Download Magnetic fluctuations and formation of large-scale inhomogeneous magnetic structures

PHYSICAL REVIEW E 76, 056307 共2007兲
Magnetic fluctuations and formation of large-scale inhomogeneous magnetic structures
in a turbulent convection
Igor Rogachevskii* and Nathan Kleeorin†
Department of Mechanical Engineering, The Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel
共Received 21 August 2007; revised manuscript received 16 October 2007; published 12 November 2007兲
Magnetic fluctuations generated by a tangling of the mean magnetic field by velocity fluctuations are studied
in a developed turbulent convection with large magnetic Reynolds numbers. We show that the energy of
magnetic fluctuations depends on the magnetic Reynolds number only when the mean magnetic field is smaller
than Beq / 4Rm1/4, where Beq is the equipartition mean magnetic field determined by the turbulent kinetic energy
and Rm is the magnetic Reynolds number. Generation of magnetic fluctuations in a turbulent convection with
a nonzero mean magnetic field results in a decrease of the total turbulent pressure and may cause the formation
of large-scale inhomogeneous magnetic structures even in an originally uniform mean magnetic field. This
effect is caused by a negative contribution of the turbulent convection to the effective mean Lorentz force. The
inhomogeneous large-scale magnetic fields are formed due to excitation of the large-scale instability. The
energy for this instability is supplied by small-scale turbulent convection. The discussed effects might be useful
for understanding the origin of solar nonuniform magnetic fields: e.g., sunspots.
DOI: 10.1103/PhysRevE.76.056307
PACS number共s兲: 47.65.Md
I. INTRODUCTION
Magnetic fields in astrophysics are strongly nonuniform
共see, e.g., 关1–8兴兲. Large-scale magnetic structures are observed in the form of sunspots, solar coronal magnetic loops,
etc. There are different mechanisms for the formation of the
large-scale magnetic structures: e.g., the magnetic buoyancy
instability of stratified continuous magnetic fields 关2,9–11兴,
magnetic flux expulsion 关12兴, topological magnetic pumping
关13兴, etc.
Magnetic buoyancy applies in the literature for different
situations 共see 关11兴兲. The first corresponds to the magnetic
buoyancy instability of stratified continuous magnetic fields
共see, e.g., 关2,9–11兴兲, and the magnetic flux tube concept is
not used there. The magnetic buoyancy instability of stratified continuous magnetic fields is excited when the scale of
variations of the initial magnetic field is less than the density
stratification length. On the other hand, buoyancy of discrete
magnetic flux tubes has been discussed in a number of studies in solar physics and astrophysics 共see, e.g., 关11,14–18兴兲.
This phenomenon is also related to the problem of the storage of magnetic fields in the overshoot layer near the bottom
of the solar convective zone 共see, e.g., 关19–22兴兲.
A universal mechanism of the formation of the nonuniform distribution of magnetic flux is associated with a magnetic flux expulsion. In particular, the expulsion of magnetic
flux from two-dimensional flows 共a single vortex and a grid
of vortices兲 was demonstrated in 关12兴. In the context of solar
and stellar convection, the topological asymmetry of stationary thermal convection plays a very important role in magnetic field dynamics. In particular, topological magnetic
pumping is caused by topological asymmetry of the thermal
convection 关13兴. The fluid rises at the centers of the convective cells and falls at their peripheries. The ascending fluid
*[email protected]
†
[email protected]
1539-3755/2007/76共5兲/056307共15兲
elements 共contrary to the descending fluid elements兲 are disconnected from one another. This causes a topological magnetic pumping effect, allowing downward transport of the
mean horizontal magnetic field to the bottom of a cell, but
impeding its upward return 关4,13,23兴.
Turbulence may form inhomogeneous large-scale magnetic fields due to turbulent diamagnetic and paramagnetic
effects 共see, e.g., 关3,24–28兴兲. Inhomogeneous velocity fluctuations lead to a transport of mean magnetic flux from regions with a high intensity of velocity fluctuations. Inhomogeneous magnetic fluctuations due to the small-scale dynamo
cause a turbulent paramagnetic velocity; i.e., the magnetic
flux is pushed into regions with a high intensity of magnetic
fluctuations. Other effects are the effective drift velocities of
the mean magnetic field caused by inhomogeneities of the
fluid density 关26,27兴 and pressure 关29兴. In a nonlinear stage
of the magnetic field evolution, inhomogeneities of the mean
magnetic field contribute to the diamagnetic or paramagnetic
drift velocities depending on the level of magnetic fluctuations due to the small-scale dynamo and level of the mean
magnetic field 关30兴. The diamagnetic velocity causes a drift
of the magnetic field components from regions with a high
intensity of the mean magnetic field.
The nonlinear drift velocities of the mean magnetic field
in a turbulent convection have been determined in 关31兴. This
study demonstrates that nonlinear drift velocities are caused
by three kinds of inhomogeneities: i.e., inhomogeneous turbulence, the nonuniform fluid density, and the nonuniform
turbulent heat flux. The nonlinear drift velocities of the mean
magnetic field cause the small-scale magnetic buoyancy and
magnetic pumping effects in the turbulent convection. These
phenomena are different from the large-scale magnetic buoyancy and magnetic pumping effects which are due to the
effect of the mean magnetic field on the large-scale density
stratified fluid flow. The small-scale magnetic buoyancy and
magnetic pumping can be stronger than these large-scale effects when the mean magnetic field is smaller than the equipartition field determined by the turbulent kinetic energy
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©2007 The American Physical Society
PHYSICAL REVIEW E 76, 056307 共2007兲
IGOR ROGACHEVSKII AND NATHAN KLEEORIN
关31兴. The pumping of magnetic flux in three-dimensional
compressible magnetoconvection has been studied in direct
numerical simulations in 关32兴 by calculating the turbulent
diamagnetic and paramagnetic velocities.
Turbulence may affect also the Lorentz force of the largescale magnetic field 共see 关33–36兴兲. This effect can also form
inhomogeneous magnetic structures. In this study a theoretical approach proposed in 关33–36兴 for a nonconvective turbulence is further developed and applied to investigate the
modification of the large-scale magnetic force by turbulent
convection and to elucidate a mechanism of formation of
inhomogeneous magnetic structures.
This paper is organized as follows. In Sec. II we discuss
the physics of the effect of turbulence on the large-scale
Lorentz force. In Sec. III we formulate the governing equations, the assumptions, and the procedure of the derivations
of the large-scale effective magnetic force in turbulent convection. In Sec. IV we study magnetic fluctuations and determine the modification of the large-scale effective Lorentz
force by the turbulent convection. In Sec. V we discuss the
formation of large-scale magnetic inhomogeneous structures
in the turbulent convection due to excitation of the largescale instability. Finally, we draw conclusions in Sec. VI. In
the Appendix we perform the derivation of the large-scale
effective Lorentz force in the turbulent convection.
II. TURBULENT PRESSURE AND EFFECTIVE MEAN
MAGNETIC PRESSURE
In this section we discuss the physics of the effect of
turbulence on the large-scale Lorentz force. First, let us examine an isotropic turbulence. The Lorentz force of the
small-scale magnetic fluctuations can be written in the form
共m兲
共m兲
F共m兲
i = ⵜ j␴ij , where the magnetic stress tensor ␴ij is given
by
␴共m兲
ij = −
具b2典
␦ij + 具bib j典,
2
共1兲
b are the magnetic fluctuations, and ␦ij is the Kronecker
tensor. Hereafter we omit the magnetic permeability of the
fluid ␮ and include ␮−1/2 in the definition of the magnetic
field. The angular brackets in Eq. 共1兲 denote ensemble averaging. For isotropic turbulence 具bib j典 = ␦ij具b2典 / 3, and the
magnetic stress tensor reads
␴共m兲
ij = −
具b2典
W
␦ij = − m ␦ij ,
6
3
共2兲
where Wm = 具b2典 / 2 is the energy density of the magnetic fluctuations. The magnetic pressure pm is related to the magnetic
stress tensor: ␴共m兲
ij = −pm␦ij, where pm = Wm / 3. Similarly, in an
isotropic turbulence the Reynolds stresses 具uiu j典 read 具uiu j典
= ␦ij具u2典 / 3, and
␴共ijv兲 = − ␳0具uiu j典 = −
2Wk
␳0具u2典
␦ij = −
␦ij ,
3
3
共3兲
where u are the velocity fluctuations, Wk = ␳0具u2典 / 2 is the
kinetic energy density of the velocity fluctuations, and ␳0 is
the fluid density. Equation 共3兲 yields the hydrodynamic pressure pv = 2Wk / 3, where ␴共ijv兲 = −pv␦ij. Therefore, the equation
of state for the isotropic turbulence is given by
1
2
pT = Wm + Wk
3
3
共4兲
共see also 关37,38兴兲, where pT is the total 共hydrodynamic plus
magnetic兲 turbulent pressure. Similarly, the equation of state
for an anisotropic turbulence reads
pT =
2
4 + 3AN
Wm +
Wk ,
3共2 + AN兲
3共2 + AN兲
共5兲
2
where AN = 共2 / 3兲关具u⬜
典 / 具uz2典 − 2兴 is the degree of anisotropy
of the turbulent velocity field u = u⬜ + uze. For an isotropic
2
典 = 2具uz2典 and the parameter
three-dimensional turbulence 具u⬜
AN = 0, while for a two-dimensional turbulence 具uz2典 = 0 and
the degree of anisotropy AN → ⬁. Here e is the vertical unit
vector perpendicular to the plane of the two-dimensional turbulence.
