Plasma and Fusion Research: Regular Articles

Volume 19, 1403019 (2024)

Implication of Parallel Velocity Gradient-Driven Instability with
Hydrodynamic Electrons to SOL Width
Itsuki OYAMA and Yusuke KOSUGA1)Interdisciplinary Graduate School of Engineering Science, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
1)
Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka 816-8580, Japan(Received 29 September 2023 / Accepted 29 February 2024)
Herein, a new aspect of the parallel velocity gradient (PVG)- driven instability is explored. We present its
linear stability analysis and investigate the transport properties of the instability, focusing on a specific electron
motion called hydrodynamic. In the realm of hydrodynamic electrons, electron motions across the magnetic field
are much faster than those along the magnetic field. This electron motion plays an important role in fluctuationtransport. This analysis reveals that the PVG convective cell is newly excited, and its feature of particle transport
is favorable, since the particle pinch by PVG with adiabatic electrons disappears.
c 2024 The Japan Society of Plasma Science and Nuclear Fusion Research⃝

Keywords: PVG instability, transport, hydrodynamic electron, scrape-oﬀ layer
DOI: 10.1585/pfr.19.1403019

1. Introduction
Today, heat load toward divertors is one of the most
crucial problems in fusion sciences. This heat load manifests in two types of phenomena. One is an abrupt heatload, well known as edge localized modes (ELMs) [1–3],
and the other is the steady heat load. As the studies
on ELM suppression progress [4], steady heat loads havegained attention these days in the context of the protection
for the plasma-facing components via SOL broadening.
Recently, the Eich scaling law satisfactorily described the
SOL heat load width [5, 6], and Goldston [7] propoundeda heuristic drift (HD) model for it, which perfectly fits thedata of H-mode discharges. As is well known, this model
is solely based on the drift motion by magnetic fields, thegrad B and curv B drifts [7]. In this scaling, the power decay length is poorly estimated as small. For example, thelength for ITER is approximately 1mm, which prompted us
to consider its handling. As previously mentioned, turbulence is not considered in the HD model. Therefore, turbulence should be a key SOL width handling element. Fluctuations are observed in SOL [8], which can drive fluxes
from experimental perspective [9]. In this paper, we will
demonstrate the behaviour of one of such fluctuations for
SOL broadening.
Considering local fluctuations is important because
they can play an important role in advanced operations,such as internal transport barriers with low confinementmode edge [10]. The contribution of turbulence to the
SOL width is estimated using simple heuristic considerations [11]. Additionally, theoretical considerations are necessary to further explain its physical mechanism [12].
Parallel velocity along magnetic field lines in SOL canauthor’s e-mail: oyama.itsuki.582@s.kyushu-u.ac.jp

become subsonic in several devices [13, 14]. These flows
lead to a radial gradient, which can drive instability, hereafter called as parallel velocity gradient (PVG) instability[15]. Lately, PVG has been studied theoretically [16–19],
experimentally [20, 21] and using simulations [22, 23]. In[18,21], it is reported that PVG contributes to inward particle pinch, which can be disadvantageous for SOL broadening. However, PVG may change its transport feature in the
future devices, such as in ITER. This is because ITER has
longer major and minor radii, which aﬀect the time scalesof electron motions. Considering resistive drift wave as an
example, excited fluctuations can be changed using electron motions based on the size of devices [24]. Particle
transport changes its intensity, and especially, heat transport can change its direction. This alteration is governed
by electron motions, which are electron drift frequency
and electron parallel diﬀusion. Hence, we focused on PVG
with altered electron motion.
We aim to reveal the altered transport features of PVG
using specific electron motion based on the size of the devices. These features are described using a quasi-linear
framework and are discussed in Sec. 2. The discussion andsummary are presented in Sec. 3.

