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W W W N A T U R E C O M N A T U R E | 1

SUPPLEMENTARY INFORMATIONdoi101038nature12128

SUPPLEMENTARY INFORMATION

The Magnetohydrodynamic Simulation

We generated the MHD turbulence data that was analyzed in this letter by a numerical

simulation of the incompressible MHD equations in a [0 2π]3 periodic spatial

domain The simulation used a formulation of the MHD equations in terms of the

Elsasser variables zplusmn = u plusmn b

parttzplusmn = plusmn(b0middotnabla)z

plusmn minus nablap + (12)[ (z+ timesω

minus + z

minustimesω+) plusmn nablatimes(z

minustimesz+)]

1113089111308911130911113091 + (12) (ν + λ)nabla2z

plusmn + (12) (ν minus λ)nabla2

z∓ + F (S1)

with nablamiddotzplusmn = 0 and ω

plusmn=nablatimesz

plusmn The magnetic field b in the simulation has been assigned

Alfveacuten velocity units so that it is given by b=Bradic4πρ in terms of the magnetic field in

cgs units The vector b0 is an externally imposed uniform magnetic field p is the

kinematic pressure field determined by incompressibility ν is kinematic viscosity and

λ = c24πσ is magnetic diffusivity The magnetic Prandtl number is unity ν = λ =

11 times 10minus4

The body force which stirs the fluid was taken to be a Taylor-Green flow

F = f0 [sin(kf x)cos(kf y)cos(kf z)ex minus cos(kf x) sin(kf y) cos(kf z)ey]

applied at modes kf = 2 with an amplitude f0 = 025 Since the same forcing is applied

to both Alfveacuten wave modes the resulting turbulence is balanced (negligible cross-

helicity) and there is no forcing in the corresponding magnetic induction equation

Equations (S1) were integrated using a pseudospectral parallel code on a 10243

periodic grid with nonlinear terms evaluated in physical space and with the pressure

field and linear dissipation terms evaluated in Fourier space The simulation was

dealiased using phase-shift and 2radic23 spherical truncation so that the effective

maximum wavenumber is about kmax = 1024radic23 asymp 482 Time-integration was carried

out by a slaved second-order Adams-Bashforth method In this scheme the linear

terms in (S1) are solved exactly in time using an integration factor to reduce

numerical stiffness The computational time-step was ∆t = 25 times10minus4

In the simulations archived in the database there is no external field b0 = 0

The magnetic fluctuations were instead seeded with initial small-scale noise and

allowed to grow by dynamo action All data from the simulation was taken after it had

reached a statistically stationary state The Ohmic contribution to the total electric

field E=-utimesBc+Jσ is plotted for a frame of this data in Fig2a in the main text with

J=cnablatimesB4π from Ampegraverersquos law and with the Ohmic field normalized by the rms

value of the motional electric field The three-dimensional structure is seen more

clearly in the Supplementary Movie 1 which provides a rotating view Both this

movie and Fig2a in the main text present a volume-rendering of the electric field

with color associated to the vector magnitude and with partial transparency to reveal

some of the internal structure The imprint of the Taylor-Green forcing can be clearly

seen in the large-scale spatial organization of the small-scale current sheets The

small magnitudes of the normalized Ohmic field reflect the high-conductivity of

the turbulent MHD flow

From the statistical steady state of the simulation 1024 frames of data were

generated in physical space and ingested into the database including the 3

components of the velocity vector u 3 components of the magnetic field vector b and


2 | W W W N A T U R E C O M N A T U R E

RESEARCH

the pressure p Also calculated and archived were the 3 components of the magnetic

vector potential a = curlminus1

b in the Coulomb gauge nablamiddota = 0 The data were stored at

every 10 DNS time-steps ie the samples are stored at time-step δt =00025

Extensive tests showed that this provides sufficient temporal resolution to allow

accurate particle tracking The total duration of the stored data is 1024 times 00025 =

