Franck Schoefs

Morgan Boukinda

Modelling of marine growth effect on offshore structures loading using kinematics field of water particle

Structural response to extreme events or fatigue loading and structural integrity are major criteria to be quantified in a rational process of reassessment for offshore structures. It is now well established that the probabilistic mechanics approach gives an efficient means for measuring the relative changing in safety level compared to a predefined requirement. This paper proposes a probabilistic modelling of marine growth effects based on both Response Surface Methodology and the panel of available tests in laboratory. This method is here suggested to condition hydrodynamics coefficient to particle kinematics field randomness.

INTRODUCTION

The proposed paper takes place in the actual challenge for the reassessment of existing offshore structures where some uncertainties are to be quantified on the basis of inspection reports analysis. When studying the mechanical behaviour of structures placed in severe environmental conditions as offshore platforms, the structural reliability analysis offers, after decades of developments, a set of robust tools and methods. It leads to state the importance of uncertainty and sensitivity analysis especially in view to characterise the stochastic structure of the loading acting on the platforms. One must consider the intrinsic randomness of waves and the uncertainty on the modelling of fluidstructure interaction in their relative contribution to the response i.e. the load on the structure. Particularly, one must take into account the natural cluster of organisms i.e. the marine growth fouling (kelps, mussel beds ...) versus a probabilistic modelling. In fact, this settlement of the structure includes the major uncertainty source in term of diameter increment and hydrodynamic coefficients level. Its corresponding uncertainty evolves upon several factors (seasons, depth, ... ) and during the life of the structures and is greater at the design step when no inspection reports are available. For steel fixed offshore structures like Jacket or Jack-up platforms, the hydrodynamic overloading caused by marine growth is major in terms of percentage on each component and consequently on their summation for the external global loading assessment. Only effects on these platforms are considered here. Because of the cost of under-water inspections, it is not realistic to research a complete description of the marine growth during the life of the structure and to require a cleaning program based on this complete knowledge. In fact, database are very poor, and assumption are to be specified on the basis of rational criteria. The proposed paper reviews the actual practice of marine growth modelling in terms of loading and test procedures. In a second time, probabilistic modelling assumptions are analysed in their relative influence on the load response surface. The response surface methodology comes to be a versatile approach to perform sensitivity studies through Monte Carlo simulations or differential analysis. It leads to a ranking of basic variables and underlines the most important to be modelled. A probabilistic modelling is proposed in the form of a discrete probability density function of hydrodynamics coefficients based on uncertainties on diameter and also on a conditioning to random variables of kinematics fields of water particle. This conditioning is assessed when introducing the hydraulic parameters, the Reynolds and Keulegan-Carpenter numbers, which allows to link the in-site randomness of wave kinematics field to the assessment of hydrodynamic coefficients in laboratories. For the illustration, the site of the North Sea is selected. Data are taken in the bibliography. The so-called jacket platforms, which are steel framed structures, are considered and subjected to a quasi-static loading during severe conditions (storms).

REVIEW OF MARINE GROWTH MODELLING AND TESTING.

It is instructive to firstly mention the relevant contributions in the literature, which can be distinguished in two families: hydrodynamic modelling from experiments and marine growth physical modelling from in-situ measurements. In the first way, lot of works have been developed on the basis on sensitivity studies of overloading due to the nature and thickness of marine growth and hydrodynamic structure of flow. As the techniques of flow velocity and acceleration measurements are particularly indepth, complete studies are mostly recent. In fact, the equipments in water tunnels or large basin are generally costly considering the fact that large-scale experiments must be consider for similitude conservation. Some of them are usefully reviewed and detailed by [START_REF] Sarpkaya | On the effect of roughness on cylinders[END_REF]. The pioneer works due [START_REF] Sarpkaya | In-line and transverse forces on smooth and sand roughened circular cylinders in oscillating flow at high Reynolds numbers[END_REF][START_REF] Sarpkaya | In-line and transverse forces on cylinders on cylinders in oscillatory flow at high Reynolds numbers[END_REF] Paper No. 2004-JSC-344

Schoefs

Page number 2 of 8 concern fixed roughened cylinders placed in a sinusoidally oscillating flow. Others consider roughened cylinders subjected to sinusoidal oscillation in a large basin with intention of promoting two-dimensional flow [START_REF] Rodenbusch | Forces on cylinders in twodimensional flow[END_REF]. After this period, another way of experimentation concerns the carriage technique i.e. forced sinusoidally oscillations of a cylinder in a basin [START_REF] Kasahara | Wave forces acting on rough circular cylinders at high Reynolds numbers[END_REF][START_REF] Jusoh | Uncertainty in the estimation of the fluid loading due to the effects of marine growth[END_REF]Justesen, 1989, Sell). The relative thickness i.e. thickness k divided by diameter D of clean cylinder tested varies from 1/240 to 1/50. These works contributed largely to a better understanding of relations between the nature and thickness of fouling and relative change of loads. Systematic measurements of Reynolds, Keulegan-Carpenter and Sarpkaya numbers allows to asses precisely the hydrodynamic conditions of the tests. In spite of "controlled laboratory conditions" significant differences in drag, inertia and lift coefficients data are found especially for Keulegan-Carpenter exceeded 8 with no predominant reason [START_REF] Sarpkaya | On the effect of roughness on cylinders[END_REF]. Effects of three-dimensional vortex instability may affect them.

A specific work carried out by Theophanatos pointed out that the oneparameter characterization of roughness i.e. k/D is not suitable to embrace all the complexity of fluid-structure interaction and to quantify loading (Théophanatos, 1988). In fact the shape of roughness, effects of soft seaweed, limitation of experimental scales and percentage of covertures have been shown to have a great influence on the results and to be taken into account. Note that in some sites like in the Guinea gulf for instance, waves and currents coexist with a similar contribution to loading and special effects of their combination should be studied (Asao, Tagaki and Uekita, 1986).

In the second way, in situ observation after survey campaigns are analyzed to purpose a modelling of marine growth effects. It is first interesting to define what should be a good inspection i.e. the inspection with the right tool, on the right places at the right time. It is generally advocated to select a pile, or a face of a specific element of which the data are supposed to be representative of the entire level (Wolfram, [START_REF] Jusoh | Uncertainty in the estimation of the fluid loading due to the effects of marine growth[END_REF]. Near the splash area, horizontal members which are generally totally covered are also inspected. The area corresponding to the first thirty meters of depth are generally preferred because of their dominant contribution on hydrodynamic loading. Locally, it is necessary to note with precision the location of the colonized face [START_REF] Rodenbusch | Forces on cylinders in twodimensional flow[END_REF]. Lot of works (see section 1) give a review of marine species present on offshore structures implanted in North Sea [START_REF] Jusoh | Uncertainty in the estimation of the fluid loading due to the effects of marine growth[END_REF]Picken, 1985), near India [START_REF] Sankalpa | Marine growth on offshore structures in Indian offshore waters and removal strategy[END_REF], near the west coast of USA (Sharma, 1983). Most of these works define the way to assess the marine growth thickness from circumference measure with an uncertainty depending of the nature of species. Considering now, the impact on hydrodynamic coefficient of Morison equations [START_REF] Eashwar | The interrelation of cathode protection and marine macro-foulind[END_REF], it is generally assumed (Wolfram, [START_REF] Jusoh | Uncertainty in the estimation of the fluid loading due to the effects of marine growth[END_REF]) that drag coefficient is obtained following a linear regression with the thickness upon (k/D). Authors generally concluded that data are sparse and that more inspection are needed. Even with these data, there will still be non reducible uncertainties due to intrinsically variable patterns and measurement errors. In attempt to optimize cleaning programs there is a way to introduce this uncertainty through Risk Based Inspections strategy.

These studies allows to advocate guidelines for a probabilistic modelling at a given time : generally the stochastic process of marine growth colonization is unknown. In particular the process of colonization depends strongly on phenomena short in time as seasons ore severe storms during winter which are able to clean part of the structure and on some electrochemistry effects as cathode protection [START_REF] Eashwar | The interrelation of cathode protection and marine macro-foulind[END_REF]. Recent works show how to introduce seasonal variation for changing the Global Design Number level [START_REF] Faber | Reliability-based management of marine fouling[END_REF]. It leads to less conservative combination values of marine growth and waves. Finally, it leads to mix a pure reliable approach based on the relative change of safety index levels due to increase of random loads with a deterministic one specifying conservative procedures and implementing sensitivity studies to hypothesis.

The next section deals with the Morison equations and response surface methodology which is well adopted to sensitivity and uncertainty analysis.

RESPONSE SURFACE OF MARINE LOADS

The previous sections give the main trends in view to quantify marine growth effects for overloading purposes. It appears that sources of uncertainty are various. First the data, based on circumferential measurement or R.O.V's picture interpretation, leads to uncertain estimation on thickness and percentage of cover. Then the force measurements in laboratory and the signal treatment lead to versatile results and to uncertain evaluation of loading. Finally, the use of Morison equations (Morison, O'Brien and Schaff, 1950) for all orientations of components is an expansion of the standard case where the formula as been developed and lead us to introduce uncertainty on the model. These uncertainties are introduced in the Morison equations. Their expression, translated in local coordinates, indicates that the force per unit span can be separated into drag (1) and inertial ( 2) components (Fig. 1),

Figure 1. Local coordinates on a beam.

(M) V (M) V 0 0 0 0 0 0 D 2 1 (M) f mg D → → ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ θ ρ = D D X C C C π (1) f (M) 1 4 D C 0 0 0 C 0 0 0 C A (M) I 2 X M M = ʹ′ → ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ρ π θ mg (2)
where ρ is the water density, D the cylinder diameter, -→ V (M) and → A (M) are respectively the instantaneous flow velocity and the acceleration at M point, -C X , C D , C' X , C M , are hydro-dynamic coefficients (respectively two drag and two inertia coefficients) for modelling the fluidstructure interaction. There are introduced as model uncertainty variables.

