applicable and for the probing angle of 35° used in experiment and the threshold value of δn/n c is roughly 0.2, which is higher than the one required for saturation in this case. Therefore, our threshold (24) predicts that nonlinear regime can not be obtained under these experimental conditions, though curvature effects not included in the model might be significant in this case.

Another example is ASDEX Upgrade Doppler reflectometry experiment [18] with simultaneous measurement of l cy and l cx . In this case L=10 cm, f~ 30 GHz. The correlation lengthes defined from experiment are l cy ~1cm and l cx ~ 3 cm. Authors also propose corrected values l cy, l cx ~0.6 cm, which are based on numerical calculations performed to account for broadening of correlation in radial correlation reflectometry measurements predicted by linear theory [19] and similar effect in poloidal correlation reflectometry. For either pair or values, the relevant formula is (24) and normal probing used in experiment does not provide a possibility for nonlinear regime. In the case of probing under similar conditions, but with a tilted antenna, formula (24) would be the most relevant. Assuming that l cy and l cx value obtained in [18] are relevant to another L-mode ASDEX Upgrade Doppler reflectometry experiment, we can consider diagnostic setup with tilted antenna [20]. In this case for f= 60 GHz corresponding L is roughly 20 cm which means formula (24) is relevant again. For a probing angle ϑ= 20° formula (24) provides threshold value of δn/n c of 0.04, which is significantly higher than the threshold δn/n c ~0.003 provided by physical optics.

To conclude, formulae (24) seem to be the most relevant for experimental parameters, in particular (24c) describes the situation where it is possible to observe nonlinear scattering regime between linear and saturated regimes.

Qualitative picture

The term δn(-2K) in formula (10) corresponds to the face that the main contribution to the scattering signal in the linear (Born) approximation is provided by the fluctuation fulfilling the Bragg rule q=-2K. In the next order of perturbation theory the Bragg rule should be also fulfilled which means that the total poloidal wavenumber of the fluctuations taking part in the scattering is also -2K, which is evident from formula (11). However, having multiple scattering events allows a certain degree of freedom in selecting the poloidal wavenumber of each separate fluctuation. The combination of the Gaussian spectra in formula ( 14) and ( 17) leads to q=-K being the poloidal wavenumber providing dominant contribution to the scattered signal in the case of quadratic approximation. This is also represented by the exponents in formulae (23) and (24).

These arguments woud also apply for further orders of perturbation theory, which means that for j-th order the dominant contribution to the scattering signal would be provided by fluctuations with q=-2K/j. That in turn means, that scattered signal power would be proportional to exp(-K 2 l cy 2 /j)×(δn/n c ) j .

Taking this into account, one can speculate, that with the growth of δn consecutively higher orders would dominate the scattered signal, as they have weaker exponential supression. This speculation explains the results of numerical modelling presented further as well as numerical results of [10].

Numerical calculations

To validate obtained theoretical results, a full-wave numerical modelling with IPF-FD3D code [START_REF] Lechte | [END_REF] was performed. Two types of calculation were carried out.

In the first one, fluctuations of density uniform along radial direction were generated and applied to linear density profile uniform along the poloidal direction. This calculation is the closest to analytical model considered. An example of input density for FD3D computation can be seen on figure 2.

Second type of calculations was performed for more realistic 2D turbulence applied to the same background profile.

Like analytical analysis, calculations were performed in the slab geometry. To generate random turbulence the density fluctuation Fourier transform amplitude used. Random phaseswere added to the amplitude of each harmonic and then Fourier transformation was performed. Each random turbulence field obtained this way was set to ha r.m.s. of 1 and then scaled to a certain fraction of n c . Each calculation was performed for 6000 separately generated turbulence fields. The calculation results were then averaged to produce the final result presented in this paper. This way a realistic random turbulence was used in calculation while it's spectral properties were still in agreement with analytical model.

All the calculations were performed for a "frozen" density without temporal dependence in agreement with analytical model.