2. Model and Linear Analysis
With the aim to describe transport phenomena in
the SOL region in fusion plasmas, we use the extended
Hasegawa-Wakatani equation with parallel flow [17–19,
25]. As previously proposed, to demonstrate the fluctuation that we focus on, this model is represented as a simplified fluid description.
()
d 2 2 eϕne2 eϕρ∇= −D∥ ∇∥−,
dt s ⊥ T e
T e n01403019-1

c 2024 The Japan Society of Plasma⃝
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles
(
)
d ne
eϕ ne
+ ∇∥ v∥ = −D∥ ∇2∥
−
,
dt n0
T e n0dnev∥ = −c2s ∇∥ ,dt
n0Volume 19, 1403019 (2024)

where d/dt = ∂t + (c/B)ẑ × ∇ϕ · ∇ is the total derivative including the nonlinear E × B convection, c s is the ion soundvelocity, ρ s is the ion sound larmor radius, D∥ = v2the /νei isthe parallel electron diﬀusion, ϕ is the electrostatic potential, n is density, and v∥ is the parallel flow. The system isdefined as a cylindrical system. The correspondence ∥↔ zand ⊥↔ (r, θ) ↔ (x, y) in the cylindrical coordinate system
is understood hereafter.
To elucidate the instability caused by parallel flow
shear, we obtain a set of equations by linearization andFourier analysis, as shown below:iωρ2s k⊥2 ϕ̂ k = −D∥ k∥2 (n̂ k − ϕ̂ k ),(1)

− iωn̂ k + iω∗e ϕ̂ k + ik∥ ṽi,k = −D∥ k∥2 (n̂ k − ϕ̂ k ),

(2)

c
e
− iωṽi,k − iky ϕ̃ k ⟨vz ⟩′ = − ikz ϕ̃ k .
B
mi

(3)

Here, it is useful to denote normalized quantities byñe /n0 → n̂ and eϕ̃/T e → ϕ̂. ⟨. . .⟩ denotes the averaged
quantities over θ and ∥ direction. ω∗e = ky ρ s (c s /Ln ) is theelectron drift frequency and Ln−1 = −⟨n⟩′ /n0 is the scale
length of the density. Here . . .′ represents the radial differential. We assume that the wavelength of the fluctuation is shorter than the typical scale length of equilibrium
quantities. Subsequently, the density and parallel velocity
responses are obtained as follows:
)
(ω
k∥
∗e
− ρ2s k⊥2 ϕ̂ k + ṽ∥,k ,(4)n̂ k =
ω(ω)′
c s k∥ ky ρ s ⟨vz ⟩ϕ̂ k .
(5)ṽi,k = c s−ωω
Substituting Eq.(4) and Eq.(5) into Eq.(1), we obtainthe following general dispersion relation:

Fig. 1 Schematic graph : region corresponding to each fluctuation. Vertical axis corresponds to the ratio of ⟨vz ⟩′ toc s /Ln and parallel axis corresponds to the ratio of ω∗e tok∥2 D∥ .

k∥2 D∥

p = −i 2 2 ω∗e + 4 4 (1 + ρ2s k⊥2 )2 ,
ρ s k⊥
3ρ s k⊥k∥2 D∥
A = i 2 2 (1 + ρ2s k⊥2 ),ρ s k⊥
andk∥4 D2∥ ω∗e
2 2 3(1 + ρ2s k⊥2 )
(1
+
ρ
k
)
−
s ⊥
3ρ4s k⊥4
27ρ6s k⊥6()
k∥2 D∥ k∥2 c2s
ky ρ s−i 2 2
1−⟨vz ⟩′ .
k∥ c sρ s k⊥

q = − 2i)2 2 (
ρ s ky ⟨vz ⟩′ω∗e k∥ c s+ 2 1−.
ωk∥ c s
ω

A
p
−
3u 3
2
i1/3 k∥ D∥≈(1 + ρ2s k⊥2 )3 ρ2s k⊥2


√


± 3 − 3i ρ2s k⊥2ω∗e× 1 +
22 222k∥ D∥ (1 + ρ s k⊥ )

ωDW = u −(6)