In a two-dimensional turbulence AN → ⬁ and the total turbulent pressure pT → Wk. Note that the magnetic pressure in a
two-dimensional turbulence vanishes. Indeed, for isotropic
magnetic fluctuations in a two-dimensional turbulence
共m兲
2 共2兲
具bib j典 = 共1 / 2兲具b2典␦共2兲
ij , and therefore, ␴ij ⬅ −共1 / 2兲具b 典␦ij
+ 具bib j典 = 0, where ␦共2兲
ij = ␦ij − eie j.
The total energy density WT = Wk + Wm of the homogeneous turbulence with a mean magnetic field B is determined
by the equation
⳵ WT
WT
= IT −
+ ␩T共⵱ ⫻ B兲2
⳵t
␶0
共6兲
共see, e.g., 关36兴兲, where ␶0 is the correlation time of the turbulent velocity field in the maximum scale l0 of turbulent
motions, IT is the energy source of turbulence, ␩T is the
turbulent magnetic diffusion, and the mean magnetic field B
is given 共prescribed兲. The second term WT / ␶0 on the righthand side of Eq. 共6兲 determines the dissipation of the turbulent energy. For a given time-independent source of turbulence IT the solution of Eq. 共6兲 is given by
冋 冉 冊册
WT = ␶0关IT + ␩T共⵱ ⫻ B兲2兴 1 − exp −
t
␶0
冉 冊
+ W̃T exp −
t
,
␶0
共7兲
where W̃T = WT共t = 0兲. For instance, a time-independent
source of the turbulence exists in the Sun. The mean nonuniform magnetic field causes an additional energy source of
turbulence, IN = ␩T共⵱ ⫻ B兲2. The ratio IN / IT of these two
sources of turbulence is of the order of
冉 冊
l0
IN
⯝
IT
LB
2
B2
1,
␳0具u2典共0兲
共8兲
where LB is the characteristic scale of the spatial variations
of the mean magnetic field. Since l0 LB and B2 ␳0具u2典共0兲,
we can neglect the small magnetic source IN of the turbulence. Thus, for t ␶0 the total energy density of the turbulence reaches a steady state WT = const= ␶0IT. Therefore, the
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MAGNETIC FLUCTUATIONS AND FORMATION OF LARGE-…
total energy density WT of the homogeneous turbulence is
conserved 共the dissipation is compensated by a supply of
energy兲; i.e.,
共9兲
Wk + Wm = const.
A more rigorous derivation of Eq. 共9兲 is given in the Appendix 关see Eq. 共A16兲兴. Equation 共9兲 implies that the uniform
large-scale magnetic field performs no work on the turbulence. It can only redistribute the energy between hydrodynamic and magnetic fluctuations.
Combining Eqs. 共4兲 and 共9兲 we can express the change of
turbulent pressure, ␦ pT, in terms of the change of the magnetic energy density, ␦Wm, for an isotropic turbulence, ␦ pT
= −共1 / 3兲␦Wm 共see 关33,34,36兴兲. Therefore, the turbulent pressure is reduced when magnetic fluctuations are generated
共␦Wm ⬎ 0兲. Similarly, for an anisotropic turbulence, the generation of magnetic fluctuations reduces the turbulent pressure; i.e.,
␦ pT = −
2 + 3AN
␦Wm .
3共2 + AN兲
The total turbulent pressure is decreased also by the tangling of the large-scale mean magnetic field B by the velocity fluctuations 共see, e.g., 关1–4兴, and references therein兲. The
mean magnetic field generates additional small-scale magnetic fluctuations due to a tangling of the mean magnetic
field by velocity fluctuations. For a small energy of the mean
magnetic field, B2 ␳0具u2典, the energy of magnetic fluctuations, 具b2典 − 具b2典共0兲, caused by a tangling of the mean magnetic field can be written in the form
具b2典 − 具b2典共0兲 = am共B,Rm兲B2 + O„B4/共␳0具u2典兲2…,
共10兲
2 共0兲
where 具b 典 are the magnetic fluctuations with a zero mean
magnetic field generated by a small-scale dynamo. Equation
共10兲 allows us to determine the variation of the magnetic
energy ␦Wm. Therefore, the total turbulent pressure reads
pT = pT共0兲 − q p
B2
,
2
共11兲
where pT共0兲 is the turbulent pressure in a flow with a zero
mean magnetic field and the coefficient q p ⬀ am共B , Rm兲. Here
we neglect the small terms ⬃O(B4 / 共␳0具u2典兲2). The coefficient q p is positive when magnetic fluctuations are generated,
and it is negative when they are damped. The total pressure
is
Ptot ⬅ Pk + pT + PB共B兲 = Pk + pT共0兲 + 共1 − q p兲
B2
,
2
共12兲
where Pk is the mean fluid pressure and PB共B兲 = B2 / 2 is the
magnetic pressure of the mean magnetic field. Now we examine the part of the total pressure Ptot that depends on the
mean magnetic field B; i.e., we consider
Pm共B兲 = PB共B兲 − q p共B兲
B2
B2
= 共1 − q p兲
2
2
共13兲
共see 关33,34,36兴兲, where now Ptot = P + Pm共B兲 and P = Pk
+ pT共0兲. The pressure Pm共B兲 is called the effective 共or com-
bined兲 mean magnetic pressure. Note that both the hydrodynamic and magnetic fluctuations contribute to the combined
mean magnetic pressure. However, the gain in the turbulent
magnetic pressure pm is not as large as the reduction of the
turbulent hydrodynamic pressure pv by the mean magnetic
field B. This is due to different coefficients multiplying by
Wm and Wk in the equation of state 共4兲 关see also Eq. 共5兲兴.
Therefore, this effect is caused by a negative contribution of
the turbulence to the combined mean magnetic pressure.
We consider the case when P B2 / 2, so that the total
pressure Ptot is always positive. Only the combined mean
magnetic pressure Pm共B兲 may be negative when q p ⬎ 1,
while the pressure PB共B兲 as well as the values Pk, pv, pm, and
pT are always positive. When a mean magnetic field B is
superimposed on an isotropic turbulence, the isotropy breaks
down. Nevertheless Eq. 共13兲 remains valid, while the relationship between q p and am may change.
In this section we use the conservation law 共9兲 for the
total turbulent energy only for elucidation of the principle of
the effect, but we have not employed Eq. 共9兲 to develop the
theory of this effect 共see for details 关34–36兴兲. In particular,
the high-order closure procedure 关34,36兴 and the renormalization procedure 关35兴 have been used for the investigation
of the nonconvective turbulence at large magnetic and hydrodynamic Reynolds numbers.
In this study we investigate the modification of the largescale magnetic force by turbulent convection. We demonstrate that turbulent convection enhances modification of the
effective magnetic force and causes a large-scale instability.
This results in the formation of large-scale inhomogeneous
magnetic structures.
III. GOVERNING EQUATIONS AND THE PROCEDURE
OF DERIVATION
In order to study magnetic fluctuations and the modification of the large-scale Lorentz force by turbulent convection
we use a mean-field approach in which the magnetic and
velocity fields and entropy are decomposed into the mean
and fluctuating parts, where the fluctuating parts have zero
mean values. We assume that there exists a separation of
scales; i.e., the maximum scale of turbulent motions l0 is
much smaller than the characteristic scale LB of the mean
magnetic field variations. We apply here an approach which
is described in 关30,31,39兴 and outlined below.
We consider a nonrotating turbulent convection with large
Rayleigh numbers and large magnetic Reynolds numbers.
We use the equations for fluctuations of the fluid velocity u,
entropy s⬘, and the magnetic field b. The equations for velocity and entropy fluctuations are rewritten in the new variables v = 冑␳0u and s = 冑␳0s⬘. We also use the new variable
H = B / 冑␳0 for the mean magnetic field B. On the other hand,
we do not use a new variable for magnetic fluctuations b.
Equations for fluctuations of fluid velocity, entropy, and
magnetic field are applied in the anelastic approximation,
which is a combination of the Boussinesq approximation and
the condition div 共␳0u兲 = 0. The turbulent convection is regarded as a small deviation from a well-mixed adiabatic reference state. This implies that we consider the hydrostatic
nearly isentropic basic reference state.
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IGOR ROGACHEVSKII AND NATHAN KLEEORIN
Using these equations for fluctuations of fluid velocity,
entropy, and magnetic field written in a Fourier space we
derive equations for the two-point second-order correlation
functions of the velocity fluctuations 具viv j典, the magnetic
fluctuations 具bib j典, the entropy fluctuations 具ss典, the cross
helicity 具biv j典, and the turbulent heat flux 具svi典 and 具sbi典. The
equations for these correlation functions are given by Eqs.
