Turbulent transport of alpha particles in tokamak plasmas
We investigate the diffusion of fusion born particles in tokamak plasmas. We determine the transport regimes for a realistic model that has the characteristics of the ion temperature gradient (ITG) or of the trapped electron modes (TEM) driven turbulence. It includes a spectrum of potential fluctuations that is modeled using the results of the numerical simulations, the drift of the potential with the effective diamagnetic velocity and the parallel motion. Our semi-analytical statistical approach is based on the decorrelation trajectory method (DTM), which is adapted to the gyrokinetic approximation. We obtain the transport coefficients as a function of the parameters of the turbulence and of the energy of the particle. According to our results, significant turbulent transport of the particles can appear only at energies of the order of 100KeV. We determine the corresponding conditions.
The turbulent transport of the fast particles was considered negligible in tokamak plasmas Zweben et al. (2000) due to the fast gyration motion with a Larmor radius much larger than the correlation length, which leads to a very small amplitude of the gyro-average potential. However, this problem was reconsidered in the last decade Zhou et al. (2011, 2010); Dewhurst et al. (2010); Heidbrink et al. (2009); Angioni et al. (2009); Hauff and Jenko (2006, 2008); Angioni and Peeters (2008); Chowdhury et al. (2012); Zhang et al. (2010, 2008); Günter et al. (2007); Gingell et al. (2014), in preparation of the Tritium experiments in JET and ITER. The conclusions are rather dispersed, from completely negligible turbulent transport Pace et al. (2013) to diffusion coefficients that can be larger than those of plasma ions Estrada-Mila et al. (2006); Vlad et al. (2005).
Most of the theoretical studies of particle turbulent transport are self-consistent numerical simulations of turbulence and particles turbulent fluxes. Other studies use the test particle approach in numerical simulations based on constructed potentials or on the results of turbulence simulations. The characteristics of turbulence and the particle fluxes are determined in the first case as functions of the macroscopic conditions (gradients, heating power). The second approach obtains the diffusion coefficient of the particles as a function of their energy and of the characteristics of the turbulence. It allows to find if the turbulent loss of particles can be significant and to identify the corresponding conditions.
This paper is included in the last category. We determine the transport regimes of the particles as a function of their energy for a realistic model of turbulence. The spectrum of the stochastic potential has the shape of the ion temperature gradient (ITG) or of the trapped electron modes (TEM) driven turbulence. The drift of the potential with the effective diamagnetic velocity and the parallel motion of the fast particles are included in the model. The main result consists in the evaluation of the energy corresponding to the maximum turbulent transport We show that does not always appear when the particles reach the energy of the plasma ions (for the ashes), but it can correspond to larger energies.
The article is organized as follows. Section II contains the basic equations and the statistical approach based on the DTM for the Lorentz transport in the gyro-kinetic approximation for a Maxwellian distribution of Larmor radii. In section III we present the transport regimes for different ranges of the parameters and obtain the energy dependence of the particles diffusion coefficient. The conclusions are summarized in section IV.
Ii Model and statistical method
ii.1 Equations of motion for fast ions
The turbulent transport of the fast ions is studied in the test particle approach starting from the Newton-Lorentz equations of motion in a stochastic potential and a constant magnetic field oriented along the axis
Where is the ion displacement in the plane perpendicular to , are the components of the velocity in this perpendicular plane, is the velocity along the magnetic field, is the ion mass, is its charge, is the cyclotron frequency and is the antisymmetric tensor ( .
The potential is modeled as a stationary and homogeneous Gaussian stochastic function. Its spectrum corresponds to the general characteristics of the ITG or TEM turbulence. Since the spectrum at saturation is mainly determined by the ion dynamics, its shape is similar for both types of turbulence. The only difference is given by the value of the typical wave numbers ( for ITG, for TEM). The spectrum has two symmetrical maxima for and zero amplitude for We use the simple analytical expression of that was found in Vlad and Spineanu (2015) to be in agreement with numerical and experimental results of Hauff and Jenko (2007); Shafer et al. (2012)
The parameters of this function are the amplitude of the potential fluctuations the correlation lengths along each direction (radial), (poloidal), (parallel), and the correlation time The Fourier transform of is the Eulerian correlation (EC) of the potential.
The change of coordinates leads to
where the time dependent Larmor radius is defined by is the guiding center position , is the particle energy, is the magnetic moment, and is the gyrophase angle.
