Negative magnetic eddy diffusivities from test-field method and multiscale stability theory

4.2 MB·English

by  Alexander Andrievsky

Negative magnetic eddy diffusivities from test-field method and
multiscale stability theory
Alexander Andrievsky1, Axel Brandenburg2,3, Alain Noullez4, and Vladislav Zheligovsky1,4
1Institute of earthquake prediction theory and mathematical geophysics Russian Ac. Sci.,
84/32 Profsoyuznaya St., 117997 Moscow, Russia
2Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-10691 Stockholm, Sweden
3Department of Astronomy, Stockholm University, AlbaNova University Center, SE-10691 Stockholm, Sweden
4Laboratoire Lagrange, Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur,
CNRS, Blvd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France
5
1
0
2

l ABSTRACT
u
J The generation of a large-scale magnetic field in the kinematic regime in the absence of an α-effect

9 is investigated by following two different approaches: the test-field method and the multiscale stability

theory relying on the homogenisation technique. Our computations of the magnetic eddy diffusivity
]
n tensor of the parity-invariant flow IV of G.O. Roberts and the modified Taylor–Green flow confirm the
y findings of previous studies, and also explain some of their apparent contradictions. The two flows have
d
large symmetry groups; this is used to considerably simplify the eddy diffusivity tensor. Finally, a new
-
u analytic result is presented: upon expressing the eddy diffusivity tensor in terms of solutions to auxiliary
l problems for the adjoint operator, we derive relations between the magnetic eddy diffusivity tensors that
f
. arise for mutually reverse small-scale flows v(x) and −v(x).
s
c
i Subject headings: MHD – magnetic fields – turbulence – dynamo
s
y
1. Introduction Since the 1960s, German scientists (Steenbeck,
h
Krause&R¨adler1966;seealsoKrause&R¨adler1980)
p
It is well-known that at sufficiently high Reynolds
[ weredevelopingthetheoryofmean-fieldelectrodynam-
number turbulence is characterised by a hierarchy of
ics (MFE), a first attempt supposed to advise how to
3 fluctuations interacting on a wide range of space and
dothis. Perhaps,thebestintroductiontotheideason
v
time scales. When this happens in a flow of conduct-
5 whichthistheoryisbuiltisbyoneofitsfoundersKarl-
ing fluid, magnetic field generation commences if the
6 Heinz R¨adler (2007). The three-dimensional magnetic
4 magnetic Reynolds number is sufficiently high (Mof- and flow velocity fields, b and v, are decomposed into
4 fatt 1978). As predicted by the magnetic induction “mean”, b and v, and “fluctuating”, b(cid:48) and v(cid:48), fields:
0 equation governing the process of generation, small
.
1 scales also develop in the generated magnetic field. b=b+b(cid:48), v=v+v(cid:48).
0 Theinteractionoffinestructuresofflowandmagnetic
5 fieldusuallyinfluencestheevolutionoftheirlarge-scale Any averaging procedure is deemed acceptable pro-
1
parts. In particular, by Parker’s hypothesis, such an vided it satisfies the Reynolds rules (see R¨adler 2007),
:
v interactionmaygiverisetoameanelectromotiveforce e.g., planar averaging over any pair of Cartesian vari-
Xi (e.m.f.), parallel to the large-scale magnetic field. ables, one-dimensional averaging along any given di-
r Inastrophysics,whenthegenerationofthegeomag- rection,orensembleaveragingforturbulentflows. The
a neticorsolarmagneticfieldisunderinvestigation,fine equations for mean magnetic field and fluctuations
structures are generally of lesser interest than global take the form
ones. With present-day computers, it is impossible
∂b
to resolve structures over the whole range of interact- =η∇2b+∇×(v×b+v(cid:48)×b(cid:48)), (1)
∂t
ing scales; by choosing the domain of integration of
the equations of magnetohydrodynamics, we can only ∂b(cid:48) =η∇2b(cid:48)+∇×(cid:0)v×b(cid:48)+v(cid:48)×b+(v(cid:48)×b(cid:48))(cid:48)(cid:1).
∂t
focus on the large or small scales. However, in simu-
(2)
lations of the global picture it is desirable to take into
account the integral influence of physical processes at
Heref(cid:48) ≡f−f denotesthefluctuatingpartofavector
small scales.
field f. The problem then reduces to the use of (2) for
1
expressing the mean e.m.f. v(cid:48)×b(cid:48) in terms of b and relates v(cid:48)×b(cid:48) through the unknown coefficients of α
v. For simplicity, we henceforth assume that v = 0 and η to b. This system can be solved to obtain α
and v(cid:48) is steady. In MFE, for homogeneous station- and η. Similarly, the temporal dependence of the ker-
aryturbulence,themeane.m.f.isusuallyexpressedin nels in (3) can be “probed” in Fourier space by con-
terms of the mean magnetic field as sidering the test fields
(cid:90)(cid:90)
v(cid:48)×b(cid:48) = (cid:0)Kα(x−ξ,t−τ)b(ξ,τ) (3) b=cos(k·x)e−iωten and b=sin(k·x)e−iωten.
(7)
(cid:1)
−Kη(x−ξ,t−τ)∇×b(ξ,τ) dξdτ In kinematic dynamo problems, where the evolu-
tion of a weak magnetic field is studied (so that its
whenaveragingisplanar(K andK donotdependon
α η
influence on the flow via the Lorentz force can be ne-
the spatial variables over which the e.m.f. is averaged
glected), the flow velocity, v, is known a priori. It
in the l.h.s.) — in general, η should be defined as a
can be a stationary field, often supposed to have a
rank 3 tensor acting on ∇b. Our task is to determine
vanishing average (v = 0), as have the flows that
the kernels. In Fourier space, (3) implies
we consider in this paper. Alternatively, it can be a
F (v(cid:48)×b(cid:48))=α(k,ω)F b−η(k,ω)F (∇× b). time-dependent flow, for instance, supplied by an in-
k,ω k,ω k,ω
(4) dependent hydrodynamic simulation. The kinematic
Here, following Brandenburg et al. (2008b), we have dynamo problem is an instance of the full magneto-
denoted hydrodynamic (MHD) stability problem that focuses
(cid:90)(cid:90) on the stability of non-magnetic states; the flow and
Fk,ωf ≡ e−i(k·x−ωt)f(x,t)dxdt, (5) magnetic field perturbations then decouple since the
Lorentz force is quadratic in the magnetic field. In a
α(k,ω)=Fk,ωKα(x,t), η(k,ω)=Fk,ωKη(x,t). general setup, one considers the stability of an MHD
regime featuring a non-vanishing magnetic field that
In the limit k → 0 and ω → 0, α and η describe the
(magnetic1) α-effect and eddy diffusivity correction2 affects the flow, and therefore perturbations involve
both the flow and magnetic field that cannot be dis-
tensors.
entangled.