共A4兲–共A9兲 in the Appendix. We split the tensor 具bib j典 of
magnetic fluctuations into nonhelical, hij, and helical, h共H兲
ij ,
parts. The helical part h共H兲
depends
on
the
magnetic
helicity
ij
共see below兲. We also split all second-order correlation functions M 共II兲 into symmetric and antisymmetric parts with re共a兲
spect to the wave vector k—e.g., hij = h共s兲
ij + hij , where the
共s兲
tensor hij = 关hij共k兲 + hij共−k兲兴 / 2 describes the symmetric part
of the tensor and h共a兲
ij = 关hij共k兲 − hij共−k兲兴 / 2 determines the antisymmetric part of the tensor.
The second-moment equations include the first-order spatial differential operators N̂ applied to the third-order moments M 共III兲. A problem arises as to how to close the equations for the second moments—i.e., how to express the thirdorder terms N̂M 共III兲 through the second moments M 共II兲 共see,
e.g., 关40–42兴兲. We will use the spectral ␶ approximation
which postulates that the deviations of the third-moment
terms N̂M 共III兲共k兲 from the contributions to these terms afforded by the background turbulent convection,
N̂M 共III,0兲共k兲, are expressed through similar deviations of the
second moments, M 共II兲共k兲 − M 共II,0兲共k兲:
N̂M 共III兲共k兲 − N̂M 共III,0兲共k兲 = −
1
关M 共II兲共k兲 − M 共II,0兲共k兲兴
␶共k兲
共14兲
共see, e.g., 关30,34,36,40,43兴兲, where ␶共k兲 is the scale-dependent relaxation time. In the background turbulent convection
the mean magnetic field is zero. The ␶ approximation is applied for large hydrodynamic and magnetic Reynolds numbers and large Rayleigh numbers. In this case there is only
one relaxation time ␶ which can be identified with the correlation time of the turbulent velocity field. A justification of
the ␶ approximation for different situations has been performed in numerical simulations and theoretical studies in
关44–49兴 共see also review 关8兴兲. The ␶ approximation is also
discussed in Sec. VI.
We apply the spectral ␶ approximation only for the nonhelical part hij of the tensor of magnetic fluctuations. The
helical part h共H兲
ij depends on the magnetic helicity, and it is
determined by the dynamic equation which follows from the
magnetic helicity conservation arguments 共see, e.g., 关50–56兴
and the review 关8兴, and references therein兲. The characteristic
time of evolution of the nonhelical part of the tensor hij is of
the order of the turbulent time ␶0 = l0 / u0, while the relaxation
time of the helical part of the tensor h共H兲
ij of magnetic fluctuations is of the order of ␶0 Rm, where Rm= l0u0 / ␩ is the
magnetic Reynolds number, u0 is the characteristic turbulent
velocity in the maximum scale of turbulent motions l0, and ␩
is the magnetic diffusivity due to electrical conductivity of
the fluid.
In this study we consider an intermediate nonlinearity
which implies that the mean magnetic field is not strong
enough in order to affect the correlation time of the turbulent
velocity field. The theory can be expanded to the case of a
very strong mean magnetic field after taking into account a
dependence of the correlation time of the turbulent velocity
field on the mean magnetic field.
We assume that the characteristic time of variation of the
mean magnetic field B is substantially larger than the correlation time ␶共k兲 for all turbulence scales. This allows us to
get a stationary solution for the equations for the secondorder moments given by Eqs. 共A10兲–共A14兲 in the Appendix.
For the integration in k space of the second moments we
have to specify a model for the background turbulent convection 共with a zero mean magnetic field, B = 0兲. Here we use
the model of the background shear-free turbulent convection
with a given heat flux 关see Eqs. 共A17兲–共A20兲 in the Appendix兴. In this model velocity and magnetic fluctuations are
homogeneous and isotropic.
This procedure allows us to study magnetic fluctuations
with a nonzero mean magnetic field and to investigate the
modification of the large-scale Lorentz force by turbulent
convection 共see Sec. IV兲.
IV. MAGNETIC FLUCTUATIONS AND LARGE-SCALE
EFFECTIVE LORENTZ FORCE
A. Magnetic fluctuations with a nonzero mean magnetic field
Let us study magnetic fluctuations with a nonzero mean
magnetic field using the approach outlined in Sec. III. Integration in k space in Eq. 共A11兲 yields an analytical expression for the energy of magnetic fluctuations, 具b2典 关see Eq.
共A21兲 in the Appendix兴. The energy of magnetic fluctuations
versus the mean magnetic field B / Beq is shown in Fig. 1,
where Beq is the equipartition mean magnetic field determined by the turbulent kinetic energy. The asymptotic formulas for 具b2典 are given below. In particular, for a very weak
mean magnetic field, B Beq / 4Rm1/4, the energy of magnetic fluctuations is given by
4
具b2典 = 具b2典共0兲 + 关具v2典共0兲 − 具b2典共0兲兴B2 ln Rm
3
+
8a* 2 共0兲 2
具v 典 B 共2 − 3 cos2␾兲,
5
共15兲
where the quantities with the superscript 共0兲 correspond to
the background turbulent convection 共with a zero mean magnetic field兲, and 具v2典共0兲 and 具b2典共0兲 are the velocity and magnetic fluctuations in the background turbulent convection.
Here the magnetic field B is measured in units of Beq and ␾
is the angle between the vertical unit vector e and the mean
magnetic field B. The unit vector e is directed opposite to the
gravity field. The parameter a* characterizing the turbulent
convection is determined by the budget equation for the total
energy, and it is given by
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MAGNETIC FLUCTUATIONS AND FORMATION OF LARGE-…
2
<b2> /B2
<b >
25
a
0.6
20
0.4
15
0.2
10
a
0
0.5
1
1.5
B/Beq
<b2>
0
0.02
2
2
0.04
0.06
0.08 B/Beq
<b > /B
0.4
b
1
0.3
0.9
0.2
0.8
0.1
0
b
0.5
1
1.5
0.7
B/Beq
0.6
FIG. 1. 共a兲 The energy of magnetic fluctuations 具b2典 versus the
mean magnetic field B / Beq for a nonconvective turbulence 共a* = 0兲,
Rm= 106, and different values of the parameter ⑀ ⬅ 具b2典共0兲 / 具u2典共0兲:
⑀ = 0 共solid line兲, ⑀ = 0.3 共dashed line兲, and ⑀ = 0.5 共thin dashed line兲.
共b兲 The energy of magnetic fluctuations 具b2典 versus the horizontal
共dashed line兲 and vertical 共thin solid line兲 mean magnetic field for a
convective turbulence 共a* = 0.7兲 and for Rm= 106, ⑀ = 0.
a−1
*
where ␯T is the turbulent viscosity, U is the mean fluid velocity, and F* = 具uzs⬘典共0兲 is the vertical heat flux in the background turbulent convection. The energy of magnetic fluctuations for a very weak mean magnetic field,
B Beq / 4Rm1/4, depends on the magnetic Reynolds number:
具b2典 ⬀ ln Rm. This is an indication of that the spectrum of
magnetic fluctuations is k−1 in the limit of a small yet finite
mean magnetic field 关34–36,57兴 共see also discussion in 关58兴兲.
When the mean magnetic field Beq / 4Rm1/4 B Beq / 4, the
energy of magnetic fluctuations is given by
B/B
eq
␲a* 2 共0兲
具v 典 共1 − 3 cos2␾兲.
40B
共17兲
The normalized energy of magnetic fluctuations 具b2典 / B2 versus the mean magnetic field is shown in Fig. 2 for a nonconvective and convective turbulence. Inspection of Figs. 1 and
2 shows that turbulent convection increases the level of magnetic fluctuations in comparison with the nonconvective turbulence. It follows from Eqs. 共15兲–共17兲 that in the case of
Alfvénic equipartition, 具u2典共0兲 = 具b2典共0兲, a deviation of the energy of magnetic fluctuations from the background level is
caused by the turbulent convection.
16 2 共0兲
关具v 典 − 具b2典共0兲兴B2兩ln共4B兲兩
3
and for B Beq / 4 it is given by
0.75
1
␲
具b2典 = 关具v2典共0兲 + 具b2典共0兲兴 −
关具v2典共0兲 − 具b2典共0兲兴
2
24B
+
8a* 2 共0兲 2
具v 典 B 共2 + 3 cos2␾兲,
+
5
0.7
FIG. 2. The normalized energy of magnetic fluctuations 具b2典 / B2
versus the mean magnetic field B / Beq for a nonconvective turbulence 共a* = 0兲 共solid line兲 and for a convective turbulence 共a* = 0.7兲
with the horizontal 共dashed line兲 and vertical 共thin solid line兲 mean
magnetic field, where the cases B Beq 共a兲 and B ⬃ Beq 共b兲 are
shown. Here Rm= 106 and ⑀ = 0.