The very large value of the cyclotron frequency enables a strong simplification of the equation of motion by using the gyrokinetic approximation Hahm et al. (1988); Littlejohn (1982). The first term in the right hand side term of Eq. (4) is negligible compared to the second one, and the solution of this equation is , where and Thus, the Larmor radius is constant, and the time dependence is contained in the uniform gyration motion. Moreover, the time variation of the potential is slow. Its characteristic time is very large compared to the gyration period Since the displacement of the guiding center during is small, Eq. (3) can be averaged over the cyclotron period at constant and One obtains
where the gyro-averaged potential was normalized with Thus, the motion of the guiding centers of the fast ions obeys the same equation as in the limit of zero Larmor radius, but with the modified potential (7). Using the Fourier representation of the potential, and performing the time integral
where the wave number is perpendicular on and is the Bessel function of the first kind. This shows that the gyro-average of the potential determines the multiplication of its Fourier transform with which corresponds to the gradual attenuation of the large wave number components of the spectrum as the Larmor radius increases.
The EC of the averaged potential for a Maxwellian distribution of particle velocities is
where the velocity is given in units of the thermal velocity of the fast particles with the temperature and the modified Bessel function of the first kind. The limit corresponds to the EC of the potential The EC (9) is represented in Figure 1 for several values of One can see that the amplitude of is a monotonically decreasing function of and that the effective correlation lengths in the perpendicular plane increase with . The general shape of the EC is not changed at large In particular, the positive and the negative parts compensate, and the integral over is zero for any This property is due to the spectrum (2), which cancels for We plot in Figure 2 the EC of the gyro-averaged potential for , and
ii.2 The decorrelation trajectory method
The transport of fast particles in turbulent plasmas is described by equations that are similar to those for the drift. Moreover, the shape of the EC of is similar to that of We introduce normalized quantities Vlad and Spineanu (2013) using as units for the perpendicular displacements, for parallel displacements, for the drift velocity and for time. The equations of motion for the normalized quantities (designed by the same symbols as the physical ones) are
where is the decorrelation time induced by the parallel motion, , and , with being the effective diamagnetic velocity. The equation is written in a frame that moves with the potential, in which the normalized effective diamagnetic velocity appears as an average velocity.
This type of statistical problem was analyzed in several papers. The time dependent diffusion coefficients were determined using the decorrelation trajectory method (DTM) Vlad et al. (1998); Vlad and Spineanu (2004). This is a semi-analytical method, which shows that can be approximated using a set trajectories obtained from the EC of the potential, the decorrelation trajectories (DTs). The main idea of this method is to group together trajectories that are similar by imposing supplementary initial conditions. Each group corresponds to a subensemble of realizations of the stochastic potential that is defined by the supplementary initial conditions. One obtains a DT for each subensemble, which then is used to evaluate as weighted sums of the contributions of all subensembles.
We use here the fast DTM introduced in Vlad and Spineanu (2015), which imposes only two supplementary initial conditions: the potential in the starting point of the guiding centers trajectories and the orientation of the normalized initial velocity. The advantage of this method is that the number of DTs is strongly reduced.
The time dependent diffusion coefficients are obtained from
where is the DT in the subensemble that is the solution of
The subensemble average velocities are obtained from the average potential
where is the space derivative. An example of the subensemble potential is show in Figure 3. The transport at large space and time scales is described by the asymptotic value
A computer code was developed for determining using Eqs. (11-15). It calculates the EC of (9) and its derivatives that appear in the subensemble average velocity (15) using a fast Fourier Transform subroutine. The latter links an uniform grid representation in the space to a two-dimensional real space mesh on which the velocity field is computed. The spectrum (2) is used for all the calculations presented in this paper, but it can easily be replaced by other models. The DTs are determined using high order interpolation techniques for the velocity field. The time step automatically adapts so that only the space steps along and have to be optimized. The condition is provided by the trajectories with which represent periodic motions on the contour lines of They have to remain close to these lines during the whole integration time that can be of hundreds of periods.
Iii Fast particle diffusion regimes
iii.1 Basic physical processes
The diffusion regimes of the electrons Vlad and Spineanu (2015) and of the ions Vlad and Spineanu (2013) in the realistic model of the spectrum (2) were studied for the limit of zero Larmor radius. The EC of the gyro-average potential (9) contains eight physical parameters, six from the spectrum of the turbulence (2): the amplitude of the potential fluctuations, , the maximum wave number , the spatial decorrelation lengths and the temporal decorrelation length plus the Larmor radius, and the effective diamagnetic velocity . The latter appears as the drift of the potential in the poloidal direction. We have found that the transport regimes are determined by four dimensionless parameters or equivalently and which are defined by
The values of and are related to the presence of trajectory trapping or eddying in the structure of the stochastic potential.
