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J. Plasma Phys. (2020), vol.86, 175860402 © The Author(s), 2020.
Published by Cambridge University Press
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doi:10.1017/S0022377820000872
LETTER
Enstrophy non-conservation and the forward
cascade of energy in two-dimensional
electrostatic magnetized plasma turbulence
G. G. Plunk 1,†
1Max Planck Institute for Plasma Physics, Greifswald 17491, Germany
(Received 2 March 2020; revised 15 July 2020; accepted 16 July 2020)
A fluid system is derived to describe electrostatic magnetized plasma turbulence at
scales somewhat larger than the Larmor radius of a given species. It is related to the
Hasegawa–Mima equation, but does not conserve enstrophy, and, as a result, exhibits
a forward cascade of energy, to small scales. The inertial-range energy spectrum is
argued to be shallower than a −11/3power law, as compared to the −5law of the
Hasegawa–Mima enstrophy cascade. This property, confirmed here by direct numerical
simulations of the fluid system, may help explain the fluctuation spectrum observed
in gyrokinetic simulations of streamer-dominated electron-temperature-gradient driven
turbulence (Plunk et al.,Phys. Rev. Lett., vol. 122, 2019, 035002), and also possibly some
cases of ion-temperature-gradient driven turbulence where zonal flows are suppressed
(Plunk et al.,Phys.Rev.Lett., vol. 118, 2017, 105002).
Key words: fusion plasma, plasma dynamics, plasma nonlinear phenomena
1. Introduction
The turbulent cascade, a mechanism for the nonlinear transfer of energy across scales,
is a key idea for understanding kinetic magnetized plasma turbulence. By considering
simplified models, in a uniform magnetic geometry, one can obtain a theoretical prediction
for the spectrum of fluctuations, valid across an ‘inertial range’ of scales, free from
energy sources and sinks. Although such a theory is not able to fully describe the
behaviour of realistic turbulence, which hosts instabilities, damped modes, complicated
magnetic geometries, etc., it nevertheless constitutes a quantitative prediction of nonlinear
behaviour of the underlying gyrokinetic equation, an equation which generally governs
actual systems of practical interest. The existence of such theoretical test cases is valuable
for validating the solution methods employed by gyrokinetic codes, and as a foundation
for physical interpretation of the volumes of data they produce.
† Email address for correspondence: gplunk@ipp.mpg.de
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2G. G. Plunk
Here, a novel quasi-two-dimensional fluid system is derived from the electrostatic
gyrokinetic system, to describe fluctuations that predominantly vary in the directions
perpendicular to the mean magnetic field, i.e. in the ‘drift plane’, at scales
larger than the Larmor radius ρ, corresponding to a species of interest. The notion
that quasi-two-dimensional behaviour might underly magnetized plasma turbulence is
intuitively justified by the fact that the magnetic guide field renders the dynamics
inherently anisotropic. Furthermore, instabilities that drive electrostatic turbulence in
fusion plasmas, e.g. the ion-temperature-gradient (ITG) and electron-temperature-gradient
(ETG) modes, exhibit a kind of localization along the field line, accompanied by the
domination of perpendicular dynamics over parallel dynamics. The fluid limit ρis
of particular importance, because the energy of the fluctuations is predominantly found at
such scales – these are the scales of importance, most directly affecting the performance
of fusion devices. Furthermore, the reduction of complexity afforded by fluid limits can
reveal important features of the dynamics, that do not manifest in the analysis of the
general gyrokinetic equations.
Although similar systems as the one presented here have been proposed and studied
in the past, most notably the Hasegawa–Mima (HM) equation (Hasegawa & Mima
1978) the present derivation takes special care in considering the consequences of the
appearance of nonlinear finite-Larmor-radius (FLR) terms that appear in the dynamical
equation for the electrostatic potential – i.e. the ‘vorticity’ equation. Such terms introduce
a closure problem in the fluid moment hierarchy, where lower moments are coupled to
ever higher ones, generally without end. This motivates the cold ion limit that underlies
the HM equation, which eliminates the inconvenient terms, but is, however, not generally
appropriate for application to fusion plasmas. In the present work, it is noted that the
presence of these terms introduces rapid dynamics, and a multiple-scale analysis is
proposed in which the fluid moment hierarchy closes at the pressure moment, without
using an ad hoc closure scheme, leading to a relatively simple system involving only two
fields.
