| In tokamak plasma,anomalous transport caused by turbulence has an important impact on the evolution of plasma density,temperature and rotational profile.The understanding and prediction of anomalous transport is of great significance for the realization of commercial controlled fusion.An important feature in turbulent transport is the presence of off-diagonal transport effects,that is,the transport fluxes contains not only the diffusion fluxes driven by the corresponding thermodynamic forces,but also the pinch fluxes driven by other generalized forces.The off-diagonal effects cause the coupling between the particle,momentum and energy transport,resulting in anomalous transport phenomena such as density peaking and spontaneous rotation.Obtaining a full transport matrix containing off-diagonal terms is important for resolving these anomalous transport phenomena and for better prediction of profile evolution.At present,the main research methods of turbulent transport are quasi-linear theory and nonlinear gyrokinetic simulation.Quasi-linear theory transforms the Vlasov equation into the convectivediffusive Fokker-Planck equation to obtain the phase space diffusion tensor,from which the full transport matrix can be calculated.The limitation is that the nonlinear effect in turbulence cannot be fully considered,therfore such method is not suitable for researching strong turbulence.The nonlinear gyrokinetic simulation directly solves the nonlinear Vlasov-Maxwell equations numerically,which can well include nonlinear physical effects,but it cannot distinguish the diagonal part and the pinch part in the transport fluxes and therefore cannot give full nonlinear transport matrix.On the other hand,gyrokinetic simulations cannot directly calculate the phase space diffusion tensor,and therefore cannot be directly compared with quasi-linear theory.In order to solve the above problems,in the first work of this paper,based on the phase-space nonlinear Fokker-Planck equation,we developed the a numerical codeNumerical Diagnosis of Transport Matrix(NDTM)for calculating the nonlinear phasespace diffusion tensor and transport matrix of electrostatic turbulence.The NDTM code can be used as an auxiliary diagnostic tool for the nonlinear gyrokinetic turbulence simulation code,which reads the equilibrium magnetic field and the self-consistent perturbation electric field obtained in the turbulence simulation,and calculates the phase space diffusion tensor and Lagrangian correlation function by numerically evolving the true orbit of the gyrocenter in the turbulent field,and the nonlinear transport matrix is further obtained by integrating the diffusion tensor in the velocity space.As a preliminary application of the code,we calculate the ion transport matrix and the corresponding transport fluxes in ion temperature gradient(ITG)turbulence obtained from the gyrokinetic turbulence simulation code NLT.We find that the non-diagonal effect plays a role in reducing the transport in ITG turbulence,and there are inward pinch particle flux driven by temperature gradient and inward heat flux driven by density gradient.The particle and heat fluxes driven by velocity inhomogeneity are both inward.These conclusions agree with the previous theories.In order to verify the results of the NDTM code,we use the transport matrix to calculate the ion effective thermal conductivity and ion effective diffusivity,and compare with the results of NLT code,the numerical results are in good agreement.Based on the first work,we further study the phase space structure of the nonlinear ion diffusion tensor in ITG turbulence and compare it with the quasi-linear theory.We find that when nonlinear ITG turbulence reaches saturation and the linear mode structure is sheared by the zonal electric field,the phase-space structure of ion diffusion tensor is still consistent with the prediction of quasi-linear theory:the ion diffusion tensor clearly presents magnetic drift resonant structure,namely elliptic band with axial ratio(?)in velocity space,the numerical results agree with that predicted by quasi-linear theory;the relation between Drr component and DrK component of the ion diffusion tensor is approximately DrK=-m(v‖2+v⊥2/2)/RDrr;the ion diffusion tensor presents strong-ballooning struture in θ direction,namely relatively strong in weak field side and relatively weak in strong field side.In the quasi-linear theory,the phase-space strcuture is formed by the resonance between ion and ITG eigen mode,however through further analysis of the structure of diffusion tensor in single n mode,we find that the ion resonance is significantly changed by the slowly changing zonal field which is stimulated by nonlinear interaction.Under the influence of zonal field,the single n ITG eigen mode is restructured to multiple Child-Ballooning-Modes(CBMs)with small radial scale and time variant mode frequency.The ions resonate with the CBMs through electric-magnetic drift motion under the poloidal acceleration induced by zonal field,the phase-space structure of nonlinear diffusion tensor is formed by such nonlinear resonance. |
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