The effective Kubo number (16) is a measure of the decorrelation of the trajectories from the potential, which is determined by the time variation of the potential and/or by the parallel motion of the particles. Trajectory trapping exists when the decorrelation is weak such that the characteristic time is larger than the time of flight
The diamagnetic parameter (17) is a dimensionless measure of the effective velocity which determines the characteristic time It is equivalent with an average potential , which adds to the stochastic potential and changes the configuration of the total potential. Bunches of opened contour lines appear on a fraction of the surface that increases from zero (for to one (for Trajectory trapping is possible only when the bunches of open lines fill only a fraction of the surface, and islands of closed contour lines exists between them. This configuration corresponds to the condition
The special shape of the spectrum (2) and of the EC (Figure 1) leads to similar dependences of on for the quasilinear regime ( and for the nonlinear regime ( Vlad and Spineanu (2015). In both cases, for small then it has a maximum at and eventually it decays as depends on the transport regime: for and for The power also depends on the regime and on the EC of the potential. We note that, for an usual decreasing EC without negative minima, the diffusion coefficient in the quasilinear regime ( has significantly larger values at large since it saturates at the maximum value instead of decaying.
The first question addressed in this paper concerns the basic physics of the transport regimes of the fast particles. We examine the dependence of the diffusion coefficient on the four parameters at a large value of the Larmor radius ( Examples of the time dependent diffusion coefficients obtained using the DTM in the nonlinear regime are shown in Figures 4 and 5.
The effects of the decorrelation and of the average velocity in the nonlinear regime are presented in Figure 4. The decorrelation leads to the transition from the subdiffusive to diffusive transport by saturating The process is similar with the case of small energy particles. The saturation time increases with the increase of and the asymptotic diffusion coefficient decreases (Figure 4b). This is due to the fraction of non-trapped trajectories that decreases when increases, leading to the decrease of The effect of the average velocity for a large value of the decorrelation parameter ( (Figure 4a) consists of the continuous decrease of the radial diffusion when increases.
The effect of the dominant wave number is shown in Figure 5a. The increase of determines the decrease of the radial diffusion coefficient in the nonlinear regime, although it leads to the increase of the amplitude of the radial velocity ().
Thus, at large Larmor radius, the time dependent diffusion coefficients are qualitatively similar to those for This is also suggested by the shape of the gyro-averaged EC (9), which is not much changed compared to that for
iii.2 Fast particle transport regimes
The dependence of the diffusion coefficient on is shown in Figure 6. The shapes of the curves are roughly similar for different values of The differences between the curves with different are practically independent on time, except for the range where a stronger dependence on time can be seen (Figure 6a).
The asymptotic diffusion coefficients are represented in Figure 7 in the nonlinear regime. Figure 7a shows the dependence on the Larmor radius One can see that both and do not depend on for and that they decay at large as This decay is the same as in the quasilinear regime. Fast particle diffusion coefficient can be evaluated analytically in the case of the quasilinear (Gaussian) transport. It decreases as The similar dependence on in the nonlinear and quasilinear regimes is rather surprising because several works have found a weaker decay in the nonlinear regime (as in Hauff and Jenko (2007)).
The cause of the faster decay with found here is the special shape (2) of the spectrum of the drift type turbulence. The average over the gyro motion leads to the attenuation of the large components of the spectrum (, while the small ( part of is not affected. The spectrum (2) decays in the small range because the modes are stable for Due to this property, all the components of the spectrum are attenuated at large enough vales of because the condition applies for all components that correspond to significant (not close to zero) values of This leads at large values of to the change of the effective EC that consists only in the decay of the amplitude but not in the modification of the shape (increase of the correlation lengths like for monotonically decaying spectra).
From Figure 7b one can see that the asymptotical value of the diffusion coefficients does not depend on the value of the wave vector for and that is has a weak exponential decay at large .