An expamples of densities for two types of calculations can be seen on figures 2 and 3 respectively. For the first figure, l cy =1 cm and turbulence r.m.s. is set to 0.1n c . Probing frequency f is 40 GHz and L is 20 cm. In the second figure radial correlation length of the turbulence is 1cm and the rest of parameters are same as in figure 2.

An issue that was adressed during calculation was the fact that uniform turbulence shown in figures 2 and 3 displays a very high turbulence level at plasma edge. To exclude this effect the turbulence was suppresed at the edge by multiplying the initial tubtulent function to a hyperbolic tangent with a step located at L/2. It should be noted, however, that in most cases, including the ones presented in this paper, this suppresion did not change the outcome of full-wave computation leading us to a conclusion that strong edge turbulence did not significantly influence the scattered signal in presented cases in the first place.

Modelling results for nonlinearity criterion

For the calcualtion with radially homogenous turbulence L=20 cm, ϑ= 40°, l cy =1.3 cm and ρ= 4 cm were used.

On the figure 4, scattered signal power depending on turbulence amplitude is plotted. Turbulence amplitude characterized by as follows: r.m.s.(δn)=a×n c . Lower half of the graph also shows the power scaling value n i =ln(P i+1 /P i )/ ln(a i+1 /a i ). subscript i in this formula refers to a specific value of tubulence r.m.s. a i used in a calculation. The value n i is a characteristic of the power dependance of scattered field power with the growth of turbulnce amplitude: P ∝ a n

In the linear regime scattered field amplitude depends linearly on turbulence r.m.s. which means that dependence of scattered signal power is quadratic and the value of n i is 2, which is indeed the case for small values of a. As the fluctuations become stronger, higher regime on nonlinearity can be seen with n i increasing to higher numbers. After that the transition to the strong phase modulation regime occurs which leads to saturation of the scatterred power, which can be seen in n i decrease.

Vertical lines on the picture correspond to different criteria derived previously. Solid green line corresponds to approptiate for this case (20), dashed purple line is saturation criterion (22) and dotted orange line corresponds to physical optics result which is described by (21). It can be seen that for these parameters the difference between our consideration and physical optics results is quite significant. It seems that in the considered case of radially uniform turbulence our criteria predicts transition to nonlinear regime more precisely than physical optics model. An example of a more realistic calculation with 2D turbulence is plotted on the figure 5. Probing beam parameters in this case were also L=20 cm, ϑ= 40°, l cx =1.3 cm, l cy =1.3 cm and ρ=4 cm.

The vertical lines in this case correspond to generalized version of the same criteria -solid green line is formula (24), the dash-dotted orange line is the criterion obtained in [10] and dashed purple line is saturation criterion

( ) ln / c cx cx n c n l L L l δ ω

>>

. Just as in the case of 1D turbulence, in 2D calculations physical optics model underestimates the threshold density fluctuation amplitude for transition to nonlinear regime while our criterion provides a correct value. As can be seen from figure 5, the difference between the two is more than of an order of magnitude which makes physical optics criterion significantly more limiting. However in the cases when (24) predicts nonlinear transition at density perturbation amplitudes so high that saturation would happen before nonlinear transition occurs, numerical results are not in agreement with (24). An example of such a calculation can be seen at figure 6. The possible cause of this discrepancy could be that in such situation Kl cy ~1, which is the limit of validity of formula (24). In this case formula (24) becomes relevant, however numerical coefficient associated with it is unclear since the integral estimation in that case was rather rough, which is why we neglected it. That omitted numerical coefficient is what we assume was the cause for discrepancy. However, an exact estimation of the coefficient and consequently validation of this explanation would require further study of analytical formulae derived in part 1 with numerical methods.

