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J. Plasma Phys. (2020), vol.86, 175860402 © The Author(s), 2020.

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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

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

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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