The dependence of of the fast ions on the decorrelation parameter is shown in Figure 7c. One can see that the diffusion coefficients are smaller than in the zero Larmor radius limit (dashed curves). The dependence on of the fast particle diffusion coefficient is the same as at for both limits of small and large (the curves are parallel in these limits). The maximum of is displaced to larger values of at large In Figure7d is plotted the dependence of on the normalized effective diamagnetic velocity . The behaviour is similar as in the case of vanishing Larmor radius (dashed lines) both in the nonlinear and in the quasilinear regimes.
iii.3 Alpha particle turbulent loss as function of the energy
Numerical simulations have shown that the main decorrelation mechanism in ITG turbulence is the parrallel ion motion. The turbulent transport of the fast particles, which have much smaller parallel time is completely dominated by the parallel decorrelation. The decorrelation parameter (16) becomes in these conditions.
Both the decorrelation time and the Larmor radius are functions of the energy of the particles. The cooling of the fast particles determine the decrease of and the increase of but the product of these parameters does not depend on particle energy. It essentially depends on plasma size through In terms of the dimensionless parameters that characterize fast particle transport
where is the ratio of the and diamagnetic velocities. This show that and (actually cannot be modified independently for ITG turbulence. Their product is a dimensionless parameter that depends on the amplitude of the turbulence (through on its normalized poloidal and parallel correlation lengths and on the gradient length of the ion temperature. Taking the typical parameters of the ITG instabilities for ITER size plasmas similar to those of the present tokamaks (as obtained in numerical simulations), one finds (for and
The dependence of the diffusion coefficient on particle energy is shown in Figure 8 for several values of . One can see an important difference between these results and those in Figure 7a, which shows that the decrease of the energy (of ) determines a monotonous increase of the diffusion coefficient. A maximum radial diffusion appears in Figure 8, which has the amplitude and the location dependent on the parameter The shape of these curves is determined by the simultaneous variation of and with the energy. The decrease of the energy determines the increase of which moves from the small decorrelation time regime in Figure 7c toward the maximum and further to the nonlinear regime with decaying The maxima of the curves in Figure 8 correspond to the maximum of in Figure 7c. This maximum is a decreasing function of the energy, and this is reflected in Figure 8, which shows that the maximum is smaller when it appears at larger energies. The diffusion coefficient in Figure 8 is small for all the range of particle energy at but at smaller values of it can reach significant values (comparable to the ion diffusion coefficient) at energies much larger than plasma ion energy (see the curve for
Thus, particle turbulent transport strongly depends on the parameter which is determined by the characteristics of the turbulence. Using typical values for the physical quantities in Eq. (20), the main contribution appears to be determined by the poloidal correlation length of the turbulence A maximum diffusion coefficient appears in Figure 8 for particle energy of 100KeV if
A detailed study of the turbulent transport of the fast particles was performed based on the development of the DTM. A realistic model of the turbulence was considered. It was necessary to determine numerically the gyro-averaged Eulerian correlation (EC) and to adapt the DTM code to the discretized EC. The code calculates the time dependent diffusion coefficient from the EC and its derivatives represented on a space mesh, using an interpolation procedure.
The dependence of the diffusion coefficient on the five dimensionless parameters of the model () was determined and analyzed.
The special shape of the spectrum of the ITG turbulence leads to the decay of the diffusion coefficient as for for both quasilinear and nonliniar regimes. The decay in the nonlinear regime is faster than in the case of monotonically decreasing Eulerian correlations, as is the one considered in Hauff and Jenko (2006). The difference between the quasilinear and the nonlinear regime is given by a factor that depends on the decorrelation parameter and is smaller at large in the nonlinear regime.
The parallel motion of the particles provides the main decorrelation mechanism in ITG turbulence. The characteristic time for this motion depends on particle energy. It is very small when particles are born in the nuclear fusion reaction, and it increases by a factor of the order during the cooling process. The dependence of the asymptotic diffusion coefficient on the energy of the particles was obtained taking into account both the parallel motion and the variation of the Larmor radius. The combined action of these effects leads to the existence of a maximum diffusion coefficient (Figure 8). We have identified a parameter (20), which includes the characteristics of the turbulence and the plasma size. The maximum turbulent loss rate and the corresponding energy are functions of Depending on the specific values of in a range that is relevant for JET and ITER plasmas, the turbulent transport of the cooling particles can be negligible or significant. The significant transport appears only when most of the energy of the particles is lost and their energy is in the range of (see Figure 8).
This work was supported by the Romanian Ministry of National Education under the contract 1EU-10 in the Programme of Complementary Research in Fusion. The views presented here do not necessarily represent those of the European Commission.
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