What is immediately apparent is that the presence of the additional field (the pressure
perturbation) breaks the nonlinear conservation of enstrophy that is famously satisfied by
the HM equation, and there causes an ‘inverse cascade’ of energy to large scales. The
new system, we argue, should exhibit distinct nonlinear behaviour, including a shallower
energy spectrum when the effect of the nonlinear FLR terms is sufficiently strong. Direct
numerical simulation of the fluid model gives some confidence in these predictions. The
results of this work may help to interpret observations of turbulence in parameter regimes
where the dynamics tends towards the quasi-two-dimensional limit. We discuss possible
examples, including cases explored in previous gyrokinetic turbulence simulations in
tokamak and stellarator geometries.
2. Equations and definitions
We assume uniform magnetic geometry, where the magnetic guide field is constant and
points in the ˆz-direction. One species is assumed to be kinetic, with the other species
satisfying a simple Boltzmann response model. We begin with a non-dimensional form
of the gyrokinetic system (Plunk et al. 2010), normalized relative to the kinetic species:
v⊥/vth →v(with vth =√T/m,andTand mare the temperature and mass of the kinetic
species) is the normalized perpendicular velocity and the normalized wavenumber is
k⊥ρ→kwhere the thermal Larmor radius of the kinetic species is ρ=vth/Ωcand
Ωc=qB/m,Bis the magnitude of the magnetic field and mis the particle mass. The
two-dimensional gyrokinetic equation is written as follows in terms of the perturbed
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Enstrophy non-conservation and the forward cascade of energy 3
gyrocentre distribution function g(R,v,t),whereR=ˆxX+ˆyYis the gyrocentre position:
∂g
∂t+{ϕR,g}=C[h]R,(2.1)
where C[h]Ris the collision operator (not treated here explicitly); the Poisson
bracket is {A,B}=ˆz×∇A·∇B=∂xA∂yB−∂yA∂xBand the gyro-average is defined
A(r)R=(1/2π)2π
0dϑA(R+ρ(ϑ)), where the Larmor-radius vector is ρ(ϑ) =ˆz×
v=v⊥(ˆycos ϑ−ˆxsin ϑ) and ϑis the gyro-angle. (Note that the quantity inside of the
collision operator is h=g+ϕRF0. Note also that the spatial coordinate is Rin the
gyrokinetic equation and, formally, the spatial derivatives are to be interpreted in this
variable, but for simplicity we avoid making the distinction explicit.) We mostly ignore the
collision operator but note that some mechanism of coarse graining will be necessary to
get sensible solutions out of the equation. Quasi-neutrality yields the electrostatic potential
ϕ(r,t),wherer=ˆxx+ˆyyis the position-space coordinate
2π∞
0
vdvgr=(1+τ)ϕ −Γ0ϕ, (2.2)
where the gis implicitly assumed to be integrated over parallel velocity so that 2π∞
0vdv
completes the integration over three-dimensional velocity space. The angle average is
defined as A(R)r=(1/2π)2π
0dϑA(r−ρ(ϑ)), and the term τϕ is the adiabatic density
response, with τ=Ti/(ZTe)for the case of ion scales and τ=ZTe/Tifor the case of
electron scales. For the ion case, this Boltzmann response might be considered reasonable
if zonal flows are strongly suppressed. The operator Γ0φ=2π∞
0vdvF0(v) φRris
more naturally expressed in Fourier space, assuming a Maxwellian background F0=
exp[−v2/2]/(2π),i.e.Γ0ϕ=kexp(ik·r)ˆ
Γ0ˆϕ,with
ˆ
Γ0(k)=∞
0
vdve−v2/2J2
0(kv) =I0(k2)e−k2,(2.3)
where I0is the zeroth-order modified Bessel function.
3. Fluid limit
We expand in the limit
δ=k21,(3.1)
i.e. we assume that the scales of interest are larger than the Larmor radius of the species
of interest. For electron scales, the limit is considered subsidiary to the adiabatic ion limit,
so scales of the turbulence must remain much smaller than the ion Larmor scale, i.e.
ρe/ρik1. Note that there may also be a minimum applicable kimposed by the
dynamics parallel to the magnetic field, but treating this explicitly is outside the scope of
this work. We will only need the first two orders of the expansion in δ. The gyrokinetic
equation, henceforth omitting explicit collisional effects, is written as
∂g
∂t+®Ç1+v2
4∇2åϕ, g´≈0,(3.2)
and (2.2) becomes
τϕ −∇2ϕ≈2π∞
0
vdvÇ1+v2
4∇2åg.(3.3)
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4G. G. Plunk
We will denote v-moments of gas
Gn=2πvdvÅv
2ãn
g.(3.4)
3.1. Naive expansion
To give a taste for the issues that arise in the expansion, let us take an initial informal look
at the moments of the gyrokinetic equation. We first examine the density moment. We
include only the ostensibly dominant nonlinear terms. We stress that this equation is given
only for illustrative purposes, and is not to be taken as a basis for the later derivations of
the paper
∂
∂tÄτϕ −∇2ϕ−∇2G2ä+¶ϕ, −∇2ϕ−∇2G2©+¶G2,−∇2ϕ©=0.(3.5)
Note that the term −∂t∇2ϕ, which appears in the HM equation, should be neglected here
because it is formally smaller than ∂tϕby one power of the ordering parameter δ. Likewise,
the term −∂t∇2G2must be considered negligible if the ordering G2∼ϕand ∂tG2∼∂tϕ
holds. The situation is, however, a bit more subtle. The above equation couples to the
v2moment of g,G2and the equation for this and other such moments can be written,
neglecting higher-order FLR terms, as
∂Gn
∂t+{ϕ, Gn}=0.(3.6)
What we now notice, examining these two equations, is that the density equation
is driven by nonlinear terms that appear to be much smaller than those controlling
the higher moments of the distribution function – that is, the equations for Gnhave
dominant contributions from the ‘E×B’ nonlinearity, while the potential evolves under
the influence of terms like the ‘polarization drift’ nonlinearity, which is smaller by a factor
of δ=k2. One possible resolution of this apparent imbalance is to consider Gnitself to be
large, as for instance in the non-resonant limit of the ITG/ETG mode, i.e. Gn∼δ−1ϕ.In
this case, the polarization drift nonlinearity can be neglected, and we see the justification
for retaining the additional time derivative term above, since ∂tϕ∼∂t∇2G2. This term can
be evaluated from the Laplacian of (3.6), yielding
τ∂ϕ
∂t+∇2{ϕ, G2}+¶ϕ, −∇2G2©+¶G2,−∇2ϕ©=0.(3.7)
Equations (3.6)–(3.7) demonstrate a consistent fluid limit, but cannot describe ITG
or ETG turbulence in the resonant limit, where ϕ∼G2. This corresponds the more
physically reasonable scenario of a modest turbulence drive – i.e. not very far from
the linear critical gradient, or considering the weakly unstable, large-scale modes that
dominate the turbulence spectrum. To treat this limit properly, we must account for the
fact that ϕevolves much more slowly than Gn. Physically, it can be argued that, in a
turbulent state, (3.6) will then describe rapid mixing of Gnby E×Bvortices, so that
any initial variation along streamlines of constant ϕwill decay on a fast time scale (with
the help of some explicit dissipation), leaving Gnto be constant along those streamlines
(S.C. Cowley, private communication 2008). To account for such processes more carefully,
we abandon conventional perturbation theory in favour of the method of multiple scales
(see, i.e. Bender & Orszag 1978). We will henceforth disregard the equations presented
here, in § 3.1, and proceed to derive equations that contain only terms justified by a set of
explicitly stated ordering assumptions.