256 ie about one large-eddy turnover time The energy spectra of the velocity and

magnetic fields time-averaged over this interval are plotted in Fig2b in the main text

Other statistical parameters of the archived flow are listed here

Velocity (w=u) Magnetic (w=b)

Total energy

euro

Ew = Ewint (k)dk =1

2|w |

2 Eu=77 times 10-2

Eb=85 times 10-2

Dissipation

euro

εw

= 2ν k2int Ew(k)dk εu= 11times 10

-2 εb= 22times 10

-2

Rms field component

euro

prime w = |w |23

1 2

uprime= 023 b = 024

Taylor microscale

euro

λw

= (15ν εw)1 2 prime w λu= 89times10

-2 λb= 66times 10

-2

Taylor-scale Reynolds Reλw=wprimeλwν Re u= 186 Re b= 144

Kolmogorov time scale τw=(νεw)12

u =01 b = 007

Kolmogorov length scale ηw=(ν3εw)

14 u= 33times 10

-3 b= 28times 10

-3

Integral scale

euro

Lw =π

2 prime w 2

kminus1int Ew(k)dk Lu= 056 Lb= 035

Large eddy turnover time Tw=Lwwprime Tu=243 Tb=146

Cross-helicity HC=〈umiddotb〉 HC= 13times10-3

Magnetic-helicity HM=〈amiddotb〉 HM= -07times10-3

Description of the JHU Turbulence Database Cluster

The JHU Turbulence database cluster stores the output of high-resolution direct

numerical simulations (DNS) of turbulent flows in a cluster of relational databases

At present it archives two distinct datasets The first dataset is the entire 10244 space-

time history of a DNS of an isotropic turbulent flow in an incompressible fluid in

3D The second is the 10244 space-time history of a DNS of the incompressible

magneto-hydrodynamic (MHD) equations In total the database clusters stores

over 80 Terabytes of data across 8 database nodes Users of the database may write

and execute analysis programs on their host computers while the programs make

subroutine-like calls (getFunctions) requesting desired variables from the archived

datasets over the network Built-in 1st- and 2nd-order space differentiation as well as

spatial and temporal interpolations are implemented on the database

We describe the architecture of the database cluster its user interface the data

analysis functionality that it supports and provide some implementation details of the

query execution framework The database cluster makes extensive use of database

technology to partition index and query multi-Terabyte simulation data Archiving

these data in a database cluster serves several purposes ndash it preserves the

computational effort it provides for easy verification and repeatability of the results

of experiments and provides public access to high-resolution simulations

Additionally it allows for new types of experimentation that are either not possible or

difficult to execute in the traditional high-performance computing (HPC) setting


SUPPLEMENTARY INFORMATION RESEARCH

employed to perform DNS of turbulent flows One such example presented in this

letter is tracking particles or the evolution of fields backward in time This process is

naturally supported by the database cluster and can be performed just as easily as

iterating forward in time This is due to the fact that experiments that move from

time-step to time-step are executed by only accessing stored data in a localized region

in space and do not require performing calculations associated with the dynamic

advancement of the DNS (which may require operating over large portions of the data

volume)

The user interface provides public access to the complete 10244 space-time histories

of the turbulence simulations It is based on a Web-services model which allows

users to make subroutine-like requests (getFunctions) that are automatically

transferred over the network and executed on the database cluster The Web-services

methods are implemented using the standard SOAP protocol (Simple Object Access

Protocol) Invoking the methods with modern programming languages such as Java or

C can be done easily For FORTRAN and C codes we provide client wrapper

interfaces to the gSOAP library and sample code that includes calls to the Web-

services methods Additionally we provide a MATLAB interface that uses the

MATLAB-Fast-SOAP package with several routines with calls to the Web-services

methods The list of getFunctions that are currently implemented is available at

httpturbulencephajhueduserviceturbulenceasmx This includes evaluating each

SupplementaryFig1ArchitectureoftheJHUTurbulencedatabasecluster



RESEARCH

of the simulated fields (velocity u pressure p magnetic field b and vector potential a)

