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Geophysical & Astrophysical Fluid Dynamics
ISSN: 0309-1929 (Print) 1029-0419 (Online) Journal homepage: http://www.tandfonline.com/loi/ggaf20
Magnetic helicity generation in the frame of
Kazantsev model
Egor V. Yushkov & Alexander S. Lukin
To cite this article: Egor V. Yushkov & Alexander S. Lukin (2017): Magnetic helicity generation
in the frame of Kazantsev model, Geophysical & Astrophysical Fluid Dynamics, DOI:
10.1080/03091929.2017.1388376
To link to this article: http://dx.doi.org/10.1080/03091929.2017.1388376
Published online: 23 Oct 2017.
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Download by: [UAE University]
Date: 25 October 2017, At: 15:52
GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS, 2017
https://doi.org/10.1080/03091929.2017.1388376
Magnetic helicity generation in the frame of Kazantsev model
Egor V. Yushkova,b and Alexander S. Lukina,b,c
Downloaded by [UAE University] at 15:52 25 October 2017
a Faculty of Physics, Department of Mathematics, Lomonosov University, Moscow, Russia; b Space Research
Institute RAS (IKI), Moscow, Russia; c National Research University Higher School of Economics, Moscow, Russia
ABSTRACT
ARTICLE HISTORY
Using a magnetic dynamo model, suggested by Kazantsev (J. Exp.
Theor. Phys. 1968, vol. 26, p. 1031), we study the small-scale
helicity generation in a turbulent electrically conducting fluid. We
obtain the asymptotic dependencies of dynamo growth rate and
magnetic correlation functions on magnetic Reynolds numbers.
Special attention is devoted to the comparison of a longitudinal
correlation function and a function of magnetic helicity for various
conditions of asymmetric turbulent flows. We compare the analytical
solutions on small scales with numerical results, calculated by
an iterative algorithm on non-uniform grids. We show that the
exponential growth of current helicity is simultaneous with the
magnetic energy for Reynolds numbers larger than some critical value
and estimate this value for various types of asymmetry.
Received 8 June 2017
Accepted 2 October 2017
KEYWORDS
Small-scale dynamo;
Kazantsev model;
asymptotic and numerical
analysis; large magnetic
Reynolds number
1. Introduction
The motion of a conducting fluid with frozen-in magnetic field can support the growth of
average field characteristics such as magnetic energy B 2 and current helicity B·[∇ ×
B] (Landau et al. 1960, Berger 1999, Brown et al. 1999). This process, usually called
a magnetic dynamo, is assumed as the most probable mechanism for the formation of
the magnetospheres of planets, stars and galaxies (see e.g. Molchanov et al. 1985). The
traditional method of dynamo analysis for these astronomical objects is connected with
spatial averaging on a typical correlation length of a random velocity field. Of course,
such averaged problems can describe only large-scale effects, however a general analysis
without spatial averaging is also possible (Kraichnan 1968, Moffatt 1978, Subramanian
1997, Subramanian 2003). This approach allows us to study the dynamo not only on large
scales but also for small dissipation. However, as this is more complex and problematical,
in our work we consider only the simplest case of turbulent isotropic flows with very short
velocity correlation times.
The use of a delta-correlated velocity field model with small, but equal (for all scales)
correlation time intervals was suggested by Kazantsev (1968). The equation derived within
this framework, describes the small-scale turbulent dynamo. For a long time it was considered as fundamental and was studied by both analytical and numerical methods (see
e.g. Novikov et al. 1983, Molchanov et al. 1985). In particular, for a mirror-symmetric
CONTACT Egor V. Yushkov
yushkov.msu@mail.ru
© 2017 Informa UK Limited, trading as Taylor & Francis Group
2
E. V. YUSHKOV AND A. S. LUKIN
isotropic velocity field the exponential temporal growth of the magnetic energy had
been demonstrated for large enough Reynolds numbers (see Zeldovich et al. 1990, and
references therein). Meanwhile, the question about the small-scale generation of the
averaged magnetic field in asymmetric flows had not been considered until Vainshtein
and Kichatinov (1986) (and a little bit later, but independently by Subramanian 1997),
who deduced the system, which describes not only symmetric but also an asymmetric part
of correlation tensor:
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where
Mt = 2r −4 (r 4 ηMr )r + 2Mr −4 (r 4 ηr )r − 4αK ,
Kt = r −4 r 4 (αM + 2ηK)r r ,
(1b)
α(r) = G0 − G(r) .
