The nonlinear global gyrokinetic (GK) simulation is critical in understanding the nonlinear behavior of the drift-wave turbulence, which mainly determines the plasma confinement in a tokamak fusion plasma. Recently, the Numerical Lie-Transform (NLT) method used in nonlinear global GK simulations has been developed in the past decade. To simplify the gyrocenter equations of motion with a perturbed field, the NLT method, based on the Lie-transform perturbation theory, transforms the phase-space gyrocenter coordinates to the new coordinates in which the equations of motion are simply identical to the ones in the equilibrium field. The effects of the perturbation on the motion are given by the Lie-transform generating vectors, which are determined by the scalar gauge function. The gauge function is found by integrating the perturbed Hamiltonian along the unperturbed orbit. The NLT method avoids the secularity problem of the perturbative method by applying itself in a short time interval, usually one time step in practical numerical computation. In solving the GK Vlasov equation to advance the gyrocenter distribution function in a time step, the NLT method evolves the distribution function along the equilibrium orbit, and makes the pull-back Lie-transform. Compared to the full-f simulation, the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\delta f\)</EquationSource> </InlineEquation> simulation has higher efficiency, but it incurs the well-known secularity problem in a long-time simulation, since it is based on the perturbative method. Recently, the Neighboring Equilibrium Update (NEU) method has been developed for the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\delta f\)</EquationSource> </InlineEquation> simulation; when the perturbation evolves to a large enough level in a long-time simulation, the NEU re-partitions the system defined by the particle distribution and field, into the updated equilibrium and perturbation. The operation of NEU does not change the system, but keeps the perturbation level low enough through moving the slowly-varying part of perturbation into equilibrium, with the updated equilibrium self-consistently constructed, e.g. the updated equilibrium distribution function is a constant of motion in the updated equilibrium field. When the NEU method is used, a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\delta f\)</EquationSource> </InlineEquation> simulation is equivalent to a full-f simulation. By using the NEU method, the self-organized evolution of the internal transport barrier in the ion-temperature-gradient turbulence has been successfully observed in a long-time nonlinear GK <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\delta f\)</EquationSource> </InlineEquation> simulation, with the simulation results well consistent with experiments in various aspects.

Numerical Lie-transform and neighboring equilibrium update in nonlinear simulation of the drift-wave turbulence in a tokamak fusion plasma

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2026年1月20日

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Yingfeng Xu,
Shaojie Wang

摘要

The nonlinear global gyrokinetic (GK) simulation is critical in understanding the nonlinear behavior of the drift-wave turbulence, which mainly determines the plasma confinement in a tokamak fusion plasma. Recently, the Numerical Lie-Transform (NLT) method used in nonlinear global GK simulations has been developed in the past decade. To simplify the gyrocenter equations of motion with a perturbed field, the NLT method, based on the Lie-transform perturbation theory, transforms the phase-space gyrocenter coordinates to the new coordinates in which the equations of motion are simply identical to the ones in the equilibrium field. The effects of the perturbation on the motion are given by the Lie-transform generating vectors, which are determined by the scalar gauge function. The gauge function is found by integrating the perturbed Hamiltonian along the unperturbed orbit. The NLT method avoids the secularity problem of the perturbative method by applying itself in a short time interval, usually one time step in practical numerical computation. In solving the GK Vlasov equation to advance the gyrocenter distribution function in a time step, the NLT method evolves the distribution function along the equilibrium orbit, and makes the pull-back Lie-transform. Compared to the full-f simulation, the \(\delta f\) simulation has higher efficiency, but it incurs the well-known secularity problem in a long-time simulation, since it is based on the perturbative method. Recently, the Neighboring Equilibrium Update (NEU) method has been developed for the \(\delta f\) simulation; when the perturbation evolves to a large enough level in a long-time simulation, the NEU re-partitions the system defined by the particle distribution and field, into the updated equilibrium and perturbation. The operation of NEU does not change the system, but keeps the perturbation level low enough through moving the slowly-varying part of perturbation into equilibrium, with the updated equilibrium self-consistently constructed, e.g. the updated equilibrium distribution function is a constant of motion in the updated equilibrium field. When the NEU method is used, a \(\delta f\) simulation is equivalent to a full-f simulation. By using the NEU method, the self-organized evolution of the internal transport barrier in the ion-temperature-gradient turbulence has been successfully observed in a long-time nonlinear GK \(\delta f\) simulation, with the simulation results well consistent with experiments in various aspects.