Modeling result to validate qualitative picture

To validate the explanation for a continuous growth of nonlinearity with the growth of δn proposed in 2.3 a calculation with a step-like poloidal turbulence spectrum was performed: 0 0 0 0 0 , ( , ) 0 or/and n q k k n q q k k δ κ δ κ κ

< < ⎧ = ⎨ > > ⎩
The value of k 0 was selected so that K<k 0 <2K. For this spectrum according to our speculation, higher orders of nonlinearity would lose the exponential advantage provided by the Gaussian spectrum, while linear scattering would be suppressed due to absence of turbulence with high enough poloidal wavenumbers. The numerical values for the calculation were:

f= 30 GHz, L= 20 cm, ϑ= 40° (which means that K=4.4 cm -1 ), ρ=4 cm and k 0 =5 cm -1 . The result of the calculation is presented at figure 7: The value n i in this case does not exceed 4, which is in agreement with our expectations, as it means higher orders of scattering are no longer dominating scattered signal. This confirms our speculation that higher orders of scattering only become viable when fast decrease of turbulence amplitude with polodal wavenumber is present. At low turbulence amplitude linear scattering regime is observed despite the absence of fluctuations capable of producing linear scattering signal in the turbulence spectrum. This can be explained by the finite width of the probing beam which makes linear backscattering at lower poloidal wavenumbers possible. Such backscattering is suppressed by antenna diagram, but can dominate a scattered signal when turbulence amplitude is low enough.

In this paper we have analytically investigated Doppler reflectometry nonlinear scattering regime in the framework of perturbation theory. The analytical results were obtained for radially uniform plasma turbulence and then generalized to describe turbulence with arbitrary radial correlation length. Depending on the set up and plasma parameters different thresholds for a transition to nonlinear regime were established and compared to experimentvalues. Analytical results were validated with full-wave numerical modelling for both radially uniform and fully 2D turbulence and an agreement between analytical nonlinear thresholds and full-wave simulation results has been shown. This agreement shows that the thresholds obtained can be applied to experiments according to the mentioned restrictions. An explanation for scattered signal power dependance has been deduced from the extension to higher orders of perturbation method and validated numerically.

Figure 1 .

 1 Figure 1. The scheme of geometry used in model. Background density profile is linear along x direction.

Figure 2 .

 2 Figure 2. Density with random fluctuations uniform in the radial direction. Grey line designates cut-off density for the probing frequency used in FD3D calculations.

Figure 3 .

 3 Figure 3. 2D density with random fluctuations. Grey line designates cut-off density for the probing frequency used in FD3D calculations.

Figure 4 .

 4 Figure 4. Scattered signal power calculated in case of radially uniform turbulence with lcy=1.3 cm. The dashed red line corresponds to quadratic power dependence. Vertical lines correspond to nonlinearity thresholds provided by our analysis (green line), physical optics (orange dash-dotted line) and strong phase modulation condition (purple dashed line).

Figure 5 .

 5 Figure 5. Scattered signal power and its exponential scaling in case of 2d turbulence with lcx=lcy=1.3 cm. The dashed red line corresponds to quadratic power dependence. Vertical lines correspond to nonlinearity thresholds provided by our analysis (green line), physical optics (orange dash-dotted line) and strong phase modulation condition (purple dashed line).

Figure 6 .

 6 Figure 6. Scattered signal power calculated in case of radially uniform turbulence with f=30 GHz and lcy=0.6 cm. The dashed red line corresponds to quadratic power dependence. Vertical lines correspond to nonlinearity thresholds provided by our analysis (green line), physical optics (orange dash-dotted line) and strong phase modulation condition (purple dashed line).

Figure 7 .

 7 Figure 7. Scattered signal power and its exponential scaling in case of step-like spectrum. The dashed and dash-dotted red lines correspond to quadratic and quartic power dependencies respectively.

Conclusions

Acknowledgements

Financial support of RFBR grant 18-32-00031 is acknowledged by O.K., whereas support of RSF grant 17-12-01110 is acknowledged by E.G.

Appendix A. Saturation criteria.

A condition of strong phase modulation, which has been derived in [12], can be obtained for the assumption, used in this paper. Formula describing probing beam phase perturbation in [START_REF] Lechte | [END_REF] in our notation and under the assumption of 1D density fluctuations and linear background profile can be written as: Assuming the turbulence to be statistically stationary and homogeneous:

( )

Now, assuming the turbulence has the Gaussian poloidal wavenumber spectrum and taking into account narrow antenna angular diagram, this integral can be estimated as: