Statistical properties of ideal three-dimensional Hall ...

PHYSICS OF PLASMAS 15, 042314 共2008兲

Statistical properties of ideal three-dimensional Hall magnetohydrodynamics: The spectral structure of the equilibrium ensemble Sergio Servidio,1 William H. Matthaeus,1 and Vincenzo Carbone2,3 1

Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA 2 Dipartimento di Fisica, Università della Calabria, Ponte P. Bucci Cubo 31C, 87036 Rende (CS), Italy 3 Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia (CNISM), Università della Calabria, Ponte P. Bucci Cubo 33B, 87036 Rende (CS), Italy

共Received 2 January 2008; accepted 20 March 2008; published online 24 April 2008兲 The nonlinear dynamics of ideal, incompressible Hall magnetohydrodynamics 共HMHD兲 is investigated through classical Gibbs ensemble methods applied to the finite Galerkin representation. The spectral structure of HMHD is derived in a three-dimensional periodic geometry and compared with the MHD case. This provides a general picture of spectral transfer and cascade by the assumption that ideal Galerkin HMHD follows equilibrium statistics as in the case of Euler 关U. Frisch et al., J. Fluid Mech. 68, 769 共1975兲兴 and MHD 关T. Stribling and W. H. Matthaeus, Phys. Fluids B 2, 1979 共1990兲兴 theories. In HMHD, the equilibrium ensemble is built on the conservation of three quadratic invariants: The total energy, the magnetic helicity, and the generalized helicity. The latter replaces the cross helicity in MHD. In HMHD equilibrium, several differences appear with respect to the MHD case: 共i兲 The generalized helicity 共and in a weaker way the energy and the magnetic helicity兲 tends to condense in the longest wavelength, as in MHD, but also admits the novel feature of spectral enhancement, not a true condensation, at the smallest scales; 共ii兲 equipartition between kinetic and magnetic energy, typical of Alfvénic MHD turbulence, is broken; 共iii兲 modal distributions of energy and helicities show minima due to the presence of the ion skin depth. Ensemble predictions are compared to numerical simulations with a low-order truncation Galerkin spectral code, and good agreement is seen. Implications for general turbulent states are discussed. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2907789兴 I. INTRODUCTION

The application of classical Gibbs ensemble methods to the nonlinear dynamics of Euler and ideal magnetohydrodynamics 共MHD兲 theory is well known in the literature.1–13 This technique can provide further details on the spectral transfer properties of turbulence, such as, for example, the direction of turbulent energy cascades. In principle, Gibbs ensemble methods can be properly applied to nondissipative systems that have global dynamical invariants. The obtained equilibrium spectrum can give a qualitative picture of all measurable quantities and can predict the general behavior of turbulence. This theory assumes that fields are Gaussian and that the system is at equilibrium among Fourier modes. However, as shown by Kraichnan,3,10 quadratic invariants, averaged over the Gibbs ensemble, have a tendency to peak in different wavenumbers of the k-vectors space when more than one such invariant is present. This tendency corresponds to cascade properties when nonlinear turbulent nonequilibrium dynamics is taken into account. Direct comparison of equilibrium predictions with turbulent systems is not appropriate because viscous effects dissipate energy at small scale, preventing relaxation to true equilibrium. Furthermore, dissipative turbulence necessarily involve departures from the Gaussian statistics.14 Nevertheless, using the equilibrium approximation, many aspects of spectral properties can be qualitatively predicted: The inverse cascade of the mean 1070-664X/2008/15共4兲/042314/15/$23.00

square potential and magnetic helicity in two-dimensional 共2D兲 and 3D MHD, respectively;11 the direct cascade of energy in Navier–Stokes turbulence;1 the dynamic alignment between velocity and magnetic fields at small scales;11,15 as well as the spectral properties of guiding center plasmas,9,16 electron fluids,17 and drift wave turbulence.18 In this paper, we extend the work of previous authors on MHD1,2,19 and we obtain the Hall magnetohydrodynamics 共HMHD兲 absolute equilibrium ensemble. The focus is on the departure from equipartition in wavenumber space, which is believed to be influential for inverse and direct spectral transfer of ideal rugged invariants. The key physical feature in HMHD is the introduction of two-fluid effects on fluctuations at scales comparable to the ion skin depth ␭i = c / ␻ pi 共␻ pi is the plasma frequency, c is the speed of light兲. We will also refer to this below as the Hall scale. Two-fluid effects are commonly observed in a variety of space plasmas, such as, for example, in the solar wind20,21 and in the Earth magnetosphere.22,23 A direct consequence of the Hall correction to Ohm’s law is that the magnetic flux is no longer frozen into the ions, but rather in the electron fluid. The presence of ion skin depth introduces additional nonlinear couplings between velocity and magnetic field, giving rise to interesting plasma phenomena such as filamentations,24,25 an increase of the magnetic reconnection rate,26 relaxation processes,27 and accelerated dynamo action.28,29 The effect of

15, 042314-1

© 2008 American Institute of Physics

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042314-2

Phys. Plasmas 15, 042314 共2008兲

Servidio, Matthaeus, and Carbone

␭i on the direction of turbulent cascades in not yet clearly understood. Notable features of the present work involve the influence of total magnetic and generalized helicity on the modal wavenumber distribution, leading to departures from the MHD case. A central motivation of this work is to provide a basis for better understanding of HMHD spectral transfer for arbitrary ratios of conserved invariants and the influence on nonlinear processes as selective decay and dynamic alignment. We also provide a starting point for future numerical experiments of dissipative 3D HMHD. The outline of this paper is as follows. In Sec. II, we will discuss the dimensionless HMHD equations, the global invariants of the system, and the equilibrium ensemble. In Sec. III, the equilibrium distribution of HMHD is obtained and then compared to the MHD case. In Sec. IV, we will examine the infinite space dimension limit. Numerical simulations of ideal HMHD will be shown in Sec. V. In Sec. VI, possible dynamical implications of the ensemble spectral predictions for dissipative HMHD are discussed.

where N is a set of 3D wave vectors k such that kmin 艋兩k兩艋 kmax. Moreover, the reality condition must be satisfied, v共k,t兲 = v*共− k,t兲,

In this paper, we will denote as N the number of Fourier modes 共vectors兲 that belong to N. In a Cartesian representation, N = 共2Nbox + 1兲3. When relations 共5兲 are inserted into Eqs. 共1兲–共4兲, a system of nonlinear coupled ordinary differential equations for the Fourier coefficients is produced. This truncated ideal 3D HMHD model conserves three quadratic invariants for every set, N, which are referred to as the rugged invariants.2 The known rugged invariants of Eqs. 共1兲–共4兲 are the energy E, the magnetic helicity Hm, and the generalized helicity Hg,

E= II. THE EQUILIBRIUM ENSEMBLE OF IDEAL HALL MHD

The equations of incompressible, ideal, constant density 3D HMHD in the usual dimensionless variables are given by30

⳵v = − 共v · ⵱兲v + 共b · ⵱兲b − ⵱p* , ⳵t

共1兲

⳵b = 共b · ⵱兲v − 共v · ⵱兲b − ⑀ ⵱ ⫻ 关共⵱ ⫻ b兲 ⫻ b兴, ⳵t

共2兲

⵱ · v = 0,

共3兲

⵱ · b = 0.

共4兲

共6兲

b共k,t兲 = b*共− k,t兲.