␯T共ⵜU兲2 + ␩T共ⵜB兲2/␳0
=1+
,
gF*
具b2典 = 具b2典共0兲 +
0.65
B. Large-scale effective Lorentz force
共16兲
The effective 共combined兲 mean magnetic force, which
takes into account the effect of turbulence on magnetic force,
eff
can be written in the form Feff
i = ⵜ j␴ij , where the effective
eff
stress tensor ␴ij reads
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PHYSICAL REVIEW E 76, 056307 共2007兲
IGOR ROGACHEVSKII AND NATHAN KLEEORIN
q
p
q
s
a
a
15
10
8
10
6
5
4
2
0
0.1
0.2
B/B
0
eq
P
0.05
0.1
0.15
0.2
0.25 B/Beq
σB
m
0.04
0.03
0.02
0.02
0
0.01
−0.02
0
b
−0.04
0
−0.01
0.1
0.2
0.3
B/Beq
0
FIG. 3. 共a兲 The nonlinear coefficient q p共B兲 for different values
of the magnetic Reynolds numbers Rm: Rm= 103 共thin solid line兲,
Rm= 106 共dash-dotted line兲, and Rm= 1010 共thick solid line兲 for a
nonconvective turbulence 共a* = 0兲 and at Rm= 106 共dashed line兲 for
a convective turbulence 共a* = 0.7兲. 共b兲 The effective 共combined兲
mean magnetic pressure Pm共B兲 = 共1 − q p兲B2 / B2eq at Rm= 106 for a
nonconvective turbulence 共a* = 0兲 共thick solid line兲 and for a convective turbulence 共a* = 0.7兲 for the horizontal field 共dashed line兲
and for the vertical field 共thin solid line兲.
1 2
1 2
␴eff
ij = − B ␦ij + BiB j − 具b 典␦ij + 具bib j典 − ␳0具uiu j典.
2
2
共18兲
The last three terms on the right-hand side 共RHS兲 of Eq. 共18兲
determine the contribution of velocity and magnetic fluctuations to the effective 共combined兲 mean magnetic force. Using
Eqs. 共A10兲 and 共A11兲 for 具uiu j典 and 具bib j典 after the integration in k space we arrive at the expression for the effective
stress tensor:
0.1
B/Beq
0.2
FIG. 4. 共a兲 The nonlinear coefficient qs共B兲 for different values of
the magnetic Reynolds numbers Rm: Rm= 103 共thin solid line兲,
Rm= 106 共thin dash-dotted line兲, and Rm= 1010 共thick solid line兲 for
a nonconvective turbulence a* = 0 and at Rm= 106 共dashed line兲 for
a convective turbulence a* = 0.7. 共b兲 The effective 共combined兲 mean
magnetic tension ␴B共B兲 = 共1 − qs兲B2 / B2eq at Rm= 106 for a nonconvective turbulence 共a* = 0兲 共thick solid line兲 and for a convective
turbulence 共a* = 0.7兲 for the horizontal field 共dashed line兲 and for the
vertical field 共thin solid兲.
netic Reynolds numbers in the range of weak mean magnetic
fields 共B ⬍ 0.1Beq兲. On the other hand, the turbulent convection reduces these nonlinear coefficients in comparison with
the case of a nonconvective turbulence. The asymptotic formulas for the nonlinear coefficients q p共B兲 and qs共B兲 are
given below. In particular, for a very weak mean magnetic
field, B Beq / 4Rm1/4, the nonlinear coefficients q p共B兲 and
qs共B兲 are given by
冋
册
4
4
8a*
−
共11 − 13 cos2␾兲,
q p共B兲 = 共1 − ⑀兲 ln Rm +
5
45
35
2
␴eff
ij = − 关1 − q p共B兲兴
b
B
␦ij + 关1 − qs共B兲兴BiB j + a*␴Aij共B兲,
2
共20兲
共19兲
where the analytical expressions for the nonlinear coefficients q p共B兲 and qs共B兲 are given by Eqs. 共A23兲 and 共A24兲 in
the Appendix, and the tensor ␴Aij共B兲 is the anisotropic contribution caused by turbulent convection to the effective stress
tensor 关which is given by Eq. 共A22兲 in the Appendix兴.
The nonlinear coefficients q p共B兲 and qs共B兲 in Eq. 共19兲 for
the effective stress tensor are shown in Figs. 3共a兲 and 4共a兲 for
different values of the magnetic Reynolds numbers. The nonlinear coefficients q p共B兲 and qs共B兲 increase with the mag-
qs共B兲 =
冋
册
8
2
24a*
共1 − ⑀兲 ln Rm +
,
−
15
15
35
共21兲
where ⑀ ⬅ 具b2典共0兲 / 具u2典共0兲. For Beq / 4Rm1/4 B Beq / 4 these
nonlinear coefficients are given by
056307-6
q p共B兲 =
16
共1 − ⑀兲关5兩ln共4B兲兩 + 1 + 32B2兴
25
−
8a*
共11 − 13 cos2␾兲,
35
共22兲
PHYSICAL REVIEW E 76, 056307 共2007兲
MAGNETIC FLUCTUATIONS AND FORMATION OF LARGE-…
qs共B兲 =
冋
册
q
32
1
24a*
共1 − ⑀兲 兩ln共4B兲兩 +
+ 12B2 −
, 共23兲
15
30
35
e
6
and for B Beq / 4 they are given by
q p共B兲 =
1
␲a*
共1 − 5 cos2␾兲,
2 共1 − ⑀兲 +
6B
80B
a
4
共24兲
2
qs共B兲 =
3␲a*
␲
共1 − 3 cos2␾兲.
共1 − ⑀兲 +
48B3
160B
共25兲
The effective 共combined兲 mean magnetic pressure Pm共B兲
2
= 共1 − q p兲B2 / Beq
is shown in Fig. 3共b兲. Inspection of Fig. 3共b兲
shows that the combined mean magnetic pressure Pm共B兲
2
= 共1 − q p兲B2 / Beq
vanishes at some value of the mean magnetic field B = B P ⬃ 共0.2− 0.3兲Beq. This causes the following
effect. Let us consider an isolated tube of magnetic field
lines. When B = B P, the combined mean magnetic pressure
Pm共B兲 = 0, the fluid pressure, and fluid density inside and
outside the isolated tube are the same, and therefore, this
isolated tube is in equilibrium. When B ⬎ B P, the combined
mean magnetic pressure Pm共B兲 ⬎ 0; the fluid pressure and
fluid density inside the isolated tube are smaller than the
fluid pressure and fluid density outside the isolated tube. This
results in an upward floating of the isolated tube. On the
other hand, when B ⬍ B P, the combined mean magnetic pressure Pm共B兲 ⬍ 0, the fluid pressure, and fluid density inside
the isolated tube are larger than the fluid pressure and fluid
density outside the isolated tube. Therefore, this isolated
magnetic tube flows down.
The effective 共combined兲 mean magnetic tension ␴B共B兲
2
= 共1 − qs兲B2 / Beq
is shown in Fig. 4共b兲. The combined mean
magnetic tension ␴B共B兲 vanishes at some value of the mean
magnetic field B = BS ⬃ 0.2Beq 关see Fig. 4共b兲兴. When B ⬎ BS,
the combined mean magnetic tension ␴B共B兲 ⬎ 0, and
Alfvénic and magnetosound waves can propagate in the isolated tubes. On the other hand, when B ⬍ BS, the combined
mean magnetic tension ␴B共B兲 ⬍ 0, and Alfvénic and magnetosound waves cannot propagate in the isolated tubes.
The anisotropic contributions ␴Aij共B兲 to the effective stress
tensor determine the anisotropic mean magnetic tension due
to the turbulent convection. The tensor ␴Aij共B兲 is character2
ized by the function ␴A共B兲 = ␴Aij共B兲eij = qeB2 / Beq
. The nonlinear coefficient qe共B兲 and the anisotropic mean magnetic tension ␴A共B兲 are shown in Figs. 5共a兲 and 5共b兲. In the next
section we show that the anisotropic mean magnetic tension
␴A共B兲 caused by the turbulent convection strongly affects the
dynamics of the horizontal mean magnetic field.
In this section we demonstrate that turbulent convection
strongly modifies the large-scale magnetic force. Let us discuss a possibility for a study of the effect of turbulence on
the effective 共combined兲 mean Lorentz force in the direct
numerical simulations. Consider the mean magnetic field
which is directed along the x axis; i.e., B = Bex. Let us introduce the functions ␴x共B兲 and ␴y共B兲,
1
␴x共B兲 = − 具b2典 + 具b2x 典 − ␳0具u2x 典,
2
共26兲
0
0.2
0.4
0.6
B/Beq
σ
A
0.15
0.1
0.05
b
0
0.2 0.4 0.6 0.8
1
1.2 B/Beq
FIG. 5. 共a兲 The nonlinear coefficient qe共B兲 versus the mean
magnetic field and 共b兲 the anisotropic mean magnetic tension
␴A共B兲 = ␴Aijeij = qeB2 / B2eq for a convective turbulence 共a* = 1兲 with
the horizontal mean magnetic field 共solid line兲 and vertical mean
magnetic field 共dashed line兲.
1
␴y共B兲 = − 具b2典 + 具b2y 典 − ␳0具u2y 典,
2
共27兲
which allow us to determine the coefficients q p共B兲 and qs共B兲
in the effective stress tensor:
q p共B兲 =
2
关␴y共B兲 − ␴y共B = 0兲兴,
B2
1
1
qs共B兲 = q p共B兲 − 2 关␴x共B兲 − ␴x共B = 0兲兴.