(1c)
(1a)
The system of Equations (1a,b) unequivocally defines the evolution of a longitudinal
correlation function M(r, t) and a helicity correlation function K(r, t) depending on the
magnetic viscosity η(r) = 1/Rm + F0 /3 − F(r)/3 and on the correlation functions F(r)
and G(r) for a turbulent incompressible liquid:
i
rirj
rirj v (r, t) v j (0, 0) = F + 12 rFr δ ij − 2 + F 2 + G ijk r k δ(t) .
r
r
(2)
However its analysis had encountered many obstacles, which have not been resolved
yet. For example, we are unsure whether or not growth of anti-symmetric part of magnetic
tensor is possible. Does the critical value of Reynolds number (when field a generation
begins) change significantly with growth of asymmetry? Is the current helicity comparable
with magnetic energy on small-scales or can it be neglected? etc. Here we show that by
solving the system (1a,b) and calculating the magnetic energy characteristics B 2 and
B·[∇ × B] from the functions M(r, t) and K(r, t) that define
i
rirj
rirj B (r, t) Bj (0, 0) = M + 12 rMr δ ij − 2 + M 2 + K ijk r k ,
r
r
(3)
we can obtain answers not only to the question about the possibility of magnetic helicity
generation, but also about its dependence on the flow helicity and magnetic viscosity.
The cornerstone of our approach is the large magnetic Reynolds number realised by
almost all physical dynamo-systems (e.g. the value of Rm for the Earth is about 102 −104 , for
the galaxies about 105 −106 , and for the Sun 107 −108 ). Therefore, the direct solution of the
Kazantsev system can be accomplished by standard large parameter asymptotic analysis.
In the mirror-symmetric case such asymptotics has been undertaken by Molchanov et al.
(1985), Artamonova and Sokolov (1986) and Rogachevskii and Kleeorin (1997), while for
asymmetric motion the problem had been studied by Boldyrev et al. (2005), Malyshkin
and Boldyrev (2007) and Yushkov and Lukin (2017). Without recalling the specific details
of the previous work, we simply note that earlier results do not allow analysis of the
functions M(r, t) and K(r, t) in the regime, where their behaviour is the most interesting
and obscure. It was shown by Yushkov and Lukin (2017) that for large Rm the transformed
correlation functions can grow on the interval r ∈ [Rm−1/2 , 1], but near zero and for large
scales the answer to the question remained unclear. Moreover we did not analyse exactly
GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS
3
the functions M(r, t) and K(r, t) (only transformed ones), although some ideas about their
behaviour had been proposed. Below we show that the developed ideas and asymptotic
accuracy of the previous studies can be improved significantly, and analytical analysis
of magnetic characteristics near r = 0 can be achieved. Moreover, corrected numerical
schemes on non-uniform grids can confirm the constructed asymptotics. They can also be
used in the cascade models and for comparison with experimental data for each particular
turbulent conditions.
2. Transformation of Kazantsev system and previous results
We rewrite the Kazantsev system (1a,b), using new notation and by assuming the exponential ∼ exp (2γ t) time dependence for the correlation functions:
Downloaded by [UAE University] at 15:52 25 October 2017
M(r, t) = e2γ t r −2 η−1/2 φ(r)
and K(r, t) =
1 2γ t −4
r
2e
4 −2
r r θ(r) r r .
(4a,b)
This traditional approach leads us to the Sturm–Liouville problem (see, e.g. Hazewinkel
2001) for the system
γ φ = ηφrr + U(r)φ − δη θrr − 2r −2 θ ,
γ θ = η θrr − 2r −2 θ + δφ + C1 r 2 + C2 r −1 ,
(5a)
(5b)
where
U(r) =
2ηr
2η
η2
ηrr
+
− 2 + r
2
r
r
4η
and δ(r) = G0 − G(r) η−1/2 .
(5c,d)
The solution of (5) defines the non-zero eigenvalue functions φ(r), θ(r) and correlation
functions M(r, t), K(r, t), which increase in time with the maximum rate 2γ and dominate
within magnetic correlation tensor after a sufficiently long time. Note that the (5b) follows
from (1b) after double integration, and the constants of integration C1 and C2 can be
removed by adding contributions to the function θ(r) proportional to r 2 and r −1 , which
cannot change the helicity function K(r, t) (see the transformation (4b)).