=

1 2V

冕

共兩v兩2 + 兩b兩2兲d3x

1 兺关兩v共k兲兩2 + 兩b共k兲兩2兴 2 k苸兵N其

=

兺

关Ev共k兲 + Eb共k兲兴,

1 2V

冕

共7兲

k苸兵N其

Hm = =

a · bd3x

1 i 兺兵关k ⫻ b共k兲兴 · b*共k兲 − 关k ⫻ b*共k兲兴 · b共k兲其, 4 k苸兵N其 k2 共8兲

The velocity and magnetic field v and b are both written in Alfvén speed units. The magnetic vector potential a is defined by b = ⵱ ⫻ a and ⵱ · a = 0. The total pressure p* is obtained from the divergence of Eq. 共1兲 using the solenoidal condition Eq. 共3兲. The Hall effect enters through the parameter ⑀, which is the ratio between the ion skin depth ␭i and the unit length scale L0. The time unit is the Alfvén time at L 0. By considering a 3D Cartesian box with sides of dimensionless length 2␲, and assuming periodicity, the fields can be written as follows:

Hg = =

兺

兺

k苸兵N其

⑀

册

v · b + 共 ␻ · v 兲 d 3x 2

1 兺兵v共k兲 · b*共k兲 + v*共k兲 · b共k兲其 4 k苸兵N其

The generalized helicity Hg, which involves the vorticity ␻ = ⵜ ⫻ v, is obtained from the hybrid helicity,31,32 1 2V

冕

共a + ⑀v兲 · 共b + ⑀␻兲d3x,

共10兲

v共k,t兲exp共ik · x兲,

k苸兵N其

b共x,t兲 =

冕冋

⑀ + i 兵关k ⫻ v共k兲兴 · v*共k兲 − 关k ⫻ v*共k兲兴 ·关v共k兲兴其. 共9兲 2

Hh = v共x,t兲 =

1 2V

共5兲 b共k,t兲exp共ik · x兲,

introduced by Turner in 1986.31 The hybrid helicity is an ideal Hall quadratic invariant. Hh and Hg are similar and not independent of each other. The relationship between Hg and the Hh can be obtained by expanding the hybrid helicity as


042314-3

Hh =


Statistical properties of ideal…

1 2V

冕

兵a · 共⵱ ⫻ a兲 + ⑀a · 共⵱ ⫻ v兲 + ⑀v · 共⵱ ⫻ a兲

+ ⑀2v · 共⵱ ⫻ v兲其d3x,

共11兲

where we use the following identity:

冕

a · 共⵱ ⫻ v兲d3x =

冕冕

⵱ · 共a ⫻ v兲d3x

+

v · 共⵱ ⫻ a兲d3x.

共12兲

The first term on the right-hand side of Eq. 共12兲 vanishes because of the periodic geometry. Substituting Eq. 共12兲 into Eq. 共11兲, and using the definition of Eqs. 共8兲 and 共9兲, a linear relation between Hg and Hh is readily found, Hh = Hm + 2⑀Hg .

共13兲

The ideal conservation of Hh, proven in Ref. 31, along with the ideal conservation of Hm, imply immediately that Hg is a rugged invariant of HMHD. For ⑀ → 0, the generalized helicity reduces a MHD invariant, namely the cross helicity.2,11 The MHD limit of Hh must of course be equivalent, but is a bit more subtle algebraically. In this paper, we prefer to work with the generalized helicity Hg instead of Hh, because, as can be seen from Eq. 共13兲, the latter depends on the other invariant Hm. The integrals are evaluated over a box of volume V = 共2␲兲3. For a given space N, it can be shown that the quantities defined in Eqs. 共7兲–共9兲 are exactly conserved for every triad of wave vectors 兵k , p , q其 that satisfy the geometric relation p + q = k. The Fourier–Galerkin approximation of

Z=

冕

e−共␤E+␥Hg+␪Hm兲d␰ =

phase space

兿

1 P = e−共␤E+␥Hg+␪Hm兲 , Z

dv1共k兲dv2共k兲db1共k兲db2共k兲.

共14兲

where ␤, ␥, and ␪ are generalized temperatures corresponding to E, Hg, and Hm, respectively. Z is the partition function and is given by

共2␲兲4k2

兿 k2 ␤2 − 1 ␥2兲2 − ␪2␤2兴 + 1 ⑀k2␥2共⑀␪2 − k2␤2⑀ − ␥␪兲 = 兿 Zk , 4 4 k苸兵N⬘其关共 k苸兵N⬘其

共15兲

关k2共␤2 − 41 ␥2兲2 − ␪2␤2兴 + 41 ⑀k2␥2共⑀␪2 − k2␤2⑀ − ␥␪兲 ⬎ 0,

where the differential phase space volume element is d␰ =

Eqs. 共1兲–共4兲 provides a very accurate representation of the interactions among retained triads 兵k , p , q其共see also Sec. V兲. As was pointed out by Lee for hydrodynamics,33 the Fourier representation of Eqs. 共1兲–共4兲 satisfies a Liouville theorem in a phase space built with the real and imaginary part of velocity and magnetic fields, taking care to enforce Eqs. 共3兲, 共4兲, and 共6兲. Again, the components of the independent real and imaginary parts of the coefficients v共k兲 and b共k兲 are used to label the axes of a multidimensional phase space; the corresponding dynamical system is described by a single point in this phase space, a point that moves about as the system evolves in time. The probability that the system point is in any part of the space can be described by a canonical probability density P, which depends only on a set of conserved quantities. These are the isolating integrals of the motion and they are usually identified with the quadratic so-called rugged invariants. Once P has been found, then the equilibrium energy spectrum can be determined, even though v共k兲 and b共k兲 are random variables. Using this procedure, one can apply classical equilibrium statistical mechanics to investigate the properties of the equilibrium spectra.1–4,6 Consequently in the HMHD, the generalized 共Gibbs兲 canonical distribution of states is

共20兲

共16兲

k苸兵N⬘其

Note that N⬘ is the subset of independent Fourier modes taking into account Eq. 共6兲, while the complex components 共v1 , v2兲 and 共b1 , b2兲 satisfy solenoidal constrictions of Eqs. 共3兲 and 共4兲.2 In a Cartesian representation, N⬘ = N / 2. The requirement that Z is finite and real implies the following relations for the inverse temperatures:

␤ ⬎ 0,

共17兲

2 ␤2 − 41 kmax ␥2⑀2 ⬎ 0,

共18兲

2 ␥2⑀2 ⬎ 0, 共␤2 − 41 ␥2兲 − 41 kmax

共19兲

where kmax is the largest wavenumber of the truncation N, namely, kmax = Nbox冑3. Given a generic quantity g, the ensemble-averaged value 具g典 can be found by integration over the phase space,

具g典 =

1 Z

冕

g exp关− 共␤E + ␥Hg + ␪Hm兲兴d␰ .

共21兲

It is interesting to notice that the equilibrium spectra can be obtained in other alternative ways, such as, for example, by differentiation of Z. This procedure is easier than solving the multidimensional integral defined in Eq. 共21兲, and corresponds to evaluate


042314-4



具E共k兲典 = −

⳵ ln Zk , ⳵␤

具Hg共k兲典 = −

具Hm共k兲典 = − 2

共22兲

d共k兲 = 关k2共␤2 − 41 ␥2兲2 − ␪2␤2兴 + 41 ⑀k2␥2共⑀␪2 − k2␤2⑀ − ␥␪兲

⳵ ln Zk , 具Hm共k兲典 = − ⳵␪

共31兲

and E共k兲 = Ev共k兲 + Eb共k兲. An additional relation that must be satisfied is

where Zk is defined by Eq. 共15兲. Another elegant way to compute ensemble averages is to use the properties of n-dimensional normal distributions.34–36 The joint distribution of l random variables 共x1 , x2 , . . . , xl兲共in our case the imaginary and real parts of velocity and magnetic fields give l = 8 for each value of k兲 has a probability function of the form

冑共2␲兲

共30兲

where

⳵ ln Zk , ⳵␥

P共x1,x2, . . . ,xl兲 =

␤2␪ + 81 k2␥2⑀共␥ − 2⑀␪兲 , d共k兲

1 l

冋

det关␭ jk兴 l

⫻exp −

冏

k具Hm共k兲典 ⬍ 1. 具Eb共k兲典

1 兺兺 ⌳ij共xi − ␺i兲共x j − ␺ j兲 , 2 i=1 j=1

共32兲

In the limit of ⑀ → 0, it is easy to show that the HMHD equilibrium is equivalent to the MHD case, obtained in Ref. 2. In the limit of vanishing ␪, a difference with respect to the MHD case is given by the following relation: 具Hm共k兲典␪=0 = −

册

l

冏

4共␤ − 2

␥ 3⑀ . 兲 − k 2␤ 2␥ 2⑀ 2

1 2 2 4␥

共33兲

In this case, the magnetic helicity still has a dependence on the k vector, while in MHD, 具Hm共k兲典␪=0 = 0.

共23兲 where 共␺1 , ␺2 , . . . , ␺l兲 are the mean values and 关␭ij兴 = 关⌳ij兴−1 is the moment matrix.35 Equation 共23兲 is of the same form of our generalized Gibbs canonical distributions of states Eq. 共14兲. By using the properties of Eq. 共23兲, the partition function is given by Z = 冑共2␲兲l det关␭ jk兴,

共24兲

and supposing that every mode has zero mean value 共␺ j = 0兲, all the ensemble averages are simply obtained as34–36

III. COMPARISON BETWEEN HMHD AND MHD EQUILIBRIUM SPECTRA

The inverse temperatures can be numerically obtained by solving the system of nonlinear polynomial equations, E=

兺

具E共k兲典,

共34兲

k苸兵N其

Hg =

兺

具Hg共k兲典,

共35兲

兺

具Hm共k兲典.

共36兲

k苸兵N其

具xixk典 = ␭ik .

共25兲

Relations 共23兲–共25兲 will be useful for some later discussions 共Sec. V兲. By applying the definition given in Eq. 共21兲, or differentiating as in Eqs. 共22兲, or, moreover, by using the procedure described by Eqs. 共23兲–共25兲, the following equilibrium spectra for HMHD are obtained: 具Ev共k兲典 = 2␤

k2共␤2 − 41 ␥2兲 − ␪2 d共k兲

具Eb共k兲典 = 2k2␤

共␤2 − 41 ␥2兲 − 41 k2␥2⑀2 d共k兲

2k 共␤ − 2

具E共k兲典 = 2␤

共26兲

,

2

1 2 4␥

兲−␪

2

−

1 4 2 2 4k ␥ ⑀

d共k兲

共27兲

,

共28兲

,

⑀

具Hg共k兲典 = − k2␥

共␤2 − 41 ␥2兲 + 2 共␤2k2⑀ + 23 ␥␪ − ⑀␪2兲 d共k兲

, 共29兲

Hm =

k苸兵N其

A useful expression can be found that relates all three inverse temperatures and the three rugged invariants,

␤E + ␥Hg + ␪Hm = 4N.

共37兲

This will be employed below to reduce the number of unknown variables when we solve for the temperatures for given values of the invariants. A closely related equation is valid also for each Fourier mode,

␤具E共k兲典 + ␥具Hg共k兲典 + ␪具Hm共k兲典 = 4.

共38兲

The analogy between the fluid equilibrium ensemble37 and classical thermodynamics can be seen in comparing Eq. 共37兲 and thermodynamic relations that define the internal energy and Helmholtz free energy. In the HMHD ensemble, there are two additional invariants 共Hg and Hm兲 that are postulated to be isolating integrals of the dynamical system. Along with the energy, these determine the ensemble distribution Eq. 共14兲. If we let T = 1 / ␤ and define a generalized energy-like quantity E⬘ = E + ␥THg + ␪THm, then the above




global relation Eq. 共37兲 is simply E⬘ = 4NT and the modal equivalent is E⬘共k兲 = 4T. This implies that in equilibrium, the value of E⬘ is T / 2 in each of the system’s 8N real degrees of freedom. 共Degrees of freedom are counted as follows: There are two fields v and b, each described by N wave vectors, and at each wave vector there are two polarizations, each with a real and imaginary amplitude.兲 Thus it is E⬘ that attains an equipartitioned distribution at equilibrium. Therefore, E⬘ = 4NT defines in effect the thermodynamic internal energy of the system. The thermodynamic analogy may be extended further if we assert, as was done previously for MHD,37 that the entropy in the most probable state38 is given by ⌺ = −兰P ln Pd␰. Then we find readily from Eq. 共14兲 that ⌺ = − 具ln P典 = ␤E + ␥Hg + ␪Hm + ln Z,

-6

10

10-8 10

-10

1

共39兲

共41兲

In HMHD, there is no such analogous equation that relates temperatures with quadratic invariants. This makes it more difficult to find the numerical value of temperatures. In fact, in HMHD the system of equations for the unknown temperatures cannot be reduced to an equation for a single temperature 共as in the MHD case兲. However, using Eq. 共37兲 in the system 共34兲–共36兲, and then solving it as a set of two nonlinear equations in two variables, inverse temperatures ␤, ␥, and ␪ can be found for every given value of E, Hg, and Hm. From these, one constructs the equilibrium spectra Eqs. 共26兲–共30兲, which constitute a zero free-parameter prediction for the time asymptotic or average spectral distributions of energies and helicities in HMHD. We now examine some properties of these equilibrium spectra. In Fig. 1, examples of energy and helicities spectra are given for a particular choice of parameters 共see caption兲 for both MHD and HMHD. The comparison reveals that in HMHD there is a condensation at small k vectors, driven by the presence of magnetic helicity. Thus the magnetic helicity, in HMHD as in MHD, exhibits in equilibrium an affinity for the longest allowed scales. If driving and dissipation are reinstated, this tendency is expected to drive a dual cascade in the k-vector space, in which energy is cascaded to small scales and magnetic helicity 共along with the required fraction of energy兲 cascade toward larger scales 共inverse cascade兲. The generalized helicity also exhibits an affinity for the largest allowed wavelengths and, moreover, can be expected to participate in an inverse cascade, provided that magnetic helicity is present. The analogous phenomenon, referring to cross helicity in the MHD case, has been called conditional condensa-

10

10-2 Modal Spectra

共40兲

where Z is the sum over states. It follows that in the present case, the quantity occupying a role analogous to the equilibrium Helmholtz free energy is F* ⬅ −T ln Z = E⬘ − T ⌺. For the case in which Hm = Hg = 0, the parallel to thermodynamics becomes very close. Note that in MHD, in addition to Eq. 共37兲, the following relation is satisfied:

␥␤E + 4␤2Hc − ␥␪Hm = 0.

10-4

MHD

which is equivalent to E⬘ − T ⌺ = −T ln Z. Now we recall that in traditional thermodynamics,38 the 共Helmholtz兲 free energy Fh is defined by Fh = − kT ln Z = U − TS,

-2

10 Modal Spectra

042314-5

kh

100

-4

10

10-6 10-8 HMHD 10

-10

1

10 k

100

FIG. 1. Comparison of spectra between HMHD 共lower panel兲 and MHD 共upper panel兲 for given values of parameters, namely, E = 1, Hm = 0.3, Hc = 0.25 共MHD兲 or Hg = 0.25 共HMHD兲 and Nbox = 32. The vertical bar in the lower panel indicates the ion skin depth wave vector 共kh = ⑀−1 = 3兲.

tion 共or, for the driven case, conditional inverse cascade兲.2 The novel feature for HMHD is that generalized helicity manifests a weak enhancement at small scales as well. The degree of condensation at the longest wavelengths for the two cases is an important issue. In Fig. 2, the ratio between MHD and HMHD spectra for energies and magnetic helicity is shown. The Hall effect causes less condensation of rugged invariants to the smallest k vectors, especially for kinetic energy, and a slight increase of excitations at the highest wavenumbers. A typical picture of magnetic and kinetic distribution is shown in Fig. 3, where a comparison with the MHD case is shown. Apart from the differences cited above in the largest scale excitations, there are two subtle but distinctive effects that occur at the smaller scales in HMHD but not MHD. One is that, close to the ion skin depth, the kinetic energy becomes bigger than the magnetic energy, in contrast with the typical and well-known equipartition of the MHD case. In particular, there is a crossover in the ratio of kinetic to magnetic energies. The k vector kvb at which kinetic-magnetic energy equipartition is found can be easily evaluated from Eqs. 共26兲 and 共27兲, and is given by