This dispersion relation Eq.(6) has a general form, which
includes several fluctuations, drift wave (lower left), drift
convective cell (lower right), and PVG (upper left), as
shown in Fig. 1. Because Eq.(6) is a cubic equation for
ω, we can use a well-known algebraic method to solve thiscubic equation, which is called the Cardano’s fomula [26].
Solving Eq.(6) using the Cardano’s formula, we will systematically obtain the following:
ω=u−
where

p
A
− ,
3u 3


1/3
√



4 3
 q 1

2
− ±
q +
,
u=
p

 2 2

27 

k∥6 D3∥Equation (7) obviously contains other fluctuations shownin Fig. 1. Taking drift wave as an example, we can derive the dispersion relation of drift wave from Eq.(7) usingω∗e /D∥ k∥2 ≪ 1 as the expansion parameter and neglecting
the flow shear. Then, we obtain:ρ2 k2−i s ⊥2 ω = − (1 + ρ2s k⊥2 )D∥ k ∥+k∥4 D2∥2
1 k∥ D∥(1 + ρ2s k⊥2 )3i1/3 ρ2s k⊥2


√


± 3 + 3i ρ2s k⊥2ω∗e× 1 −
2k∥2 D∥ (1 + ρ2s k⊥2 )2 −

2i k∥ D∥(1 + ρ2s k⊥2 )3 ρ2s k⊥2ω∗e=,
1 + ρ2s k⊥2−(7)where O(ω∗e /D∥ k∥2 ) ≪ 1 is assumed. With flow shear,
the dispersion relation of PVG (upper left in Fig. 1) can
be clearly obtained from Eq.(7).
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Plasma and Fusion Research: Regular Articles

Volume 19, 1403019 (2024)

2.1 PVG convective cellIn this section, we discuss the fluctuation with the hydrodynamic electron response, i.e., ω∗e /D∥ k∥2 ≫ 1. In thiscase, the electron motion resembles that of a quasi-twodimensional vortex perpendicular to the toroidal plane.
The expansion parameter in this case is D∥ k∥2 /ω∗e ≪ 1.
When this expansion parameter becomes valid, this limitation corresponds to a small k∥ . The assumption regarding
the wave number parallel to the magnetic field line is regarded as k∥ = 1/qR. qR is the product of the safety factor
q and major radius R. Taking the safety factor as q95 ∼ 3,with some simplifications, the electron response shall be
proportional to the size of devices. In [24], this hydrodynamic electron regime can be easily found in large devices,e.g., ITER. Accordingly, these limitations relevant to theparallel flow shear and electron motion laid the foundation
for the desired discussion.
Further, we show the dispersion relation for the PVG
convective cell. The PVG convective cell is the new regimeof this system. In addition, we demonstrate modes in
the PVG convective cell. Using the hydrodynamic electron regime k∥2 D∥ /ω∗e ≪ 1 as the expansion parameter inEq.(7), we obtain:
1/3
 2
 k∥ D∥ k∥ c s ky ρ s ⟨vz ⟩′  .
ω ≈ −iρ2s k⊥2(8)

Furthermore, we can directly derive Eq.(8) from
Eq.(6) using the same assumption as above. In a directcomparison between the first term in the LHS and last termin the RHS of Eq.(6), we can obtain the same dispersion
relation as Eq.(8).
k∥ c s ρ s ky ⟨vz ⟩′ρ2 k2−i s ⊥2 ω ∼ −.
ω2D∥ k ∥
Moreover, we can derive this dispersion relation from
Eq.(1), Eq.(2) and Eq.(3). Assuming ⟨vz ⟩′ is suﬃciently
large, the velocity fluctuation is activated only by the parallel flow shear in Eq.(5).
ṽi,k ∼ −c s ky ρ s ⟨vz ⟩′
ϕˆk .
ω

(9)

Substituting this into Eq.(2), the density fluctuation isfound to be strongly activated by the parallel flow shear.
n̂ k ∼ −k∥ c s ky ρ s⟨vz ⟩′ ϕ̂ k .
ω2contraction of the trigonometric function.
 √


3 i ω = ±+  σ,
2
2

(11)

and
ω = iσ,

(12)

where
σ=

k∥2 D∥ k∥ c s ky ρ s ⟨vz ⟩′ρ2s k⊥2

1/3

.

