S7-4
Theoretical Prediction of the Isotopic Effect of Turbulent Transport
T. S. Hahm
Dept. of Nuclear Engineering, Seoul National University, Seoul, Korea
1 Gwanak-ro, Gwanak-gu, Seoul 151-744, Korea
We briefly review a few theoretical attempts to explain isotopic dependence of tokamak anomalous transport
in the past. Then, we introduce more recent effort in the context of ExB zonal flows driven by turbulence.

1. Introduction
Isotopic dependence is one of key unsolved
issues in magnetic confinement research.
Deuterium (D) plasmas have been observed to bebetter confined than hydrogen (H) plasmas overmany years of tokamak research as reviewed in[1,2]. Addition of Tritium has made confinement
even better in TFTR[3] and in JET[4].
Furthermore, success of ITER relies on expectedenhancement of H-mode access and confinement
as experiments proceed with working gas in theorder of H, He, DD, and DT.
In this manuscript, we discuss possible
nonlinear theoretical mechanisms behind isotopic
dependence of tokamak confinement.
2. Partial Survey of Theories on Isotopic
Dependence
There have been many theoretical attempts toexplain the isotopic dependence of confinement in
the past. It is not possible to give a thoroughreview here, and the author can only cover a few
of them. Some proposed mechanisms are based onassumptions which can only be satisfied at theplasma edge. For dissipative drift wave turbulence,
the isotopic dependence may exist through nonadiabatic electron dynamics which depend on 𝑘𝑘∥
and magnetic shear[5]. This needs to be further
confirmed with global simulations with results
which are independent of simulation box size andshape. There has also been work on the isotopic
dependence of the energy content of the H-mode
pedestal region[6]. More recent work seems toreport, however, the absence of this trend[7]. For
the current-driven ballooning mode(CDBM)[8],
the isotopic dependence comes from the Alfvén
speed. That model uses electron dynamics which
are hydrodynamic, rather than near-adiabatic (orBoltzmann response).
Gyro-Bohm scaling has an isotopic dependence
which is opposite to experimental observations.
From local calculations of ITG instabilities
without an explicit consideration of nonlinearly
self-generated zonal flows, one usually obtains a

Gyro-Bohm scaling, for instance[9]. Furthermore,quasi-local nonlinear gyrokinetic simulations oftokamak core turbulence for parameters from
actual experiments have reported trends close to
Gyro-Bohm scaling[10].
3. Isotopic Dependence of Zonal Flows
Almost all previous theoretical attempts basedon models which are free from other significant
inconsistencies with experimental results have notsucceeded in obtaining the isotopic dependence
observed in experiments. One commonality of
those attempts is that ExB zonal flows driven by
turbulence has not been explicitly included.
There’s a theoretical work related to zonal flows
in stellarator geometry[11]. The effect consideredin that work disappears in tokamak geometry
which is axisymmetric. Isotopic dependence ofstellarator plasma confinement was observed to beweaker than that of tokamak plasmas.
While the well-known Rosenbluth-Hinton
residual zonal flows with radial scale greater thanthe magnetically trapped ion radial width 𝜌𝜌𝑏𝑏𝑏𝑏
have no isotopic dependence [12], a recent
study[13] finds that shorter radial scale (shorterthan 𝜌𝜌𝑏𝑏𝑏𝑏 ) but larger than the magnetically trappedelectron radial width 𝜌𝜌𝑏𝑏𝑏𝑏 residual zonal flows
exhibit isotopic dependence. These finer scale
zonal flows in deuterium (D) plasmas can bestronger than those of hydrogen (H) plasmas, andpossibly lead to lower turbulence and transport andbetter confinement in qualitative agreement withexperimental results.
Based on analytic calculations utilizing
neoclassical and classical polarization densities
which appear in modern bounce-kinetic[14] andmodern gyrokinetic theories[15,16], a usefulconnection formula covering a wide range of radial
scales of ZFs from 𝜌𝜌𝑏𝑏𝑏𝑏 down to 𝜌𝜌𝑒𝑒 scale has beenobtained in [17]. The residual zonal flow level
𝑅𝑅𝑍𝑍F from a calculation including both ion andelectron dynamics can be written as,𝜏𝜏𝜒𝜒 +𝜒𝜒
𝑅𝑅𝑍𝑍𝑍𝑍 = 𝜏𝜏𝜒𝜒 +𝜒𝜒 𝑐𝑐𝑐𝑐,𝑖𝑖+𝜏𝜏𝜒𝜒𝑐𝑐𝑐𝑐,𝑒𝑒 +𝜒𝜒(1)
𝑐𝑐𝑐𝑐,𝑖𝑖