θ mg acts as a multiplying coefficient for taking into account the "marine growth" screen effects on beam diameter uncertainty.

These equations generally predict the main trends in measured data quite well, once appropriate joint distribution for hydro-dynamics coefficients depending on sea-state parameters and distribution of θ mg can be provided. This factor itself depends of the marine growth nature (hard or soft) and structure (percentage of curvature). Nevertheless Morison equations are not able to translate all the complexity of the Keeping in mind these short comings as source of response problems, they are relevant to introduce non linear effects and their complexity is consistent with respect to the large uncertainty in marine growth or fluid interaction modelling caused by the lack of investigations. In fact higher complexity should be lost in this uncertainty. A matrix response surface has been developed [START_REF] Labeyrie | Matrix Response Surfaces For Describing Environmental Loads[END_REF] and is shown to be very tractable for sensitivity studies. Such a response enables us to introduce the main uncertainties in flow modelling versus C d (drag coefficient) and also the increasing screen effects due to marine growth (θ mg ). Sensitivity studies have shown (schoefs, 2002) that the variance of this coefficient Cd contributes up to 61 % to the variance of external nodal forces when the extreme wave height and and the corresponding period have respectively influence ratios of 35 and 3 % in the wave area. From a probabilistic point of view, these variables are not dependent. This paper proposes a conditioning of Cd to the wave kinematics. By extension, a similar approach can be followed for the other hydrodynamic coefficients.

SENSIBILITY OF THE HYDRAULIC PARAMETERS AND THE DRAG COEFFCIENT

Hydrodynamics coefficients depend mainly on hydraulic parameters (number of Reynolds and Keulegan-Carpenter) and of the relative roughness of the beam element. For our application, a platform in the North Sea is considered and submitted to storm conditions. The water depth is 100 meters and table 1 presents the extreme wave specifications and probabilistic hypotheses considered for numerical applications.

Table 1. Extreme sea state parameters in North Sea (Schoefs, 1995). where U stands for the velocity intensity of the particles of the fluid ( m/s ) in extreme conditions, T is the extreme wave period in such sea states (s), ν denotes the kinematic viscosity (m 2 /s) and D the diameter of cylinders (m). This Response Surface will be presented in a further paper. For simulations and sensitivity studies, computing is performed for several cylinder diameters (0.5 m, 1m and 1.5 m). The range on water depth considered here is the first 50 meters were marine growth is mainly present. Only vertical and horizontal orientations of components are considered for illustration. For vertical cylinders, the wave velocity introduced in hydraulic parameters are in the wave plane and in the wave direction (Ux) as for horizontal cylinders, they are in the wave plane, perpendicular to the wave direction (Uy). It leads that the corresponding hydraulic parameters are complementarity, in the sense of the velocity intensity U (Eqs. 5~7).

( ) ( ) 2 2 2 2 2 * Re Re ν D U U y x h v + = + (5) 2 1 2 2 * Re Re U C h v = + ; 2 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ν D C (6) ( ) ( ) 2 2 2 2 2 2 2 * * U C D T U U Kc Kc y x h v = + = + ; ( ) 2 2 D T C = (7)
Where

h h v v Kc Kc , Re , , Re
are respectively the Reynolds and Keulegan carpenter numbers for vertical (index v) and horizontal (index h) components.

Figure 2. Spatial variation of the Reynolds number for a vertical cylinder of diameter 1 meter.

Figure 3. Spatial Variation of the number of Reynolds for a horizontal cylinder of 1 metre diameter.

Figures 2 and3 present the variation of the trajectories of the Reynolds number expectation with depth. To describe this stochastic field indexed by (x,z,t), it is equivalent to consider the index time for a specific location of the structural member and several location of this member in the wave direction. This last option is chosen here and explain the term of abscissa x placed in the figures. For a vertical cylinder (Fig. 2) the Reynolds number is maximal when it is minimal for an horizontal cylinder located at the same depth (Fig. 3). From these figures, well known results can be proved : the velocity intensity and the Reynolds number decreases with depth. Moreover, the statistics of the hydraulic parameters show a very strong variability according to the orientation and the diameter of the cylinder [START_REF] Boukinda | Caractérisation mécanique des chargements hydrodynamiques dus aux bio salissures marines[END_REF].

In view to model the marine growth overloading effect from wave kinematics field, there is a need to characterize flow regime. The statistical data obtained on the various common diameters [START_REF] Boukinda | Caractérisation mécanique des chargements hydrodynamiques dus aux bio salissures marines[END_REF] lead to the conclusion that for the field studied here, three types of regime are present : sub-critical (Re<10 5 ), super-critical and postcritical (Re>3.5*10 6 ). They have been determined from Achenbach definitions (Theophanatos, pp. 18-21, 1988). As only a little number of Reynolds numbers are in the sub-critical area, the experimental studies of Théophanatos ( 1988), being driven for most part in sub and postcritical, will be used to quantify marine growth overloading effect. A comparative study could be based on American Petroleum Institute advice (API recommended practice, pp. [113][114][115][116][117][118][119]1993). This one, for modelling of offshore structures in severe environmental condition as in our case, refers essentially to post-critical regime. However to determine drag coefficient they introduce the ratio CD / CDS (CDS for steady-flow drag coefficient at post-critical Reynolds number) which take into account wake amplification in post-critical regime. The use of CDS would be conservative for the statistical calculations of wave forces (API recommended practice, p.114, 1993). The figure below summarizes the area of experimental conditions of both approaches API advice and Théophanatos tests.

Analysis of hydraulic parameters random field

Statistical distributions of the hydraulic parameters. The use of an algebric programming, adapted to the matrix writing of the response surface, presents the advantage to control the properties studied through interface of visualizations [START_REF] Schoefs | Response Surface of wave loading for reliability of marine structures[END_REF]. The discrete distributions plotted on figures 4, 5 and 6 represent the fluctuation of the Reynolds number. They are obtained from the stochastic field describe upon by an integrating over the wave length. These results have been obtained from a Latin Hypercube Sampling of extreme wave height and period in the North Sea. Similar results have been obtained for the Keulegan-Carpenter number, both for vertical and horizontal cylinders and whatever the diameter. -an exponential distribution, -a distribution of dexter or left-hand asymmetrical type, -a distribution of type normal, A more complete analysis shows that the distributions of asymmetrical type are dominant in surface, near the mean sea level. The distributions following an exponential or normal distribution are more important in depth. However one can find some cases of asymmetrical or normal distributions for several depths [START_REF] Boukinda | Caractérisation mécanique des chargements hydrodynamiques dus aux bio salissures marines[END_REF]. According to the hypotheses supplied to characterize the sea state (table 1), a normal distribution implies a major influence of the wave period. An asymmetrical distribution indicates a dominant effect of the wave height. In surface, the velocity field is influenced by the height and the period of the swell. The period plays a more dominant role in depth, influencing the intensity statistics of the water particle velocity of the fluid. This result is well known and has been proved in previous paper [START_REF] Schoefs | Response Surface of wave loading for reliability of marine structures[END_REF].

Graphs of iso-values contour. Graphs of iso-values contour allow to

analyze the variation of hydraulic parameters statistics with depth and time. Figure 8 represents contours of the mean of Reynolds number for a vertical cylinder. The abscissa has been divided in subsection of 10 meters. It is too crude to analyze distribution effects -3 meters is better- (schoefs, 1996) but sufficient to evaluate variations of the mean value. It is interesting to underline that the variation of this parameter is significant during a wave flow -from 1.7 to 5.6 10 6 -(abscissa in the graph) and with depth too -from 2.7 to 5.6 10 6 -. Another point must be underlined and is particularly interesting for the reduction of computational time : the symmetry of the graph. In fact, the wave motion being periodic, the information on intensity of velocity projection is periodic. In particular, in the case where the point is always in water, the values can be treated on a quarter of the period or the wave length. In others cases, non suitable values must be removed. The Figure 9 presents mean values of the Keulegan-Carpenter number for a horizontal cylinder. A similar interpretation can be made. The symmetry versus a vertical axis is obtained for an abscissa of 80 and Kc takes value in a very large range from 15 to 75. We denote that the symmetry is not perfect in depth for the Keulegan-Carpenter number. This can be due to the role of variables H and T which varies with depth and can affect the symmetry of distribution and thus the mean value [START_REF] Labeyrie | Matrix Response Surfaces For Describing Environmental Loads[END_REF].

Estimation of drag coefficient for a vertical cylinder.

On the basis of hypothesis presented in table 1, simulations can be performed starting from the response surface of velocity field and expressions on hydraulic parameters. By introducing the factor θmg (see Eqs 1~2) an exhaustive study can be driven to measure the influence of particle kinematics field and marine growth on the hydrodynamic coefficients. This study focuses on a point of a vertical member at a water depth of 10 meters, in the wave area where the marine growth is of major importance. The present work starts from available data reported by Wolfram (1993) about a colonization of mussels in the North Sea. Mussels are considered in the paper for illustration only and the modelling can be extended to all type of rigid marine growth. From these data we choose the case of a colonization coverage 100 %, average thickness 50 mm, situated between 0 and 10 m of depth. The hydraulic parameters formula in presence of marine growth are given :

D De mg × θ = (8) D t mg 2 1+ = θ (9) Re De U Re mg mg × θ = ν × = ( 10
) mg mg Kc De T U Kc θ = × = (11)
De stands for effective diameter of the colonized cylinder, θ mg for factor which allows to take into account the screen effect of impact of marine growth, t for average thickness of marine growth, Re mg and Kc mg for hydraulic parameters associated to the presence of marine growth.