2
B
共28兲
共29兲
Therefore, Eqs. 共26兲–共29兲 allow us to determine the effective
共combined兲 mean magnetic force in the direct numerical
simulations.
V. LARGE-SCALE INSTABILITY
The modification of the mean magnetic force by turbulent
convection causes a large-scale instability. In this study we
investigate the large-scale instability of continuous magnetic
fields in small-scale turbulent convection and we do not consider buoyancy of the discrete magnetic flux tubes. In order
to study the large-scale instability in a small-scale turbulent
convection we use the equation of motion 共with the effective
eff
magnetic force Feff
i = ⵜ j␴ij determined in Sec. IV兲, the induction equation, and the equation for the evolution of the
mean entropy 关see Eqs. 共A25兲–共A27兲 in the Appendix兴. We
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PHYSICAL REVIEW E 76, 056307 共2007兲
IGOR ROGACHEVSKII AND NATHAN KLEEORIN
γ
estimate the growth rate ␥ of this instability and the frequencies ␻ of generated modes, neglecting turbulent dissipative
processes for simplicity’s sake. We also neglect a very small
Brunt-Väisälä frequency based on the gradient of the mean
entropy. We seek a solution of these equations in the form
⬀exp共␥t + i␻t − iK · R兲.
0.4
0.2
0
0
A. Horizontal mean magnetic field
冉 冊
K yCA
␴ =
KL␳
2
共Q + iD兲,
共30兲
where ␴ˆ = ␥ + i␻, L␳ is the density stratification length, LB is
the characteristic scale of the mean magnetic field variations,
CA = B / 冑␳0 is the Alfvén speed,
再
冋 冉
Q = 1 − q p共y兲 − yq⬘p共y兲 + a* y ␴A⬙ 共y兲 1 − 2
冉
1
L␳
+ ␴A⬘ 共y兲 1 − 4
2
LB
冊册冎 冉 冊
冉 冊
y=B2
D = a*KzL␳关␴A⬘ 共y兲兴y=B2
L␳
LB
冊
L␳
−1 ,
LB
L␳
−1 ,
LB
␴A共y兲 = qe共y兲y, ␴A⬘ 共y兲 = d␴A共y兲 / dy, B is measured in units of
the equipartition field Beq = 冑␳0u0, and K = 冑Kz2 + K2y .
When Q ⱖ 0 the growth rate of perturbations with frequency
␻=
Ky
冑2K
0.5
B/Beq
1
ω
1
First, we study the large-scale instability of a horizontal
mean magnetic field that is perpendicular to the gravity field.
Let the z axis of a Cartesian coordinate system be directed
opposite to the gravitational field, and let the x axis lie along
the mean magnetic field B. Consider the case Kx = 0, which
corresponds to the interchange mode. The dispersive relation
for the instability reads
ˆ2
a
0.6
冉 冊
CA
D共冑Q2 + D2 + Q兲−1/2
L␳
共31兲
0.5
b
0
0
0.2
0.4
0.6
B/B
eq
FIG. 6. The growth rate 共a兲 of the large-scale instability and
frequency of the generated modes 共b兲 for the horizontal mean magnetic field with L␳ / LB = 0.2 versus B / Beq for different values
Kz : KzL␳ = 3 共dash-dotted line兲 and KzL␳ = 5 共solid line兲 for turbulent convection with a* = 1 共thick curves兲 and a* = 0.3 共thin curves兲
and for a nonconvective turbulence a* = 0 共dashed line兲. Here ⑀ = 0
and Rm= 106.
magnetic field versus B / Beq for different values Kz and
L␳ / LB for a nonconvective and convective turbulence are
shown in Figs. 6 and 7. Here ␥ and ␻ are measured in units
of t−1
* , where t* = 共L␳ / u0兲共K / K y 兲. In turbulent convection
there are two ranges for the large-scale instability of the horizontal mean magnetic field 共when Q ⬎ 0 and Q ⬍ 0兲, while in
a nonconvective turbulence 共a* = 0兲 there is only one range
for the instability. The first range for the instability is related
to the negative contribution of turbulence to the effective
magnetic pressure for the case of L␳ ⬍ LB, while the second
range is mainly caused by the anisotropic contribution ␴A共B兲
due to turbulent convection.
In the absence of turbulence 共small Reynolds numbers兲 or
turbulent convection 共small Rayleigh numbers兲 the coeffiγ
1.5
is given by
␥=
冉 冊
CA
1/2
冑 2 2
冑2K L␳ 共 Q + D + Q兲 .
Ky
1
共32兲
0.5
a
When Q ⬍ 0 the growth rate of perturbations with frequency
冉 冊
CA
1/2
冑 2 2
␻ = − sgn共D兲
冑2K L␳ 共 Q + D + 兩Q兩兲
Ky
0
0
0.2
0.4
0.6
B/B
eq
ω
1.5
共33兲
1
0.5
is given by
冉 冊
CA
−1/2
冑 2 2
␥=
冑2K L␳ 兩D兩共 Q + D + 兩Q兩兲 .
Ky
b
0
0
共34兲
Therefore, in small-scale turbulent convection this largescale instability causes excitation of oscillatory modes with
growing amplitude. In a nonconvective turbulence 共a* = 0兲
this large-scale instability is aperiodic.
The growth rate ␥ of the large-scale instability and frequency ␻ of the generated modes for the horizontal mean
0.5
1
B/Beq
FIG. 7. The growth rate 共a兲 of the large-scale instability and
frequency of the generated modes 共b兲 for the horizontal mean magnetic field with L␳ / LB = 10 versus B / Beq for different values
Kz : KzL␳ = 3 共dash-dotted line兲 and KzL␳ = 5 共solid line兲 for turbulent convection with a* = 1 共thick curves兲 and a* = 0.3 共thin curves兲
and for a nonconvective turbulence a* = 0 共dashed line兲. Here ⑀ = 0
and Rm= 106.
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PHYSICAL REVIEW E 76, 056307 共2007兲
MAGNETIC FLUCTUATIONS AND FORMATION OF LARGE-…
cients Q = L␳ / LB − 1, D = 0, and the criterion for the largescale instability, L␳ ⬎ LB, coincides with that of the Parker’s
magnetic buoyancy instability 关2,9,10兴. In this case, ␻ = 0;
i.e., the oscillatory modes with growing amplitude are not
excited. On the other hand, in a developed turbulent convection the effective 共combined兲 magnetic pressure becomes
negative and the Parker’s magnetic buoyancy instability cannot be excited. However, the instability due to the modification of the mean magnetic force by small-scale turbulent
convection can be excited even when L␳ ⬍ LB—i.e., even in a
uniform mean magnetic field.
The instability mechanism due to the modification of the
mean magnetic force consists in the following. An isolated
tube of magnetic field lines moving upwards turns out to be
lighter than the surrounding plasma. This is due to the fact
that the decrease of the magnetic field in the isolated tube
caused by its expansion is accompanied by an increase of the
effective magnetic pressure inside the tube. Since the effective 共combined兲 magnetic pressure is negative, this leads a
decrease of the density inside the tube. The arising buoyant
force results in the upwards floating of the isolated tube; i.e.,
it causes excitation of the large-scale instability 共see also
discussion in 关36兴兲.
B. Vertical uniform mean magnetic field
Consider a vertical uniform mean magnetic field; i.e., the
magnetic field is directed along z axis. The growth rate of
perturbations is given by
冋
␥ = CAKz qs共y兲 − 1 − 2yqs⬘共y兲
2
K⬜
K2
册
1/2
,
y=B2
共35兲
where K⬜ is the component of the wave vector that is per2
. It follows from Eq.
pendicular to the z axis and K = 冑Kz2 + K⬜
共35兲 that the large-scale instability of the vertical uniform
mean magnetic field is caused by modification of the mean
magnetic tension by small-scale turbulent convection. When
qs ⬎ 1 共i.e., when B ⬍ 0.2Beq兲 the instability occurs for an
arbitrary value K⬜, while for qs ⬍ 1 the necessary condition
for the instability reads K⬜ ⬎ K冑␹, where
␹=
冋
1 − qs共y兲
2y兩qs⬘共y兲兩
册
,
y=B2
and we take into account that qs⬘共y兲 ⬍ 0. The growth rate of
the instability is reduced by the turbulent dissipation ⬀␯T K2.
The maximum growth rate ␥max at a fixed value of the wave
number K 共i.e., at a fixed value of the turbulent dissipation兲
is attained at Kz = Km = K 关共1 − ␹兲 / 2兴1/2, and it is given by
␥max = KCA共1 − ␹兲
冋
y兩qs⬘共y兲兩
2
册
1/2
,
y=B2
共36兲
where ␹ ⬍ 1. The maximum growth rate ␥max of the instability for the vertical uniform mean magnetic field versus B / Beq
is plotted in Fig. 8. Here ␥max is measured in units of u0K.
The value of ␥max is larger for a nonconvective turbulence,
but the range for the instability is wider for the turbulent
convection 共see Fig. 8兲.
γmax
0.12
0.1
0.08
0.06
0.04
0.02
0
0.1
0.2
0.3
0.4
0.5
B/Beq
FIG. 8. The maximum growth rate of the large-scale instability
for the vertical mean magnetic field versus B / Beq for turbulent convection with a* = 1 共solid line兲 and for a nonconvective turbulence
a* = 0 共dashed line兲. Here ⑀ = 0 and Rm= 106.