042314-6



1.2 1

10-2 Modal Spectra

E Ev Eb

0.8 Ratio

Ev Eb

MHD

kh

0.6 0.4

-4

10

0.2

10

0 1

10

-6

10

100

1

10

100

1.1 Hm

1.08

10-2 Modal Spectra

1.06 Ratio

Ev Eb

HMHD

1.04 1.02 1 0.98 0.96

kh kvb

10-4 10

0.94 1

10 k

100

10-6 1

10

100

k FIG. 2. Ratio between HMHD and MHD spectra. Upper panel: 具E典HMHD / 具E典MHD 共solid line兲, 具Ev典HMHD / 具Ev典MHD 共dot-dashed line兲, and 具Eb典HMHD / 具Eb典MHD 共dashed line兲. In the lower panel is represented 具Hm典HMHD / 具Hm典MHD. Values of parameters and rugged invariants are the same as in Fig. 1; the vertical bar indicates again kh.

k vb =

冑冏

冏

2␪ . ⑀␥

共42兲

This is a distinct departure from the MHD case, for which it is easily shown that 具Ev共k兲典艋具Eb共k兲典共Refs. 2 and 12兲关this follows from comparing Eqs. 共26兲 and 共27兲 with ⑀ = 0兴. A second point of comparison is that in HMHD, at the same wavenumber kvb, there is a minimum in the modal total energy spectrum, whereas this is a monotonic decreasing function of k in MHD. In Fig. 4, one sees that the HMHD kineticmagnetic ratio is smaller than 1 at k ⬍ kvb, while it becomes ⬎1 at higher k vectors. This effect is strongly related to the presence of generalized helicity in the system. In Fig. 4, we report the ratio 具Ev共k兲典 / 具Eb共k兲典 for different values of generalized and magnetic helicities. For Hg → 0, the typical MHD equipartitioned at high k spectrum is found. From Eq. 共42兲, it can be seen that the influence of the Hall effect on the spectrum is not localized in k space, because it depends on the temperatures, i.e., on the rugged invariants. In a case with Hg = Hm ⯝ 0, this effect cannot be seen. As indicated above, there is a minimum of energy in k space also at kvb. This effect is subtle in Figs. 1 and 3. There are also found minima 共denoted with the symbol k쐓兲 in the

FIG. 3. MHD and HMHD 共upper and lower panel, respectively兲 power spectra for the kinetic and magnetic energy. In the insets is represented a zoom of the equilibrium spectra at high k vectors. In the HMHD case 共inset of the lower panel兲, a crossover between the kinetic and the magnetic energy appears; kvb indicates the k vector at which the equipartition occurs. The spectra are evaluated for the same rugged invariants as in Figs. 1 and 2.

kinetic energy 共see Table I, at k쐓Ev兲, and in the modal helicity spectra 共at k쐓Hg兲 for HMHD. These effects also do not occur in the ensemble predictions for MHD. The positions of these minima in k space can be found easily from Eqs. 共26兲–共30兲 by setting the appropriate derivatives to zero. Table I summarizes these results for several cases with varying globals and parameters. In some cases, the minima k쐓 are found outside the retained k space boundaries—unless kmin ⬍ k쐓 ⬍ kmax, the result is unphysical. Such cases are marked with the symbol “—.” IV. INFINITE PHASE SPACE DIMENSION LIMIT

In this section, we consider the case N → ⬁ while E, Hg, and Hm remain fixed, which differs somewhat from the conventional thermodynamic limit. In the thermodynamic limit, one lets the volume of the system and the particle number go to infinity while maintaining constant energy and density per unit volume. That is, a nonzero energy per degree of freedom is maintained in the limiting process. This leads to a total energy that diverges while the temperature remains finite, a limit not of relevance for the present purposes. The limiting


042314-7



1.2 /

1.15

(a)

1.1 1.05 1 0.95

Hm = 0.01

0.9

Hm = 0.25

0.85

Hm = 0.35

0.8 10

100

1.2 /

1.15

(b)

1.1 1.05 1 0.95

Hg = 0.01

0.9

Hg = 0.25

0.85

Hg = 0.35

0.8 10

100 k

FIG. 4. The ratio between kinetic and magnetic energy for several values of rugged invariants and kh = 3: 共a兲 Hg = 0.3 and Hm = 0.01, 0.25, 0.35; 共b兲 Hm = 0.3 and Hg = 0.01, 0.25, 0.35. As can be seen, the generalized helicity affects the ratio of kinetic and magnetic energy, and, in particular, when Hg → 0, this phenomenon disappears and the MHD kinetic-magnetic equipartition is recovered.

process of interest here is the one for which rugged invariants are held constant, with constant system length, while the number of Fourier modes diverges, by increasing kmax. In our notation, kmax = 冑3Nbox. Since the values of wave vector act in the role of conventional particles, one has a limit of zero energy per particle, corresponding to divergent inverse temperatures. The two cases have in common that they describe the statistical properties of the system as the number of phase space coordinates goes to infinity, while maintaining constant average spatial densities of the invariant energies. Kraichnan5 has suggested that the infinite k-space volume

limit at constant total energy is the limit of interest for hydrodynamic statistical ensembles, occupying a role analogous to the well known classical thermodynamic limit. Specifically, we will be able to draw inferences from this limit concerning tendencies of spectral quantities to be preferentially transferred to the long or short wavelengths 共or both兲. Such tendencies, in the equilibrium ensemble statistics, have often contributed to the body of evidence supporting related conjectures for dissipative, high Reynolds number turbulence,1,6,7 which also involves an extremely large number of dynamically involved Fourier modes. In ordinary MHD, the only length scales involved are connected with kmin = 2␲ / L0 for arbitrary “box size” L0, and also kmax, which sets the smallest scale represented in the Galerkin approximation. As kmax → ⬁, only one scale remains. However, in HMHD, as kmax → ⬁, the Hall scale remains, and in computing the modified thermodynamic limit one must decide how the Hall wavenumber kh = ⑀−1 should behave. There are several possibilities. In particular, kmin⑀ = O共1兲 appears to be unphysical in that as kmax → ⬁ it implies an infinite dimensional space at k ⬎ kh, and this would ignore the existence of physics at the electron scales. This case requires a two-fluid or kinetic treatment. Alternatively, one could require that kh → ⬁ faster than kmax, but this reduces immediately to ordinary MHD. A remaining possibility is that kmax⑀ tends to a constant. This is the only novel and physical possibility for a modified thermodynamic limit in HMHD, and it is the case that we explore here. For HMHD, we consider N → ⬁ by increasing kmax and keeping Nbox⑀ = 2, where, as before, N = 共2Nbox + 1兲3. By using this limit, once N is defined, the Hall scale will be fixed at around half of the maximum k vector. Figure 5 illustrates the spectra for several N, suggesting the asymptotic behavior for this case: Spectral ensemble averages accumulate at smallest k. In this case, we imposed Hg = Hm ⯝ 0.3. As shown in Fig. 6, as the number of Fourier modes goes to infinity, the energy condenses in kmin. This large scale back-transfer of energy is observed also in the MHD case. However, HMHD exhibits in addition a subtle condensation of energy and helicities at small scale. Even if this cannot be interpreted as a pure condensation 共see Fig. 6, lower panel兲, it is strong evidence that the Hall effect modifies high wavenumber interactions. As can be seen in Fig. 7, the generalized helicity has a very similar behavior in the limit of high N. In analogy with the energy distribution 共see Fig. 6兲, the generalized helicity appears to condense at small k, and the amount remain-

TABLE I. Values of kvb and k쐓 for several values of globals and parameters. Case 1 2 3 4 5 6