(13)

Briefly, we refer to Eq.(11) as the oscillatory mode andEq.(12) as the pure growth mode. These are named after
the physical pictures of these waves in a real space. In theoscillatory mode, the amplitudes of waves increase withoscillation in the real space and propagate as time passes;
this represents the travailing mode. Furthermore, the direction of vz and its gradient have a certain meaning for its
propagation direction. This is because the direction of ⟨vz ⟩′imposes constraints over k∥ and ky . For example, when thesign of ⟨vz ⟩′ is positive, the sign of the product k∥ ky must be
negative to meet the condition to be excited. Considering
the phase velocities ω/k∥ and ω/ky , it can be easily seenthat the sign of ⟨vz ⟩′ determines the propagation directionof this traveling mode. In the pure growth mode, the amplitudes of waves increase exponentially without propagatingin the real space.
The excitation of these modes depends on the sign ofσ in Eq.(13), and the component k∥ ky ⟨vz ⟩′ is responsiblefor its sign. Considering the imaginary parts of Eq.(11)and Eq.(12), for exciting the oscillatory mode, the signof k∥ ky ⟨vz ⟩′ must be negative. Meanwhile, when the signof k∥ ky ⟨vz ⟩′ is positive, the pure growth mode can be excited. In summary, if k∥ ky ⟨vz ⟩′ , 0, either the oscillatory or
pure growth mode can be chosen. Compared with otherregimes, especially PVG [19], we can evaluate this as abroader condition. This is because a PVG with adiabatic
electrons demands k∥ ky ⟨vz ⟩′ > 0.
In addition, k∥ ky ⟨vz ⟩′ is the component that determines
the time evolution of vorticity. Especially, k∥ ky ⟨vz ⟩′ is excited from the compression term in Eq.(2). The excitation of these distinctive modes is regulated by the compressibility, which has also been discussed previously, e.g.,in [17, 27].

(10)

2.2 Fluctuation

Further, we determine the behavior of the PVG convective
cells from the vorticity equation, Eq.(1). Its expression corresponds to Eq.(8). Thus, vorticity is strongly aﬀected by
the parallel flow shear.
We now focus on the value of (±i)1/3 . Here, the sign
is determined by the sign of k∥ ky ⟨vz ⟩′ . We take positive
sign for k∥ ky ⟨vz ⟩′ < 0 and vice versa. Note that the Euler’s formula, Eq.(8), takes the following forms due to theIn each mode, the density and momentum fluctuations
are derived from Eq.(9) and Eq.(10), respectively. In theoscillatory mode, we obtain density fluctuation as:ñosc,k = −√k∥ c s ky ρ s ⟨vz ⟩′3)ϕ̃ k .
(1
∓
i
2σ2

(14)

In the above equation, phase diﬀerences exist between thedensity and electrostatic potential fluctuations. Thus, we

1403019-3

Plasma and Fusion Research: Regular Articles

Volume 19, 1403019 (2024)⟩⟨√
c s ky ρ s · ky(−1 ∓ i 3)|ϕ̃ k |2 ,=2σ · B0∑ c2s ky2 ρ2sRe[Πrz,osc ] = −|ϕ̂ k |2 ⟨vz ⟩′2σkcan expect finite particle transport in this mode. Furthermore, we can obtain the momentum fluctuation in the oscillatory mode as:ṽ∥,osc,k = −√ky c s ρ s ⟨vz ⟩′(i ∓ 3)ϕ̃ k .
2σ= −Dv,osc ⟨vz ⟩′ .