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Enstrophy non-conservation and the forward cascade of energy 5
3.2. Multiscale expansion
We introduce the fast and slow time variables tf,andts, such that ∂tf∼{ϕ, .}and ∂ts∼
¶∇2ϕ, .©∼δ∂tfand expand the fields as
ϕ=ϕ(0)(ts,tf,x,y)+ϕ(1)(ts,tf,x,y)+..., (3.8)
Gn=G(0)
n(ts,tf,x,y)+G(1)
n(ts,tf,x,y)+..., (3.9)
where ϕ(m+1)/ϕ(m)∼O(δ), etc. We reiterate that the assumptions we have made are δ1,
the above multi-scale expansion, and the quasi-two-dimensional approximation, whereby
variation in the direction along the magnetic field is neglected, and the non-kinetic species
is assumed to follow a Boltzmann distribution, implying (2.2); no further approximations
will be made in this section. We proceed to examine the moments of gyrokinetic equation,
order by order. The density moment at dominant order in δis
∂ϕ(0)
∂tf=0,(3.10)
from which we formally establish that ϕ(0)depends only on the slow time variable. At next
order in δwe obtain
τ∂ϕ(0)
∂ts+τ∂ϕ(1)
∂tf−∂
∂tf∇2G(0)
2+¶ϕ(0),−∇2ϕ(0)−∇2G(0)
2©+¶G(0)
2,−∇2ϕ(0)©=0.
(3.11)
We introduce a time-average operator to extract the smooth-time behaviour from this
equation
Û
A=1
Δttf+Δt/2
tf−Δt/2
dt
fA(t
f). (3.12)
This time average extends over a period of time much longer than the short time scale
(Δt−1{ϕ, .}). Applying this average to (3.11), we obtain
τ∂ϕ(0)
∂ts+¶ϕ(0),−∇2ϕ(0)−∇2Û
G(0)
2©+¶Û
G(0)
2,−∇2ϕ(0)©=0.(3.13)
The dominant-order equation for Gnis
∂G(0)
n
∂tf+¶ϕ, G(0)
n©=0,(3.14)
from which, upon time averaging, we conclude that Û
G(0)
nis constant along closed
streamlines of constant ϕ. Informally, we will say that Û
G(0)
nis a function of ϕ, although
it can be multi-valued. For n=2 we adopt the notation
Û
G(0)
2=χ(ϕ,ts). (3.15)
Note that, formally, we must exclude special points and lines where ∇ϕ=0 (o-points, and
the ‘separatrices’ that include x-points) but these should occupy negligible volume in the
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6G. G. Plunk
x–yplane. At the next order, we will obtain the smooth evolution of Gn,
∂G(0)
n
∂ts+∂G(1)
n
∂tf+¶ϕ(0),G(1)
n©+¶ϕ(1),G(0)
n©+¶∇2ϕ(0),G(0)
n+2©=0.(3.16)
To this equation we apply two averages, the time average, and also an average along
streamlines of constant ϕ. To define this average, we introduce a coordinate swhich
parameterizes these streamlines and satisfies ˆz×∇ϕ·∇s=1 for convenience. Then we
define
As=dsA(s)
ds.(3.17)
The integral over sis either closed in the sense that the streamlines are closed,
or effectively closed by periodic boundary conditions. The second term of (3.16)is
annihilated by the time average. Noting that {F(ϕ), A}=∂s(AF), the third term is zero
under the s-average, as is the last term after time average, since Û
G(0)
n+2is a function of ϕ(0)
by (3.14). This is a crucial cancellation since the fluid moment hierarchy is consequently
showntobeclosed.
It is convenient to now introduce notation for the part of a field that varies on the
fast time scale, i.e. the ‘fluctuating part’, complementing the mean component defined
by (3.12)
˜
A=A−Û
A.(3.18)
What results from the double average of (3.16) can then be expressed
∂Û
G(0)
n
∂tss+ˇ
¶˜ϕ(1),˜
G(0)
n©s=0.(3.19)
To evaluate the nonlinear term of (3.19), we must obtain dynamical equations for the
fluctuating fields ˜ϕ(1)and ˜
G(0)
n. These come from (3.11) and (3.14), respectively. The
fluctuating part of (3.14)is
∂˜
G(0)
n
∂tf+¶ϕ(0),˜
G(0)
n©=0.(3.20)
Subtracting (3.13)from(3.11), and using the Laplacian of (3.20)toevaluate∂tf∇2˜
G(0)
n,we
find
τ∂˜ϕ(1)
∂tf+∇2¶ϕ(0),˜
G(0)
2©+¶ϕ(0),−∇2˜
G(0)
2©+¶˜
G(0)
2,−∇2ϕ(0)©=0.(3.21)
Finally, noting that ¨∂tsϕ(0)∂s=0 (from (3.13)), we obtain, from (3.19), an expression
determining the explicit time dependence of χ
Å∂χ
∂tsãϕ+ˇ
¶˜ϕ(1),˜
G(0)
2©s=0,(3.22)
where the partial time derivative is taken at constant ϕ(0). To summarize, (3.20) and (3.21)
are the fast-time equations that determine ˜ϕ(1)and ˜
G(0)
n, which can be substituted into the
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Enstrophy non-conservation and the forward cascade of energy 7
slow-time equation (3.22)forχ, and coupled with the following equation (a repetition of
(3.13) written in terms of χ) to close the system
τ∂ϕ(0)
∂ts+¶ϕ(0),−∇2ϕ(0)−∇2χ©+¶χ, −∇2ϕ(0)©=0.(3.23)
Noting that Û
ϕ≈ϕ(0)and ˜ϕ≈˜ϕ(1), we can, without introducing ambiguity, simply drop
the superscripts in what follows.