Let us consider the results obtained by Théophanatos on vertical cylinders. From an hydraulic point of view, the artificial roughness and the mussels allow to make a direct work on curves presented by Théophanatos (p.129, 1988). For cylinders covered with sand, the relative roughness is equal to 0.023. For cylinders covered with pyramids, it is equal to 0. p.114, 1993). This point leads to distinguish between two studies depending of Kc value (Kc < 12 and Kc > 12).

A comparative study on simulation based on these two approaches -Theophanatos tests and APT advice-shows that the drag coefficient differs. The effect of marine growth on the drag coefficient seems to be more important on cylinders of large diameters. In every case, this study shows the interest to lead an analysis more precise on marine growth. This would allow to establish a methodology of inspection and more rigorous maintenance by a qualification of data base.

PROBABILISTIC MODELLING OF DRAG COEFFICIENT

The previous section leads to conclude that the water particle kinematics field play a dominant role when analyzing hydraulic parameters and then hydrodynamic coefficients. Moreover, this stochastic field governs the randomness on C d . Besides, test results show that there is a correlation between hydrodynamic coefficients and hydraulic parameters. Measures of the hydrodynamic coefficients are thus subjected to uncertainty which should be introduced in the probabilistic approach. We propose a probabilistic modelling which takes into account the several sources of randomness or uncertainties in the problem :

-randomness on the numbers of Reynolds and Keulegan-Carpenter, due to the randomness on the sea state parameters in severe conditions.

-uncertainties on the diameter of colonized cylinders ( θ mg ) due to the uncertainty on the model of marine growth evolution or on measurements. Size measurements is probably the most ambiguous aspect of marine growth. The use of a tapemeasure, often employed to estimate the increase of the average thickness, stays a technique very useful but also very subjected to uncertainties on the thickness ( hard fouling) or the length (soft fouling) of the various type of marine growth and generally cannot quantify the variations which occur among member faces (Théophanatos, 1988).

-uncertainties on C d assessment in tests (in our case, tests made by several authors).

Scheme of probabilistic modelling. The figure below summarizes the approach developed here.

First the wave height and period, respectively H and T, and the marine growth screen effect θ mg are introduced as basic variables in view to assess the hydraulic parameters as stochastic field. This transfer can be modeled through a response surface. Then a discrete distribution of Cd is obtained by using the transfer of test results and graphs. Results obtained by several authors (A 1 and A 2 on Fig. 12) can be mixed to increase the number of available data. If some authors seems to have made their test in better conditions or in a more realistic way compared to the reality, a weighting can be introduce (w 1 and w 2 on Fig. 12). The results depend on the subdivision of hydraulic parameter axis and C d axis and an optimization should be proposed. This question is not treated in this paper. In view to simplify following expression of probabilities, we adopt the notation :

) x X x ( P ) X ( P 1 i i 1 i i + - < < = (12)
According to the uncertainties connected to the tests and the reliability of the measures, the calculations are balanced. It leads to the probability : Elsewhere a study on correlation between the drag coefficient and the non-dimensional relative roughness (k/De) show clearly that the drag coefficient is a function of k/De (Wolfram J., Jusoh I. et Sell D., ASME 1993). The increase of the drag coefficient depends of both relative roughness and hydraulic parameters. These two parameters having uncertain evaluation due to the experimental conditions, they introduce uncertainties on the forces coefficients of Morison equation. The probabilistic approach used here to estimate drag coefficient, allow to select hypothesis when introducing these variables in reliability analysis. The results of probabilistic calculations following the different intervals of C D are given in the tables 3 -5. All tests results are not brought together to assess statistics of Cd knowing Kc. According to the reliability of the experimental test procedures, there is a way to advocate a weight depending of the procedure. The coefficients of accommodation (table 6) are given according to the adaptation of the test procedures to represent the mussel colonization and to well measure forces. The experimental data of Sarpkaya for a sand roughness agree reasonably well with those of Théophanatos (rough cylinder with pyramids). Those of Rodenbusch are very closed to those obtained by Théophanatos. Nath's results diverge with the results described previously. This difference is probably due to marine growth modelling and to the procedures of forces measurement, filtering and corrections of the deep parameters (Théophanatos, 1988, pp. 130-132).

) Kc ( P ) Kc / Cd ( P w ) Kc ( P ) Kc / Cd ( P w ) Cd ( P i i 2 A 2 i i 1 A 1 × + × = ( 13
) ) Cd ( P ) Cd ( P n 1 i i ∑ = = ) Cd ( P w ) Cd ( P w i n 1 i 2 A 2 i n 1 i 1 A 1 ∑ ∑ = = + = =1 (
According to this comment a value close to 1 is selected when the model represent very well the complexity of fooling and hydraulic conditions and when the sequences of signal treatment are robust. This method allows to introduce in the modelling all the available data. The conditional probability of the drag coefficient shows an asymmetrical distribution (Fig. 13).

Figure 13. Drag coefficient probability conditioned to Kc.

The discrete probability density function is deduced from this previous results and the probability of Kc being in the tree ranges (Fig. 14). Ranges of Cd refers to those given in table 3~5.

  interaction with enough accuracy (i. e. high frequency content of the flow, gross vortex shedding effects, ...).

  Hs : significant wave height : Gumbel distribution : : u = 15.5 m ; ν = 1.2 Tstat : sea state duration ; Exponential distribution with expectation 3hrs H given (Hs, Tstat) : Gumbel distribution : Hs= 15.5m ; Tstat = 3hrs ; u = 28.5 m ; ν = 1.9 T given H : Normal (Log-normal) distributed, E(T/H)= φ(H)= 39 H 6 +Hydraulic parameters computingThe Response Surface mentionned in the previous section is based on a matrix Response Surface of the kinematics field. It allows to determine the hydraulic parameters from the stochastic field of particle velocity U following the equations: Number of Reynolds:

Figure 4 .

 4 Figure 4. Hydraulic experimental conditions of Théophanatos and the API advice.

Figure 5 .

 5 Figure 5. Example of exponential distribution obtained for a vertical cylinder of diameter 1 meter.

Figure 6 .

 6 Figure 6. Example of asymmetrical distribution obtained for a vertical cylinder of diameter 1 meter.

Figure 7 .

 7 Figure 7. Example of normal distribution obtained for a vertical cylinder of diameter 1 meter.

Figure 8 .

 8 Figure 8. Contour of iso-values for mean of Reynolds number (vertical cylinder).

Figure 9 .

 9 Figure 9. Contour of iso-values for mean of Keulegan-Carpenter number (horizontal cylinder).

  038. These values correspond to the relative Paper No. 2004-JSC-344 Schoefs Page number 6 of 8 roughness we are needed for respective cylinders of 1.1 m and 0.6 m of equivalent diameter. Let us consider now API regulations, the factor of development of the swell (CD / CDS) is introduced. The drag coefficient of the rough and smooth cylinder follow the same evolution when the number of Keulegan-Carpenter is normalized by the CDS (Kc upper than 12). For Kc values lower than 12, it is similar if the number of Keulegan-Carpenter is not normalized by the CDS (API recommended practice,

  Figure 10 presents three statistics of Cd, the minimum Cd min , the maximum Cd max and the mean values obtained for two representative diameter of O.6 and 1.1 meter. The data collected from the graphs supplied by Théophanatos show a fluctuation less important than that coming from the regulation API. Ones can also note taht the drag coefficients are, altogether, more important when they are established by Théophanatos's curves. A reasoning from the studies of Théophanatos are more conservative in our case.

Figure 10 .

 10 Figure 10. Comparison API/ Théophanatos (vertical rough cylinders)

Figure 12 .

 12 Figure 12. Plan of the probabilistic modelling of the drag coefficient.

  tests. The two ranges of Kc values selected for fitting regression curves are presented in table 2. First, the dispersion of the data obtained by different authors is measured to analyze if there is a simple relationship (polynomial of low order) between the drag coefficient and the number ofKeulegan-Carpenter (Fig. 9.5 p. 131, Théophanatos, 1988). For Keulegan-Carpenter numbers between 10 and 21, the coefficient of correlation of 0.5 shows a large dispersion around the linear relationship coming from measurements and different tests procedures used by authors(Théophanatos, 1988). The coefficient of determination obtained for Keulegan-Carpenter numbers between 21 and 35 indicates a rather satisfactory linear relationship.

Table 3 .

 3 Probabilistic calcuations for 10 < KC < 15

	10< Kc <15	Théophanatos Sarpkaya Rodenbusch Nath
	P(Cd1<Cd<Cd2)	0	0	0	1
	P(Cd2<Cd<Cd3)	0.12	0	0	0
	P(Cd3<Cd<Cd4)	0.75	0.29	1	0
	P(Cd4<Cd<Cd5)	0.12	0.71	0	0
	Table 4. Probabilistic calcuations for 15 < KC < 21
	15< Kc <21	Théophanatos Sarpkaya Rodenbusch Nath
	P(Cd1<Cd<Cd2)	0	0	0	0.5
	P(Cd2<Cd<Cd3)	0	0.6	0	0.5
	P(Cd3<Cd<Cd4)	1	0.4	1	0
	P(Cd4<Cd<Cd5)	0	0	0	0
	Table 5. Probabilistic calcuations for 21 < KC < 35
	21< Kc <35	Théophanatos Sarpkaya Rodenbusch Nath
	P(Cd1<Cd<Cd2)	0	0	0	0.67
	P(Cd2<Cd<Cd3)	0	0.09	1	0.33
	P(Cd3<Cd<Cd4)	1	0.9	0	0

Table 6 .

