title: Ð§Ð¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ Ð½Ð¸Ð¶Ð½ÐµÐ³Ð¸Ð±ÑÐ¸Ð´Ð½Ð¾Ð³Ð¾ ÑÐ¾ÐºÐ° ÑÐ²Ð»ÐµÑÐµÐ½Ð¸Ñ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¾Ð³Ð¾ ÐºÐ¾Ð´Ð° FRTC Ð¸ ÐºÐ¾Ð´Ð° ASTRA
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: Numerical modelling of the lower hybrid current drive using the dynamic FRTC and ASTRA codes
Creators: name: Ð¥Ð°Ð²Ð¸Ð½, ÐÐ°ÑÐ¸Ð»Ð¸Ð¹
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2022
Subjects: ÐÐ°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
Ð­Ð»ÐµÐºÑÑÐ¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ¾Ðº
ÐÐ»Ð°Ð·Ð¼Ð° (ÑÐ¸Ð·.)
ÑÐ¾ÐºÐ°Ð¼Ð°Ðº
Ð½Ð¸Ð¶Ð½ÐµÐ³Ð¸Ð±ÑÐ¸Ð´Ð½ÑÐµ Ð²Ð¾Ð»Ð½Ñ
ÑÑÐ½ÐºÑÐ¸Ñ ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ñ ÑÐ»ÐµÐºÑÑÐ¾Ð½Ð¾Ð²
ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ Ð¤Ð¾ÐºÐºÐµÑÐ°-ÐÐ»Ð°Ð½ÐºÐ°
ÑÐµÐ»ÑÑÐ¸Ð²Ð¸ÑÑÑÐºÐ¾Ðµ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ Ð¤Ð¾ÐºÐºÐµÑÐ°-ÐÐ»Ð°Ð½ÐºÐ°
Ð¼ÐµÑÐ¾Ð´ Ð§ÐµÐ½Ð³Ð°-ÐÑÐ¿ÐµÑÐ°
tokamak
lower hybrid waves
electron distribution function
Fokker-Planck equation
relativistic Fokker-Planck equation
Chang-Cooper method

description: Ð¢ÐµÐ¼Ð° Ð²ÑÐ¿ÑÑÐºÐ½Ð¾Ð¹ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ: Â«Ð§Ð¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ Ð½Ð¸Ð¶Ð½ÐµÐ³Ð¸Ð±ÑÐ¸Ð´Ð½Ð¾Ð³Ð¾ ÑÐ¾ÐºÐ° ÑÐ²Ð»ÐµÑÐµÐ½Ð¸Ñ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¾Ð³Ð¾ ÐºÐ¾Ð´Ð° FRTC Ð¸ ÐºÐ¾Ð´Ð° ASTRAÂ». ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð°Ð½Ð°Ð»Ð¸Ð·Ñ ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÑ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² ÑÐµÑÐµÐ½Ð¸Ñ Ð¾Ð´Ð½Ð¾Ð¼ÐµÑÐ½Ð¾Ð³Ð¾ Ð½ÐµÑÐµÐ»ÑÑÐ¸Ð²Ð¸ÑÑÑÐºÐ¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ Ð¤Ð¾ÐºÐºÐµÑÐ°-ÐÐ»Ð°Ð½ÐºÐ° Ñ ÑÐµÐ»ÑÑ ÑÐ°ÑÑÐµÑÐ° Ñ ÐµÐ³Ð¾ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð½Ð¸Ð¶Ð½ÐµÐ³Ð¸Ð±ÑÐ¸Ð´Ð½Ð¾Ð³Ð¾ ÑÐ¾ÐºÐ° ÑÐ²Ð»ÐµÑÐµÐ½Ð¸Ñ Ð´Ð»Ñ Ð°Ð½Ð°Ð»Ð¸Ð·Ð° ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½ÑÐ¾Ð² Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ ÑÐ¸ÑÑÐµÐ¼Ñ ÐºÐ¾Ð´Ð¾Ð² FRTC Ð¸ ASTRA. ÐÑÐ¸Ð²ÐµÐ´ÐµÐ½Ð¾ Ð¾Ð¿Ð¸ÑÐ°Ð½Ð¸Ðµ Ð¼ÐµÑÐ¾Ð´Ð° Ð§ÐµÐ½Ð³Ð°-ÐÑÐ¿ÐµÑÐ° Ð´Ð»Ñ ÑÐµÑÐµÐ½Ð¸Ñ Ð´Ð°Ð½Ð½Ð¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ. Ð ÑÐ°Ð±Ð¾ÑÐµ Ð¿ÑÐ¾Ð´ÐµÐ¼Ð¾Ð½ÑÑÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ Ð¾Ð±ÑÑÐ½Ð¾Ð³Ð¾ Ð½ÐµÑÐ²Ð½Ð¾Ð³Ð¾ ÑÐ°Ð·Ð½Ð¾ÑÑÐ½Ð¾Ð³Ð¾ Ð¼ÐµÑÐ¾Ð´Ð° Ð¸ Ð¼ÐµÑÐ¾Ð´Ð° Ð§ÐµÐ½Ð³Ð°-ÐÑÐ¿ÐµÑÐ°. ÐÑÐµ ÑÐ°ÑÑÐµÑÑ Ð²ÑÐ¿Ð¾Ð»Ð½ÐµÐ½Ñ Ñ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸ÐµÐ¼ Ð²ÑÐ¾Ð´Ð½ÑÑ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð², Ð²Ð·ÑÑÑÑ Ð¸Ð· ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½ÑÐ¾Ð², Ð´Ð»Ñ ÑÑÑÐ°Ð½Ð¾Ð²Ð¾Ðº Ð¤Ð¢-2 Ð¸ ÐÐ»Ð¾Ð±ÑÑ-Ð2. Ð¡ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð°Ð½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¸Ñ Ð¸ ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÑ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð¿Ð¾ÐºÐ°Ð·Ð°Ð½Ð¾ Ð¿ÑÐµÐ¸Ð¼ÑÑÐµÑÑÐ²Ð¾ Ð¼ÐµÑÐ¾Ð´Ð° Ð§ÐµÐ½Ð³Ð°-ÐÑÐ¿ÐµÑÐ° Ð² Ð²Ð¸Ð´Ðµ Ð±Ð¾Ð»ÐµÐµ ÑÐ¾ÑÐ½Ð¾Ð³Ð¾ ÑÐ°ÑÑÑÑÐ° ÑÑÐ½ÐºÑÐ¸Ð¸ ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ñ Ð¸ ÑÐ¾Ð¾ÑÐ²ÐµÑÑÑÐ²ÐµÐ½Ð½Ð¾ Ð±Ð¾Ð»ÐµÐµ ÑÐ¾ÑÐ½Ð¾Ð³Ð¾ ÑÐ°ÑÑÐµÑÐ° Ð½Ð¸Ð¶Ð½ÐµÐ³Ð¸Ð±ÑÐ¸Ð´Ð½Ð¾Ð³Ð¾ ÑÐ¾ÐºÐ° ÑÐ²Ð»ÐµÑÐµÐ½Ð¸Ñ. Ð¢Ð°ÐºÐ¶Ðµ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð°Ð½Ð°Ð»Ð¸Ð·Ñ Ð´Ð²ÑÐ¼ÐµÑÐ½Ð¾Ð³Ð¾ ÑÐµÐ»ÑÑÐ¸Ð²Ð¸ÑÑÑÐºÐ¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ Ð¤Ð¾ÐºÐºÐµÑÐ°-ÐÐ»Ð°Ð½ÐºÐ°. ÐÐ¾ÐºÐ°Ð·Ð°Ð½Ð¾, ÑÑÐ¾ Ð² ÑÐ»ÑÑÐ°Ðµ Ð½ÐµÑÐµÐ»ÑÑÐ¸Ð²Ð¸ÑÑÑÐºÐ¾Ð¹ ÑÐµÐ¾ÑÐ¸Ð¸ Ð±ÑÑÑÑÑÐµ ÑÐ°ÑÑÐ¸ÑÑ Ð² ÑÐ²Ð¾ÑÑÐµ ÑÑÐ½ÐºÑÐ¸Ð¸ ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ñ ÑÐ°Ð·Ð³Ð¾Ð½ÑÑÑÑÑ Ð´Ð¾ ÑÐºÐ¾ÑÐ¾ÑÑÐµÐ¹ Ð¿Ð¾ÑÑÐ´ÐºÐ° ÑÐºÐ¾ÑÐ¾ÑÑÐ¸ ÑÐ²ÐµÑÐ°. Ð Ð² ÑÐ¾Ð´Ðµ Ð°Ð½Ð°Ð»Ð¸Ð·Ð° Ð´ÐµÐ¼Ð¾Ð½ÑÑÑÐ¸ÑÑÐµÑÑÑ, ÑÑÐ¾ ÑÐ²ÐµÐ´ÐµÐ½Ð¸Ðµ Ð´Ð°Ð½Ð½Ð¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ Ðº Ð¾Ð´Ð½Ð¾Ð¼ÐµÑÐ½Ð¾Ð¼Ñ Ð²Ð¾Ð·Ð¼Ð¾Ð¶Ð½Ð¾, Ð½Ð¾ Ð½ÐµÐ¿ÑÐ¸Ð¼ÐµÐ½Ð¸Ð¼Ð¾ Ð´Ð»Ñ ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ð³Ð¾ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ. Ð¢Ð°ÐºÐ¶Ðµ Ð¿ÑÐ¸Ð²ÐµÐ´ÐµÐ½ Ð²ÑÐ²Ð¾Ð´ ÑÑÐ¾Ð»ÐºÐ½Ð¾Ð²Ð¸ÑÐµÐ»ÑÐ½Ð¾Ð³Ð¾ Ð¾Ð¿ÐµÑÐ°ÑÐ¾ÑÐ° Ð´Ð»Ñ Ð´Ð²ÑÐ¼ÐµÑÐ½Ð¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ Ð² Ð¿ÑÐµÐ´ÐµÐ»Ðµ Ð±Ð¾Ð»ÑÑÐ¸Ñ Ð¸Ð¼Ð¿ÑÐ»ÑÑÐ¾Ð².
description: The subject of the graduate qualification work is âNumerical modelling of the lower hybrid current drive using the dynamic FRTC and ASTRA codesâ. This given work is devoted to analyzing the numerical methods for solving the one-dimensional nonrelativistic Fokker-Planck equation to calculate with it the lower hybrid current drive for the analysis of experiments purposes with the FRTC and ASTRA code system. The Chang-Cooper method for solving this equation is described. A comparison of the implicit finite difference method and the Chang-Cooper method is demonstrated in the paper. All calculations were performed using the input parameters taken from the experiments for the FT-2 and Globus-M2 tokamaks. Using analytical and numerical methods, the advantage of the Chang-Cooper method is described. It allows us to calculate the distribution function more accurately and consequently to calculate the lower hybrid current drive more accurately. The work is also devoted to the analysis of the two-dimensional relativistic Fokker-Planck equation. In the case of the nonrelativistic theory, fast particles in the tail of the distribution function are accelerated to velocities of the order of the speed of light. During the analysis, it is shown that reducing this equation to a one-dimensional one is possible but not applicable for numerical modelling. A derivation of the collision operator for the two-dimensional equation in the large momentum limit is also given.