Nbox

kh

Hg

Hm

k vb

k쐓Ev

k쐓Eb

k쐓Hg

k쐓Hm

32 32 32 32 32 16

3 3 3 3 3 8

0.4 0.01 0.01 0.25 0.35 0.3

0.01 0.8 0.01 0.6 0.3 0.3

15.00 — — 12.47 13.63 6.36

2.002 2.000 1.999 2.004 2.004 2.97

— — — — — 13.60

2.29 2.29 2.289 2.29 2.29 3.44

— — — — — 27.06


042314-8



0.11

/E

10-4

10-6

10-9

0.105 0.1 0.095 0.09

-11

10

0 1

10

0.05

100

0.1 0.15 1/Nbox

0.2

0.25

0.1 0.15 1/Nbox

0.2

0.25

0.0025 0.002 /E

10-4

10-6

-9

10

0.0015 0.001 0.0005

-11

0

10

1

10

100

0.05

FIG. 6. Fraction of HMHD energy in the smallest 共top panel兲 and in the highest 共bottom panel兲 k vector as a function of 1 / Nbox. In this case, we have chosen Hg = Hm ⯝ 0.3. The dashed line in the upper figure represents the ensemble limit given by Eq. 共54兲. The large-scale condensation of energy is a process similar to that observed in the MHD case 共Ref. 2兲 At small scales 共k = kmax兲, when Nbox → ⬁, the energy converges toward zero, therefore the high wavenumber enhancement of energy in HMHD is not a true condensation, even though there remains a local maximum at kmax as suggested in Fig. 5.

10-4

0

10-6

10-9

10-11 1

10 k

100

ing at high k is small. Therefore, the high wavenumber enhancement of Hg is not a true condensation. These conclusions are based so far on the numerical experiments, but in fact the nature of the N → ⬁ limit can also be examined analytically. We begin this by noting that Eq. 共37兲 demands that at least one of ␤, ␪, ␥ must→ ⬁ as N → ⬁. Therefore, we normalize the inverse temperatures as

␥ ␥ˆ = , N

␪ and ␪ˆ = . N

Equation 共44兲 becomes

共44兲

For this to be valid, as N → ⬁, requires that

FIG. 5. Magnetic helicity, generalized helicity, and energy spectra for Nbox = 4 共solid兲, 16 共dot-dashed兲, and 64 共dashed兲, suggesting the nature of the N → ⬁ limit. Vertical arrows indicate the value of kh for each truncation Nbox.

␤ ␤ˆ = , N

␤ˆ E + ␥ˆ Hg + ␪ˆ Hm = 4.

共43兲

␤ˆ , ␥ˆ , ␪ˆ ⬃ O共1兲.

共45兲

This reasoning is verified by several numerical tests. As can be seen from Fig. 8, the normalized temperatures tend to a constant value as the number of Fourier modes diverges. We now substitute Eq. 共43兲 into Eq. 共30兲, and take into account that

⑀⬃

q 兵q = O共1兲其, N1/3

共46兲

which reflects our earlier assumption that Nbox⑀ = O共1兲. It then follows that


042314-9



-7

0.022

-6.8

0.02 0.019

θ/N

/Hg

0.021

0.018

-6.6 -6.4

0.017 0.016

-6.2

0.015 0

0.05

0.1 0.15 1/Nbox

0.2

0.25

-6 102 103 104 105 106 107 N -2

0.0015

-2.2 0.001

γ/N

/Hg

0.002

0.0005

-2.4 -2.6

0 0

0.05

0.1 0.15 1/Nbox

0.2

-2.8

0.25

-3 102 103 104 105 106 107 N

FIG. 7. Fraction of generalized helicity in the smallest 共top panel兲 and in the highest 共bottom panel兲 k vector as a function of 1 / Nbox. The dashed line in the upper figure represents the ensemble limit given by Eq. 共57兲. In analogy with the energy distribution 共see Fig. 6 for comparison兲, the generalized helicity condenses at small k vectors, while there is only a small amount of helicity at high k. The high wavenumber enhancement of Hg 共see middle panel of Fig. 5兲, even if is not negligible, cannot be interpreted as a true condensation. This behavior is in accord with prediction given by Eq. 共60兲.

7 6.8

2␤ˆ 2␪ˆ 1 , N k2 ␤ˆ 2 − 1 ␥ˆ 2 2 − ␪ˆ 2␤ˆ 2 4

共

兲

共47兲

which coincides precisely with the MHD case in the limit of large N.2 For the limit of N → ⬁ and a fixed k, the last equation gives 具Hm共k兲典 ⯝ 0 if the denominator of the fraction does not vanish. If this were true for every k, then, because of Eq. 共36兲, the total magnetic helicity would scale as N−2/3 → 0. This contradicts our assumption of nonzero magnetic helicity. We can escape from this dilemma, as in the MHD case, by examining the possibility that the magnetic helicity condenses in a single k⬘, in such a way that

冉

1 k⬘2 ␤ˆ 2 − ␥ˆ 2 4

冊

2

const − ␪ˆ 2␤ˆ 2 ⬃ . N

共48兲

To discuss this possibility, we will need to note that as N → ⬁, Eqs. 共17兲–共20兲 become

␤ˆ ⬎ 0,

共49兲

␤ˆ 2 − 41 ␥ˆ 2 ⬎ 0,

共50兲

6.6

β/N

具Hm共k兲典 ⯝ −

6.4 6.2 6 102 103 104 105 106 107 N FIG. 8. Dependence of the inverse temperatures on the number of Fourier modes N. ␪ 共upper panel兲, ␥ 共middle panel兲, and ␤ 共lower panel兲 are normalized to the number of modes itself. These ratios tend to constant values as N → ⬁, suggesting that temperatures diverge as linear functions of N.

共

2 kmin ␤ˆ 2 − 41 ␥ˆ 2

兲2 − ␪ˆ 2␤ˆ 2 ⬎ 0.

共51兲

To proceed, first let us suppose that k⬘ ⬎ kmin. Then Eq. 共48兲 implies a violation of the realizability condition Eq. 共51兲.


042314-10



Therefore, the only possibility is that the magnetic helicity condenses into a single Fourier mode with k⬘ = kmin. In that case, substituting Eq. 共47兲 into Eq. 共36兲, we find that 2 kmin

冉

1 ␤ˆ 2 − ␥ˆ 2 4

冊

2

2␤ˆ 2兩␪ˆ 兩nmin , − ␪ˆ 2␤ˆ 2 ⬃ N兩Hm兩

共52兲

where nmin indicates the wave-vector degeneracy at kmin. From the previous discussion, for all other modes, with k ⬎ kmin, 具Hm共k兲典 → 0 as N → ⬁. This proves the condensation of magnetic helicity in the smallest k vector, in the sense that Hm = nmin具Hm共kmin兲典.

具E共kmin兲典 ⯝

具Ev共kmin兲典 ⯝

兩␪ˆ 兩 ␤ˆ

2kmin −

冊

nmin

冉

kmin −

兩␪ˆ 兩 ␤ˆ

nmin

冊

兩Hm兩,

共54兲

兩Hm兩,

共55兲

kmin 兩Hm兩, 具Eb共kmin兲典 ⯝ nmin 具Hg共kmin兲典 ⯝ −

共56兲

kmin␥ˆ 兩Hm兩. 2␤ˆ n

共57兲

min

As can be seen from Eqs. 共54兲–共57兲, if a finite amount of total magnetic helicity is present, there will also be a partial condensation of energy and generalized helicity. As in the MHD case, the HMHD longest wavelength condensation depends on the presence in the system of a non-null global magnetic helicity. As can be seen from Figs. 6 and 7 共dashed lines in the upper panels兲, Eqs. 共54兲–共57兲 give an accurate ensemble prediction of the longest wavenumber behavior, as N → ⬁. In order to examine a possible condensation at high k vectors, let now study the case in which N → ⬁ and k = kmax. In this case, the Hall effect will become important. In a periodic Cartesian geometry, we can use the following substitution: kmax = kminCN1/3 兵C = O共1兲其.