We assume that a symmetric part of correlation tensor for the velocity field in a random
flow has the classical Gaussian form (for more details see Novikov et al. 1983, Artamonova
and Sokolov 1986):
(r 3 F)r
(6a)
f (r) =
3r 2
with either
(6b,c)
f (r) = exp ( − r 2 ) or F(r) = exp ( − 3r 2 /5) .
Then on only retaining logarithmic accuracy order, we consider problem (1a) on three
√
√
distinct intervals r ∈ [0, ε ], r ∈ [ ε, 1] and r ∈ [1, ∞), where ε = 1/Rm. Such intervals
are convenient for asymptotic expansion of the functions F(r) and η(r). Indeed, for large
r 1 (6) define rapid decreasing functions F(r) and, consequently, η ∼ 1/3. By contrast,
for small r 1 the function η(r) increases and can be expanded near r = 0 by the Taylor
series
3r 4
r2
−
+ o(r 4 ) .
(7)
η = ε+
5
50
4
E. V. YUSHKOV AND A. S. LUKIN
Using this expansion, we rewrite the potential U(r) (see (5c)) in the form
U(r) = −
2ε 3 48r 2
1/5 − 2r 2 /50
−
+
+
+ o(r 4 ) .
r2
5
50
1 + 5ε/r 2
For a random isotropic and reflection-invariant (δ = 0) velocity field the system
(5a,b) decays into two independent equations and the problem becomes much easier.
Equation (5a) containing the potential U(r) can be considered equivalent to the stationary
Schrodinger problem with variable mass. Its solution gives the asymptotics of the longitudinal correlation function M(r, t) (studied earlier by Artamonova and Sokolov 1986):
Downloaded by [UAE University] at 15:52 25 October 2017
2π ln r
exp (2γ t)
.
M(r, t) = 3/2 1/2 sin
r η
ln ε
(8)
Solving (8) subject to zero boundary conditions, we find that the dynamo rate γ depends
only on magnetic Reynolds number Rm = ε−1 in the form
3 1
γ = −
4 5
2π
ln ε
2
with Rmcr = exp 2π
4
15
.
(9)
We obtain the growth rate γ and the critical value of Reynolds number Rmcr , from which
the dynamo-process begins (γ = 0) within the asymptotic framework of weak logarithmic
convergence. However, we have successfully verified this asymptotic result by numerical
methods, which use the algorithm proposed by Novikov et al. (1983) for this problem.
Note also that the obtained asymptotic function (8) is non-zero only in the middle interval
√
r ∈ [ ε, 1] (outside this interval φ(r) ≡ 0). Thus the correlation function M(r, t) is
undefined for small r, and so the function B(0)·B(r) can not be constructed and the
accuracy of solution is insufficient for magnetic energy analysis. Besides, the solution of
the second Equation (5b), with δ = 0 and for positive γ , is zero. Therefore both the helicity
function K(r, t) and current helicity B·[∇ × B] in the mirror-symmetric flow decays.
Below we show that, even for a weakly-asymmetrical flows δ = 0, the situation totally
reverses.
3. Solutions of Kazantsev system at three intervals
We consider the large r interval r ∈ [1, +∞). We assume that γ 2r −2 , η = 1/3, and
rewrite the system (5a,b) in the form
φrr = 3(γ − δ 2 )φ + 3γ δθ ,
(10a)
θrr = − 3δφ + 3γ θ .
(10b)
For each particular random velocity field the flow helicity function G(r) has its own specific
form. However the
√ idea of rapid decay for r 1 seems reasonable, and so
√the asymmetry
parameter δ ∼ 3(G0 − G(r)) can be equated here to the constant 3G0 . Then the
solution of the system (10) can be written as a combination of exponentials exp (λr), in
GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS
5
which the values of λ are the roots of the characteristic equation (see, e.g. Kamke 1976)
λ4 + (3δ 2 − 6γ )λ2 + 9γ 2 = 0 .
(11)
It has four roots, two with positive and two with negative real parts:
3δ 2
λ2 =
2
2γ
−1±i
δ2
4γ
−1
δ2
=⇒
⎞
⎛
2
2
δ
δ
⎠.