Thetest-field method3 (TFM)forcomputingαand
MHD perturbations involving much larger spatial
η was developed within the MFE paradigm. To the
andtemporalscalesthanthoseoftheperturbedMHD
bestofourknowledge,itwasfirstproposedbySchrin-
regimes (which, e.g., can be periodic or quasi-periodic
ner et al. (2005, 2007). Perhaps, the most detailed
in space, and steady or periodic in time) can also be
description of the TFM procedure applied by Devlen
explored by an approach known as the multiscale sta-
et al. (2013) is found in Brandenburg et al. (2008a).
bility theory (MST). It originates from the studies of
Therecipeistosolveequation(2)forzero-meanmag-
netic perturbation b(cid:48), where b is a test field. The ini- hydrodynamic stability (Dubrulle & Frisch 1991) and
tial condition for b(cid:48) can be any solenoidal small-scale kinematic dynamo (Lanotte et al. 1999) and relies on
mathematically precise asymptotic methods for ho-
zero-mean field (for instance, 0). For space-periodic
mogenisation of elliptic operators. An introduction to
magnetic fields, the test fields
MST can be found in Zheligovsky (2011); the linear
b=cos(k·x)en and b=sin(k·x)en, (6) MHD stability problem for large-scale perturbations
wasconsideredbyZheligovsky(2003)(seealsoChap.6
are chosen. By using sufficiently many independent
of Zheligovsky 2011). Here we will only consider the
test fields, we obtain a linear system of equations that
kinematic dynamo problem, and focus on the gener-
ation of a magnetic field involving large scales by a
1This paper is devoted to the study of magnetic α-effect and
small-scale fluid flow. For a steady flow, the dynamo
magnetic eddy diffusivity exclusively — as opposed to the hy-
drodynamicα-effectknownastheAKA-effect(seeFrischetal. problem can be reduced to the eigenvalue problem for
1987; Dubrulle & Frisch 1991), or combined α-effect and eddy the magnetic induction operator:
diffusivity emerging in large-scale perturbations of magnetohy-
drodynamicregimes(seeChaps.6–9inZheligovsky2011). Note η∇2b+∇×(v×b)=λb (8)
thattheexpression“magneticα-effect”issometimesusedwith
adifferentmeaning,designatingatermproportionaltocurrent
helicity that quenches against the kinetic α-effect. With this (hereη denotesthemagneticmoleculardiffusivityand
disclaimerinmind,weomittheattribute“magnetic”fromnow λ is the eigenvalue).
onwhenreferringtotheα-effectandeddydiffusivity.
We assume that a large-scale magnetic mode
2We use here the terminology of the multiscale stability theory.
b(X,x) depends on fast, x, and slow, X=εx, spatial
Infact,the“corrections”canbemuchlargerthanthemolecular
diffusivitywhichthey“correct”—theturbulentdiffusivitycan variables, the flow depends only on x, and the scale
bebyordersofmagnitudelargerthanthemoleculardiffusivity. ratio ε is small. We proceed by expanding a mode
3NottobeconfusedwithKraichnan’s“test-fieldmodel”ofturbu- b(X,x) and the associated eigenvalue λ (its real part
lence(Kraichnan1971),usedbySulemetal.(1975)asamethod
forclosureofthehierarchyofmomentequations.
2
is the growth rate of the mode) in power series in ε, ume averaging in the generic case, when the kernel of
themagneticinductionoperatorcomprisesthreemag-
∞ ∞
(cid:88) (cid:88) netic modes with non-vanishing linearly independent
b= b (X,x)εn, λ= λ εn, (9)
n n averages. However, (3) does not necessarily hold for
n=0 n=0
othertypesofaveraging,orwhenthedimensionofthe
and deriving a hierarchy of equations that the eigen- kernel is higher — in the latter case, amplitudes of
value equation yields in successive orders εn. As it all neutral modes are involved in (3), as this happens,
turns out, we can find each term of the expansions by e.g.,fortranslation-invariantconvectivedynamos(see,
solvingsuccessivelyequationsfromthishierarchy. For e.g., Chertovskih & Zheligovsky 2015). For MHD tur-
parity-invariant flows, that we will mostly consider, bulence, (3) is likely to stem, for various averaging
the series for the eigenvalue involves only even powers procedures, from the ergodic properties of the respec-
of ε (see Section 3.5 of Zheligovsky 2011). tive MHD dynamical system, but, to the best of our
The first equation in the hierarchy shows that the knowledge, thisequalitywasneverfullydemonstrated
leading terms b and λ in the expansion (9) are, re- in the context of MFE at the mathematical level of
0 0
spectively, a small-scale eigenfunction and the associ- rigour; itremainsaphenomenologicalpropertyoftur-
ated eigenvalue of the operator of magnetic induction. bulence (such as, for instance, the Kolmogorov law).
The asymptotic expansion can be developed for any The standard α-effect and eddy diffusivity, arising
eigenvalue λ . For small scale ratios ε, the growth in the limits k → 0 and ω → 0, are an idealisation in
0
rate may exceed Re(λ ) due to the interaction of the which nonlinear terms, higher spatial derivatives and
0
fluctuating components of the magnetic field and of temporal derivatives of the magnetic field are omitted
the small-scale flow, but the corrections are at best in the expression (3) for the e.m.f. This is justified if
linear in the small parameter ε and hence small. We the mean fields vary sufficiently slowly in space and
are mostly interested in the case where no small-scale time, i.e., on scales much larger and longer than those
magnetic field is generated and λ = 0, since then of the fluctuations. While this simplification may be
0
the presence of large spatial scales can, in principle, permissible in some cases, e.g., for forced turbulence
result in the onset of magnetic field generation, i.e., with sufficient scale separation, for certain flows, such
in a qualitative change in the behaviour of the MHD as the Roberts and Otani flows, it is not (Hubbard &
system. (The case of an oscillatory small-scale kine- Brandenburg 2009). A particularly striking example
matic dynamo occurring for imaginary λ was consid- are flows II and III of G.O. Roberts (1972); for de-
0
ered in Section 3.8 of Zheligovsky 2011; it is, actu- scribing the nature of the dynamo in those flows, it is
ally, algebraically much simpler.) For λ =0, the first crucial to retain the convolution in time in the inte-
0
term b is a linear combination of neutral small-scale gral operators in (3) (Rheinhardt et al. 2014). Then
0
magnetic modes with coefficients depending on the theelectromotiveforceatagiventimedependsonthe
slow variable. These coefficients, called amplitudes, magnetic field also at earlier times, so the system pos-
are determined from the solvability conditions for the sesses “memory”. It is important to realise that the
higher-order small-scale equations from the hierarchy. memory effect does occur even for steady flows such
Whentheproblemisconsideredinathree-dimensional as those considered here. Excluding the memory ef-
periodic domain, the kernel of the magnetic induc- fectfromconsiderationmoreoftenresultsinquantita-
tionoperatorcomprisesthreeneutralmagneticmodes tive distortions, such as too high an estimate for the
whose averages are the unit Cartesian coordinate vec- critical dynamo number (Rheinhardt & Brandenburg
tors(generically,thekernelisthree-dimensional). The 2012), rather than in qualitative changes.