 6 Theoretical expertise for the drag coeffcient

		Théophanatos Sarpkaya Rodenbusch Nath
	Coefficients of accomodation	0.8	0.5	0.3	0.1
	Weight	0.47	0.29	0.18	0.06

Schoefs

Page number 8 of 8

Figure 14. Drag coefficient probability.

This modelling allows to conclude that the distribution is asymmetric.

Considering the followed methodology and the conditioning to Kc which follows a non-symmetrical distribution, this result is not surprising. The mean value µ and the standard deviation σ are obtained by the equations ( 15) and (16).

Cdi p Cd

The mean value is 1.834 and the standard deviation 0.051. This values lead to a COV of only 3 %. It is of course very fair and show that CoV of 30 % generally assumed can overestimate the variation in some cases.

CONCLUSION

The modelling of marine loads on Jacket platforms in presence of marine growth remains very complex. Randomness on Reynolds and Keulegan-Carpenter numbers due to the wave stochastic field, uncertainties on hydrodynamics coefficients and experimental tests implies a work on probabilistic modelling. The response surface methodology allow to integrate in calculations hydraulics parameters and the determination of the hydrodynamics coefficients can be accessible from the experimental tests. It leads to a discrete probability density function of hydrodynamic coefficients based on a conditioning to the wave kinematics basic variables. Results show that usual hypothesis on CoV for hydrodynamic coefficients are mainly overestimated and as a consequence make the reliability less.


R B Morales
email: morales@ukaea.uk

S Heuraux

R Sabot

S Hacquin

F Clairet

Tore Supra

The reconstruction of hollow areas in density profiles from frequency swept reflectometry

Introduction

The FM-CW (frequency-modulated continuous-wave) reflectometry diagnostic is a well established technique for density profile measurement with successful implementations on various medium and large size tokamaks, such as DIII-D [START_REF] Wang | [END_REF], Tore Supra [2,3], ASDEX Upgrade [4,[START_REF] Varela | the CFN Team and the ASDEX Upgrade Team[END_REF] and JET [6]. Even though there have been significant improvements in the reflectometry hardware design [2,7] and data extraction techniques [START_REF] Varela | the CFN Team and the ASDEX Upgrade Team[END_REF]8,9] over the last two decades, the measured density profiles on fusion experiments still require further improvements in the data analysis front in order to improve the accuracy of the reconstructed profile. As an example of a demanding application, the LFS (Low Field Side) reflectometer being built for ITER has as its first operation priority to achieve a minimum radial accuracy of 5 mm [START_REF] Doyle | Status of the system design and component testing for the ITER low-field side reflectometer system[END_REF]. Improving the accuracy on the reconstructed density profile also improves the accuracy of extracted parameters for physical studies such as transport, MHD instabilities and turbulence.

The data analysis for FM-CW profile reflectometry can be divided into three topics: the initialization technique as initially investigated in [START_REF] Heuraux | An X-mode reflectometry study on the reflection point for the density profile reconstruction Talk and proceeding at the 9th International Reflectometry Workshop[END_REF]; the recursive profile reconstruction algorithm as originally proposed by Bottollier-Curtet [START_REF]Bottollier-Curtet H and Ichtchenko G[END_REF] with minor revisions in [START_REF] Shelukhin | X-mode lower cutoff high field side reflectometer for electron density profile measurements[END_REF][START_REF] Bottollier-Curtet | Réflectométrie hyperfréquence pour la détermination de la densité électronique et de ses fluctuations sur le Tokamak[END_REF] and a thorough review with some improvements in [START_REF] Morales | [END_REF]; and the description of blind areas, as focused in this contribution.

Both O-mode and X-mode density profile reconstruction techniques rely on the assumption of a monotonic cut-off frequency profile. However, there are many plasma perturbations that introduce hollow areas in the cut-off frequency profile, breaking the aforementioned assumption. Inside these hollow areas, the probing microwaves exhibit no specular reflections and thus they are referred to as blind areas. Even though no reflection occurs inside the blind areas, the higher probing frequencies that propagate through these areas carry information about them. For isolated ___________________________________________________________________________ perturbations, the perturbation signature in the reflectometer signal is related to the size of the perturbation. In this contribution, large perturbations are considered, which are out of the Born approximation validity and the probing electric field over the perturbation is no more similar in magnitude to the unperturbed case. This situation can occur during massive gas and pellet injections [16], MHD activity [17] and in hollow profiles that emerge during the initiation of heating systems [18] or even due to relativistic effects [19].

If the reconstruction method does not incorporate identification and reconstruction tools for the blind regions, big discrepancies can appear in the reconstructed profile. An example is shown in figure 1 using a simulated phase under the WKB approximation as the input signal and profiles typical of Tore Supra with a low magnetic field strength of 2 T at the plasma center. a blind area. The profile was reconstructed with the standard Bottollier-Curtet algorithm and a constant correction, as in [START_REF] Shelukhin | X-mode lower cutoff high field side reflectometer for electron density profile measurements[END_REF][START_REF] Bottollier-Curtet | Réflectométrie hyperfréquence pour la détermination de la densité électronique et de ses fluctuations sur le Tokamak[END_REF][START_REF] Morales | [END_REF], using three different treatments in the phase signal. The radial axis is defined from zero at the plasma edge and increasing towards the plasma center.

It is clear from figure 1 that the unmodified standard reconstruction algorithm is unable to reconstruct the density perturbation. Furthermore, if the oscillations are smoothed, the perturbation can be neglected entirely, or even worse, a radial shift can be introduced in the reconstructed profile after the perturbation if the time-of-flight jump is filtered out, which is the case for the dashed green line in figure 1.

Reconstruction of blind areas using database of perturbation signatures

The profiles of the blind areas can be reconstructed by inverting a database of perturbation signatures in the time-of-flight signal. Various perturbations were investigated with 1D full-wave simulations in reference [START_REF] Morales | Density profile reconstruction methods for X-mode reflectometry[END_REF], and the conclusion was that the full-wave effects simulated in 1D (tunneling, wave-trapping, scattering and interference) were restricted to a probing frequency band of about 1 GHz around the time of flight jump over the blind area. As long as a corresponding experimental signal is treated outside of the bandwidth affected by full-wave effects, the experimental time-of-flight can be assumed well described by the WKB simulated signal, which significantly simplifies the construction of a database.

An initial database was constructed for a simple sine shaped valley and a broad range of all parameters (f prob ,f ce ,∇f cut ,width,depth) including coverage for Omode since the f ce domain starts at zero. The database inversion procedure was successfully tested in a noise-free synthetic example. As covered in detail in [START_REF] Morales | Density profile reconstruction methods for X-mode reflectometry[END_REF], it also demonstrated how a single database can accommodate a broad range of plasma conditions and it was also used to infer the dependencies of the reconstruction accuracy across the full database domain. As expected, the reconstruction accuracy decreases when the perturbation signal is smaller, as is the case for the highest cutoff gradients and widths. In practice, the biggest contribution for the inversion accuracy is expected to be the precision on the estimations of the perturbation width, gradient, the unperturbed profile and the experimental noise level, rather than the intrinsic precision of the calculated database.

To demonstrate the potential of this method to reconstruct blind areas, figure 2 shows the first experimental application of a database inversion for the Tore Supra discharge 32029, where magnetic islands have been identified. The reconstructed density profiles using the standard reconstruction method on the perturbed and unperturbed measurements versus the new method to reconstruct blind areas introduced in this paper.

The perturbation being investigated builds up from previous observations of MHD activity with reflectometry [START_REF] Clairet | FM-CW reflectometry for MHD activity measurements on Tore Supra[END_REF][START_REF] Vermare | [END_REF]. In this circumstance, the focus is on the reconstruction of the magnetic island on the q = 2/1 rational surface, as located by the equilibrium code [START_REF] Vermare | [END_REF].

According to simulations of magnetic islands [23,24], they are formed by a positive perturbation followed by a negative perturbation along the probing path. Furthermore, the shape of the negative perturbation corresponds to the shape of the positive perturbation. In this case, the experimental unperturbed signal is also well determined since the island is intermittent with an directly observable frequency. In this case, a reduced database was constructed based only on the specific plasma parameters present on the application of the database, with the cutoff profile already including the positive perturbation since it's conventionally probed, and the valley shape assumed symmetrical to the positive probed section. Next, the simulated signals with different valley depths are compared to the experimental signal to find a best match.

It is clear from observing figure 2 that the new reconstruction scheme for the blind area describes much better the magnetic island and eliminates a reminiscent tail of lower density.

Final remarks and future prospects

As it was demonstrated, the standard profile reconstruction algorithm adds a significant error in the density profile around and after the blind areas if no special technique is applied around the perturbed region. This contribution showed a brief summary of the work started in [START_REF] Morales | Density profile reconstruction methods for X-mode reflectometry[END_REF] that aimed at laying the foundations for this novel technique to describe the blind areas. More detailed explanations of the reconstruction steps will also be found in the coming extended version of this proceeding in a special issue of the Plasma Science and Technology journal.

This technique was verified to be very accurate in the absence of noise and with precise input of parameters when tested on synthetic data. Thus, the final reconstruction accuracy is expected to be related to available signal-to-noise ratio and the accuracy in which the perturbation width, shape and the local parameters of ∇f cut and f ce have been determined, which will vary significantly according to each application. Based exclusively on the comparison of the database signal to the noise level of the experimental signal, the experimental application shown in figure 2 had an error bar of 33% of the estimated valley depth. A detailed consideration of the resulting error-bars propagating from the assumptions of the valley shape and estimations of the width and cut-off gradient will be the focus of future publications where the use of an ultra-fast sweeping rate [25] with the technique to stack multiple sweeps [START_REF] Varela | the CFN Team and the ASDEX Upgrade Team[END_REF] is expected to improve the extracted perturbation signal.