±i
λ = ± 3γ ⎝ 1 −
4γ
4γ
Downloaded by [UAE University] at 15:52 25 October 2017
(12a,b)
Obviously we consider the solutions decaying at infinity. For that reason,
the
asymmetry
√
parameter needs to be sufficiently small: δ 2 < 4γ . The exponents with λ = − 3γ ( · ± i · )
from (12b) determine solutions
r
4γ
r
√
+
F
,
cos
−
1
φ = E γ exp −
δ2
l
l
⎤
⎡
r
r
r ⎣ δ2
δ2
4γ
cos
+F + 1−
sin
+ F ⎦,
θ = E exp −
−1
δ2
l
4γ
l
4γ
l
(13a)
(13b)
where
l=
4
.
3δ 2
(13c)
The equalities (13a,b) show that, by contrast to the mirror-symmetric case, the
monotonic decay of function φ(r) is absent and the dynamo oscillates in throughout
entire space with the large wavelength 2πl, where l = 2/(3G0 ). Substituting the value
η = 1/3 for the magnetic diffusivity, we see that this results remarkably coincides with a
typical scale of large-scale dynamo process = 2η/G0 (for more details, see Vainshtein
1970, Moffatt 1978). However, further for simplicity we neglect terms proportional to 3δ 2
in (11) in comparison with terms proportional to γ . The approximation leaves us with
non-oscillatory exponentially decaying solutions
φ = E1 exp − 3γ r
and θ = E2 exp − 3γ r .
(14a,b)
√
We consider the middle interval r ∈ [ ε, 1]. For that we use the expansion (7) for the
magnetic diffusivity and note that r 2 /5 ε. Then from(5a,b) we obtain the system
δ
4
r2
φrr + φ − r 2 θrr − 2θ = γ φ ,
5
5
5
1 2
r θrr − 2θ + δφ = γ θ .
5
(15a)
(15b)
6
E. V. YUSHKOV AND A. S. LUKIN
Upon making the substitutions φ = ey/2 ϕ, θ = ey/2 ϑ and y = ln r the system (15)
becomes
15
ϕyy = 5γ − 5δ 2 −
ϕ + 5γ δϑ ,
(16a)
4
9
ϑyy = − 5δϕ + 5γ +
ϑ.
(16b)
4
Downloaded by [UAE University] at 15:52 25 October 2017
The coefficients in the system depend on the helicity function, because δ(r) = (G0 −
√
2
G(r))/ η. Upon taking into account
√ that G(r) ∼ G0 +√G0 r and η ∼ r /5, we find that
the asymmetry parameter δ ∼ 5 G0 − G(r) /r ∼ − 5G0 can be assumed constant.
So in the first approximation the solution of (16) can be presented as the combination of
exponentials exp (λr), where each λ is a root of the characteristic equation
45δ 2
135
3
15
λ2 + 25γ 2 − γ −
−
= 0,
λ4 − 10γ − 5δ 2 −
2
2
4
16
(17a)
which determines
5
3
λ2 = 5γ − δ 2 − ±
2
4
5 2
δ +3
2
2
− 25γ δ 2 .
(17b)
For small mirror-asymmetry γ δ 2 < (3/5 + δ 2 /2)2 the square root in (17b) can be
expanded to give two roots with different signs:
λ2 = 5γ
6
9
+ > 0
2
6 + 5δ
4
and
λ2 = 5γ
6 + 10δ 2
15 + 20δ 2
< 0.
−
6 + 5δ 2
4
(18)
From the previous result (8) and numerical solutions we know that the eigenfunction
φ(r) is nonmonotonic. So we choose λ2 < 0 and, consequently, obtain the solution of the
system (16) in the form
5
3
1/2
φ ∼ 5γ δr sin λ ln r + D ,
λ =
2 − 25γ δ 2 − 5γ + δ 2 + , (19a,b)
2
4
2
1
φ
, = 5δ + 6 .
(19c,d)
θ ∼ − 2 − 25γ δ 2
5γ δ
2
For weak asymmetry (δ 2 9/25γ ) the functions φ(r) and θ(r) can be approximated
by
φ = Cr 1/2 sin λ ln r + D ,
(20a)
5δ 1/2
θ = C r sin λ ln r + D .
(20b)
6
√
Finally, we consider the interval r ∈ [0, ε] near r = 0. Inspection of the asymptotics (20)
reveals that for small δ the ratio θ/φ is insignificant. Thus, the contribution of the last term
in (5a) can be ignored and for small r (γ 2r −2 ) the function φ(r) should satisfy the
equality
GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS
0 = εφrr − 2εr −2 φ .