amplitudesofthesemodescanclearlybeinterpretedas We note in passing that, instead of implementing
the Cartesian components of the mean magnetic field. an integral transform in both space and time, which
Furthermore,bythetheoremontheFredholmalterna- is cumbersome, it is convenient to solve an evolution
tive (see, e.g., Stone & Goldbart 2009), the solvability equation for the e.m.f. v(cid:48)×b(cid:48). Such an equation
condition consists of the orthogonality of the inhomo- was first derived by Blackman & Field (2002) using
geneous term to the kernel of the operator adjoint to the τ approximation, which captures temporal nonlo-
the operator of magnetic induction. Generically, this cality, i.e., the memory effect (Hubbard & Branden-
amounts to vanishing of the integral of the inhomo- burg 2009). This was then extended by Rheinhardt
geneous term over the periodicity box. As a result, & Brandenburg (2012) to capture also spatial nonlo-
when λ0 = 0, equations for the amplitudes can be cality. Usually this also yields a satisfactory (at least
interpreted as mean-field equations, where the respec- qualitatively) description of the unusual phenomena
tive terms describe the α-effect or the eddy diffusivity relatedtothememoryeffect,suchastheonesencoun-
effect. tered in flows II and III of G.O. Roberts (Rheinhardt
TheMSTanalysisrevealsthenon-universalcharac- etal.2014). Theseideaswillturnouttobeimportant
ter of (3). This asymptotic equality can be rigorously inSection5.4,whenwecomparethemagneticfieldfor
derived for a multiscale kinematic dynamo and vol- themodifiedTaylor–Greenflow(mTG)obtainedfrom
3
direct numerical simulations (DNS) with that found magnetic field? Such an averaging does not obey the
by TFM. Reynolds rules, namely, because averaging and taking
We have thus two independent theories: MFE, the spatial gradient do not commute, and turns the
physical in spirit, especially when making simplify- midplane into an artificial boundary. In each half-
ing assumptions regarding the kernel in the integral cell the mean field depends only on the horizontal
equation (3) for MHD turbulence, and MST, which variables. The opposite α-effect values in two adja-
yields a mathematically rigorous derivation of equa- cent half-cells force us to assume opposite mean fields
tions for similar quantities from first principles. MST over and below the midplane, in order to avoid sin-
has a narrower scope, being applicable to treat only gularities in the α-effect operator at the midplane.
linear and weakly nonlinear MHD stability problems. This inevitably implies the existence of a boundary
WhileMSTappliesspecificallytothelimitk→0and layer at the midplane. However, nothing resembling
ω → 0, TFM can be applied to non-infinitesimal |k| a boundary-layer kind of behaviour of magnetic field
andω. Itcanthereforebeusedtoassemblethekernels in the numerical solutions was reported ibid., clearly
K and K . In other words, MST strives to describe showing that averaging over a half-cell is unnatural
α η
an influence of the flow, characterised by certain tem- and incompatible with the physics of the problem,
poral and spatial scales, on magnetic fields involving and is also inappropriate from the mean-field electro-
much larger scales; TFM is more ambitious in trying dynamics perspective. The α-effect operator must be
to assess the influence of both larger and smaller hy- calculatedbyaveragingovertheentireperiodicitycell;
drodynamic scales on magnetic field of a given scale. theobserved“antisymmetryofαaboutthemidplane”
Although the limits k → 0 and ω → 0 can be numer- (Cattaneo&Hughes2006)simplyimpliesthatinthese
ically expensive for TFM, a comparison with MST is dynamos the relevant α-effect is zero (i.e., the α-effect
possible. operator is not involved in the equations for the evo-
lution of the mean field), and the essential eddy effect
RecentlyDevlenetal.(2013)appliedTFMtocom-
is eddy diffusivity. Furthermore, the convective dy-
pute the magnetic eddy diffusivity in flows previously
namos considered ibid. are translation-invariant, and
employed in the studies of Lanotte et al. (1999) and
hence some amplitudes, essential in the description of
G.O. Roberts (1972) with the use of MST and a sim-
the large-scale modulation of the generated instabil-
ilar approach. Dubrulle et al. (2007) observed in sim-
ity modes, cannot be interpreted as mean fields4 (see
ulations the beginning of magnetic field generation by
Chaps. 8 and 9 in Zheligovsky 2011); neglecting these
mTG when increasing the magnetic Reynolds num-
modes is also likely to affect the results of Cattaneo
berstartingfromsmallvalues, whichtheauthorscau-
& Hughes (2006, 2008). As a result, no sound conclu-
tiously attributed to the onset of the action of nega-
sions concerning the α-effect, intended for astrophysi-
tive magnetic eddy diffusivity investigated by Lanotte
cal or general MHD applications, can be drawn from
etal.(1999). Devlenetal.(2013)found, inagreement
the findings of those two papers.