Additionally, finite phase diﬀerences exist between the velocity and electrostatic potential fluctuations. Therefore,
we can expect finite momentum transport in this mode.
In the pure growth mode, we obtain the density fluctuation as:ñ pg,k =k∥ c s ky ρ s ⟨vz ⟩′ϕ̃ k .
σ2(15)

Considering this, we cannot expect net particle transport inthis mode because no phase diﬀerence exists between thedensity and electrostatic potential fluctuations. In addition,
we can obtain momentum fluctuation in the pure growthmode asṽ∥,pg,k = −i

(16)∑Here Dv,osc = k (c2s ky2 ρ2s /2σ)|ϕ̂ k |2 . Because the real partsof Πrz,osc can, likewise as to particle flux, contribute to
momentum transport, there is finite momentum transport.
From Eq.(16), this momentum transport can be formed as
the eddy viscosity on flow. This eddy viscosity characterizes momentum flux. When exciting the oscillatory mode,
its momentum flux is down the gradient.
In the pure growth mode, we can also calculate each
flux as follows. Beginning from the density flux in the pure
growth mode, we obtainΓ pg = ⟨ṼE×B ñ pg ⟩⟨⟩k∥ c s ky ρ s ⟨vz ⟩′iky= − ϕ̃ k ·ϕ̃ k ,B0
σ2

c s ky ρ s ⟨vz ⟩′ϕ̃ k .
σRe[Γ pg ] = 0.

Hence, we can, likewise as to the oscillatory mode, expect
finite momentum transport in the pure growth mode.

2.3 Transport
In this section, we focus on the transport featuresof PVG convective cells using quasi-linear analysis [28].
We will perform each calculation for the oscillatory andpure growth modes. In the quasi-linear analysis, the particle flux caused by fluctuating E × B drift is given byΓ = ⟨ṼE×B ñ⟩. Analogously, the momentum flux is givenby Πrz = ⟨ṼE×B ṽ∥ ⟩.
In the oscillatory mode, each flux can be calculated asfollows. Starting from the density flux in the oscillatorymode, we can getΓosc = ⟨ṼE×B ñosc ⟩
⟩
⟨
√
iky
k∥ c s ky ρ s ⟨vz ⟩′
(1
∓
i
3)
ϕ̃
= − ϕ̃ k · −
k ,
B0
2σ2
⟨
⟩
√ k∥ c s ky ρ s · ky
2′Re[Γosc ] = ± 3
|
ϕ̃
|
⟨v
⟩
k
z
2σ2 · B0

As expected from Eq.(15), no finite particle transport isobserved in this mode. This is obvious because the density fluctuations are in phase with the electrostatic potential
fluctuations in Eq.(15). The momentum flux in this modeis calculated as follows:Πrz,pg = ⟨ṼE×B ṽ∥,pg ⟩⟩⟨iky
c s ky ρ s ⟨vz ⟩′ϕ̃ k ,= − ϕ̃ k · −iB0σ
∑ c2s ρ2s ky2
Re[Πrz,pg ] = −|ϕ̂ k |2 ⟨vz ⟩′σk= −Dv,pg ⟨vz ⟩′ .

(17)∑
Here Dv,pg = k (c2s ρ2s ky2 /σ)|ϕ̂k |2 . This momentum flux hasa real part, which contributes to momentum transport. As
with the case of the pure growth mode, this flux can beformed as the eddy viscosity in Eq.(17). When the pure
growth mode is excited, its momentum flux is down thegradient.

= Γ+osc + Γ−osc

3. Summary and Discussion≈ 0.

Herein, the linear stability analysis and transport features of the PVG convective cell are explored theoretically. The PVG convective cell can be excited when thetime scale of the electron motion across the magnetic field
becomes shorter than that of electron diﬀusion along themagnetic field, in addition to a strong parallel flow shear.
This electron response is characterized by the so-called hydrodynamic response, while it gets an adiabatic response
in an opposite relation for these time scales. Comparing
PVG with adiabatic electrons, the PVG convective cell has
a broad condition for ky k∥ ⟨vz ⟩′ for instability. A PVG convective cell can be destabilized with ky k∥ ⟨vz ⟩′ , 0, while