7
(21)
After repeated integration of (21), we obtain φ = Ar 2 , where A is a constant of
integration. To determine θ, we consider (5b) by expanding the functions φ(r) and θ(r) in
terms of power series. We find that the first correction to θ(r) is proportional to ∼ r 4 , and
that the correction to φ(r) is smaller than ∼ r 5 . Thus, ignoring φ(r) in (5b) and assuming
that η ∼ ε near r = 0, we obtain the equation
γ θ = ε(θrr − 2r −2 θ) ,
which can be solved by standard methods (see, e.g. Kamke 1976, p. 401):
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θ =
1
1
γ
γ
γ
γ
−
exp
r +
+
exp −
r .
ε
r
ε
ε
r
ε
Saving only first two terms and approximating ε by η, we obtain the asymptotic result
γ r2
2
θ = Br 1 +
+ o(r ) .
10 η
2
(22)
Near r = 0 the first term dominates. So on the interval r ∈ [0,
solutions in the form
φ = Ar 2 ,
√
ε] we can rewrite the
θ = Br 2 .
(23a,b)
These solutions describe the general features of the helicity and longitudinal correlation
functions, but they contrast in an essential way with numerical results (figure 1, right
panel), as we will discussed below.
4. Conjunction at the boundaries
The simplest way to realise solution conjunction is to use the asymptotics (14), (20) and
(23). Accordingly we demand continuity of
⎧
2
⎪
⎨Ar ,
φ(r) = Cr 1/2 sin λ ln r + D ,
⎪
√
⎩
E exp − 3γ r ,
√
for r ∈ [0, ε],
√
for r ∈ [ ε, 1],
for r ∈ [1, ∞)
(24)
and its first derivations φ (r) at the boundaries of the intervals identified. The conditions
on the function θ(r) will be automatically fulfilled due to their proportionality:
θ
5
= δ
φ
6
√
[ ε,1]
B
=
A
√
[0, ε]
E2
=
E1
[1,∞)
=
5
− G0
√
6
5
.
(25)
If we take into account from our previous approximations that the constants A, E, D
and !
λ, where λ = 2!
λ + 2π/ ln ε, should be small, then from the boundary conditions we
determine the following equalities
8
E. V. YUSHKOV AND A. S. LUKIN
A = − Cε−3/4 !
λ ln ε + D , E = C exp ( 3γ )D,
4π
4π
= 3 !
= − (2 3γ + 1)D.
4!
λ+
λ ln ε + D , 4!
λ+
ln ε
ln ε
(26)
(27)
We solve (27) to obtain
−6π
, !
λ = −
D = √
√
(3 3γ + 3/2) ln ε − (4 3γ + 8)
√
3γ + 2
3
2D
.
ln ε
(28)
Note that these formulas follow strictly from the boundary conditions of function φ(r).
From (4a,b) and (5b) the correlation functions can now be expressed as
φ(r)
(29a)
√ ,
r2 η
√ φ(r) γ θ
φ(r) G0 r
5 5
K(r, t) = exp (2γ t) 2
− δ = exp (2γ t) 2
. (29b)
√ − γ G0
2r η φ
2r η
η
6
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M(r, t) = exp (2γ t)
A delicate matter arises here, because the equalities (25) show that the ratio θ/φ is
approximately constant on all intervals, while the numerical calculations demonstrate
√
different constants near r = 0 and on the interval r ∈ [ ε, 1]. This problem follows from
√
the conjunction condition at the point r = ε for two asymptotics, which are not exactly
√
√
√
correct at r = ε, but rather correct for small r ( ε) and for large r ( ε). The
disparity can be rectified by using the improved asymptotic result (22), in which the second
term will be retained. This improved result does not change the values A, C, E, λ, because
they were obtained only from φ-conditions, but it leads to complex questions about the
conjunctions for the derivatives of the function θ(r). Here, we ignore these additional
corrections due to their insignificance at large Rm and small asymmetry. Analysing (22)
√
at large r we see, that the ratio θ/φ converges to a constant for r ε. Thus we can
choose parameter B in such way to make this constant equal to the value 5δ/6 in the point
of conjunction (see the ratio of (20a,b)). In this way, we obtain the equality
√
B
γ r2
B
5γ
θ
5 5
=
1+
=⇒
1+
∼ − G0
(30)
φ
A
10 η
A
10
6
√
together with the corresponding constant B = −AG0 5 5/(6 + 3γ ) (cf. (25)). Thus, on
√
√
the both intervals r ∈ [0, ε] and r ∈ [ ε, 1] we can write the ratio in the form
√ γ r2
θ
5 5
= − G0
1+
,
(31)
φ
6 + 3γ
10 η
but disregarding it beyond these intervals due to the rapid exponential decay of θ and φ.