withG.O.Roberts(1972),thattheso-calledflowIVof
G.O. Roberts (further referred to as R-IV) does yield Our paper is organised as follows. In Section 2
negative magnetic eddy diffusivity, but they failed to we remind the reader of the MST formalism for the
reproduce the results of Lanotte et al. (1999) on the large-scale kinematic dynamo. In Section 3 we calcu-
presenceofnegativemagneticeddydiffusivityinmTG. late, in the MST framework, the operator of magnetic
We resolve this controversy in the present paper and eddy diffusivity for R-IV using its many symmetries,
showthatinasuitableparameterrangeeddydiffusiv- and state results of the computation of its two coef-
ity is negative, however, the relevant TFM averaging ficients. In Section 4 we discuss how the symmetries
is not over the horizontal plane (which is applicable of mTG reduce the number of auxiliary problems in-
for R-IV), but one along the vertical direction, or a volved in MST computations of eddy diffusivity, and
planaroneoveranyoftheothertwoCartesiancoordi- present numerical results. Despite using algorithms
nate planes such that the average still depends on one that differ drastically from those used by Lanotte et
of the two horizontal directions. al. (1999), we reproduce the results of this paper with
The need for the cross-examination stems from the
factthatsomeapplicationsoftheMFEideascanfailto 4WewillseeinSection2thatalarge-scalemagneticmodehasthe
conform with the mathematical structure of problems structureb=(cid:80)Nn Bn(X)S(cid:101)n(x)+O(ε),whereN =dimkerLis
thenumberofindependentsmall-scaleneutralmagneticmodes
under consideration. For instance, the mean e.m.f. S(cid:101)n(x). By normalizing the small-scale modes, we can impose
computed as “an average over the lower half-volume, theconditions
tdhiffeeurepnpceerohfatlhf-evsoelutwmoe”, wora,sbuestetderinsttihlle, ostnuedhieaslfooffCtahte- S(cid:101)n=(cid:26) e0n ffoorrnK≤+K1,≤n≤N.
taneo & Hughes (2006, 2008) of the α-effect in con- We then find b = (cid:80)Kn Bn(X)en; thus, for n ≤ K the am-
vective dynamo in a layer. How could these proce- plitudes Bn(X) have the sense of the mean components of the
dures possibly help to track the evolution of the mean meanfieldb;forn≥K+1noanysuchorsimilarinterpretation
ispossible.
4
4 significant digits. In Section 4.2 we explain why no isthesmall-scalemagneticinductionoperator,ande
n
large-scale dynamo was found for mTG by Devlen et are unit vectors of the Cartesian coordinate system.
al. (2013), and show that eddy diffusivities obtained S (x) are solenoidal.
n
by TFM with an alternative planar averaging quali- LetB(X)beasolenoidalspace-periodicsolutionto
tatively agree with the MST values. In Section 4.3 the eigenvalue problem (10) whose associated eigen-
we show that the growth rates of large-scale dynamo value is Λ. Then B(µX) is also a solenoidal solution
modes have the symmetry properties implied by the to (10) whose associated eigenvalue is µΛ; for any in-
structureoftheeddydiffusivityoperator. InSection5 teger µ, positive or negative, this mode possesses the
wedemonstratethattheTFMprocedurewiththespa- spatial periodicity of the original mode B(X). Thus,
tial averaging reproduces the MST α-effect and eddy a mean field, that is initially an infinite sum of modes
diffusivity tensors, and consider analytically and nu- defined by (10), grows in general superexponentially;
merically the difference of the two approaches for a consequently, the large-scale magnetic field grows and
planaraveragingusingmTGasanexample. Conclud- destabilises the MHD system on time scales that are
ing remarks end the paper. intermediatebetweenthefasttimetandtheslowtime
T =εt(unlessallmodesdefinedby(10)areassociated
2. The mathematical theory of generation with imaginary eigenvalues Λ).
of large-scale magnetic field
2. Magnetic eddy diffusivity. A field f is parity-
invariant, if
We review here the results of application of MST
f(−x)=−f(x), (13)
for the investigation of large-scale magnetic field gen-
eration by small-scale steady flow of electrically con- and parity-antiinvariant, if
ductingincompressiblefluid(Lanotteetal.1999;Zhe-
ligovsky et al. 2001; Zheligovsky 2011). We consider f(−x)=f(x).
the kinematic dynamo problem as a problem of deter-
For parity-invariant flows v, parity-invariant and
mination of the spectrum of the magnetic induction
parity-antiinvariant vector fields constitute invari-
operator, which enables us to find growing large-scale
ant subspaces of the magnetic induction operator L.
modesevenwheninadditionasmall-scaledynamoop-
Hence, vector fields S (x) are parity-antiinvariant,
erates. For the sake of simplicity, both the large-scale n
and the α-effect is absent: A = 0. The magnetic
magneticmodeb(X,x)andtheflowv(x)areassumed
field (9) is then
to be 2π-periodic in each fast spatial variable x . The
i
mode is solenoidal and satisfies the eigenvalue equa- 3 (cid:32)
(cid:88)
tion (8) for the magnetic induction operator. b(X,x)= Bn(X)(S (x)+e ) (14)
n n
1. Magnetic α-effect. Generically, the average of n=1
the leading term in the expansion (9) of a magnetic (cid:88)3 ∂Bn (cid:33)
mode, B(X) = (cid:104)b0(X,x)(cid:105), and the leading term in + ε ∂X (X)Gmn(x) +O(ε2),
m
the expansion of the associated eigenvalue, Λ = λ , m=1
1
are a solution to the eigenvalue problem for the α-
wherevectorfieldsG (x)arezero-meansolutionsto
mn
effect operator,
auxiliary problems of type II:
∇X×AB=ΛB, (10) LG =−2η∂Sn −e ×(v×(S +e )). (15)
mn ∂x m n n
m
in the subspace of solenoidal fields, ∇ ·B=0. Here
X
the tensor of magnetic α-effect, A, is the 3×3 matrix G (x) are parity-invariant.