at arbitrary locations in space and time computing their first and second derivatives

evaluating box filters of arbitrary width computing the sub-grid stress tensor and

tracking particles forward and backward in time

The set of processing functions are implemented as stored procedures on the database

servers in C using Microsoft SQL Serverrsquos Common Language Runtime Spatial

interpolation is performed using Lagrange polynomial interpolation of 4th

6th

and 8th

order Temporal interpolation is performed using Cubic Hermite Interpolation First

and second derivatives are evaluated using centered finite differencing of 4th

6th

and

8th

orders Spatial filtering is implemented with a box filter of user specified filter

width Finally fluid particle tracking is performed by means of a second order

accurate Runge Kutta integration scheme The processing functions set was designed

with careful consideration of the trader-offs between generality and efficiency The

set includes the basic common tasks described above as these tasks are the essential

building blocks of most data analysis techniques and are also the most data-intensive

in nature Hence they can be efficiently executed on the database cluster and are

general enough to support a variety of data analysis techniques specific to each client

The architecture of the database cluster is presented in Supplementary Figure 1 The

cluster consists of several database nodes each running Microsoft SQL Server and a

Web servermediator The Web servermediator hosts the frontend Website and the

Web services methods It also acts as a mediator ndash it processes user queries breaks

them down according to the spatial partitioning of the data submits each part for

execution to the appropriate database node and finally assembles the results and

returns them to the user The data are distributed across the nodes of the cluster based

on the Morton z-order space-filling curve This curve also governs the partitioning

and indexing of the data within a database node We employ the Morton z-curve for

the organization of the data because it has good spatial locality (nearby regions in the

3D space are mapped to nearby locations in the 1D index space) it is easy to compute

and the ordering that it produces is both stable and admissible (at each step at least

one horizontal and one vertical neighbor have already been encountered) The entire

data volume of each dataset is subdivided into data voxel or ldquoatomsrdquo of size 83 for a

total of 221

atoms These data atoms are stored as binary large objects (blobs) in a

database table and are indexed using a standard B+-tree database index The key for

the B+-tree clustered index is a combination of the time step and Morton z-index of

the lower left corner of each atom This data structure ensures fast access to nearby

regions in the physical space because such regions will reside in nearby locations on

the Morton z-curve and will therefore be placed close together on disk

We have developed efficient query processing techniques for the rapid execution of

the batch requests submitted by clients of the service For the evaluation of batch

queries (queries consisting of multiple target locations) that perform decomposable

kernel computations we have developed an IO streaming method of execution which

evaluates the queries incrementally and in parts by means of partial-sums

Decomposable kernel computations are computations that evaluate a linear

combination of the data values in a particular region (kernel) and a set of coefficients

In one dimension such computations can be modeled as follows



g(x ) = li (x ) f (xnN

2+i

i=1

N

)

where x is the target location xn

is the location on the grid closest to the target

location N is the kernel width f (xi ) are the data values stored at grid nodes and

l(x ) are the coefficients of the computation Such computations are decomposable

because they can be executed in parts where each part is evaluated separately and the

results are summed together Interpolation differentiation and filtering all fall in this

category We make use of this incremental evaluation to efficiently execute a batch of

individual queries at the same time We preprocess all of the target locations and their

data requirements in a dictionary (map) data structure This data structure stores key-

value pairs where each key is the index of a database atom and each value is a list of

target locations that need data from this atom When all of the target locations are

processed we create a temporary table of all of the indexes of the database atoms that

have to be retrieved We retrieve the atoms by executing a join between this

temporary table and the table storing the data As each atom is retrieved from disk it is

routed to each target location that needs data from it Subsequently for each target

location a partial sum of the computation is evaluated over the intersection of the