The final system, (3.20)–(3.23), has some noteworthy features. First, (3.22)hasthe
appearance of a heat transport equation, where the flux is carried by the rapidly varying
pressure perturbation ˜
G2and the small amplitude fluctuating potential ˜ϕ.Notealsohow
(3.20) and (3.21) bear a strong resemblance to the fluid system given by (3.6)–(3.7), where
a similar ordering is satisfied, namely ˜ϕ˜
G2.
4. Decaying turbulence
We will avoid the complications introduced by the instabilities that physically drive
turbulence, and instead now consider decaying turbulence. (We note that a linear instability
could be added to this fluid system using the non-resonant limit of the toroidal branch of
the ITG or ETG mode, but this would require some care to maintain consistency with the
ordering assumptions, as discussed in § 3.1.) Let us consider periodic boundary conditions,
and include explicit dissipation using a fourth-order hyperviscosity term. Without drive
terms, (3.20) implies the rapid decay of ˜
Gnto zero, implying (∂tsχ)ϕ=0 (i.e. it only
depends on the time via its dependence on ϕ). The variation of χin ϕ(or more formally, its
variation between distinct lines of constant ϕ) can be considered as an initial condition of
our calculation. We need only then solve a single equation, which, neglecting superscripts
for order and the subscripts of the time variable ts, becomes
τ∂ϕ
∂t+¶ϕ, −∇2ϕ−∇2χ©+¶χ, −∇2ϕ©=ν4∇4ϕ. (4.1)
The electrostatic energy
E=τ
2dxdyϕ2,(4.2)
is conserved by the nonlinearity for arbitrary χ, which can be verified by multiplying
the equation by ϕand integrating over the x–ydomain. Note that the resulting integral
of the final nonlinear term of (4.1) can be rewritten as −dxdyvE·∇(ϕχ ∇2ϕ),with
vE=ˆz×∇ϕ, which is zero using ∇·vE=0 and periodicity.
Another quantity of interest is the enstrophy, which we will define here as
Z=τ
2dxdy|∇ϕ|2.(4.3)
The enstrophy balance equation is found by multiplying (4.1)by−∇2ϕand integrating
over xand y. Note that the presence of χin the equation breaks enstrophy conservation
if χis a nonlinear function of ϕ. The nonlinear invariance of Zis associated with the
inverse cascade of energy in Hasegawa–Mima turbulence. We thus expect to recover the
spectra corresponding to the potential limit of the Hasegawa–Mima equation (i.e. where
∇2ϕϕ; see Plunk et al. 2010), if χis small, and qualitatively different cascade when χ
is sufficiently large.
The Hasegawa–Mima spectra can be derived in the rough ‘phenomenological’ style, in
terms of the fluctuation amplitude at scale , denoted ϕ, by assuming constancy of the
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8G. G. Plunk
nonlinear flux of its nonlinear invariants (see e.g. Frisch 1995; Plunk et al. 2010). For the
forward cascade, i.e. at scales smaller than the scale of energy injection, the enstrophy flux,
denoted εZ, is assumed constant (independent of scale ), which is expressed as follows:
εZ=τ−1
NL −2ϕ2
∼ϕ3
−6,(4.4)
with τNL() denoting the nonlinear turnover time. This leads to the scaling ϕ∼2ε1/3
Z,
implying a one-dimensional energy spectrum of E(k)∼k−5. The constancy of the
scale-by-scale flux of energy, expected for the inverse cascade at scales larger than the
injection scale, is expressed as
εE=τ−1
NL ϕ2
∼ϕ3
−4,(4.5)
implying ϕ∼4/3ε1/3
Eand a spectrum E(k)∼k−11/3.