Even though the main physical characteristics of the blind regions have been well described by the 1D simulations present in this contribution, future 3D simulations [START_REF] Da Silva | Introducing REFMULF, a 2D full polarization code and REFMUL3, a 3D parallel full wave Maxwell code[END_REF] will be necessary to verify any additional tridimensional geometrical aspects. After all, the probing beam area and shape, plus the shape of the perturbations, make in conjunction a system too complex to be completely described in one dimension. These geometrical aspects influence the amplitude signal across the perturbation bandwidth and the appearance and dynamics of scattering and resonances. Understanding these effects will help to better extract the signals from the blind area, apply any correction due to the 3D structures and ultimately will lead to a more accurate employment of the database inversion technique. This research on the geometrical aspects will also intersect with the application on improved initialization techniques that observe the time-of-flight and amplitude evolution in the presence of perturbations.

Figure 1 :

 1 Figure 1: Synthetic example of input versus reconstructed density profiles with

Figure 2 :

 2 Figure 2:The reconstructed density profiles using the standard reconstruction method on the perturbed and unperturbed measurements versus the new method to reconstruct blind areas introduced in this paper.

14th

  Intl. Reflectometry Workshop -IRW14 (Lausanne) 22-24 May 2019 O.105 ___________________________________________________________________________

14th Intl. Reflectometry Workshop -IRW14 (Lausanne) 22-24 May 2019 O.105

Acknowledgments

This work has been carried out with the support of the Brazilian National Council for Scientific and Technological Development (CNPq) under the Science Without Borders programme, within the framework of the French Federation for Magnetic Fusion Studies (FR-FCM) and of the EUROfusion consortium with funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053, and also been part-funded by the RCUK Energy Programme [grant number EP/P012450/1]. The views and opinions expressed herein do not necessarily reflect those of the European Commission.


S Heuraux

O Krutkin

E Gusakov

F Da Silva

Full-Wave simulations of the enhanced Upper-Hybrid Resonance Scattering

Introduction:

The aim of this work is to improve the simple linear interpretative model usually applied to extract turbulence characteristics of tokamak plasmas using data of Upper-Hybrid Resonant Scattering (UHRS) experiments [1]. To evaluate the limits of such model, full-wave simulations have to be performed. These computations are complex due to the fact that the spatial scales change a lot especially as the wave propagates in vicinity of the resonance. Close to the resonance the mode conversion should be taken into account, which requires considering both components of the wave electric field together with the thermal effects. The amplitude of the electromagnetic component E y , remains finite overall the propagative zone, which can be used in computational scheme. However the group velocity decreases drastically and can reach in principle a velocity close to zero as it was shown numerically [START_REF] Brusehaber | Piliya Plasma Physics and Control[END_REF] and experimentally [3][4]. This cold plasma approximation is inapplicable for our problem, and the thermal effects have to be included to describe the wave conversion from extraordinary mode (X-mode) to a warm plasma mode as mentioned in [1]. However the possible mode conversion described by the warm plasma model used corresponds to a warm wave propagating in an opposite direction to the one expected in tokamaks for the probing frequency close to the fundamental electron cyclotron harmonic due to constraints of the numerical scheme. Assuming that the probing wave propagates along the resonance cone which assumes to have a parallel wavenumber, ignored here, giving more or less a propagation at constant density, so 1Dstudies on the Doppler shift induced by moving structures near the resonance were performed to evaluate if the effective Doppler shift obtained is directly connected to the velocity of the scattering structure. The scattering efficiency above the limitations of the [1] model is also provided for different cases. To finish an estimation of the parameters required to simulate realistic cases is provided and corresponds to the existing devices such as FT2, WEST, and what could be expected from the full-wave simulation to interpret more accurately the UHRS diagnostic measurements.

Back to basics:

The assumptions for the X-mode including slow time evolution of the plasma parameters and the writing of the time-dependent wave equation can be found in [5]. This set of partial differential equations in one-dimension in space can be reduced to:

The numerical scheme including variable normalizations used to compute the electric field components is described in [6]. To save computing time to reach the vicinity of the upper-hybrid resonance, it is better to use a steady state configuration in the monochromatic case which permits to reduce, in the 1-D case, to an Helmholtz equations where the X-mode is used to compute ycomponent of the electric field and a polarization equation is used to deduce x-component of the wave electric field. The advantage is that E y component stays finite without collision. The question in this case is if the scattering is fully and correctly described? The index of X-mode for a given probing pulsation ω follows the expression:

where ω pe is the electron plasma pulsation and ω ce electron cyclotron pulsation. In a tokamak the way to reach the resonance is to launch the probing wave from the high field side fulfilling the condtion ω > ω ce_min (ω ce at the low field side), ω < ω ce_max and ω > ω L-max (left circular cut-off pulsation or f c-) for high density plasma ω pe >> ω ce .

Figure 1: Density and magnetic field intensity profiles of standard Tore Supra discharges and associated cut-offs and resonance frequencies (left) and (right) X-mode index square with plasma and electron cyclotron normalized frequencies versus position normalized by x UH upper-hybrid resonance position for a probing frequency of 90 GHz.

For the warm plasma description, the pressure term is added to the electron fluid equation as following:

This equation is combined with Maxwell's equations using the same normalization as before. The time-dependent set of differential equations for X-mode in the warm plasma approxiamtion is written as: where λ De is the Debye length and v Te is the electron thermal velocity. This system is solved numerially using for the 1 st equation a Numerov method of 4 th order in time where v Te 2 ∂ x 2 E x is treated as a source term and can be evaluated at the wanted order in space, for the 2 nd wave equation Colin's scheme [6] at the 2 nd order in time and space using magic step, for the last equations of the motion a Runge-Kutta (RK45) scheme where v Te 2 ∂ x 2 E x is evaluated from the first equation. These numerical methods are associated to transparent boundary conditions at the mesh border, which are different for the electromagnetic wave and for the electron acoustic wave equations. The use of UTS (Unidirectional Transparent Source) [7] permits a direct access to the scattered wave. This code permits to determine numerically the position where the mode conversion can occur, and, thus thanks to monotonous increase of the probing wave wavenumber with the radial coordinate, deduce the wavenumber cut above which the higher wavenumber can be ignored in the interpretation model of the UHRS data.

N X 2 = 1- ω pe 2 (ω 2 -ω pe 2 ) ω 2 (ω 2 -ω ce 2 -ω pe 2 ) ∂ t v = - e m e E - e m e v × B - k B T e m e
∇P e n e (x)

∂ t 2 E x -v Te 2 ∂ x 2 E x +ω pe 2 E x = -ω pe 2 v y ∂ t 2 E y -c 2 ∂ x 2 E y +ω pe 2 E y = ω pe 2 v x ∂ t v x = -ω ce (1+ λ De 2 ∂ x 2 )E x -ω ce v y ∂ t v y = -ω ce E y +ω ce v x          14th

___________________________________________________________________________

Results on the UHRS efficiency:

To evaluate the UHRS efficiency as function of density fluctuation wave number, computations have been done for a given plasma density and magnetic field intensity profiles in cold approximation with artificially introduced damping factor near UHR. However using a Helmholtz equation we need to determine the insensiblility modulation in vacuum making it impossible to reduce the computation time by squeezing part of the plasma, launching the wave into the plasma and looking at the vicinity of the resonance. But as the phase measurement is not an issue to determine the scattered power, it is possible to compress the range of radius without contribution. This compression, however, imposes some limitations. If the index gradient is too high some spurious reflection can be generated. The density profile and magnetic field must be compressed in the same way up to a smooth transition to the radius range of interest where no compression exists. An hyperbolic tangente transition function is used to ensure that no spurious reflection occurs. To avoid misinterpretations no density fluctuation is put in this area. As consequences the range of fluctuation wavenumbers is limited to values above 2 k o where k o is the vacuum wavennumber. To approach the resonance close enough, the number of points wavelength in vacuum is of the order of 10000 points per vacuum wavelength. At the same time the damping factor has to be optimized to damp the wave without generating spurious reflection. The last keypoint is not easy to fulfil. Most of the time a residual reflection imposes restriction on the scattered power detectable here P s /P o ~10 -3 . This reduces the range of density fluctuation parameters in which the linear behaviour can be studied. Taking into account these constraints the UHRS effciency was studied for the standard Tore Supra case requiring 84000 grid points. To improve the accuracy even more, more points are needed but the computing time would become too large since a minimum of 30 wavenumber values is needed to elabore a curve for one value of the density fluctuations. Another important point is the shape of the envelop used to study this efficiency, such as a Gaussian shape or a wavetrain covering all the plasma. The wavetrain has the advantage to have an unique wavenumber value but in this case the harmonic of the Bragg rule should be taken into account for the interpretation [8]. The Gaussian shape localizes more the scattering process but the wavenumber spectrum has a finite width.

Figure 2: UHRS efficiency as function of fluctuation wavenumber normalized to vacuum wave number for three envelop shapes (left) and local index squared variations (Gaussian shape for the density perturbation) and associated index k-spectrum in the vicinity of the resonance as function of normalized coordinates (right).

The impact of the shape is clearly seen on Fig. 2 where the interferences (modulation of the effciency) of scattered waves are present , except for the case of a narrow Gaussian shape. Nevertheless for the Gaussian shape there is a contribution at the low wavenumber values. The decay at high k-values is due to the damping introduced to ensure the wave absorption in the close vicinity of UHR. If instead of the density pertubations an index modulation is used, the behaviour of the effciency changes drastically due to the fact that the index k-spectrum is more well defined, howerer the width of the Gaussian covers a large range of the unperturbated index values. This fact gives a rather flat variation of the high wavenumber efficiency around the vicinity of the UHR.