locationrsquos kernel and the atomrsquos data This partial sum is added to the running total

and produces the result when all data atoms have been processed This mode of

execution allows us to stream over the data atoms in a single pass that performs IOs

to increasing offsets which is at least as efficient as a sequential pass Data atoms are

retrieved only once even if need by multiple queries and since they are small enough

to fit in cache the data is effectively reused from cache for all of the associated partial

sum computations Evaluating by partial sums also supports distributed computations

where parts of the computation can be performed on different database servers and

added together at the mediator

Stochastic Flux-Freezing

The mathematical theory of stochastic line-motion underlying the database calculation

is contained in Ref19 and is briefly reviewed here To put this work into physical

context it is important to emphasize that magnetic field lines do not really move

As has long been understood313233

magnetic line motion is just a convenient fiction

useful for intuitive understanding of the MHD solutions but without any physical

reality For smooth solutions of ideal MHD there are generally infinitely many

consistent line-motions and any of these may be used to interpret the solutions This is

analogous to the freedom to choose a gauge in electrodynamics calculations While

a particular line-motion can never be distinguished from any consistent alternative

a law of motion of field-lines has observable consequences (eg conservation of

magnetic flux through co-moving loops) that allow it to be empirically falsified

When one adds non-ideal terms to the Ohmrsquos law on the other hand there is in

general no deterministic line-motion whatsoever that is consistent with 3D MHD

(and generally no smooth line-motion in 2D) For a proof of this assertion by explicit

example for the case of resistive MHD see Ref 34 However once one realizes that

ldquoline-motionrdquo is a purely theoretical construct there is no need to restrict attention

only to deterministic ldquomotionsrdquo of lines Ref19 showed that a stochastic motion of

ldquovirtualrdquo magnetic fields is always consistent with resistive MHD and gives an

intuitive way to understand its solutions



RESEARCH

The stochastic line-motion law for resistive MHD in any dimension can be stated

precisely as follows19

the solution of the resistive induction equation

parttB=nablatimes(utimesB) +λnabla2B

with initial condition B(xt0) at time t0 is given by a stochastic Lundquist formula

euro

B(xt) = 〈B(at 0) sdot nablaa

˜ x (at)det[nablaa

˜ x (at)] |˜ a (xt)〉

where

euro

˜ x (at) is the solution of the initial-value problem for the stochastic differential

equation

euro

d˜ x (at) = u( ˜ x (at)t)dt + 2λd ˜ W (t)

euro

˜ x (at 0) = a

where

euro

˜ a (xt) is the inverse function of the flow map

euro

˜ x (at) and where the average 〈〉 is over the ensemble of Brownian motions

euro

˜ W (t) This formula can be written also as

euro

B(xt) = 〈 ˜ B (xt)〉 where the ldquovirtual magnetic fieldrdquo at (xt) is defined by

euro

˜ B (xt) = B(at 0) sdot ˜ J (att 0) |˜ a (xt)

and where

euro

˜ J (att 0) =nablaa˜ x (at)det[nabla

a˜ x (at)] As in quantum theory these virtual

fields

euro

˜ B (xt)have meaning only as intermediate states that must be summed over

(averaged) to give physical results The matrix

euro

˜ J (att 0) satisfies the differential

equation

euro

d

dt˜ J (att 0) = ˜ J (att 0)nabla

xu( ˜ x (at)t) - ˜ J (att 0)(nabla

xsdot u)( ˜ x (at)t)

euro

˜ J (at 0t 0) = I

forward in time from t0 to t along the stochastic trajectories which arrive at x at time t

Using this matrix equation together with the stochastic differential equation for the

trajectories one may in principle calculate the ensemble of virtual fields

euro

˜ B (xt)at time

t for any specified initial data

euro

B(at 0) at time t0 For more details see Refs1922

The formal similarity of the above theorems with the textbook results for ideal MHD