Because the additional nonlinear terms of (4.1), henceforth called the ‘χnonlinearity’,
formally break enstrophy conservation, we expect that if they are sufficiently strong, the
inverse cascade should be eliminated and the forward cascade of Zreplaced with a direct
cascade of E. If this flux is carried by the HM nonlinearity, one might expect to observe
the spectrum E(k)∼k−11 /3, as suggested by Plunk (2019). On the other hand, balancing
the χnonlinearity with the HM nonlinearity, scale by scale, implies the linear relation
χ∼ϕ,i.e.χ∝ϕ, which would imply that enstrophy is actually a nonlinear invariant,
preventing the forward cascade of E. For this reason, we may expect to observe an energy
spectrum distinct from k−11/3, whose steepness depends on the relationship between χ
and ϕ, which itself depends on details of the turbulence.
Providing a definitive prediction of this relationship is beyond the scope of the present
work, but a power law seems to be a reasonable possibility to explore, i.e. χ∼ϕα
.Note
that any super-linear scaling α>1 should lead to a spectrum shallower than k−11/3, while
a sub-linear scaling α<1 would imply χis not analytic in xand y. The fluctuating fields
˜
G2and ˜ϕcould be especially active in regions of low E×Bshear (see (3.20)), causing
local extrema in the function χ,via(3.22), so that a quadratic relationship prevails in
such regions, χ∼ϕ2
. Whether or not this seems plausible, assuming a simple nonlinear
relationship will allow us to make the discussion now more concrete; qualitatively similar
conclusions should apply for all α>1. Let us consider the following form for χ:
χ(ϕ) =λ
2ϕ2.(4.6)
The nonlinear energy flux by the χterms is then expressed as εE∼λϕ4
−4,implying
ϕ∼(εE/λ)1/4, and the corresponding energy spectrum
E(k)∼k−3.(4.7)
This spectrum should prevail in cases where the χ-nonlinearity dominates (e.g. large λ).
At sufficiently low λ, one expects a return to the HM behaviour, implying E(k)∼k−5for
the forward cascade.
Some sort of hybrid behaviour may also be possible, although the broad scale range
needed for clear observation of this may be not be present for realistic conditions
encountered in fusion plasmas. One might argue that, because the amplitude of
fluctuations ϕis generally expected to decrease as scales do, the cubic nonlinearity should
be dominant at large scales, and subdominant at small scales. Thus, for sufficiently large
λ, the energy cascade scaling ϕ∼(εE/λ)1/4should hold from the injection scale, down
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Enstrophy non-conservation and the forward cascade of energy 9
(a) (b)
FIGURE 1. Comparison of spectra exhibited by HM-type system (λ=0) and our
two-dimensional turbulence model (λ→∞).
to a transition scale, which can be found by balancing the HM nonlinearity with the
χ-nonlinearity, i.e. ϕ2
−4∼λϕ3
−4. Defining the outer scale oas the scale of energy
injection (or initial energy containing scale), and ϕo=ϕo, we can write the ϕscaling
as ϕ∼(/o)ϕo, so that the above balance occurs at the ‘transition’ scale t∼o/(λϕo).
Thus, if λϕo1 one might expect E∼k−3scaling for −1
o<k<
−1
tfollowed by E∼k−5
for k>
−1
t.
4.1. Direct numerical simulations
To explore the behaviour of the model, (4.1), and test the theoretical predictions, we
perform direct numerical simulations, assuming the simple quadratic form of χ(ϕ)
in (4.6). This introduces a nonlinearity that is cubic in ϕ, which can be treated
pseudo-spectrally using a padding factor of 2 for dealiasing; higher order nonlinearities
require additional padding (Hossain, Matthaeus & Ghosh 1992). The boundary conditions
for the simulations are periodic in xand y,andτ=1 for all simulations.