mn
whose n-th column is (cid:104)v×Sn(cid:105), (cid:104)·(cid:105) denotes the aver- Thesolenoidalmeanpartoftheleadingterminthe
age over the periodicity cell T3 = [0,2π]3 of the fast expansion(9)ofthemode,andtheleadingterminthe
variables, expansion of the associated eigenvalue, Λ = λ , are a
2
solution to the eigenvalue problem for the operator of
(cid:90)
(cid:104)f(cid:105)(X)=(2π)−3 f(X,x)dx, magnetic eddy diffusivity:
T3
(cid:88)3 (cid:88)3 ∂Bn
vectorfieldsSn(x)arezero-meansolutionstoauxiliary η∇2XB+∇X× Dmn∂X =ΛB. (16)
problems of type I: m
n=1m=1
∂v Here, D is the tensor of eddy diffusivity correction,
LS =− (11)
n ∂x
n D =(cid:104)v×G (cid:105). (17)
mn mn
⇔ L(S +e )=0, (12)
n n
We assume that the mean fields reside and are
Lb≡η∇2xb+∇x×(v×b) bounded in the entire space R3. Hence, solutions to
5
the eigenvalue problem (16) are Fourier harmonics5 box when periodicity conditions in space are consid-
ered),buttheydecayandareunimportantforgenera-
B(X)=B(cid:101)eiq·X, B(cid:101) ·q=0. (18) tion. By contrast, when the small-scale dynamo is in-
active,thepresenceoflargescalesinthefieldbecomes
Here, B(cid:101) = (B(cid:101)1,B(cid:101)2,B(cid:101)3) and q = (q1,q2,q3) are con- akeyingredient,withoutwhichthemechanismofneg-
stantvectorssatisfyingtheconditions|q|=1,B(cid:101)·q=0 ative eddy diffusivity cannot make a dynamo work. It
(solenoidality of the mean magnetic mode) and can also happen that the small- and large-scale mech-
anisms coexist and are acting simultaneously.
3 3
−ηB(cid:101) −q×(cid:88) (cid:88) DmnB(cid:101)nqm =ΛB(cid:101). (19) 3. Computation of the eddy diffusivity tensor. The
load of computation of the tensor of eddy diffusivity
n=1m=1
correction is halved, if instead of computing the fields
Solenoidality of the modes implies G one solves auxiliary problems for the adjoint op-
mn
erator (Zheligovsky 2011):
B(cid:101) =βtT+βpP, (20)
L∗Z =v×e , (24)
l l
where
forzero-meanfieldsZ ,1≤l≤3,theadjointoperator
T=(−q ,q ,0), P=(q q ,q q ,−(q2+q2)) (21) l
2 1 1 3 2 3 1 2 being
L∗z≡η∇2z−v×(∇ ×z),
(this is equivalent to decomposing the mode into the x x
toroidal and poloidal (this is equivalent to decompos- since, as it is easy to see from (17), (15) and (24),
ing the mode into toroidal and poloidal components).
(cid:28) (cid:18) (cid:19)(cid:29)
Substituting (20) into (19) and scalar multiplying by Dl = Z · 2η∂Sn +e ×(v×(S +e )) .
T and P, we recast (19) into an equivalent eigenvalue mn l ∂x m n n
m
problem in the coefficients β and β : (25)
t p
4. Relationsbetweentensorsofmagneticeddydiffu-
(cid:88)
− DlmnPl(βtTn+βpPn)qm =(q12+q22)(η+Λ)βt, sivity correction for mutually opposite flows. The av-
m,l,n erage(25)canbeexpressedintermsofsolutionstothe
(22) auxiliary problems for the adjoint operator. We dec-
(cid:88)Dl Tl(β Tn+β Pn)q =(q2+q2)(η+Λ)β . orate by the superscript “minus” the quantities perti-
mn t p m 1 2 p nent to the reverse flow −v:
m,l,n
(23)
L−b≡η∇2b−∇ ×(v×b),
x x
Taking into account the symmetries of the generat-
L−(S−+e )=0, (L−)∗(Z−+e )=0.
ingflowcanconsiderablysimplifytheeigenvalueprob- n n l l
lem (22)–(23) (see Sections 3 and 4). Clearly, (24) implies
Eigenvalues Λ depend on the wave vector q of the
L−(∇ ×Z +e )=0, (26)
large-scale amplitude modulation: Λ = Λ(q). If the x l l
realpartofΛ(q)isthemaximumofRe(Λ(q))overunit
(cid:101) and hence for all l,
wave vectors q, then η =−Λ(q) is called the min-
eddy (cid:101)
imum magnetic eddy diffusivity. When Re(ηeddy)>0, ∇ ×Z =S− ⇒ Z =η−1∇−2(v×(S−+e )),
generation of large-scale magnetic field by the mech- x l l l x l l
(27)
anism of negative eddy diffusivity is possible. From
where ∇−2 denotes the inverse Laplacian in the fast
a physicist’s point of view, this mechanism is impor- x
variables. Using the analogues of these relations for
tant only if the flow v does not generate small-scale
the flow v to eliminate S in (25), we obtain
n
magnetic fields (i.e., fields of the same spatial period-
icity,asthatoftheflow),becauseotherwisesmall-scale (cid:28) (cid:18) ∂Z− (cid:19)(cid:29)
Dl =η Z · 2∇ × n −e ×∇2Z− . (28)
magnetic fields grow and destabilise the MHD system mn l x ∂x m x n
m
on time scales of the order of unity, which is faster
than the growth of the large-scale field in the slow Applyingstandardvectoranalysistransformations,we
time T =ε2t. This can also be interpreted as follows: can express this average as an integral of the scalar
when only the small-scale dynamo is acting, the mag- product of Z− and a field resulting from the action of
n
netic field can involve Fourier harmonics of arbitrarily adifferentialoperatoronZ . Byself-adjointnessofthe
l
largewavelengths(compatiblewiththeboundarycon- Laplacianandthecurl,andantisymmetryofthetriple
ditions, i.e., not exceeding the size of the periodicity product with respect to permutation of its factors, we
find
5The vector εq is analogous to the wave vector k referred to in Dl =−(D−)n . (29)
mn ml
theexpositionofTFMintheIntroduction.
6
When small-scale magnetic fields are not generated type, making this choice of flow somewhat academic.)
(i.e., alleigenvaluesofthesmall-scalemagneticinduc- Hence,applyingtheoperationofshiftbyhalfaperiod
tion operator have non-positive real parts), the auxil- in the direction x , which we denote by :
1 (cid:98)
iary problems can be solved numerically by comput-
ing Sn+en and ∇x×Zl+el as small-scale dominant (cid:98)f(x1,x2,x3)≡f(x1+π,x2,x3),
eigenmodesofthemagneticinductionoperatorsLand
to the eigenvalue problem (26), we find
L−, respectively, (see (12) and (26)) in the subspace
ozefrsoo.leTnhoeidsaalmveecstmoralfil-eslcdaslewehiogseenvaavleureagceodceanisbaeppnloiend- Z−n =Z(cid:98)n. (31)
to solve all these six eigenproblems, the flow being re- Substitutingthisinto(28),usingtheself-adjointnessof
versed, v→−v, when computing ∇x×Zl. theLaplacian,thecurlandoperator(cid:98),andintegrating
by parts in x the first term in (28), we obtain
m
3. Generation of large-scale magnetic field
by R-IV Dl =−Dn , (32)
mn ml
G.O. Roberts (1972) studied how simple flows de- and Dn =0 for any flow possessing translation anti-
mn
pending on two spatial variables x and x (deemed invariance with respect to the shift by half a period in
1 2
horizontal), such as (30) (see below), generate mag- one of the spatial variables.