Therefore, for r ∈ [0, 1], the correlation functions take the asymptotic forms
φ(r)
√ ,
r2 η
√ "
#
G0 5 5
φ(r) G0 r
γ r2
K(r, t) = exp (2γ t) 2 √ − γ
1+
,
2r η
η
6 + 3γ
10 η
M(r, t) = exp (2γ t)
(32a)
(32b)
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GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS
9
Figure 1. On the left panel the eigenfunctions ψ(r) of problem (5) calculated from the initial
approximation (marked by points) on the half line [0, ∞] for Rm = 104 . On the right panel are
the longitudinal M(r) and helicity K(r) functions, following from the numerical results (marked by
points), and the analytical rough and improved asymptotics, obtained in (29), for Rm = 104 and marked
by dashed and solid lines respectively.
and much smaller oscillating functions for r ∈ [1, ∞) with periods typical for largescale dynamo. The constants in (29) and (32) are defined by the formulas (26)–(28). The
magnetic energy characteristics can be calculated as
B(r)·B(0) = 3M(r) + rM (r) ,
B(0)·[∇ × B](r) = 2 3K(r) + rK (r) .
(33a)
(33b)
Approximating λ = 2!
λ/ ln ε in (19a) and retaining accuracy up to order δ 2 , we can write
the growth rate γ in the form
6 + 5δ 2
γ =
6 + 10δ 2
3
1
+ δ2 −
4
5
2π
ln ε
2 ,
(34)
and estimate the critical Reynolds number as
2π
Rmcr = exp 15/4 + δ 2
,
where
δ 2 = 5(G0 )2 ,
(35)
Of course, these estimates are not very accurate, because they were obtained for small
values of the asymmetry parameter δ and for very large magnetic Reynolds numbers
(log Rm 5, in other words Rm 10000). Moreover, for γ close to zero the asymptotics
fails due to the loss of solution localisation and fast exponential decay for r → ∞, so
the more precise result can be obtained from (19b) numerically or from the numerical
experiments. However, the formulas (34) and (35) demonstrate the main features of the
dependence on the flow conditions: the saturation level of the dynamo rate increases and
the critical value decreases simultaneously with the mirror-asymmetry growth.
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10
E. V. YUSHKOV AND A. S. LUKIN
Figure 2. On the left panel the first and the second eigenvalues of problem (5) for various Rm obtained
for mirror-symmetric (9) and asymmetric (34) cases and marked by dashed and solid lines (points show
the numerical results). On the right panel are the first eigenvalues calculated for various helicity functions
G(r) of the velocity random field (solid and dashed lines show the mirror-symmetric (9) and asymmetric
(34) asymptotics and points show the numerical results: linear (large points), exponential (small points)
and gauss (circles).
Figure 3. On the left panel the energy B(0)·B(r) and helicity B(0)·[∇ × B(r)] characteristics
of generated magnetic field calculated by (42) from analytical (solid lines) and numerical (points)
results for Rm = 104 . On the right panel are the dependencies of the ratio of helicity and energy
B(0)·[∇ × B(0)]/B(0)2 for magnetic field plotted against the same diagnostic for the velocity field,
normalised by typical length Rm−1/2 and 1 respectively. Bold and thin solid lines show two cases of
Reynolds numbers Rm = 102 and Rm = 106 , and points show the numerical results.
5. Numerical analysis of Kazantsev system
The numerical analysis of the system (5) is based on the classical iterative schemes for
Sturm–Liouville problems. This method for dynamo models in mirror-symmetric flows
was proposed by Novikov et al. (1983), and later in Yushkov (2015), Yushkov and Lukin
(2017). It was developed for non-uniform grids and for weak mirror-asymmetry. Here
we use the reverse iteration method, presented in details in, e.g. Kalitkin et al. (2005),
GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS
11
Wilkinson (1965), and realised on the half-line with knots at the points ri∗ ∈ [0, ∞]:
ri∗ =
0.05ri
,
(1 − ri )2
where ri ∈ [0, 1] is a uniform grid.
