25F04
Study of entropy transport using global gyrokinetic simulation in open system
開放系プラズマにおけるエントロピー輸送に関するジャイロ運動論的解析
Kenji Imadera, Yasuaki Kishimoto and Jiquan Li
今寺賢志, 岸本泰明，李継全
Department of Fundamental Energy Science, Graduate School of Energy Science,
Kyoto University, Gokasho, Uji, Kyoto, 611-0011, JAPAN京都大学大学院 エネルギー科学研究科 〒611-0011 京都府宇治市五ヶ庄
We have introduced entropy balance equation in an open system with external source, sink and collisions,and investigated the entropy dynamics with global profile relaxation by performing gyrokinetic full-fVlasov simulation. It is found that the entropy decrease due to eternal heat input/output mainly balanceswith the microscopic entropy decrease, which corresponds to the turbulent generation. After the formationof stiff temperature profile, we find that the main part of microscopic entropy decrease can be generated in
the center region, and the entropy only propagates in the source-sink free region. This may show that thestiff temperature profile is kept by the turbulent generation in the center region.

1. Background and Motivations
Turbulent transport in magnetically confined
fusion plasmas exhibits various prominent featurescharacterized by different temporal and spatial
scales. Zonal modes, such as zonal flow andpressure, which are poloidally and toroidally
symmetric macro-scale structures nonlinearly
generated from micro-scale turbulence, play animportant role in regulating turbulent structure andthen transport [1]. These zonal modes are
considered to be tightly linked to non-local
characteristics of turbulent transport that are notexplained by the Gaussian statistics, such asintermittent transport, turbulent spreading [2], andavalanche dynamics, etc. Self-organized critical(SOC) transport [3] is also an example in which theturbulence provides a strong constraint on
relaxation, leading to a self-organized stiff
temperature profile.
To characterize such transport dynamics, phasespace entropy, which connects micro-scale
turbulent structure to macro-scale thermodynamic
quantities, is introduced [4]. However, the entropy
has been generally treated as a global quantity thatis integrated over phase space, so that the effects ofzonal flow and turbulent spreading do not appear
explicitly in the entropy balance equation. As theresult, the non-local nature of transport is hardlydiscussed.
Recently, we extended the entropy balance
equation keeping the dynamics in radial direction sothat it can describe the spatio-temporal dynamics
not only of the entropy production related to zonal
flow and local heat flux, but also of perturbed
entropy convection [5].
In this work, we have developed a global

gyrokinetic Vlasov code, which is based on a full-f
approach and includes source, sink and collisions.
Source and sink models are necessary to simulate
open system, whereas collisions work as a physicaldissipation mechanism of fine scale velocity spacestructures, which are produced by mixing due to theballistic mode. From the viewpoint of the entropy
balance relation [6], such a dissipation mechanism
is essential for reaching statistically steady states inlong time gyrokinetic turbulence simulations. We
investigate entropy evolution in such an open
system and study which components balance withthe entropy decrease due to external heat
input/output.
2. Entropy Balance Equation in Open System
We here employ a gyrokinetic model forelectrostatic ITG/ETG turbulence in slab geometry.
The entropy balance equation is derived bymultiplying the gyrokinetic Vlasov equation by− (1+ log f ) and integrating it over whole phasespace, we obtain the entropy balance equation in an
open system as
dds (1)dV + ∫ s (2 )dV = ssrc + ssnk + scoll , (1)
∫dtdt
(1)
and
are defined aswheress ( 2)(1)s ≡ −δf (1 + log f 0 ) and s ( 2 ) ≡ −δf 2 / 2 f 0 , whichcorresponds to the first and second order quantities
for Taylor series of s ≡ − f log f with respect toδf / f 0 . In the absence of external source and sink,
macroscopic entropy s (1) increases due to profile
relaxation, which is related to the principle of themaximum entropy. On the other hand, themicroscopic entropys ( 2 ) decreases, whichcorresponds to the self-organization of themicro-scale turbulence. ssrc , ssnk and scollcorrespond to the entropy production due to source,
sink and collision.
3. Numerical Results and Discussions
We investigate the entropy dynamics with global
profile relaxation by performing gyrokinetic full-f
Vlasov simulation. We here employ a shear-lessslab geometry for simplicity. We use the models fora heat source S src at center and a heat sink S snk inboundary region:
S src (x, v|| ) = Asrc ( x ){f M (v|| ,2T0 ) − f M (v|| , T0 )}, (2)S snk (x, v|| ) = − Asnk ( x ){f M (v|| ,2T0 ) − f M (v|| , T0 )}. (3)Here Asrc and Asnk are deposition profiles, whichsatisfy the conservation of particle, momentum andenergy, but decreases entropy in whole phase space
due to the difference of temperature between centerand edge. Fig. 1 shows the schematic picture ofentropy dynamics in an open system.

Fig. 1: Schematic picture of entropy dynamics in opensystem. Heat input at the core increases entropy whereas
the output at the edge decreases entropy.

Fig. 2 shows the time evolution of entropy
production related to macroscopic, microscopic
quantities and external heat input/output. Note thatthe sum of them is well conserved, following Eq.
(1). It is also found that the entropy decrease due toeternal heat input/output mainly balances with themicroscopic entropy decrease which corresponds tothe turbulent generation.
Fig. 3 shows the spatial profiles of these entropy
productions after the formation of stiff temperature
profile. We can see that the main part of
microscopic entropy decrease can be generated in
the center region, and the entropy only propagatesin the source-sink region. This may show that thestiff temperature profile is kept by the turbulentgeneration in the center region.
We will discuss their details and the contribution
from zonal flows and collisions in this presentation.

Fig. 2: Time evolution of entropy production related tomacroscopic (red), microscopic (green) quantities andexternal heat input/output (blue). Black line shows thesum of them as a reference.

Fig. 3: Spatial profiles of entropy production. Each colorline corresponds to same one in Fig. 2.

Acknowledgments
This work is supported by Grant-in-Aid for
Research Activity Start-up (Grant No. 23840025)
from JSPS. This is also supported partly byCollaborative Research Program for Efficient
Parallel
Programming
andLarge-Scale
Computation of ACCMS and IIMC, Kyoto
University.
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