netic fields, whose dependence on x3 enters via the 2. Symmetry in x2 of R-IV:
factor eiεx3. Here, ε is a small parameter; thus this
work is clearly in the multiscale spirit, although he v1(x1,−x2,x3)=v1(x1,x2,x3),
didnotpresentthecompletemultiscaleformalism,nor v2(x ,−x ,x )=−v2(x ,x ,x ), (33)
1 2 3 1 2 3
derived the operator of eddy diffusivity. His flow IV
v3(x ,−x ,x )=v3(x ,x ,x );
(labelled here R-IV) lacks the α-effect; it is the first 1 2 3 1 2 3
knownexampleofadynamoexploitingthemechanism
antisymmetry in x , is defined by changing here the
2
of negative eddy diffusivity, as was suggested previ-
signs in the r.h.s. to the opposite ones. Clearly, the
ouslyongeneralgrounds(Zheligovskyetal.2001). To
curlorvectormultiplicationbyR-IVmapsfields,sym-
the best of our knowledge, Devlen et al. (2013) were
metric in x , to fields, antisymmetric in x , and vice
2 2
thefirsttoidentifyandstudyindetailthismechanism
versa. Consequently, fields symmetric and antisym-
forR-IV.ItshouldbeemphasisedthatflowsIIandIII
metric in x constitute invariant subspaces of the op-
2
arealsonon-helicaldynamos, thusindicativeofaneg- erators of magnetic induction L and L−. It follows
ative eddy diffusivity effect; however, later those flows
from (11) that S are symmetric in x for odd n and
n 2
turned out to have positive eddy diffusivity, and their
antisymmetricinx forn=2;(24)impliesthatZ are
2 l
dynamoactionwasidentifiedasbeingduetoturbulent
antisymmetric in x for odd l and symmetric in x for
2 2
pumping with a time delay (Rheinhardt et al. 2014).
l=2.
WefollowDevlenetal.(2013)ininvestigatinglarge-
Vector multiplication by e also maps symmetric
m
scale generation by R-IV. In the spatial variables in-
in x fields to antisymmetric ones and vice versa for
troduced by Tilgner (2004) (rotated by 45◦ about the 2
odd m, and does not change the symmetry and anti-
vertical axis with respect to the variables used by
symmetry of a field in x for m = 2. Therefore, (25)
2
G.O. Roberts 1972), its Cartesian components are
implies
√
v = 2sinx cosx ,
1 √ 1 2 Dlmn =0, if l+m+n is odd. (34)
v =− 2cosx sinx , (30)
2 1 2
v =sinx . 3. Wave vector parity. We call “even” a three-
3 1
dimensionalvectorfielddependingontwospatialvari-
It is clearly incompressible and parity-invariant (see ablesx andx ,whenitisalinearcombinationofhar-
1 2
(13)), thus lacking an α-effect. monicsB(cid:101)qeiq·x suchthatB(cid:101)3 =0ifq1+q2 isevenand
3.1. The effect of symmetries B(cid:101)1 = B(cid:101)2 = 0 if q1+q2 is odd; we call a field “odd”,
when it is a linear combination of harmonics B(cid:101)qeiq·x
The symmetries of the flow control the structure of such that B(cid:101)3 =0 if q1+q2 is odd and B(cid:101)1 =B(cid:101)2 =0 if
the tensor of eddy diffusivity correction D. q +q is even. Clearly, in this terminology R-IV (30)
1 2
1. Translation antiinvariance with respect to the is even.
shift by half a period in x1 of R-IV: Taking the curl or calculating the vector product
with R-IV transforms an even field into an odd one,
v(x ,x ,x )=−v(x +π,x ,x ).
1 2 3 1 2 3 and vice versa. Thus, even and odd fields constitute
(NotethatthenonlinearityintheNavier–Stokesequa- invariant subspaces of the magnetic induction opera-
tion is not invariant for the antisymmetry of this torsLandL−. Byvirtueof(11)and(24),S areeven
n
7
for n = 1,2 and odd for n = 3, while Z are odd for involving β and β in the l.h.s. of (22) and (23), re-
l p t
l = 1,2 and even for l = 3. Vector multiplication by spectively, vanish, and therefore these equations yield
e maps odd fields into even ones and vice versa for the same eigenvalue
m
m=1,2, and it does not change this type of “parity”
(cid:88)
for m=3. Using this, it is easy to show that Λ=−η+(q12+q22)−1 DlmnTlPnqm.
m,l,n
D2 =D3 =0. (35)
11 32 By virtue of (21), (32), (34), (35) and (37),
4. Swapping of the horizontal coordinates x1 ↔x2. Λ=−η+D312(q12+q22)+D231q32.
Since the flow and solutions S and S to auxiliary
1 2
problemsoftypeIareindependentoftheverticalcoor- Actually, we have calculated the symbol of the eddy
dinate, equationsforhorizontalcomponentsofS and diffusivity operator acting on mean fields (defined by
1
S involvetheverticalcomponentsneitheroftheflow, the l.h.s. of (16)); hence this operator for R-IV (30) is
2
nor of the respective Sn. We establish by inspection (cid:18) ∂2 ∂2 (cid:19) ∂2
that the field (S2(x ,x +π),S1(x ,x +π)) satisfies (η−D3 ) + +(η−D2 ) . (38)
2 2 1 2 2 1 12 ∂X2 ∂X2 31 ∂X2
the same equation as (S1(x),S2(x)), and hence 1 2 3
1 1
The minimum eddy diffusivity is
S1(x ,x )=S2(x ,x +π),
1 1 2 2 2 1
S2(x ,x )=S1(x ,x +π). (36) ηeddy =η+min(−D312,−D231).
1 1 2 2 2 1
3.2. Numerical results
We use the second of these relations to show that
Thecoefficientsη−D3 andη−D2 oftheeddydif-
D3 =D2 . (37) 12 31
21 13 fusivity operator (38) have been computed using (25).