共58兲

By using Eqs. 共43兲, 共46兲, and 共58兲 to Eqs. 共26兲–共30兲, the following asymptotic limits can be obtained:

1 具E共kmax兲典 ⯝ N

冉

冋冉

冊

册

␥ˆ 2 ␤ˆ 2 − − ␥ 4 8 fe ⬟ , 2 2 2 2 2 Nd C kminq 2 ˆ 2 ˆ␤2 − ␥ˆ − ␥ˆ ␤ 4 4

4␤ˆ

冊

2 q2 2 C2kmin ˆ

共59兲

1 N

冉

册

冊

冊

共60兲

具Hm共kmax兲典 ⯝

共53兲

By using Eq. 共52兲 in Eqs. 共26兲–共30兲, and taking into account the constrains 共49兲–共51兲, the following relations can be found:

冉

具Hg共kmax兲典 ⯝

冋冉

2 q2 ˆ 2 C2kmin ␥ˆ 2 ␤ˆ 2 − + ␤ 4 2 fg , ⬟ 2 2 2 2 2 Nd ˆ k q C ␥ min 2 2 ˆ␤2 − ˆ − ␥ˆ ␤ 4 4

− ␥ˆ

1 N4/3 ˆ 2 ␥ˆ 2 ␤ − 4

冉

−

冊

2

q␥ˆ 3 4

2 q2 2 ˆ 2 C2kmin − ␥ˆ ␤ 4

⬟

fm , N4/3d 共61兲

where f e, f g, f m, and d are constant functions of normalized temperatures and geometric factors. Moreover, the constrains given by Eqs. 共17兲–共20兲 now become

␤ˆ ⬎ 0,

共62兲

2 q2 2 C2kmin ␤ˆ 2 − ␥ˆ ⬎ 0, 4

共63兲

冉冉

冊冊

2 q2 2 C2kmin ␥ˆ 2 ␤ˆ 2 − − ␥ˆ ⬎ 0, 4 4

共64兲

␥ˆ ␤ˆ 2 − 4

共65兲

2 2

−

2 q2 2 ˆ 2 C2kmin ␥ˆ ␤ ⬎ 0. 4

The question is whether Eqs. 共59兲–共61兲 can give a high-k condensation. Recall that these are discrete Fourier modes, and the denominator d is the same in all three cases, and, due to Eq. 共65兲, is a finite, nonzero, constant. Note that null numerators in Eqs. 共59兲–共65兲共f i = 0兲 are forbidden by constrains 共62兲–共65兲. Moreover, because of Eq. 共65兲, every denominator d must be a positive, finite, number. Therefore, taking the limit of Eqs. 共59兲–共61兲 as N → ⬁, the only solution is that every ensemble-averaged spectrum, evaluated at kmax, is zero. As shown numerically 共see Figs. 6 and 7, lower panels兲, this discussion proves that in HMHD, there is no condensation at high k. V. NUMERICAL SIMULATIONS OF DYNAMICS

By introducing Eq. 共5兲 in Eqs. 共1兲–共4兲, one can obtain

⳵vk = i 兺 ⬘关共k · bp兲bq − 共k · vp兲vq兴, ⳵t p+q=k ⳵bk = 兺 ⬘兵i共k · bp兲vq − i共k · vp兲bq ⳵t p+q=k

共66兲

− ⑀关k · 共p ⫻ bp兲bq + 共k · bp兲共q ⫻ bq兲兴其, where vk and bk are abbreviations for the velocity and magnetic Fourier coefficients given by Eq. 共5兲. Here the prime indicates that the sum is extended on every possible p and q that satisfy p + q = k, where 兵k , p , q其苸 N. Numerical simulations of Eq. 共66兲 were performed by using a Galerkin spectral method code that exactly conserves


042314-11


Statistical properties of ideal… -1

10

0.25

theory simulation

helicities

0.2

0.15

-2

kh

10

0.1 Hc (ε/2)Hv Hg

0.05

(a) 10-3 -2

10

1

10

1

10

0 0

200

400

600

800

1000

-3

10

(b) 10-4 -1

10

-2

10

-3

E, Hm, and Hg, apart from round-off and time integration errors. Here we present results for a Galerkin simulation of a system with N = 5, kmax = 5冑3, and kh = ⑀−1 = 3. The initial conditions were chosen with flat E and Hc spectra and an Hm spectrum ⬃k−1. All undetermined phases were randomly chosen. Time integrations were performed out to 1900 eddy turnover times using a time step ⌬t = 5 ⫻ 10−3. This was sufficient to allow the system to go through many transient fluctuations so that adequate time-averaged spectra are obtained. The integration error for the energy after 1900 turnover times was ⯝0.005. Based on comparisons of spectral moments computed from the canonical ensemble, the microcanonical ensemble, and time averages of similar low-order truncations of 2D hydrodynamics,39 the number of modes retained here might be expected to give a reasonable level of agreement with the absolute equilibrium rugged invariant spectra. The behavior of cross helicity Hc, kinetic helicity Hv = 共2V兲−1兰v · ␻ d3x, and the generalized helicity Hg = Hc + ⑀ / 2Hv is shown in Fig. 9. The cross helicity is not an invariant of HMHD, but rather is balanced by the kinetic helicity. In such cases, the nonconserved quantity may be expected to oscillate.15 From Fig. 10 it can be seen that there is good agreement between simulated and theoretical values of the timeaveraged 具E共k兲典, 具Hg共k兲典, and 具Hm共k兲典. This supports the belief that the only quantities ruggedly conserved by Galerkin representations of HMHD equations are the energy, generalized helicity, and magnetic helicity. Evidence such as Fig. 10 points to the suggestion that the solutions may be at least approximately treated as a weakly ergodic, in the sense that second-order moments 共rugged invariant spectra兲 are accurately given by the ensemble predictions. It is possible that higher-order moments such as fourth-order moments may require a microcanonical treatment;39 however, we defer this investigation to a later study. As shown in Fig. 11 and as predicted by the ensemble, the spectrum of Ev / Eb strongly departs from the MHD case. At wavenumbers smaller than kh, this ratio is bigger in the MHD case, while at k ⬎ kh for the Hall case this ratio is bigger than 1. The modal distribu-

10

10-4 10-5

(c) 10-6

1

k

10

FIG. 10. Comparison of time-averaged simulated rugged invariant spectra 共full squares兲 and ensemble prediction 共full line兲 for a Galerkin simulation with Nbox = 5. The rugged invariants are the same as Fig. 9. 共a兲 Total energy, 共b兲 generalized helicity, 共c兲 magnetic helicity.

1.2 /

FIG. 9. Cross helicity Hc, kinetic helicity Hv 共multiplied by ⑀ / 2兲, and generalized helicity Hg as a function of time. The rugged invariants are E = 1.0, Hg = 0.25, and Hm = 0.17. For this simulation, Nbox = 5 and ⑀ = 1 / 3.

time

HMHD

1

MHD

0.8 0.6 0.4 0.2 0 1

10 k

FIG. 11. The ratio between kinetic and the magnetic spectra: HMHD 共solid line兲 vs MHD 共dashed line兲. For the HMHD run, ⑀ = 1 / 3.


042314-12



-2

helicities spectrum

10

Gibbs ensemble predictions. The essence of the argument is that some of the ideal invariants have different symmetry properties: Some are scalars 共energy兲 and some are pseudoscalars 共helicities兲. These thus divide the total system’s phase space into noncommunicating 共disconnected兲 parts, while the equilibrium ensemble assumes ergodicity of the phase space.40,41 The Gibbs ensemble, in fact, assumes that the phase point of the system may go anywhere in the phase space, so long as it preserves the ideal invariants’ numerical values. The crucial point is that the ensemble average for single modes gives

kh

-3

10

-4

10

具v j共k兲典 = 具b j共k兲典 = 0.