VI. DISCUSSION
In the present study we investigate magnetic fluctuations
generated by a tangling of the mean magnetic field in a developed turbulent convection. When the mean magnetic field
B Beq / 4Rm1/4, the energy of magnetic fluctuations depends on the magnetic Reynolds number. We study the modification of the large-scale magnetic force by turbulent convection. We show that the generation of magnetic
fluctuations in a turbulent convection results in a decrease of
the total turbulent pressure and may cause the formation of
large-scale magnetic structures even in an originally uniform
mean magnetic field. This phenomenon is due to a negative
contribution of the turbulent convection to the effective mean
magnetic force.
The large-scale instability causes the formation of inhomogeneous magnetic structures. The energy for these processes is supplied by the small-scale turbulent convection,
and this effect can develop even in an initially uniform magnetic field. In contrast, the Parker’s magnetic buoyancy instability is excited when the density stratification scale is
larger than the characteristic scale of the mean magnetic field
variations 共see 关2,9,10兴兲. The free energy in the Parker’s
magnetic buoyancy instability is drawn from the gravitational field. The characteristic time of the large-scale instability is of the order of the Alfvén time based on the largescale magnetic field.
We study an initial stage of formation of large-scale magnetic structures for horizontal and vertical mean magnetic
fields relative to the vertical direction of the gravity field. In
turbulent convection there are two ranges for the large-scale
instability of the horizontal mean magnetic field. The first
range for the instability is related to the negative contribution
of turbulence to the effective magnetic pressure for the case
of L␳ ⬍ LB, while the second range for the instability is
mainly caused by the anisotropic contribution of the turbulent convection to the effective magnetic force. The largescale instability of the vertical uniform mean magnetic field
is caused by modification of the mean magnetic tension by
small-scale turbulent convection. The discussed effects in the
present study might be useful for understanding of the origin
of sunspot formation.
Since in the present study we neglect the very small
Brunt-Väisälä frequency based on the gradient of the mean
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PHYSICAL REVIEW E 76, 056307 共2007兲
IGOR ROGACHEVSKII AND NATHAN KLEEORIN
entropy, we do not investigate the large-scale dynamics of
the mean entropy. This problem was addressed in 关59兴
whereby the modification of the mean magnetic force by the
turbulent convection was not taken into account.
In order to study magnetic fluctuations and the modification of the large-scale Lorentz force by turbulent convection
we apply the spectral ␶ approximation 共see Sec. III兲. The ␶
approach is a universal tool in turbulent transport that allows
one to obtain closed results and compare them with the results of laboratory experiments, observations, and numerical
simulations. The ␶ approximation reproduces many wellknown phenomena found by other methods in the turbulent
transport of particles and magnetic fields, in turbulent convection, and in stably stratified turbulent flows 共see below兲.
In turbulent transport, the ␶ approximation yields correct
formulas for turbulent diffusion, turbulent thermal diffusion,
and turbulent barodiffusion 共see, e.g., 关60,61兴兲. The phenomenon of turbulent thermal diffusion 共a nondiffusive streaming
of particles in the direction of the mean heat flux兲 has been
predicted using stochastic calculus 共the path integral approach兲 and the ␶ approximation. This phenomenon has been
already detected in laboratory experiments in oscillating grid
turbulence 关62兴 and in a multiple-fan turbulence generator
关63兴 in stably and unstably stratified fluid flows. The experimental results obtained in 关62,62兴 are in a good agreement
with the theoretical studies performed by means of different
approaches 共see 关60,64兴兲.
The ␶ approximation reproduces the well-known k−7/3
spectrum of anisotropic velocity fluctuations in a sheared
turbulence 共see 关65兴兲. This spectrum was found previously in
analytical, numerical, and laboratory studies and was observed in atmospheric turbulence 共see, e.g., 关66兴兲. In the turbulent boundary layer problems, the ␶ approximation yields
correct expressions for turbulent viscosity, turbulent thermal
conductivity, and the classical heat flux. This approach also
describes the counterwind heat flux and the Deardorff’s heat
flux in convective boundary layers 共see 关65兴兲. These phenomena have been studied previously using different approaches 共see, e.g., 关41,42,67兴兲.
The theory of turbulent convection 关65兴 based on the ␶
approximation explains the recently discovered hysteresis
phenomenon in laboratory turbulent convection 关68兴. The results obtained using the ␶ approximation allow us also to
explain the most pronounced features of typical semiorganized coherent structures observed in the atmospheric convective boundary layers 共“cloud cells” and “cloud streets”兲
关69兴. The theory 关65兴 based on the ␶ approximation predicts
realistic values of the following parameters: the aspect ratios
of structures, the ratios of the minimum size of semiorganized structures to the maximum scale of turbulent motions,
and the characteristic lifetime of semiorganized structures.
The theory 关65兴 also predicts excitation of convective-shear
waves propagating perpendicular to the convective rolls
共“cloud streets”兲. These waves have been observed in atmospheric convective boundary layers with cloud streets 关69兴.
A theory 关70兴 for stably stratified atmospheric turbulent
flows based on the ␶ approximation and the budget equations
for the key second moments, turbulent kinetic and potential
energies, and vertical turbulent fluxes of momentum and
buoyancy is in good agrement with data from atmospheric
and laboratory experiments, direct numerical simulations,
and large-eddy simulations 共see the detailed comparison in
Sec. V of 关70兴兲.
A detailed verification of the ␶ approximation in direct
numerical simulations of turbulent transport of passive scalars has been recently performed in 关46兴. In particular, the
results on turbulent transport of passive scalars obtained using direct numerical simulations of homogeneous isotropic
turbulence have been compared with those obtained using a
closure model based on the ␶ approximation. The numerical
and analytical results are in good agreement.
In magnetohydrodynamics, the ␶ approximation reproduces many well-known phenomena found by different
methods: e.g., the ␶ approximation yields correct formulas
for the ␣ effect 关3,28,71,72兴, the turbulent diamagnetic and
paramagnetic velocities 关24–28兴, the turbulent magnetic diffusion 关3,25,28,30,73兴, the ⍀ ⫻ J and ␬ effects 关3,28兴, and
the shear-current effect 关30,39,74兴.
Generation of the large-scale magnetic field in a nonhelical turbulence with an imposed mean velocity shear has been
recently investigated in 关75兴 using direct numerical simulations. The results of these numerical simulations are in good
agreement with the theoretical predictions based on the ␶
approximation 共see 关30,39,74兴兲 and with the numerical solutions of the nonlinear dynamo equations performed in
关76,77兴 共see the detailed comparison in 关39兴兲.
The validity of the ␶ approximation has been tested in the
context of dynamo theory in direct numerical simulations in
关47兴. The ␣ effect in mean-field dynamo theory becomes
proportional to a relaxation time scale multiplied by the difference between kinetic and current helicities. It is shown in
关47兴 that the value of the relaxation time is positive and, in
units of the turnover time at the forcing wave number, it is of
the order of unity. Kinetic and current helicities are shown in
关47兴 to be dominated by large-scale properties of the flow.
Recent studies in 关48兴 of the nonlinear ␣ effect showed that
in the limit of small magnetic and hydrodynamic Reynolds
numbers, both the second-order correlation approximation
共or first-order smoothing approximation兲 and the ␶ approximation give identical results. This is also supported by simulations 关49兴 of isotropically forced helical turbulence
whereby the contributions to kinetic and magnetic ␣ effects
are computed. The study performed in 关49兴 provides an extra
piece of evidence that the ␶ approximation is a usable formalism for describing simulation data and for predicting the
behavior in situations that are not yet accessible to direct
numerical simulations.
ACKNOWLEDGMENTS
We have benefited from stimulating discussions with Axel
Brandenburg and Alexander Schekochihin. We thank the
Isaac Newton Institute for Mathematical Sciences 共Cambridge兲 for support during our visit in the framework of the
programme “Magnetohydrodynamics of Stellar Interiors.”