√ SolutionsS toauxiliaryproblemsoftypeI,andsolu-
n
Denote ψ = 2sinx sinx ; clearly,
1 2 tionsZ toauxiliaryproblemsfortheadjointoperator
l
havebeencomputedbyoptimisediterations(Zheligov-
(v1,v2,0)=∇ ×(ψe ).
x 3 sky 1993) as the dominant (associated with the zero
eigenvalue)eigenfunctionsoftheoperatorsofmagnetic
Since the flow is independent of x , for n = 3 the
3 induction L (12) and L− (26). Iterations were termi-
source term in the r.h.s. of (11) vanishes, and hence
S− = 0. Therefore, by virtue of (27) for l = 3, nated when the estimate of the dominant eigenvalue
3 was below 10−10 in absolute value and the norm of
Z = (2η)−1∇ ψ. Since gradients are orthogonal
3 x
the discrepancy for the normalised associated eigen-
to solenoidal fields in the Lebesgue space, in expres-
vectorwasbelow5·10−11. Aresolutionof642 Fourier
sion (25) for D3 the term involving the derivative
21 harmonics was used before dealiasing, that was per-
∂S /∂x is zero. Hence, on the one hand,
1 2
formed by discarding harmonics with wave numbers
D3 =(2η)−1(cid:104)∇ ×(ψe )·(v×(S +e ))(cid:105) over 28. With this resolution, energy spectra of solu-
21 x 2 1 1
tionstoauxiliaryproblemsdecayby30ordersofmag-
=(2η)−1(cid:104)ψe ·∇ ×(v×(S +e ))(cid:105)
2 x 1 1 nitude for η = 0.2 and still by 4 orders for η = 0.01.
=−(2η)−1(cid:10)ψe2·η∇2xS1(cid:11) Plots of η−D312 and η−D231 are shown in Fig. 1 for
=(cid:10)ψS2(cid:11). 0.01 ≤ η ≤ 0.2; in this range of molecular diffusivi-
1
ties no generation of small-scale magnetic fields takes
Wehaveusedheretheself-adjointnessofthecurl,(12) place.
forn=1andtheself-adjointnessoftheLaplacian. On Figure 1 implies that a large-scale magnetic field
the other, by the self-adjointness of the curl and by is not generated for horizontal wave vectors q of the
virtue of (25), (31) for l = 2, (12) for n = 2 and the harmonic large-scale modulation, but it is generated
relation S− =0, for the vertical wave vector. We did not check if gen-
3
eration of small-scale fields starts on further decreas-
D2 =(cid:104)Z ·(e ×(v×e ))(cid:105)
13 2 1 3 ing the molecular diffusivity; the behaviour of plots
(cid:68) (cid:69)
=−η−1 ∇−x2(v×(S(cid:98)2+e2))·(e1×∇xψ) iηn−FDig3. 1→su−g∞gesatnsdthηa−t Dit2m→ay∞takwehpenlacηea,papnrdoatchheens
(cid:68) (cid:69) 12 31
=η−1 ∇−x2(η∇2xS(cid:98)2)·ψe1 the critical value for the onset of small-scale magnetic
fieldgeneration. Ifthishappens,thetypeofthegener-
=(cid:104)ψS(cid:98)1(cid:105). atedlarge-scalefieldchanges: forsmallerη,generation
2
of large-scale magnetic field for the vertical wave vec-
Thus, (37) follows from (36).
tor q replaces the one for horizontal wave vectors.
5. Eddydiffusivity. Wecalculatenoweigenvaluesof Plotsofthetwoentries,η−D3 andη−D2 ,ofthe
12 31
the eddy diffusivity operator (16). By (32), the sums
eddy diffusivity tensor are shown in Fig. 1 for a range
8
from that of η−D2 presented in Fig. 1.
31
4. Generation of large-scale magnetic field
by the modified Taylor–Green flow
As Lanotte et al. (1999) and Devlen et al. (2013),
wenowconsiderlarge-scalegenerationbythemodified
Taylor–Green flow (mTG), whose components are
v =sinx cosx cosx +asin2x cos2x
1 1 2 3 1 3
+ bcosx (sinx cos3x +csin3x cosx ),
3 1 2 1 2
η v =−cosx sinx cosx +asin2x cos2x (39)
2 1 2 3 2 3
−bcosx (cos3x sinx +ccosx sin3x ),
3 1 2 1 2
v =−asin2x (cos2x +cos2x )
3 3 1 2
+ dsinx (cosx cos3x −cos3x cosx ).
3 1 2 1 2
Theflowisincompressibleford=b(3c−1),whichwill
be henceforth assumed. We now consider its symme-
tries relevant for simplification of the eigenvalue prob-
lem (22)–(23) and calculate the eigenvalues.
4.1. The effect of symmetries
1. Symmetries in x . A field f = (f1,f2,f3) is
i
called symmetric in x , if for all i and j such that
i
1≤i,j ≤3
η
fj((−1)δi1x1,(−1)δi2x2,(−1)δi3x3)=(−1)δijfj(x)
(cf. (33)), and antisymmetric in x , if for all such i
Figure 1. Entries η−D3 (upper panel) and η−D2 i
12 31 and j
(lower panel) of the eddy diffusivity operator (38) for
R-IV (30). fj((−1)δi1x1,(−1)δi2x2,(−1)δi3x3)=(−1)1−δijfj(x),
whereδj istheKroneckersymbol. SincemTGissym-
of molecular diffusivities over the critical value for the i
metric in all x , it is parity-invariant and lacks an
onset of generation of the small-scale magnetic field. i
α-effect.
The form (38) of the operator of eddy diffusivity cor-
roborates the conclusions of Devlen et al. (2013) that When a flow is symmetric in xi, vector fields pos-
theeddydiffusivitytensorforR-IVisdiagonalandhas sessingthesymmetryorantisymmetryinxi constitute
a double eigenvalue, i.e., its action on fields depend- invariantsubspacesoftheoperatorsofmagneticinduc-
ingontheverticalslowvariable(whichwastheobject tionLandL−. Sinceallthethreesymmetriesinxiare
of the studies of G.O. Roberts 1972 and Devlen et al. independent, there are eight such invariant subspaces.