14th Intl. Reflectometry Workshop -IRW14 (Lausanne) 22-24 May 2019 O.116

___________________________________________________________________________

Mode conversion X-mode to electrom warm plasma mode:

A previous article [9] shown the possiblility to describe the wave propagation in warm plasma approximation in the time-domain without using Particle In Cell code. Here we have used a gradient pressure perturbation corresponding to the linear pertubation inducd by the probing wave. The goal of such computation is to determine, in a time-dependent problem, the k-value for which the mode conversion occurs. The evaluation of the scattered power should take of the mode conversion power for the wavenumber values above this k-value, which means a reduction of the Bragg backscattering efficiency on the X-mode above the conversion point. As shown in [1] knowing the k-range becomes and the effciency, permits to interpret the frequency spectrum obtained by a UHRS diagnostic on WEST the upgrade of Tore Supra tokamak [START_REF] Bourdelle | WEST Physics Basis[END_REF]. On Figure 3, the E x component exhibits two typical wavelengths, the longest one is linked to X-mode, and the shortest wavelength is identified as the warm mode. The warm mode group velocity is then connected to the electron thermal velocity, which propagates slowly than the X-mode except when the X-mode wave approaches the resonance where both propagates more or less at the same speed. Unlike the X-mode the warm wave can cross the resonance without change, and carry the wave energy outside the resonance zone. The conversion X-mode to warm mode is forward conversion. The direct backward conversion cannot be computed due to the fact the scheme is numerical instable for backward warm mode. The tests performed do not show any influence on Bragg rules by a possible coupling to the warm mode. This tool is able to describe the mode conversion from X-mode and warm in time-dependent description, and can be used to studied the effects of the density fluctuations on this process in a dynamic way as it was done in the cold plasma plasma approximation to explain why the frequency spectrum is so large. However it is clear that an accurate electron temperature is required to apply such tools for the UHRS data interpretation.

Conclusion:

The efficiency of the Bragg backscattering was obtaned for relevant tokamak plasma parameters. Depending on the envelop of density fluctuations used to evaluate this efficiency, different contributions appear at low k-values. The upper bound of the integration in collisionless plasma can be accessed determining at the position of the conversion X to Warm mode using the code XtoW.

Figure 3 :

 3 Figure 3: X-mode (lower branch) Warm mode conversion (right) and X-mode (upper branch) to Warm mode conversion (left) for the same plasma and wave parameter profiles.

  Intl. Reflectometry Workshop -IRW14 (Lausanne) 22-24 May 2019 O.116

14th Intl. Reflectometry Workshop -IRW14 (Lausanne) 22-24 May 2019 O.116

14th Intl. Reflectometry Workshop -IRW14 (Lausanne) 22-24 May 2019 O.116 ___________________________________________________________________________


J Vicente

F Da Silva

T Ribeiro

S Heuraux

G D Conway

B Scott

C Silva

Impact of plasma edge and scrape-off layer turbulence levels on conventional O-mode reflectometry investigated with REFMUL and GEMR codes

Introduction

The relative amplitude of electron density fluctuations δne/ne, i.e. the plasma turbulence level, may determine both the analytical modelling possibilities and the measuring capabilities of reflectometry diagnostics. On one hand, high fluctuation levels may break down the Wentzel-Kramers-Brillouin (WKB) and Born approximations in modelling the scattering processes [1]. On the other hand, the experimental measuring capabilities, including of density fluctuations themselves, may become limited above a given turbulence level due to non-linearity and saturation of the reflectometry signals (e.g. see [2][3]). This is particularly relevant in the edge and scrape-off layer (SOL) plasma regions where density fluctuation amplitudes are usually high due to high turbulence levels (δne/ne > 10%) and/or the abundant presence of high density perturbations (δne/ne ~ 100%) known as blobs or filaments (e.g. see [4][5]). In this work, the impact of the turbulence level on conventional O-mode reflectometry is studied with a synthetic reflectometer, implemented on the wellestablished 2D full-wave code REFMUL [6], embedded on numerical plasma descriptions given by the 3D gyro-fluid code GEMR [7][8].

The REFMUL code

REFMUL is a full-wave code that solves the Maxwell equations in 2D for O-mode wave propagation. It employs the finite-difference time-domain (FDTD) technique using the Yee schema [9]. For O-mode, the cut-off condition nc = fo 2 (4π 2 εome/e 2 ), where fo is the probing frequency, e the electron charge, me the mass of the electron and εo the permittivity of free space, sets the critical electron density nc where reflection occurs in the plasma. In this 14th Intl. Reflectometry Workshop -IRW14 (Lausanne) 22-24 May 2019 O.201 ___________________________________________________________________________ work, the case of plasma probing at fixed frequency in conventional set-up (i.e. normal incidence) is treated, since such systems have been traditionally devoted to study density fluctuations. A monostatic design is employed together with a unidirectional transparent source, allowing separation of the emitted probing wave from any returning waves [6]. An H-plane horn antenna with half power beam diameter of ~ 6.5 cm at the plasma entry was chosen, with the on-axis line-of-sight at the low magnetic field side (LFS) mid-plane. An I/Q detection scheme, providing synthetic in-phase

I(t) = A(t)cos[φ(t)] and quadrature Q(t) = A(t)sin[φ(t)] signals is included, similarly to what is typically measured in experiments.
Details on how the I/Q signals can be combined to compute the returned amplitude A(t) and phase φ(t) reflectometry signals, in particular for the case where the background and turbulent terms of density fluctuations are probed independently as in the case of GEMR output data, will be published elsewhere [10].

The GEMR code

To obtain the characteristics of edge and SOL plasma turbulence, the evolution of density fluctuations and profile gradient was computed using a 3D electromagnetic gyrofluid model. This model solves the first six moments of the gyrokinetic equation in global geometry. The geometry is global in the sense that both poloidal and radial dependencies of geometrical quantities are kept, without using the flux tube approximation. GEMR simulations were carried out assuming a simplified magnetic equilibrium with circular plasma cross-section. The radial domain included both closed and open field line regions (edge and SOL regions, respectively) and local plasma parameters representative of a typical L-mode discharge at the ASDEX Upgrade (AUG) tokamak [11]. In particular, the major radius R = 1.65m, minor radius a = 0.5m, magnetic field Bt = 2.6T, ion and electron temperatures Ti = Te = 60eV, ion and electron densities ni = ne = 1.2×10 19 m -3 , temperature and density gradient scale lengths LT = Ln/2 = 3.5 cm, nuclear mass of deuterium MD = 3670me, effective nuclear charge number Zeff = 2, and safety factor q = 4.6. The evolution of electron density data, along the GEMR simulation, was recorded at several sequential time iterations. After post-processing the data, it was rendered on a poloidal plane thus taking into account the cross-section of the magnetic flux surfaces. Further details about the GEMR simulation run, the required steps for coupling of GEMR and REFMUL, and the first results obtained from using the two codes together will also be reported in [10]. A linear relation between reflectometry phase fluctuations and the amplitude of small density perturbations is expected in simplified 1D theory, where Bragg backscattering (BBS) is accounted as the main scattering mechanism [12][13]. given wavenumber component of the density fluctuation spectrum, was given in [13]. This theoretical estimate is also shown in Fig. 2, computed with an effective turbulence wavenumber kf = 2.5cm -1 estimated from the inverse radial correlation length 1/lc, where lc was computed at the cut-off location directly from the GEMR data [10]. In previous studies, with different plasma models, an excellent agreement between the simulated data and expected linear response was observed only for low turbulence levels (up to δne/nc ~ 2-4%) followed by a transition to a non-linear regime and eventual saturation (e.g. [14][START_REF] Vicente | JINST 14th Intl. Reflectometry Workshop -IRW[END_REF]). In Fig. 2, the simulated data oscillates around the theoretical estimate, without departing considerably from the linear trend predicted by the 1D BBS theory. It appears that a better agreement is found across the whole turbulence range, at the cost of lower accuracy in the low turbulence levels. However, several GEMR runs would be required to assess the error associated with the simulated data. ___________________________________________________________________________ also be able to use the amplitude data or remain closer to linear phase regimes at higher δne/ne values than previously expected.

14th

  Intl. Reflectometry Workshop -IRW14 (Lausanne) 22-24 May 2019 O.201 ___________________________________________________________________________ An example of a GEMR density snapshot for AUG-like parameters is shown in Fig. 1. Also displayed in the figure is a zoom in a typical region of interest on the equatorial LFS used for REFMUL simulations. The region is chosen ensuring that it remains away from the boundaries where GEMR source and sinks act.Turbulence level scanIn this work, for the purpose of scanning a comprehensive range, of several orders, of plasma turbulence levels, the turbulent components of the GEMR density data, taken directly from the GEMR output, are scaled prior to being used in REFMUL. Note that the original GEMR data considered here displayed only a very narrow range of turbulence levels (~ 3-6%) across the region of interest. Simply, a scaling factor is applied in order to increase or decrease the fluctuation level. This may not guarantee a fully realistic description of turbulence along the conditions that set different turbulence levels in a tokamak, for instance w.r.t spatial correlation lengths. However, to characterize the reflectometer response it is useful to study the impact of δne/ne variations without further effects. Equivalently, the same background density (profiles) is taken as it was obtained from the self-consistent evolution of GEMR, for all turbulence levels. Only one probing frequency fo = 30 GHz was selected. The density gradient length at the corresponding cutoff location was Ln = 10 cm. The root mean square of the density fluctuations at the cut-off position (i.e. turbulence level at the cut-off δne/nc) was thus varied covering the range ~ 0.07-70% on a set of different REFMUL simulation runs. The phase fluctuations measured by the synthetic reflectometer across the entire turbulence level scan are displayed in Fig.2.