10-5 1

10 k

FIG. 12. Cross 共solid兲 and kinetic 共dashed兲 helicities for the HMHD run. The cross-helicity spectra are very similar to the MHD case; the kinetic helicity is peaked at smaller scales with a minimum close to the ion skin depth. The result of these two effects is a double-peaked spectrum of generalized helicity.

tion of cross and kinetic helicity is reported in Fig. 12; the kinetic helicity is peaked essentially at the smallest scales with a minimum close to the ion skin depth. The result of these two effects is a spectrum of generalized helicity with a strong 共condensation兲 peak at large scales and a weaker 共enhancement兲 peak at small scales. A. Time averages of single Fourier modes

In the past decade, some authors 共see Refs. 40 and 41 and references therein兲 have raised interesting questions 共and doubts兲 about the robustness of the equilibrium ensemble predictions for fluid and MHD turbulent systems. It has been shown, by using ideal MHD numerical simulations, that in some particular cases, and for approximately 1000 nonlinear times, time averages of single Fourier modes do not follow

共67兲

If large parts of the phase space are inaccessible to it, the time average of every Fourier mode is non-null, contradicting Eq. 共67兲. Thus, averaging over the whole phase space, the Gibbs ensemble may not be expected to give an accurate picture of the system. This phenomenon has been called broken ergodicity.40,41 It is interesting to note that one could in principle violate Eq. 共67兲 but retain a Gaussian distribution of the form Eq. 共23兲. One would need then to specify the value of 兵␺i其 as constraints. That is, additional invariants would enter. However, as we discuss presently, we see no need for this extension of the Gibbs ensemble. In order to verify the robustness of Eq. 共67兲, we did several numerical tests, listed in Table II, in which we analyze the time average of single Fourier modes. First we look at the phase space trajectory of the real and imaginary part of field components. In Fig. 13, this trajectory for several timesnapshots is shown. In this case 共run IX兲, substantial nonzero amounts of the pseudoscalar magnetic and generalized helicities are imposed at t = 0. In the initial stage of the time evolution, the phase-space trajectory appears to be trapped in a quadrant of the space. However, this is only an ephemeral effect. In fact in every case, as t → ⬁, our simulations show that the allowed phase space is filled entirely. We have found

TABLE II. Table of HMHD and MHD runs. The MHD runs are the ones with ⑀ = 0 共Hg → Hc兲. The column “⌬t” lists the time step of the Runge–Kutta technique; “RK” indicates the order of the Runge–Kutta time splitting; the last column indicates the error on global quantities at t = 10000. The error is evaluated as 关F共t兲 − F共t = 0兲兴 / F共t = 0兲共being F = E , Hg , Hm兲, and is reported in percent. Even after many eddy-turnover times, this numerical error, due to the precision of the Galerkin spectral technique, is very small. Run

Hm

Hg

⑀

⌬t

RK

Error 共%兲

I II II IV V VI VII VIII IX X XI XII XIII

0.17 0.16 7.0⫻ 10−6 −1.25⫻ 10−7 0.17 0.16 7.0⫻ 10−6 −1.25⫻ 10−7 0.16 0.155 2.0⫻ 10−7 −1.2⫻ 10−5 0.16

0.25 5.0⫻ 10−3 0.4 4.1⫻ 10−6 0.25 5.0⫻ 10−3 0.4 4.1⫻ 10−6 0.23 8.0⫻ 10− 3 0.4 2.0⫻ 10−6 0.23

1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 0 0 0 0 0

5 ⫻ 10−3 5 ⫻ 10−3 5 ⫻ 10−3 5 ⫻ 10−3 2 ⫻ 10−3 2 ⫻ 10−3 2 ⫻ 10−3 2 ⫻ 10−3 2 ⫻ 10−3 2 ⫻ 10−3 2 ⫻ 10−3 2 ⫻ 10−3 2 ⫻ 10−3

2nd 2nd 2nd 2nd 2nd 2nd 2nd 2nd 2nd 2nd 2nd 2nd 3rd

2.0 3.0 0.3 0.85 0.5 0.9 0.18 0.25 0.13 0.043 0.035 0.13 0.015


042314-13


Statistical properties of ideal… 0.3 t=300÷1000

Imaginary

0.2 0.1 0 -0.1 -0.2 -0.3 -0.3 -0.2 -0.1

0

0.1 0.2 0.3

Real 0.3 t=300÷3000

Imaginary

0.2 0.1 0 -0.1 -0.2 -0.3 -0.3 -0.2 -0.1

0

0.1 0.2 0.3

Real 0.3 t=300÷6000

Imaginary

0.2

具共xi − ␺i兲共xk − ␺k兲典 = ␭ik .

共68兲

If ␺ j ⫽ 0, then every ensemble prediction that was made under the assumption that ␺ j = 0 could be wrong. The agreement of the numerical results with the ␺ = 0 case supports the view that these additional invariants are not present. We would like to remark that Galerkin simulations are very accurate, free of aliasing errors, and conserve all quadratic invariants very accurately. The only error is due to round-off and this is very small, as is reported in Table II. We imposed several initial conditions 共different values of helicities兲 and, moreover, we varied the time step of the integration, and the time-integration scheme, going from second- to third-order Runge–Kutta 共RK兲 technique. In all the cases mentioned above, the results remain unchanged and confirm the ergodicity assumed in the Gibbs ensemble predictions with zero mean values for all Fourier coefficients.

0.1 0

VI. DISCUSSION -0.1 -0.2 -0.3 -0.3 -0.2 -0.1

0

0.1 0.2 0.3

Real 0.3 t=300÷10000 0.2 Imaginary

techniques 共not shown here兲. As can be seen from Fig. 13, the trajectory is trapped in a subspace for some thousands of nonlinear times, but as time goes on, the phase space is ergodically filled. It is important to note that if Eq. 共67兲 is not satisfied, the mean values enter Eq. 共23兲 through nonzero values of 兵␺ j其. In this case, the partition function, and therefore every ensemble-averaged quantity, is changed. The relation given by Eq. 共25兲 becomes

0.1 0 -0.1 -0.2 -0.3 -0.3 -0.2 -0.1

0 Real

0.1 0.2 0.3

FIG. 13. Time trajectory of a single Fourier mode, imaginary vs real part, for bx共0 , 1 , 0 , t兲共run V兲. Other components of the magnetic and velocity field have the same behavior. Initially the trajectory seems to be trapped in a quadrant of the phase space 共t = 300– 1000兲, but when t → ⬁ the allowed phase space is ergodically full 共lower panel兲. In other words, the time average of every Fourier mode is consistent with the ensemble prediction of Eq. 共67兲.

this behavior to be statistically the same for every component of both magnetic and velocity fields, for both MHD and HMHD, for every set of initial conditions, for different choices of simulation parameters, and for different numerical

In the previous sections, we have described the statistical properties of ideal, inviscid, three-dimensional, Hall MHD in periodic geometry. Our approach is based entirely on the absolute equilibrium ensemble method, which has previously been used to develop inverse cascade and selective decay conjectures for three-dimensional MHD.1,2 Supporting numerical experiments are based on a Fourier–Galerkin representation. Our main extension is to elaborate on the effects of magnetic and generalized helicity in the equilibrium statistics, leading to a number of conclusions regarding the behavior of the full set of three known rugged invariants 共E, Hm, and Hg兲 and their spectra. This first examination of HMHD Galerkin absolute equilibrium ensemble statistics leads to several general conclusions, apparently unnoticed previously, about the relevance of ion skin depth 共or Hall scale兲 in the system. Notably, the equilibrium statistics, which are in MHD scale free except for the Galerkin cutoff and box size, are considerably complicated in HMHD by the presence of the Hall electric field. In equilibrium, the Hall electric field leads to a redistribution of energy and helicity spectra relative to the MHD case. In particular, the excess of excitations at the longest wavelengths, which is characteristic of helical MHD turbulence, is weaker in HMHD equilibrium than in MHD. Accordingly, there is more small-scale excitation in HMHD equilibrium than in MHD. Through detailed examination of the spectral distribution of the invariants over wavenumber, we have seen that properties of the equilibrium state depend on the helicities, the Hall scale, and the number of Fourier modes in the system N.