(eg the standard Lundquist formula) suggests that the usual formulas should be

recovered in the limit λrarr0 For example in the stochastic equation for

euro

˜ x (t) if

one simply drops the term involving λ then it reduces to the deterministic equation

dxdt=u(xt) which describes the standard line-motion law of Alfveacuten (ie field-lines

ldquofrozen-inrdquo to the bulk plasma fluid velocity) It is rigorously correct that stochastic

flux-freezing reduces to standard flux-freezing in the limit of infinite conductivity for

smooth laminar MHD solutions Eg if the velocity field is Lipschitz continuous

| u(xt) ndash u(xt) | le K |x-x|

(corresponding to Houmllder exponent h=1) then it is not hard to show35

that



euro

˜ x (at) minus x(at)2le

2dλ

K(e

Ktminus1) (S2)

where the average 〈〉 is over the ensemble of Brownian motions and d is the space

dimension In that case the ensemble of stochastic flows

euro

˜ x (at) converges with

probability 1 to the deterministic flow x(at) as λrarr0 With a somewhat stronger

differentiability assumption on the velocity field the gradients

euro

nablaa˜ x (at) also

converge to

euro

nablaax(at) and the standard Lundquist formula of ideal MHD

euro

B(xt) = B(at 0) sdot nablaax(at)det[nabla

ax(at)] |a(xt)

is rigorously recovered as λrarr0 Notice that this discussion includes smooth chaotic

flows in which the constant K can be taken to be the leading Lyapunov exponent and

the inequality (S2) becomes an equality asymptotically at long times

The above results need not hold however if the Lipschitz constant K (or the norm

nablau of the velocity gradient) diverges to infinity as λ approaches zero Textbook

proofs of Alfveacutenrsquos Theorem always implicitly assume that nablau remains finite as λrarr0

However this is not true for turbulent solutions of the MHD equations in the limit

of infinite conductivity with the magnetic Prandtl number νλ fixed In that case

gradients of both velocity and magnetic field diverge so that the energy dissipation

rate stays finite in the limit (the ldquozeroth law of turbulencerdquo Ref16) Not only do the

textbook proofs of Alfveacutenrsquos Theorem not apply when the velocity andor magnetic

field become singular but it has been known for some time that the theorem itself

can then fail Already in 1978 an exact solution of the ideal induction equation

was constructed by HGrad which exhibits reconnection at an X-point where the

advecting velocity is singular36

The necessary conditions for solutions of ideal MHD

to violate standard flux-freezing were established in Ref 37 The numerical results in

this Letter indicate that MHD turbulence not only does not satisfy the assumptions of

the textbook proofs but that the standard flux-freezing relation actually fails to hold

even as conductivity increases without bound Note however that the stochastic form

of flux-freezing established in Ref19 holds for λ finite but arbitrarily small Very

interestingly the field-line ldquomotionrdquo seems to remain random in the limit of infinite

conductivity because of the physical phenomenon of ldquospontaneous stochasticityrdquo

Stochastic flux-freezing thus appears to be a property of the rough or singular

solutions of ideal MHD that are obtained in the limit λrarr0 We have here considered

vanishing resistivity but the same limiting behavior of turbulent MHD solutions is

expected for any sort of small non-ideal term in the Ohmrsquos law This type of

universality has been rigorously proved to hold in the Kraichnan-Kazantsev model of

kinematic dynamo13

Stochasticity of field-line motion in high Reynolds-number

MHD turbulence is not a consequence of resistive diffusion but is instead an effect of

advection by a ``roughrsquorsquo velocity field See Refs 8131522 for more discussion

The analysis above has important implications for the reconnection problem It has

generally been assumed that magnetic field lines act as if ``frozen-inrsquorsquo for ideal MHD

Thus microscopic plasma physics mechanisms are thought necessary to explain how

field-lines ldquobreakrdquo or ldquosliprdquo relative to the plasma However it turns out that

solutions of ideal MHD need not obey Alfveacutenrsquos flux-freezing relation when the

velocity and magnetic fields become ldquoroughrdquo or singular Of course ideal MHD

never strictly holds because there are always some non-ideal terms in the generalized