Fig. 1 :

 1 Fig.1: GEMR electron density data on a poloidal plane cut, defined on a polar geometry, and zoom on region employed for REFMUL simulations.

  The standard deviation of the amplitude fluctuations from the corresponding reflectometry signals are shown in Fig. 3. The fluctuations in amplitude scale linearly with the amplitude of the density perturbations up to δne/nc ~ 10% and then saturate. The possibility of inferring the turbulence level from either phase fluctuations or amplitude fluctuations appears to be improved in the case of GEMR simulations. If GEMR is more realistic than other models, experimental measurements might

Fig. 2 :

 2 Fig. 2: Standard deviation of simulated reflectometry phase φ(t) at the corresponding turbulence levels at the cut-off location δne/nc. Also shown (solid line) is the theoretical estimate of 1D Bragg backscattering contributions.

Fig. 3 :

 3 Fig. 3: Standard deviation of simulated reflectometry amplitude A(t) at the corresponding turbulence levels at the cut-off location δne/nc.

Acknowledgments

This work has been carried out 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 programme 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. IST activities also received financial support from "Fundação para a Ciência e Tecnologia" through project UID/FIS/50010/2019.


E Z Gusakov

A B Altukhov

A D Gurchenko

S Heuraux

M A Irzak

L A Esipov

T P Kiviniemi

O L Krutkin

C Lechte

S Leerink

P Niskala

G Zadvitskiy

Validation of full-f global gyrokinetic modeling results against the FT-2 tokamak Doppler reflectometry data using synthetic diagnostics

Introduction

The anomalous energy transport remains one of the main issues in magnetic confinement fusion research. According to the present day understanding [1,2] the anomalous transport is determined by the multi-scale drift-wave turbulence and the nonlinear interaction of its components. The gyrokinetic (GK) simulations [3,4] provide an efficient tool for studying the nonlinear turbulent plasma dynamics and in the recent papers [5,6] such simulations of the electron and ion distribution functions from the first principles were performed for the small research limiter tokamak FT-2. The interplay of the multi-scale turbulence components was studied and successful quantitative comparison was presented with O-mode Doppler reflectometry (DR) and enhanced microwave scattering in the upper hybrid resonance [5,6].

Unfortunately, the interpretation of DR data is complicated due to the contribution of the poorly localized small-angle scattering, which can lead to the overestimation of the turbulence radial correlation length in the linear scattering regime [7,8], and to its underestimation in the strongly nonlinear regime [9]. In the latter case the turbulence wavenumber spectrum measurements are questionable, and only the plasma turbulence mean velocity could be determined [10]. Moreover the 2D propagation effects additionally complicate the experimental data interpretation and its comparison with the theory predictions.

To overcome these issues, synthetic diagnostic can be used both for the interpretation of the experimental results and for the code benchmarking [11,12]. In this paper two versions of the Xmode DR synthetic diagnostics are compared in the framework of the ELMFIRE global GK modeling of the FT-2 tokamak ohmic discharge [6]. The X-mode DR signal is computed both in the linear theory approximation using the reciprocity theorem [13] and by the full-wave code IPF-FD3D [14]. The temporal behavior of the DR signal frequency spectra and the dependence of its amplitude, frequency shift and shape on the probing antenna position are computed and compared to those measured in the experiment at the FT-2 tokamak. In the case of multifrequency probing the radial correlation (RC) DR cross-correlation function (CCF) is also determined by both of the synthetic diagnostics and compared to that obtained in the experiment.

The experimental approach

The experiment was performed at the FT-2 tokamak (the major radius R = 0,55 m, the limiter radius a = 0,079 m) in the hydrogen ohmic discharge (with plasma current I p = 19 kA, central density n e (0) = 410 19 m -3 and electron temperature T e = 470 eV). The discharge is similar to that utilized for successful comprehensive benchmarking of the ELMFIRE GK code in [5,6], expect for the toroidal magnetic field (B t (0) = 1.7 T instead of 2.1 T). The measured electron density and temperature profiles for this discharge used in the GK modelling, as well as the ion temperature profile are shown in Fig. 1. The vertically movable (by ±2 cm) X-mode double antenna set (shown in Fig. 2) installed at high magnetic field side in the vicinity of equatorial plane allowed plasma probing at variable incidence angle with frequencies in the range f i = (50-75) GHz. It was used both in DR measurements utilizing the quadrature scheme and in the RCDR technique [15], as described in [16].

In the latter case the reference channel generator was tuned to the master frequency f 0 =70 GHz determining the measurement position in the vicinity of r = 5 cm, whereas another generator was used in the second, channel to determine the turbulence two-point CCF.

The GK computation approach and the synthetic diagnostics

The simulations are performed with the global electrostatic particle-in-cell code ELMFIRE [3]. The simulation geometry has a circular cross-section and covers the whole radial range. The time step is set to t = 30 ns and spatial grid to 1108 cells in radial (r) and toroidal () directions while the number of cells in poloidal () direction is set to increase with r up to 950 at r=7 cm. The corresponding computational grid cell sizes at the minor radius r=5 cm are 0.07cm in radial and poloidal directions. They are sufficient for an adequate description of fluctuations contributing to the experimental DR signal, however, not for the modelling of smaller-scale turbulence, in particular, belonging to the Electron Temperature Gradient (ETG).

The "fast" synthetic X-mode DR diagnostics developed for this code [13] is based on the reciprocity theorem of electrodynamics [17] relating the high-frequency current in the plasma volume and the signal radiated by it and received by the antenna. The probing and backscattering wave propagation in the plasma not perturbed by turbulence is described in the fast synthetic diagnostic by full-wave code WaveTOP2D [18]. It should be mentioned, however, that this approach neglects the perturbations of the probing wave propagation which makes it equivalent to the linear (Born) approximation in the fluctuation amplitude. The benefit of this approach is the high speed of computations, whereas the drawback is the neglecting of the nonlinear effects.

The alternative "slow" DR synthetic diagnostic is free from this drawback. It is based on the full-wave code IPF-FD3D describing the probing and scattered wave propagation in turbulent plasma [14].

Comparison of DR frequency and turbulence poloidal wavenumber spectra

The DR spectra obtained in the experiment and provided by the synthetic diagnostics are plotted in Fig. 3 for vertical antenna shifts of different sign and value and a reasonable agreement can be seen. The meaningful difference in spectral shift is only observed at the largest antenna displacement y a = ±2.0 cm. This difference is probably due to a contribution of experimental spectrum wings opposite to the frequency shift becoming especially heavy at high antenna displacement. This wing is opposite to the mean frequency shift of the spectrum and, presumably, is caused by small-scale drift modes, which rotate in the direction of the ion diamagnetic drift and are not described well by the GK code. Good agreement is confirmed by Fig. 4 and Fig. 5, where the mean frequency shift and the mean frequency width of the experimental and synthetic spectra obtained as the first and the second moment of the spectrum, respectively, are plotted as functions of the antenna vertical displacement and the turbulence poloidal wavenumber.  for the full-wave one. Fig. 5 shows the mean width of synthetic spectra to be systematically smaller than in experiment which can be attributed to the contribution of the spectral wings, which are higher in the experiment. The large backscattering spectrum width observed in Fig. 5 is partly associated with strong plasma potential oscillations caused by geodesic acoustic mode (GAM) excited in the FT-2 plasma [5,6,19]. These oscillations could be extracted from the temporal behavior of the DR spectrum frequency shift [20].

Utilizing experimental and synthetic DR signals oscillations of the plasma poloidal velocity v θ (t) caused by the GAM are obtained (Fig. 6). The velocity oscillation amplitudes provided by the experimental measurement, fast and full-wave synthetic diagnostics, direct GK code results are close (1.7±0.3 km/s; 1.6±0.3 km/s, 1.4±0.2 km/s and 1.3±0.5 km/s respectively). The standard deviations of the velocity fluctuations determined using DR data are also similar: the values 1.1 km/s; 0.9 km/s, 1.0 km/s and 0.8 km/s were obtained in experiment, given by the two synthetic diagnostics and directly provided by the GK code. Finally GAM frequencies provided by the experimental measurement, fast and full-wave synthetic diagnostics, direct GK code results are close too (40 kHz, 45 kHz, 36 kHz and 40 kHz respectively).

FIG. 6 Poloidal velocity temporal behavior. Thick line -computed directly by GK code, circlesmeasured; triangles -fast synthetic diagnostics; squares -full-wave synthetic diagnostics

Another comparison made was the DR signal power dependence on the antenna vertical displacement (Fig. 7). As it is seen in Fig. 7, the decrease of the DR signal power with growing fluctuation poloidal wavenumber is substantially faster than in experiment for both synthetic diagnostics. The difference could be attributed to at least three possible reasons. The first one is related to incorrect modeling of ETG mode-scale fluctuations nonlinearly leading to an underestimation of the turbulence poloidal wavenumber spectrum in the intermediate-scale domain

1 10cm    
. The second reason is a possible underestimation by the GK code of the total density fluctuation level. The final reason is neglecting multiple wave reflections between the cutoff and the antenna taking place at the equatorial probing in synthetic diagnostics.

RCDR CCF comparison

Inspired by a reasonable agreement of the experimental and synthetic DR spectral data we have performed the comparison of the computed and measured RCDR CCFs for antenna displacement of +2.0 cm. The corresponding CCFs as well as density fluctuations two-point CCF for frequency f s -f i = 300 kHz from GK data are plotted at Fig. 8. The slow decay of the CCF provided by fast synthetic diagnostics is most likely due to the small-angle-scattering effects [7] and neglecting strong probing wave phase modulation [9,10,21] coming into play with growing turbulence level. Both synthetic CCFs might also be wider due to the GK code spatial grid limitations leading to an underestimation of the turbulence radial wavenumber spectrum width, and thus -to the overestimation of the RCDR CCF width. The CCF provided by the full-wave synthetic diagnostics is much closer to the experimental one, which shows that the second reason mentioned above probably plays a minor role.