𝑐𝑐𝑐𝑐,𝑒𝑒𝑛𝑛𝑛𝑛,𝑖𝑖

𝑛𝑛𝑛𝑛,𝑒𝑒2where 𝜒𝜒𝑐𝑐𝑐𝑐,𝑖𝑖(𝑒𝑒) = 1 − Γ0 �𝑞𝑞𝑟𝑟2 𝜌𝜌𝑖𝑖(𝑒𝑒)� . The generalizedconnection formula for neoclassical shielding can bewritten as1
𝜒𝜒𝑛𝑛𝑛𝑛,𝑖𝑖(𝑒𝑒) = �2
⁄321.63𝜖𝜖 𝑞𝑞𝑟𝑟2 𝜌𝜌𝜃𝜃𝜃𝜃(𝑒𝑒)1√8𝜖𝜖 ′√8𝜖𝜖 ′+ �1 +Γ𝑡𝑡𝑡𝑡 + �1 −
� Γ𝑝𝑝 �2𝜋𝜋𝜋𝜋
1 + 𝑞𝑞𝑟𝑟2 𝜌𝜌𝑖𝑖(𝑒𝑒)

𝜋𝜋 3
√8𝜖𝜖
Γ
+� 𝑞𝑞𝑟𝑟 𝜌𝜌𝑖𝑖 �1 +2𝜋𝜋 𝑡𝑡𝑡𝑡
√8𝜖𝜖2𝑞𝑞𝑟𝑟2 𝜌𝜌𝑖𝑖(𝑒𝑒)+ �1 − 𝜋𝜋 � Γ𝑝𝑝 � 1+𝑞𝑞2 𝜌𝜌2 �𝑟𝑟 𝑖𝑖(𝑒𝑒)−1(2)with 𝜌𝜌𝜃𝜃𝜃𝜃(𝑒𝑒) = 𝜌𝜌𝑖𝑖(𝑒𝑒) 𝐵𝐵⁄𝐵𝐵𝜃𝜃 ,and other notations
defined in [17].
In [17], the coefficient 1.63 was 1.83. This has
been recently revised using the same bounce-kinetic
approach[18], but without using the usual deeply
trapped or strongly circulating approximations
employed in [17]. This formula exhibits a desired
isotopic dependence as explained in [13].
Since the zonal flows reduce turbulence andtransport, one can characterize this effect by a zonalflow-induced
turbulence
reduction
factor
𝑅𝑅𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 [19,20] which appears as
χ = 𝑅𝑅𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝜒𝜒0(3)where 𝜒𝜒0 is a hypothetical value of the thermaldiffusivity χ in the absence of zonal flows. While
there's no first-principle-based analytic formula for
𝑅𝑅𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 with a wide validity regime, a useful theorybased fitting formula from a series of numericalsimulations has appeared from a recent work[21].
4. Conclusion
There has been a recent experimental work[22]which shows a decrease in fluctuation structures atthe edge which are long-range correlated in toroidal
direction as working gas is changed from D to H.(werecall that zonal flows are toroidally symmetric).
Accompanying increase in transport from D to H
plasmas was higher for TEXTOR tokamak, than TJⅡ stellarator. Extending this kind of research tocore fluctuation measurements which include theHeavy Ion Beam Probe measurements of zonal
flows[23] could be highly rewarding. On theoretical
front, dedicated nonlinear gyrokinetic simulations
with a high resolution in velocity space (to properlytreat the residual zonal flows) which include shorter
wavelength fluctuations down to the electron banana
width scales can be illuminating.
Acknowledgments
The author acknowledges fruitful interaction overmany years regarding this subject with P. H.
Diamond, F. Wagner and F. W. Perkins(deceased).

This work was supported by the World ClassInstitute(WCI) Program of the National Research
Foundation of Korea(NRF) funded by the Ministryof Science, ICT & Future Planning(MSIP)(No.
WCI-2009-0001) and by National R&D Program
through the National Research Foundation ofKorea(NRF) funded by the Ministry of Science, ICT& Future Planning(No. 2012M1A7A1A02034)
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