(36)
Following the general idea of iterative approach, we chose the analytical results (8) and
(9) as the zero-step approximation (we have also tried to use the new asymptotics on the
initial step, but there were no significant differences), and then apply the iterative formula
(A − γ̄ E)ψ (s+1) = ψ (s) ,
(37)
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in which the meaning of the functions on (s+1)- and (s)-step is defined by the unit matrix
E and composite matrix of differential operator in (5a,b):
A11 A12
,
A=
A21 A22
(38a)
where
A11
d2
= η 2 + U(r) ,
dr
A21 = δ ,
A12
A22
d2
−2
= − δη
− 2r
,
dr 2
2
d
=η
− 2r −2 .
dr 2
(38b,c)
(38d,e)
The difference derivatives are used with shifted steps common for non-uniform grids and
unbounded intervals (Kalitkin et al. 2005):
un+1 − un−1
,
2(xn+1/2 − xn−1/2 )
un+1 − un
1
un − un−1
un =
−
.
2(xn+1/2 − xn−1/2 ) xn+3/4 − xn+1/4
xn−1/4 − xn−3/4
un =
Using the eigenfunction ψ (s+1) calculated by (37), we obtain the eigenvalue γ (s+1) after a
substitution of Aψ (s+1) = γ (s+1) ψ (s+1) into (37) and minimisation of the norm
$
$ (s+1)
$(γ
− γ̄ )ψ (s+1) − ψ (s) $.
(39)
γ̄ ) (see$ (39)), so to
Note that at each step we derive ψ (s) via the very small value (γ (s+1) −%$
(s+1) = ψ (s) $ψ (s) $. The norm
avoid
the
unlimited
growth,
we
make
the
normalisation
ψ
$$
$·$ can be chosen for any linear space, e.g. in Novikov et al. (1983) the space of continuous
functions was used, while here we use the norm for integrable functions:
$ $2
$u$ = (u, u ),
&
where (u, w) =
u(r)w(r)dr ,
(40)
12
E. V. YUSHKOV AND A. S. LUKIN
because for our examples this choice exhibits better numerical convergence and stability.
The minimisation (39) gives the eigenvalue at the next step
γ s+1 = γ̄ +
(ψ (s+1) , ψ (s) )
(ψ (s+1) , ψ (s+1) )
(41)
and allows us to repeat the iterative process from the beginning. Confining ourselves
only by ten iterations, we obtain and present results of numerical calculations for various
random velocity fields and wide range of Reynolds numbers on the figures 1–3 and
speculate on them in the next section.
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6. Discussion and conclusions
The studying of small-scale dynamo processes in the frame of the Kazantsev model shows
that in mirror-asymmetric flow the current helicity can grow simultaneously with the
averaged magnetic energy. This dynamo effect can be observed only for large Reynolds
numbers Rm and can be analysed by both numerical and analytical methods. By constructing the analytic asymptotics for Rm 1 and considering weak asymmetry, we verify the
analytic results by the numerical approach. The clear coincidence confirms the obtained
dependencies on random velocity field parameters.
For asymptotic analysis of Kazantsev problem we use ideas suggested for the symmetric
case by Artamonova and Sokolov (1986). Using special substitutions (4), we rewrite
√
√
the Kazantsev system (1a,b) on the three intervals [0, ε], [ ε, 1] and [1, ∞), obtain
solutions and blend them on the boundaries. This conjunction gives the asymptotics of the
dynamo growth rate γ and eigenfunctions, from which the magnetic correlation tensor
Bi (r, t)·Bj (0, 0) can be restored. Of course, the results depend on the longitudinal and
helicity correlation functions of the random velocity field, so we consider analytically a
special case, namely the longitudinal function is chosen to be of Gaussian form (this is
a general way, used by Novikov et al. (1983), Artamonova and Sokolov (1986), Yushkov
(2015), and for helicity we take a linear dependence on r as the first approximation. More
complicated and particular cases were examined only by the numerical approach.