Figure 1 compares the simulation results with the theoretical scaling laws. All
simulations are initialized with randomly phased fluctuation amplitude of ϕ∼1 around
k=1, falling off exponentially at higher k. Note that although the model assumes k1
there is no conflict in using k>1 for the simulations, as scaling symmetries of the model
allow the results to be reinterpreted for k1. The spectrum found for the λ=0 case is
roughly consistent with the theoretical power law k−5expected for the potential limit of the
HM equation. We note that similar results (not shown here) are encountered for λ0.1.
At larger λ, the breaking of enstrophy conservation is indeed observed in the time trace of
Z, as the energy fills in the spectrum at large k. For the case labelled λ→∞in figure 1,
the spectrum seems consistent with the theoretical k−3prediction at scales smaller than
the injection scale. Note that this limit is obtained by actually setting λ=1andsimply
removing the HM nonlinearity (i.e. the first term of (4.1)) from the equation, as can be
formally justified by rescaling (4.1) in the limit λ→∞. Similar behaviour is observed for
λ1. Intermediate values of λshow intermediate behaviour.
One example is shown in figure 2, which seems to show evidence of a transition
scale between the two theoretical power laws, giving some support to the predictions
of a hybrid scenario described theoretically in the previous section. A more extensive
set of simulations would be needed to test the predictions in detail, for instance the
dependence of the transition scale ton system parameters. We would like to generally
stress that the results of all of the numerical simulations presented here come at a very
modest computational expense, and larger scale computational effort, especially using a
gyrokinetic code, could offer a more extensive test of the conclusions of this work.
https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377820000872
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10 G. G. Plunk
FIGURE 2. Energy spectrum for a case of intermediate strength of χnonlinearity (λ=0.5).
5. Discussion
A novel fluid system has been derived to describe the behaviour of certain classes of
quasi-two-dimensional electrostatic magnetized plasma turbulence. A possible application
is to describe the energy cascade in cases of streamer-dominated ETG turbulence (note the
spectrum, noted to be close to k−11/3, in figure 5 of Plunk 2019), where the nonlinear
stability of elongated turbulent eddies is believed to stem from the two-dimensional
character of the dominant instabilities, e.g. the absence of sufficient variation of the
mode structure in the direction along the magnetic field (Jenko & Dorland 2002). This
turbulent state is, however, sensitive to magnetic geometry, and seems to vanish when, for
instance, the global magnetic shear is varied in such a way as to induce stronger parallel
electron flow to the ETG mode. The ensuing dynamics then depends on kinetic physics
involving the parallel streaming term, absent from two-dimensional models. A second
possible application of the present model might be to describe ITG turbulence in cases
where the zonal flows are suppressed. One candidate is a case observed with simulations
of the HSX stellarator having surprisingly steep fluctuation spectra (Plunk, Xanthopoulos
& Helander 2017), found to be close to k−10/3.
Although the presented model has limited application, it fills a significant gap in
present theories describing gyrokinetic turbulence cascades, as it accounts for the essential
nonlinear terms that arise when the cold ion approximation is invalid. These terms, it is
found, alter the conservative properties of the nonlinearity, with significant consequences
on the cascade, so that, even in the two-dimensional limit, the inverse cascade of energy
can be shut down. The numerical simulations confirm that the size of the pressure
perturbation (χ) can control the cascade type, and HM-like behaviour can be recovered
if it is sufficiently small. This may underlie the slow secular growth of large-scale zonal
flows (Guttenfelder & Candy 2011) and other coherent structures (Nakata et al. 2010)in
simulations of ETG turbulence, and the related appearance of a Dimits shift phenomenon
in near-marginal cases (Colyer et al. 2017).
Acknowledgements
This work has been carried out within the framework of the EUROfusion Consortium
and has received funding from the Euratom research and training programme 2014–2018
https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377820000872
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Enstrophy non-conservation and the forward cascade of energy 11
and 2019–2020 under grant agreement No 633053. The views and opinions expressed
herein do not necessarily reflect those of the European Commission.
Editor Alex Schekochihin thanks the referees for their advice in evaluating this article.
Declaration of interests
The author reports no conflict of interest.
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