Because the real parts of Γosc can only contribute to particle
transport, there are equal volumes of transport in the opposite direction, as shown above. Contrary to our expectationfrom Eq.(14), there was no particle transport here because
they cancel each other. Momentum flux in this mode iscalculated as follows:Πrz,osc = ⟨ṼE×B ṽ∥,osc ⟩
⟨
⟩
√
iky
c s ky ρ s ⟨vz ⟩′= − ϕ̃ k · −(i ∓ 3)ϕ̃ k
B0
2σ1403019-4

Plasma and Fusion Research: Regular Articles

Volume 19, 1403019 (2024)
magnitude between v∗e ∂y ϕ̂ and ∇∥ ṽi is obtained ask∥ c s ky ρ s ⟨vz ⟩′cs≥ ky ρ s ,
ω
Ln
√
cs
c s mi /me⟨vz ⟩′ ≥.
Ln Ln νe,iFig. 2 This is the plot of Eq.(18). Parallel axis corresponds
to the electron temperature [eV] and vertical axis corresponds to the parallel flow shear [Hz].

(18)

Here we assume ρ s k⊥ ∼ 1 and ky /k⊥ ∼ 1. Ln is taken
as minor radii, thus Ln is 2[m] in ITER. Taking mi /meas the usual physical constant, it is approximately 1.8 ×
103 . νe,i is the electron collision frequency. Considering
the case of a large device like ITER, we can assume c s /Ln
as approximately equal to 100 [kHz]. Plotting Eq.(18) inthe range of 0 to 60 [eV], the curve is proportional to T e3/2
in νei . Finally, we obtain Fig. 2. The PVG convective cell
can be excited for values above the curve in Fig. 2.

Acknowledgments
PVG with adiabatic electrons can only be destabilized withky k∥ ⟨vz ⟩′ > 0. Two distinctive modes are found when the
PVG convective cell is excited: the oscillatory and pure
growth modes. The oscillatory mode is characterized byspatial propagation, while the pure growth mode is not.
Moreover, density fluctuations exhibit phase diﬀerences in
the oscillatory mode, which is unlikely to occur in the pure
growth mode, while parallel velocity fluctuations in bothmodes demonstrate phase diﬀerences.
For transport, these two modes share their transportcharacteristics. Momentum flux is characterized by eddy
viscosity, which relaxes the parallel velocity gradient wheneither of the modes is excited. Meanwhile, particle flux
does not contribute to transport in either modes. Note thatthe mechanisms that lead to these particle transports are
diﬀerent from each other. In the oscillatory mode, because
the same amount of finite particle transport can be conveyed in the opposite direction, these fluxes cancel eachother. In contrast, in the pure growth mode, because its
density and electrostatic potential fluctuations are in phasewith each other, its particle flux is not originally generated.
Owing to these results, the PVG convective cell cannot sufficiently broaden the SOL width to protect divertor plates
from heat loads. However, compared with PVG with adiabatic electrons, a PVG convective cell can help passively
to achieve this purpose. This is because the pinch eﬀectcaused by PVG with adiabatic electrons disappears in the
PVG convective cell.
In the discussion, we mention the excitation of PVG
convective cells. As previously mentioned in Sec. 2.1, thetime scale for parallel velocity (∇∥ vi ) is dominant over thetime scale for drift motion (ω∗e ) in the PVG convective
cell. Additionally, we can identify the region where the
PVG convective cell is likely to be excited using v∗e ∂y ϕ̂ ≤∇∥ ṽi . This leads to the comparison between the second andthird terms on the LHS of (2). Concerning ∇∥ ṽi , it is obvious to obtain ∇∥ ṽi from Eq.(9). Then the comparison of the

We are grateful to the useful discussions with participants of APTWG 2023. This work was partly supported by
the Grants-in-Aid for Scientific Research of JSPS of Japan(JP21H01066), a joint research project at RIAM, Kyushu
University.
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