Our numerical method, adapted for the modified Kazantsev system (5), is based on the
iterative schemes for Sturm–Liouville problems, earlier applied by Novikov for analogous
but symmetric cases. Using the non-uniform grid (36) and shifted derivatives, we reveal
fast convergence of the process for a wide range of initial approximations. The example
of convergence (for ten iterations) in asymmetric case from the initial approximation
in the form of the old asymptotics (8) is presented on the figure 1 (left panel). Using
numerically obtained eigenfunctions, the longitudinal and helicity correlation functions,
M(r) and K(r), can be calculated by (4) and compared with the analytical curves. On
figure 1 one can see that the symmetric part of the correlation tensor ∼ M(r) monotonically
√
decreases on the typical scale r ∼ ε and the asymmetric part ∼ K(r) changes sign on this
interval; that probably follows from the closing of magnetic field line loops. Comparing
the right panel of figure 1 and the left panel of figure 2, we see that the agreement between
numerical and analytical results is quite good not only for correlation functions, but also
for the magnetic energy growth rate, especially at large Rm. Analysing a wide range of
asymmetry parameters, we see that, just as in the symmetric cases, generation is possible
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GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS
13
only at large Rm > Rmcr . This critical value Rmcr decreases with asymmetry growth
(35) and simultaneously the maximum value of the dynamo growth rate for Rm → ∞
increases (34). So we can claim that the small-scale dynamo becomes more active for more
asymmetric flows. Note also that the small-scale Kazantsev model for asymmetric flow has
many features, which are intrinsic to the mean-field dynamo. For example, the typical scale
of oscillations δ = 2η/G0 (see (13a)) and the typical growth rate γ = G02 /4η (when the
process loses its localisation near the dissipative scales, see (13b)) precisely correspond to
the typical times and distances of the large-scale Steenbeck–Krause–Radler model (more
details for this mean-field dynamo model in the 3D space can be found in Yushkov (2014)).
Thus, we can suppose that the advocated Equations (1a,b) allow us to study not only the
helicity growth near the small scales, but also to investigate the role of this helicity in
large-scale processes.
The constructed asymptotic dynamo rate (34) is defined by two parameters: the magnetic Reynolds number Rm and the inclination of flow helicity function G (0), where
G(r) ∼ V ·[∇ × V ]. In our analytical case this function was linear with constant tilt from
r = 0 until r = 1, however the numerical approach allows us to consider other cases, for
example, exponential G(r) ∼ exp ( − r) or Gaussian G(r) ∼ exp ( − r 2 ) functions and
make sure that the main contribution in helicity generation is provided by the behaviour
√
of G(r) only near zero r < ε. Indeed, there are almost no differences on the right panel
of figure 2 between linear and exponential form of G(r), which are close to each other on
√
√
the interval [0, ε], while for Gaussian form of G(r) almost zero, meaning on [0, ε] the
growth rate is close to symmetric case even for large Rm. Therefore, we must emphasise
that for small scales the dynamo should be analysed in details for each particular random
fields, and our results are correct only as the first step.
The restored correlation tensor of magnetic field (3) and its derivatives allow us to
estimate by (33) the natural physical characteristics, for example, energy and current
helicity dependencies B(0)·B(r) and B(0)·[∇ × B(r)]. The left panel of figure 3
shows that except of points of conjunction (where the small “peaks” arise from the nonideal nature of matching conditions and calculated derivatives) they repeat the correlation
√
functions M(r) and K(r), localised on the dissipative scale r ∼ ε. However the influence
of this generated current helicity can be important not only in the vicinity of the dissipative
scales, but also for the large-sale dynamo. The results obtained show that the ratio of
magnetic helicity and energy is quite significant
√
B·[∇ × B]
−1/2
5 5γ
.
(42)
= − G0
Rm
6 + 3γ
B 2 Indeed, the value (42), multiplied by the scale of the correlation function localisation
√
ε = Rm−1/2 , is dimensionless and it is only twice smaller than the same ratio for velocity
field normalised on the typical scale of the velocity random field r = 1 (see the curves on
the right panel of figure 3). Moreover, knowing that there are no limitations on the helicity
transport toward the larger scales, we should suspect that the small-scale contribution in
the total magnetic dynamo can play a serious role in the large-scale helicity balance and
nonlinear mean-field dynamo suppression, so it should be additionally studied both in the
numerical cascade models and by the real laboratory experiments.
14
E. V. YUSHKOV AND A. S. LUKIN
Acknowledgements
The authors are grateful to D.D. Sokoloff for valuable help in the discussion of general ideas and
technical details. Also we appreciate the attempts of T. Postnikova to solve the problem.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work was supported by RFBR [grant number N15-02-01407].
Downloaded by [UAE University] at 15:52 25 October 2017
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