APPENDIX: VELOCITY AND MAGNETIC
FLUCTUATIONS IN TURBULENT CONVECTION
In order to study the velocity and magnetic fluctuations
with a nonzero mean magnetic field and to derive the effec-
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PHYSICAL REVIEW E 76, 056307 共2007兲
MAGNETIC FLUCTUATIONS AND FORMATION OF LARGE-…
tive stress tensor in the turbulent convection, we use a meanfield approach in which the magnetic and velocity fields and
entropy are decomposed into the mean and fluctuating parts,
where the fluctuating parts have zero mean values. The equations for fluctuations of the fluid velocity, entropy, and the
magnetic field are given by
冉冊
冋
⳵ gij共k兲
= i共k · H兲关f ij共k兲 − hij共k兲兴 + gen P jn共k兲Gi共− k兲 + N̂gij ,
⳵t
共A6兲
⳵ Fi共k兲
= − i共k · H兲Gi共k兲 + gen Pin共k兲⌰共k兲 + N̂Fi ,
⳵t
1
1 ⳵ v共x,t兲
p
g
=−⵱
−
冑␳0 s + 冑␳0 共b · ⵱兲H + 共H · ⵱兲b
⳵t
␳0
冑␳ 0
+
册
⌳␳
关2e共b · H兲 − 共b · e兲H兴 + vN ,
2
共A7兲
共A1兲
⳵ b共x,t兲
= 共H · ⵱兲v − 共v · ⵱兲H
⳵t
⌳␳
+
关v共H · e兲 − H共v · e兲兴 + bN ,
2
⍀2
⳵ s共x,t兲
= − b 共v · e兲 + sN ,
⳵t
g
共A2兲
共A3兲
where we used new variables 共v, s, H兲 for fluctuating fields
v = 冑␳0u and s = 冑␳0s⬘ and also for the mean field H
= B / 冑␳0. Here B is the mean magnetic field, ␳0 is the fluid
density, e is the vertical unit vector directed opposite to the
gravity field, ⍀2b = −g · ⵱S is the Brunt-Väisälä frequency, S is
the mean entropy, g is the acceleration of gravity, u, b, and s⬘
are fluctuations of velocity, magnetic field, and entropy 共we
have not used new variables for magnetic fluctuations兲, vN,
bN, and sN are nonlinear terms which include the molecular
viscous and diffusion terms, p = p⬘ + 冑␳0共H · b兲 are the fluctuations of total pressure, and p⬘ are the fluctuations of fluid
pressure.
Equations 共A1兲–共A3兲 for fluctuations of fluid velocity, entropy, and magnetic field are written in the anelastic approximation, which is a combination of the Boussinesq approximation and the condition div 共␳0u兲 = 0. The equation div u
= ⌳␳共u · e兲 in the new variables reads div v = 共⌳␳ / 2兲共v · e兲,
where ⵱␳0 / ␳0 = −⌳␳e. The quantities with the subscript 0
correspond to the hydrostatic nearly isentropic basic reference state; i.e., ⵱P0 = ␳0g and g · 关共˜␥ P0兲−1⵱P0 − ␳−1
0 ⵱␳0兴 ⬇ 0,
where ˜␥ is the specific heat ratio and P0 is the fluid pressure
in the basic reference state. The turbulent convection is regarded as a small deviation from a well-mixed adiabatic reference state.
Using Eqs. 共A1兲–共A3兲 and performing the procedure described in Sec. III we derive equations for the two-point
second-order correlation functions of the velocity fluctuations f ij = 具viv j典, the magnetic fluctuations hij = 具bib j典, the entropy fluctuations ⌰ = 具ss典, the cross-helicity gij = 具biv j典, and
the turbulent heat flux Fi = 具svi典 and Gi = 具sbi典. The equations
for these correlation functions are given by
⳵ f ij共k兲
= i共k · H兲⌽ij + N̂f ij ,
⳵t
共A4兲
⳵ hij共k兲
= − i共k · H兲⌽ij + N̂hij ,
⳵t
共A5兲
⳵ Gi共k兲
= − i共k · H兲Fi共k兲 + N̂Gi ,
⳵t
共A8兲
⍀2
⳵ ⌰共k兲
= − b Fz共k兲 + N̂⌰
⳵t
g
共A9兲
共see for details 关31兴兲, where ⌽ij共k兲 = gij共k兲 − g ji共−k兲, N̂f ij
= gen关Pin共k兲F j共k兲 + P jn共k兲Fi共−k兲兴 + N̂f̃ ij, and N̂f̃ ij, N̂hij, N̂gij,
N̂Fi, N̂Gi, and N̂⌰ are the third-order moment terms appearing due to the nonlinear terms. The terms ⬃Fi in the tensor
N̂f ij can be considered as a stirring force for the turbulent
convection. Note that a stirring force in the Navier-Stokes
turbulence is an external parameter.
We split the tensor of magnetic fluctuations into nonheli共H兲
cal, hij, and helical, h共H兲
ij , parts. The helical part hij depends
on the magnetic helicity, and it is determined by the dynamic
equation which follows from the magnetic helicity conservation arguments. We also split all second-order correlation
functions into symmetric and antisymmetric parts with re共a兲
spect to the wave vector k—e.g., f ij = f 共s兲
ij + f ij , where the
共s兲
tensor f ij = 关f ij共k兲 + f ij共−k兲兴 / 2 describes the symmetric part
of the tensor and f 共a兲
ij = 关f ij共k兲 − f ij共−k兲兴 / 2 determines the antisymmetric part of the tensor. We use the spectral ␶ approximation 关see Eq. 共14兲 in Sec. III兴. We assume also that the
characteristic time of variation of the mean magnetic field B
is substantially larger than the correlation time ␶共k兲 for all
turbulence scales. This allows us to get a stationary solution
for the equations for the second-order moments:
f 共s兲
ij 共k兲 ⬇
1
共0s兲
关共1 + ␺兲f 共0s兲
ij 共k兲 + ␺hij 共k兲
1 + 2␺
− 2␺␶gen Pin共k兲F共s兲
j 共k兲兴,
h共s兲
ij 共k兲 ⬇
1
共0s兲
关␺ f 共0s兲
ij 共k兲 + 共1 + ␺兲hij 共k兲
1 + 2␺
+ ␺␶gen Pin共k兲F共s兲
j 共k兲兴,
g共a兲
ij 共k兲 ⬇
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共A10兲
共A11兲
i␶共k · H兲 共0s兲
共s兲
关f 共k兲 − h共0s兲
ij 共k兲 + ␶gen Pin共k兲F j 共k兲兴,
1 + 2␺ ij
共A12兲
F共s兲
i 共k兲 ⬇
F共0s兲
i 共k兲
,
1 + ␺/2
共A13兲
PHYSICAL REVIEW E 76, 056307 共2007兲
IGOR ROGACHEVSKII AND NATHAN KLEEORIN
共s兲
G共a兲
i 共k兲 ⬇ − i␶共k · H兲Fi 共k兲
共A14兲
共for details see 关31兴兲, where ␺共k兲 = 2共␶k · H兲2 and we neglected terms ⬃O共⍀2b兲. In Eqs. 共A10兲–共A14兲 we neglected
also the large-scale spatial derivatives. The correlation func共a兲
共s兲
共a兲
共s兲
vanish because they are
tions f 共a兲
ij , hij , gij Fi , and Gi
proportional to the first-order spatial derivatives. Equations
共A10兲 and 共A11兲 yield
and the functions A共1兲
n 共y兲 are given by Eq. 共A34兲. The magnetic stress tensor is given by Eqs. 共19兲, where the anisotropic contribution ␴Aij to the magnetic stress tensor is determined by
1
共1兲
共1兲
␴Aij = 关eij⌿兵2C共1兲
1 + A1 + A2 其
2
共1兲
+ cos ␾共ei␤ j + e j␤i兲⌿兵2C共1兲
3 − A2 其兴,
共s兲
f 共s兲
ij 共k兲 + hij 共k兲
共0s兲
= f 共0s兲
ij 共k兲 + hij 共k兲 −
␺
␶gen Pin共k兲F共s兲
j 共k兲.
1 + 2␺
共A15兲
and the nonlinear coefficients q p共B兲 and qs共B兲 are given by
q p共B兲 =
in the background
Therefore, when the mean heat flux F共0s兲
i
turbulence is zero 共i.e., for the nonconvective turbulence兲, we
obtain
f 共s兲
ij 共k兲
+
h共s兲
ij 共k兲
=
f 共0s兲
ij 共k兲
+
h共0s兲
ij 共k兲.
2 共0兲
f 共0兲
ij 共k兲 = ␳0具u 典 W共k兲Pij共k兲,
共A17兲
2 共0兲
h共0兲
ij 共k兲 = 具b 典 W共k兲Pij共k兲,
共A18兲
共0兲
F共0兲
i 共k兲 = 3␳0具uis⬘典 W共k兲e j Pij共k兲,
共A19兲
共0兲
2 共0兲
⌰ 共k兲 = 2␳0具s⬘ 典 W共k兲,
1
共0兲
共具v2典共0兲 − 具b2典共0兲兲关6 − 3A共0兲
1 共4B兲 − A2 共4B兲兴
12
a*
+ 具v2典共0兲关2⌿兵A1其 + 共1 + 3 cos2␾兲⌿兵A2其兴,
6
共1兲
+ cos2␾⌿兵6C共1兲
3 + A2 其兴兴,
qs共B兲 = −
共A23兲
1
关共1 − ⑀兲A共0兲
2 共4B兲
12B2
共1兲
2
+ 6a*关⌿兵C共1兲
3 其 + cos ␾⌿兵C2 其兴兴.
共A24兲
The asymptotic formulas for these coefficients are given by
Eqs. 共20兲–共25兲.