Conclusions

ELMFIRE code benchmarking against the FT-2 X-mode Doppler reflectometry experimental data utilising the fast linear and full-wave DR synthetic diagnostics has demonstrated a good agreement between the measured and computed DR frequency spectra. However, the variation of the DR signal power with growing incidence angles in experiment is slower than that predicted by both of the synthetic diagnostics. Possible explanations were suggested. The difference was also found in the decay of the DR signal CCFs with growing frequency separation. The quick decrease of the RCDR coherence observed in the experiment and full-wave synthetic diagnostic compared to the fast synthetic RCDR is attributed to nonlinear effects.

211 FIG. 1 .

 2111 FIG.1. The discharge parameter profiles. Density -solid curve; safety factor -dashed curve. Electron temperature -circles; ion temperature -triangles.

fFIG. 3

 3 FIG.3 Comparison of DR spectra for different antenna vertical displacement. Circles -experiment; triangles -fast synthetic DR,squares -full-wave synthetic DR.

Fig. 4

 4 allows us to obtain estimations for the fluctuation poloidal velocity. The estimation of the mean fluctuation poloidal velocity in experiment is

FIG. 4

 4 FIG. 4 Dependence of the DR signal frequency shift on the antenna vertical displacement y a. and fluctuation wavenumber. Circlesexperiment; triangles -fast synthetic DR; squares full-wave synthetic DR. FIG. 5 Dependence of the DR signal frequency width (rms) on the antenna vertical displacement y a. Circles -experiment; triangles -fast synthetic DR; squares full-wave synthetic DR.

14th Intl. Reflectometry Workshop -IRW14 (Lausanne) 22-24 May 2019 O.211

Acknowledgements

The financial support of the Russian Science Foundation grant 17-12-01110 is acknowledged. The work has been supported by the Academy of Finland grants 278487, 296853 and 318314. CSC -IT Center for Science is acknowledged for generous allocation of computational resources for this work. The FT-2 tokamak maintenance was supported by Ioffe Institute whereas M.A. Irzak acknowledges the support of Russian Academy of Science Presidium program. References


V Bulanin

E Gusakov

S Heuraux

C Lechte

A Petrov

N Teplova

A Yashin

G Zadvitskiy

F Da Silva

Globus-M Team

Full wave modeling of Doppler backscattering from filaments

Introduction:

It is recognized that the filaments have a significant effect on the anomalous energy and particle transport in the tokamak periphery. They are actively investigated using various diagnostics in this regard. Recently studies of filaments using the Doppler backscattering (DBS) method have been performed in Globus-M [1] and ASDEX-Upgrade [START_REF] Hennequin | 44th EPS Conference on Plasma Physics[END_REF][START_REF] Trier | 45th EPS Conference on Plasma Physics[END_REF]. Backscattering from filaments manifests itself as a burst of quasi-coherent oscillations of the signals of IQ detectors. Such signals are easy to describe theoretically within the Born approximation using the diagnostic weighting function [1]. However, the filaments in tokamaks differ noticeably in their size and intensity. With an increase in the amplitude of the filaments, it is necessary to consider the transition from linear scattering to nonlinear one and further up to the transition from backscattering to reflection from a moving over dens filament. This problem can be solved only using a full wave code. In this paper we present the results of two-dimensional full wave simulations of Doppler backscattering (DBS) from filaments in slab geometry.

Numerical modeling: Instead of using complicated non-linear MHD codes for filament description we have used relatively simple model of a filament with a Gaussian cross-section

(𝛿𝛿 = 𝐴 100% 𝛿 𝑐 𝑒𝑒𝑒(-((𝑟 -𝑟 0 )/𝑙 𝑐 ) 2 )exp (-((𝑒 -𝑒 0 )/𝑙 𝑐 ) 2 )
, where A is a filament amplitude in %, n c -critical density, r -radial position, r 0 -radial position of the filament center, xpoloidal position, x 0 -poloidal position of the filament center, 𝑙 𝑐 is a half width of the filament) that allows freedom in choosing the size, amplitude and position of a filament in radial and poloidal coordinates, which are here identified in Cartesian coordinates used in our simulations. Two-dimensional full wave simulations are relevant knowing that the filament has a very long extension along the magnetic field line, and were done with finite-difference

14th Intl. Reflectometry Workshop -IRW14 (Lausanne) 22-24 May 2019 O.212 ___________________________________________________________________________ B) Size and non-linear effects: The mechanism of BQO formation can be easily explained in a frame of Born approximation: if the size of a filament exceeds the spatial period of the weighting function (spatial distribution of the non-perturbed electric field, see figure 1a), π/k i sinα (k i -free space probing wave number), the back scattering efficiency is expected to decay in Born approximation due to Fourier component suppression, as it is shown in figure 3(a). However, the BQOs can be still clearly observed in experiment. On contrary, the signal amplitude calculated using full wave code in non-linear case is only slightly depend on filament size as it is seen from figure 3(b). In figure 4 the result of signal calculation for the filament size 6cm >> π/k i sinα ≈0.7cm is shown. At small amplitudes (figure 4a), BQO is observed, similar to BQO in linear scattering. Double BQOs appear with large amplitude of the filament (figure 4b). This could be qualitatively explained with the use of raytracing, which did not take into account the density perturbation inside the filament, assuming the possibility of transition from backscattering to reflection, shown in figure 5.

Conclusions: Full-wave calculations have shown that similar BQOs are formed in case of scattering off from filaments either in case of linear or non-linear regimes. As soon as the signal amplitude is measured in experiment in arbitrary units it is not possible to extract information on filament amplitude from experimental data. Nevertheless it is possible to determine the filament size in radial direction in case of multi-frequency probing. As it was expected the signal amplitude decreases with growing filament size in linear regime. It was shown that the signal amplitude is independent on filament size in non-linear regime. The results obtained largely explain the similarity of the IQ detector data registered in different tokamaks.

The calculations have been performed in O-mode using simple slab model and have not taken into account complicated magnetic field configuration taking place in real experiment. Such a task that is difficult even in the optical approximation is particularly relevant for tokamaks with a low aspect ratio. Moreover in presented computation turbulence effects are not taken into account as well. Despite the difficulties described above the simulation result has shown in particular the amplitude at which transition to the non-linear regime occurs. Also the preliminary calculations using full-wave finite-difference time-domain code REFMULF [START_REF] Silva | Proc. of the 13th International reflectometry workshop[END_REF] are being processed currently.

14th

  Intl. Reflectometry Workshop -IRW14 (Lausanne) 22-24 May 2019 O.212 ___________________________________________________________________________ time-domain code IPF-FD3D[START_REF] Lechte | [END_REF] in slab geometry. The goal was to calculate the DBS responses depending on various filament amplitudes (0.1%, 1%, 5%, 50%, 100% and 150% of density at cut off of the probing wave) and size (0.5cm, 1 cm, 1.5cm, 3cm, 5cm and 6cm). A) Amplitude: This dependence was calculated for various filament positions in relation to the cut off and filament shapes (circle and elongated ellipse in radial direction). The parameters of the computations are as following: antenna tilt angle 13°, antenna horn mouth 5.5 cm, Gaussian beam with flat wave front in the antenna mouth. Computations have been performed for the frequencies 16GHz, 24 GHz, 32GHz, 40GHz, 48GHz, 56GHz, 64GHz in O-mode that corresponds to the frequency range used in the Globus-M tokamak DBS diagnostics (20GHz, 29GHz, 40Ghz and 48GHz). Experimental plasma density profile used for the computations close to linear, n e = 5.6•10 19 cm -3 at r = 15 cm (shot#36569 at Globus-M) is shown in figure 1 by color bar. The filament moves in poloidal direction at r=9.2cm, the direction of motion is shown in figure 1 by an arrow. The antenna horn mouth is shown by a solid line; the yellow circle shows the position and circular shape of the filament. The imitation of the filament motion was simulated by independent snapshots with a spatial step 1.3 mm. If a time interval assigned to this step the velocity of filament and IQ signals can be determined. For each snapshot IQ signal was calculated (amplitude and phase) modeling a 'time dependency'. In figure 1 the electric field distribution calculated for filament width

Figure 1 .

 1 Figure 1. Electric field distribution for linear (a) and non-linear (b) regime.

Figure 2 . 1 .

 21 Figure 2. (a) Calculated Q(t) signals for linear (blue) and nonlinear(red) regimes; (b) normalized calculated signals; (c) I(t) and Q(t) DBS signals measured in Globus-M shot#36537, f=48GHz.

Figure 3 .

 3 Figure 3. Signal amplitude on filament size dependence in linear (a) and non-linear (b) case.

Figure 4 .

 4 Figure 4. Calculated signal for 6cm filament diameter, 10% of critical density n c (a) and 70% of n c (b).

Figure 5 .

 5 Figure 5. Ray tracing of the signal scattering off the filament 6cm.

Acknowledgements. The work is supported by RSCF grant 18-72-10028 whereas the maintenance of Globus-M tokamak and standard discharge diagnostics systems was supported by the Ioffe Institute. This work has been partially carried out within the framework of the French Federation for Magnetic Fusion Studies (FR-FCM) and the EUROfusion Consortium 14th Intl. Reflectometry Workshop -IRW14 (Lausanne) 22-24 May 2019 O.212 ___________________________________________________________________________ and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement N° 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.