2013) is homogeneous. However, since the two coeffi- We label them by 3-character strings; A and S in the
cients in (38) are distinct (see Fig. 1), eddy diffusivity i-thentryofthelabelindicatethatvectorfieldsinthe
is anisotropic, differing in the vertical and horizontal invariant subspace are symmetric or antisymmetric in
directions. Comparison of the lower panel of Fig. 3 xi, respectively. For instance, SAA labels the invariant
in Devlen et al. (2013) with the plot of η − D2 in subspace, in which vector fields are symmetric in x1
31
the lower panel of Fig. 1 reveals a reasonable qualita- and antisymmetric in x2 and x3.
tiveconsistencybetweenthelarge-scalemagneticfield By virtue of (11) and (24), invariance of the fields,
growthrates,obtainedbyDevlenetal.(2013)inDNS, symmetric or antisymmetric in x implies that S for
i n
andtheMSTminimumeddydiffusivityvalues, shown n(cid:54)=i and Z are symmetric in x , while S and Z for
i i i l
in thelowerpanel of Fig.1, for roughly η <0.1. How- l(cid:54)=iareantisymmetricinx . Consequently, Dl =0
i mn
ever, while here we study eddy diffusivity in the limit when none of the indices l,n and m are equal to i. It
ε→0, Devlen et al. (2013) computed turbulent mag- follows
netic diffusivity for finite scale separations; in partic-
ular, the plot in Fig. 3 ibid. shows the diagonal entry Dlmn =0 if m=n, or l=m, or l=n. (40)
of η(ε) for ε = 1, whose behaviour is clearly different
9
intoanother,aswellasASSandSAS.Sinceitalsomaps
A(cid:48)
aneigenfunctionoftheoperatorofmagneticinduction,
L, to an eigenfunction, restrictions of L on the two
invariant subspaces, constituting any of the two pairs,
x
3
x2 have the same spectra. The subspaces AAA, AAS, SSA
O(cid:48) andSSSareinvariantundertheactionofthesymmetry
π
π A B(cid:48) γ; each of them splits into invariant subspaces of L,
ζ3 that consist of γ-symmetric or γ-antisymmetric fields.
3. Wave number parity. Inspection of (39) reveals,
ζ
1
that mTG is comprised of Fourier harmonics eik·x in
C(cid:48)
O Q B which all the three wave numbers ki have the same
π 2π x1 parity, e.g., the sum of any two wave numbers is even.
Consequently, the obvious periodicity cell T3 of the
flow,whichisacubeofsize2πwhoseedgesareparallel
ζ
2
to the Cartesian coordinate axes, is not the smallest
π one. It is easily seen that a flow possessing the parity
C
property of this kind is invariant under shifts along
any of the periodicity vectors
Figure 2. An elementary periodicity cell of mTG (39):
a prism whose edges are periodicity vectors ζ (42).
i ζ =(π,π,0), ζ =(π,−π,0), ζ =(π,0,π).
1 2 3
The vertex O(cid:48) of the upper square base O(cid:48)A(cid:48)B(cid:48)C(cid:48)
(42)
projects down along the vertical into the centre Q of
(Clearly,thistranslationinvarianceimplies2π-periodi-
the lower square base OABC of the prism.
city in any Cartesian variable x .) Therefore, elemen-
i
taryperiodicitycellsoftheflowareprismswhoseedges
are these vectors (see Fig. 2). Alternatively, one can
2. Swapping of the horizontal coordinates x ↔x .
1 2
regard the parallelepiped
ThemTGhasalsoasymmetry,whichwedenotebyγ:
a field f is γ-symmetric, if
0≤x ≤2π, 0≤x ≤π, 0≤x ≤π
1 2 3
f1(x1,x2,x3)=f2(x2,x1,x3+π), asanelementaryperiodicitycelloftheflow, assuming
f2(x ,x ,x )=f1(x ,x ,x +π), the“brickwall”tilingofspacebythesecells, inwhich
1 2 3 2 1 3
theparallelepipedsarearrangedininfinite“bars”par-
f3(x ,x ,x )=f3(x ,x ,x +π),
1 2 3 2 1 3 allel to the x -axis, and any two adjacent bars are
1
shiftedalongthex -axisbyhalfaperiodrelativeeach
and γ-antisymmetric, if 1
other. The volume of the elementary periodicity cells
f1(x ,x ,x )=−f2(x ,x ,x +π), ofbothtypesis2π3,e.g.,aquarterofthatofT3. Nev-
1 2 3 2 1 3
ertheless, by a small-scale dynamo we understand the
f2(x ,x ,x )=−f1(x ,x ,x +π),
1 2 3 2 1 3 generation of magnetic fields which are 2π-periodic in
f3(x1,x2,x3)=−f3(x2,x1,x3+π). each variable xi.
Each invariant subspace of L considered above fur-
γ-symmetric and γ-antisymmetric fields constitute in-
thersplitsintosubspacesoftheso-calledevenandodd
variant subspaces of the operators of magnetic induc-
fields that are linear combinations of Fourier harmon-
tion L and L−. This implies that S is γ-symmetric,
3 icssuchthatthesumsofthewavenumbersk +k are
Z is γ-antisymmetric, for n = 1,2 the field S i j
3 n even or odd. We therefore extend the labels of invari-
is mapped by the γ-symmetry to S , and Z is
3−n n ant subspaces by two additional characters denoting
mapped by the γ-antisymmetry to Z . (We thus
3−n the parity of the sums k + k of wave numbers in
need to compute just 4 solutions to the auxiliary 1 2
the horizontal directions (the fourth character), and
problems, say, S , S , Z and Z ; S and Z can
1 3 1 3 2 2 the sums k + k of wave numbers in directions x
then be obtained by applying the γ-symmetry and 1 3 1
and x (the fifth character); E and O indicate even or
γ-antisymmetry to S and Z , respectively.) Conse- 3
1 1 odd such sums, respectively. For instance, the invari-
quently, the remaining non-zero entries of the eddy
ant subspace SAAOE consists of vector fields that are
diffusivity correction tensor satisfy the relations
symmetric in x , antisymmetric in x and x , and are
1 2 3
D3 =−D3 , D1 =−D2 , D2 =−D1 . (41) comprised of Fourier harmonics such that the sum of
12 21 23 13 31 32
wave numbers in the horizontal directions is odd and
When the γ-symmetry acts on a vector field, the thesumk1+k3 iseven; thespectrumofListhesame
symmetryorantisymmetryinx becomesasymmetry in this subspace and in ASAOO.
1
or antisymmetry, respectively, in x , and vice versa. 4. Eddy diffusivity. For an eddy diffusivity cor-
2
Thus,theγ-symmetrymapsASAandSAAmutuallyone rection tensor with the properties (40) and (41) stem-
10