In order to study the large-scale instability we use the
equation of motion, the induction equation, and the equation
for the mean entropy:
冉 冊 冋
1
P̃tot
B2
DUi
= − ⵜi
+
关1 − q p共B兲兴 ⌳␳ei + 共B · ⵱兲
2
Dt
␳0
␳0
⫻兵关1 − qs共B兲兴Bi其 + ⵜ j关2␳0␯T共B兲共⳵ U兲ij兴 + ⵜ j␴Aij
− gS − ␯T共B兲⌳␳eidiv U,
册
共A25兲
共A20兲
where Pij共k兲 = ␦ij − kik j / k2, W共k兲 = E共k兲 / 8␲k2, ␶共k兲 = 2␶0¯␶共k兲,
E共k兲 = −d¯␶共k兲 / dk, ¯␶共k兲 = 共k / k0兲1−q, 1 ⬍ q ⬍ 3 is the exponent
of the kinetic energy spectrum 共e.g., q = 5 / 3 for Kolmogorov
spectrum兲, k0 = 1 / l0, and ␶0 = l0 / u0. Note also that g共0兲
ij 共k兲 = 0
共k兲
=
0.
and G共0兲
i
This procedure allows us to study magnetic fluctuations
with a nonzero mean magnetic field and to derive the effective stress tensor in the turbulent convection 共see Sec. III兲. In
particular, integration in k space in Eq. 共A11兲 yields the energy of magnetic fluctuations,
具b2典 = 具b2典共0兲 +
1
共0兲
共0兲
关共1 − ⑀兲关A共0兲
1 共0兲 − A1 共4B兲 − A2 共4B兲兴
12B2
共1兲
共1兲
+ 2a*关⌿兵6C共1兲
1 − 2A1 − A2 其
共A16兲
This is in agreement with the fact that a uniform mean magnetic field performs no work on the turbulence 共without
mean heat flux兲. It can only redistribute the energy between
hydrodynamic fluctuations and magnetic fluctuation. A
change of the total energy of fluctuations is caused by a
nonuniform mean magnetic field. For the integration in k
space of these second moments we have to specify a model
for the background turbulent convection 共with zero mean
magnetic field, B = 0兲. Here we use the following model of
the background turbulent convection 关denoted with the superscript 共0兲兴:
共A22兲
共A26兲
DS
= − ⵱ · F̂共s兲 ,
Dt
共A27兲
where ␩T共B兲 is the turbulent magnetic diffusion, D / Dt
= ⳵ / ⳵t + 共U · ⵱兲, and F̂共s兲 = −␬共T兲
ij 共B兲⵱S is the turbulent heat
is
the
tensor
for
the
nonlinear turbulent thermal
flux, ␬共T兲
ij
diffusivity, and
P̃tot = Pk + 关1 − q p共B兲兴
共A21兲
where ⌿兵X其 = X共1兲共2B兲 − X共1兲共4B兲, ␾ is the angle between the
vertical unit vector e and the mean magnetic field B, the
functions A共0兲
n 共y兲 are given by Eqs. 共A33兲, 共A35兲, and 共A36兲,
⳵B
= ⵱ ⫻ 关U ⫻ B − ␩T共B兲共⵱ ⫻ B兲兴,
⳵t
B2
− ␯T共B兲␳0 div U,
2
Pk is the mean fluid pressure, 2共⳵U兲ij = ⵜiU j + ⵜ jUi, and
␯T共B兲 is the turbulent viscosity. The turbulent viscosity ␯T共B兲
in Eq. 共A25兲 is given by
056307-12
PHYSICAL REVIEW E 76, 056307 共2007兲
MAGNETIC FLUCTUATIONS AND FORMATION OF LARGE-…
冋
共1兲
␯T共B兲 = ␯T 2A共1兲
1 + 共1 + ⑀兲A2 − 2共1 − ⑀兲H兵A1其
Ā1 =
1
− 关共1 − 29⑀兲C共1兲
1 − 4共7 − 8⑀兲H兵C1其
6
+ 共1 − 3⑀兲G兵C1其 − 2共1 − ⑀兲Q兵C1其兴
册
Ā2 = −
.
y=4B
共A28兲
The explicit form of the functions H兵X其, G兵X其, and Q兵X其 is
given in 关30兴. The asymptotic formula for ␯T共B兲 for a weak
mean magnetic field, B Beq / 4, is given by ␯T共B兲 = ␯T共1
+ 2⑀兲, and for B Beq / 4 it is given by ␯T共B兲 = 共␯T / 4B兲共1
共T兲
+ ⑀兲. The turbulent heat flux F̂共s兲
i 共B兲 = −␬ij 共B兲ⵜ jS and the
tensor for the nonlinear turbulent thermal diffusivity,
␬共T兲
ij 共B兲 =
␬共T兲
*
4
C̄1 =
␬共T兲
*
关2共10 − ␤2 ln Rm兲␦ij + ␤2 ln Rm␤ij兴,
20
共A30兲
␬共T兲
*
5
冑
共T兲 2␲
␬共T兲
ij 共B兲 = ␬*
4␤
冕
冕
kij sin ␪
d␪d␸ = Ā1␦ij + Ā2␤ij,
1 + a cos2␪
3␤4
␲
冕
␤
␤Rm1/4
␤
Ān共X2兲
dX,
X3
共A33兲
Ān共X2兲
dX,
X5
共A34兲
冋
冉
冋
1 + ␤2
1 + ␤2冑Rm
冊册
共A35兲
,
2
arctan ␤
9
2−
共9 + 5␤2兲 + 2 − ␤2 ln Rm
3
5
␤
␤
冉
1 + ␤2
1 + ␤2冑Rm
冊册
共A36兲
,
where ␤ = 冑8B / Beq. For B Beq / 4Rm1/4 these functions are
given by
冋
册
2 2
2
.
A共0兲
2 共␤兲 ⬃ − ␤ ln Rm +
5
15
For Beq / 4Rm1/4 B Beq / 4 these functions are given by
冋
册
2 2
16 4 2
A共0兲
+ ␤ ,
1 共␤兲 ⬃ 2 + ␤ 2 ln ␤ −
5
15 7
+ ␦in␤ jm + ␦ jm␤in + ␦ jn␤im + ␦mn␤ij兲,
Ḡijmn共a兲 =
A共1兲
n 共␤兲 =
− 2␤2 ln
kijmn sin ␪
d␪d␸
1 + a cos2␪
+ C̄2␤ijmn + C̄3共␦ij␤mn + ␦im␤ jn
冕
冕
冕
␤Rm1/4
1 2
A共0兲
1 共␤兲 ⬃ 2 − ␤ ln Rm,
5
= C̄1共␦ij␦mn + ␦im␦ jn + ␦in␦ jm兲
H̄ijmn共a兲 =
3␤2
␲
− 2␤2 ln
共A32兲
In order to integrate in Eqs. 共A10兲–共A14兲 over the angles in
k space we used the following identity:
K̄ijmn =
A共0兲
n 共␤兲 =
1
arctan ␤
6
2+2
共3 + 5␤2兲 − 2 − ␤2 ln Rm
5
␤3
␤
A共0兲
1 共␤兲 =
A共0兲
2 共␤兲 =
共␦ij + ␤ij兲.
C̄3 = Ā1 − 5C̄1
2
2
2/3
and similarly for C共m兲
n 共␤兲, where X = ␤ 共k / k0兲 = a. The ex共m兲
共m兲
plicit form of the functions An 共␤兲 and Cn 共␤兲 for m = 1,2
共0兲
are given in 关30兴, and the functions A共0兲
1 共␤兲 and A2 共␤兲 are
given by
关共5 − 2␤2兩ln ␤兩兲␦ij + ␤2兩ln ␤兩␤ij兴, 共A31兲
and when B Beq / 2, it is
册
共for details, see 关30兴兲. The functions A共m兲
n 共␤兲 are given by
where ␤ = 冑8B / Beq. When Beq / 2Rm1/4 B Beq / 2, the
function ␬共T兲
ij 共B兲 is
K̄ij =
冋
冑
5a
␲
2 arctan共 a兲
−
−1 ,
2 共a + 1兲
冑a
2a
3
共0兲
共0兲
关共2A共0兲
1 共2B兲 + A2 共2B兲兲␦ij − A2 共2B兲␤ij兴,
2
where ␬共T兲
* = u0l0 / 3 and ␤ij = BiB j / B . The asymptotic formula
共T兲
1/4
for ␬ij 共B兲 for B Beq / 2Rm reads
␬共T兲
ij 共B兲 =
2␲
arctan共冑a兲
共a + 3兲
冑a − 3
a
C̄2 = Ā2 − 7Ā1 + 35C̄1 ,
共A29兲
␬共T兲
ij 共B兲 =
册
册
冋
冋
2␲
arctan共冑a兲
共a + 1兲
冑a − 1 ,
a
冋
⳵
kijmn sin ␪
K̄ijmn共a兲,
2 2 d␪d␸ = K̄ijmn共a兲 + a
⳵a
共1 + a cos ␪兲
kijmn sin ␪
a ⳵
H̄ijmn共a兲,
2 3 d␪d␸ = H̄ijmn共a兲 +
共1 + a cos ␪兲
2 ⳵a
册
2 2
2
− 3␤2 ,
A共0兲
2 共␤兲 ⬃ ␤ 4 ln ␤ −
5
15
and for B Beq / 4 they are given by
where a = ␤ /¯␶共k兲, and
2
056307-13
A共0兲
1 共␤兲 ⬃
␲ 3
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␤ ␤2
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2 共␤兲 ⬃ −
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PHYSICAL REVIEW E 76, 056307 共2007兲
IGOR ROGACHEVSKII AND NATHAN KLEEORIN
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