042314-14



The appearance of spectral peaks at the longest and shortest wavelength in the equilibrium statistics is a distinctive feature of 3D HMHD, and one that plays an important role in establishing the connection to realistic dissipative turbulence.14 A quantity experiences condensation if, as N → ⬁, it resides entirely in modes of a particular wavelength. An example is the concentration of magnetic helicity in the longest wavelength modes in MHD.1 This condensation of Hm in the modified thermodynamic limit suggests that in the dissipative case, the magnetic helicity is back-transferred in the wavenumber space. The Hall effect seems not to qualitatively affect this typical MHD process, although the degree of accumulation is modified relative to MHD. At the smaller scales, in some circumstances, HMHD can produce a flat spectra with a local minimum near the Hall scale, and a spectral enhancement at the highest allowed wavenumbers. The role of the Hg in HMHD turbulence is crucial, as only this ideal invariant involves the Hall length scale. When Hg ⫽ 0, differences are seen in equilibrium across the entire spectrum, and in particular at small scales an energy excess relative to MHD. Generalized helicity Hg consists of both cross helicity Hc and kinetic 共hydrodynamic兲 helicity Hv, and for HMHD at the longer wavelengths, Hc acts very similar to the MHD case, while the spectrum of Hv becomes peaked at the smaller scales 共see Fig. 12兲. However, the accumulation of excess energy at the shortest scales does not attract a finite fraction of the total energy in the same limit. This characteristic feature of HMHD should probably be called an enhancement of energy to distinguish it from the stronger case of condensation of energy at large scales, which occurs for both HMHD and MHD. The large-scale condensation, which appears in both MHD and HMHD, is due entirely to the presence of a finite amount of global magnetic helicity in the system. A very interesting effect of ion skin depth in the spectra is the breaking of equipartition between kinetic and magnetic energies 共in MHD sometimes called the Alfvén effect兲. In HMHD, for Hg ⫽ 0, the kinetic energy near the Hall scale becomes greater than the magnetic energy. A similar effect has been seen in a compressible HMHD treatment of a Kelvin–Helmholtz instability 共KHI兲.36,42,43 Numerical evidence, such as that given in Sec. V, supports the belief that an arbitrary initial condition will relax toward this absolute equilibrium property. When dissipation is added to the system, the cascade of energy to high k will compete with the process that energetically equipartitions the system. As pointed out by Kraichnan and Nagarajan,44 both processes occur on times that are the same order of magnitude 共about an eddy turnover time兲. Thus qualitative theories are incapable of predicting which process will have the greater influence. Evidently, the balance between these effects leads to broadband spectra of the type found in the Kolmogorov theory.45,46 The details of this are well beyond the scope of the present equilibrium analysis. The distinctive small-scale effects in HMHD equilibrium may be related to phenomena that occur in realistic turbulence that includes dissipative and transport effects. One problem of this kind is the dynamo problem. In particular, in mean-field electrodynamics47 and related approx-

imations,48,49 the effective or “turbulent” transport coefficients can depend on the relative amount of excitations in velocity and magnetic fields, especially in the respective helicities 共see Fig. 12兲. The reversal of ordering of 具Ev共k兲典 and 具Eb共k兲典 at smaller scales seen in HMHD may in principle affect these transport coefficients, including their sign,47–49 and therefore the differences between HMHD and MHD may be influential in turbulent dynamo action. This has also been suggested previously from a very different perspective, that of nonlocality of spectral transfer.50 The results obtained in this paper have interesting support and analogies with those claimed in Ref. 51, even though the studies differ greatly 共compressible and dissipative vs incompressible and ideal兲. In particular, that work found that the Hall effect had little influence on the energy decay rate unless ⑀ becomes rather large 共⑀ ⬃ 1兲. It is important to remark that the authors51 considered only cases in which the total magnetic and generalized helicities are zero. In this case, the equilibrium prediction is for a flat equipartitioned energy spectrum regardless of the value of ⑀. From this limited perspective, one would not expect much difference in energy transfer and therefore not much difference in energy decay rate when dissipation is present. It is, however, very difficult to make a direct comparison with the ensemble prediction and dissipative turbulence, and, from the Gibbs ensemble point of view, it is hard to quantify the impact of the Hall effect on the energy decay rate. We believe that more direct numerical investigations are needed of HMHD, starting with nonzero generalized and magnetic helicities. A closely related issue in comparison of MHD and HMHD is that of the turbulent electric field at small scales, which can not only influence the dynamo, but also is of importance in understanding turbulence decay rates and turbulent reconnection.51–53 In particular, the turbulent electric field drives changes in the magnetic field, and can give rise to a turbulent or effective resistivity of fluid-scale origin. Simulations of decaying HMHD turbulence have shown previously that the small-scale total electric field is enhanced at the Hall scale and smaller.54 An enhancement of electric field at scales smaller than the Hall scale has also been reported in solar wind observations.55 It is of interest that a similar effect is seen in the HMHD ideal equilibrium case. For the present case of a nondissipative Galerkin HMHD representation, the electric field ⌶ = −v ⫻ b + ⑀共⵱ ⫻ b兲 ⫻ b in Fourier representation becomes ⌶共k兲 =

兺⬘ p+q=k

再

冋

k − vp ⫻ bq + i⑀ bq共k · bp兲 − 共bq · bp兲 2

册冎

. 共69兲

A comparison between the MHD and HMHD electric field spectra 兩⌶共k兲兩2 is shown in Fig. 14, where we observe that the small-scale electric field in HMHD is considerably larger than its MHD counterpart. We did several numerical simulations, by choosing different sets of parameters, and it seems that this effect does not depend on the global amount of Hg or Hm. One can also in principle compare these spectra to the equilibrium predictions for the 共fourth-order兲 electric field spectrum, however keeping in mind the possible requirement


042314-15



Electric field spectrum

8

kh

10-4

HMHD MHD 10-5 1

10 k

FIG. 14. The electric field spectrum 兩⌶共k兲兩2 for MHD 共dashed兲 and HMHD 共solid兲. See Ref. 54 for a comparison with direct numerical simulations. For these simulations, we imposed E = 1, Hg = Hc = 0.25, and Hm = 0.2. Moreover, we chose Nbox = 8 and kh = 4. The enhancement at high k is due to the Hall effect.

to treat this outside of the Gibbs ensemble, which we will address at later time. We found for MHD and HMHD that the time average of Fourier modes is consistent with the ensemble prediction given by Eq. 共67兲. Looking at the phase space trajectory of the system with a set of Galerkin simulations, we found that the point that describes the system is not trapped in any subsection of the volume. The system fills ergodically all the allowed space, consistent with the Gibbs ensemble. A final remark is that many of the distinctive effects seen in the HMHD equilibrium spectra require that the Hall scale be within the range of scales of interest 共that is, the effects are related to the scale ⑀兲 but also a number of these effects require the presence of generalized helicity Hg. Therefore, several types of small-scale HMHD signatures may be present in astrophysical plasmas, including those that require Hg ⫽ 0 共such as nonequipartition near the Hall scale兲, as well as others 共e.g., the enhanced electric field spectra above the Hall scale兲 that persist even when Hg = 0. The former can be found in situations in which the initial equilibrium configuration starts with the velocity and the magnetic field aligned, or any plasma that is generated in a near-Alfvénic state. We intend to explore further consequences of the Hall MHD Gibbs ensemble in future work. ACKNOWLEDGMENTS

This research is supported in part by NSF Grant No. ATM0539995, NASA Grant No. NNG05GG83G, and Subcontract No. LANL-11748-001-05. We would also like to acknowledge the anonymous referee for useful remarks and suggestions that improved the quality of the paper. 1

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10-3

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