RESEARCH

Ohmrsquos law however small Physically speaking the turbulence accelerates the tiny

violations of flux-freezing due to such non-idealities so efficiently in fact that the

violations persist in the limit of vanishing non-ideal terms and are independent of the

exact form of the non-ideality This is possible just because the ldquoroughrdquo mathematical

solutions obtained in the limit do not conserve magnetic flux

Numerical Implementation of Stochastic Flux-Freezing

The mathematical formulation of stochastic flux-freezing presented above is not

convenient for numerical implementation because of the necessity of inverting

euro

˜ x (at) to find

euro

˜ a (xt) As noted in Ref22 it is easier to find directly the stochastic

trajectories that end at x by solving the stochastic equation

euro

d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )

euro

˜ a (t) = x (S3)

backward in time from τ=t to τ=t0 The matrix

euro

˜ J ( ˜ a tτ) for each trajectory is obtained

by solving simultaneously

euro

d

dτ˜ J ( ˜ a tτ) = -nabla

xu( ˜ a (τ)τ) ˜ J ( ˜ a tτ) +(nabla

xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)

euro

˜ J ( ˜ a tt) = I (S4)

from τ=t to τ=t0 and calculating the ldquovirtual fieldrdquo

euro

˜ B (xtτ) = B(˜ a (τ )τ ) sdot ˜ J ( ˜ a tτ) for

all τ between t0 and t in a single backward integration The average

euro

B(xt) = 〈 ˜ B (xtτ )〉

so calculated is independent of time τ and coincides with the solution of equation (4)

The above-described algorithm is the same as that employed previously in Ref22 to

study the kinematic dynamo but using the new MHD turbulence database it fully

incorporates the effects of the Lorentz force

The stochastic flux-freezing calculation is implemented numerically in the database

by solving the stochastic differential equation (S3) with the Euler-Maruyama scheme

for a time-step Δτ=195times10-5

chosen conservatively so that

euro

prime u Δτ + 2λΔτ lt 01ηb

where ηb=28times10-3

is the resistive length The matrix equation (S4) is solved with a

corresponding Euler scheme for the same time-step The simultaneous backward

integration of (S3)(S4) requires calling the velocity u and the velocity-gradient

nablau from the database at each time step Note that the database MHD flow is

incompressible so that nablasdotu=0 and the term involving the velocity divergence can be

dropped from (S4) (In practice the velocity gradient matrices nablau obtained by calls to

the database are only traceless to a fraction of a percent because of errors introduced

by Lagrange interpolation and finite-difference approximation In our calculation we

thus use the velocity-gradient

euro

nablauminus1

3(nabla sdot u)I with the trace removed to make the

Lagrange-interpolated velocity-gradient more consistent with the pseudo-spectral

simulation results) An N-sample ensemble of independent stochastic trajectories

euro

˜ a n(τ) matrices

euro

˜ J n( ˜ a tτ)and virtual fields

euro

˜ B n(xtτ) is generated in this manner for

n=1hellipN and the empirical average calculated as

euro

1

N˜ B n(xtτ)

n=1

N

sum This average

should converge to the archived magnetic field B(xt) when Δτ is taken sufficiently



small and N is taken sufficiently large

The process is illustrated in Supplementary Movie 2 The movie begins showing the

stochastic trajectories

euro

˜ a n(τ) n=1hellipN going backward in time from spacetime point

(xt) The magnetic fields at the locations of the ldquocloudrdquo of particles at time τ are

those which contribute significantly to the magnetic field B(xt) After reaching an

(arbitrarily) chosen time τ=t0 the physical fields

euro

B( ˜ a n(t 0)t 0) n=1hellipN in the

ldquocloudrdquo are then taken as initial conditions to calculate an ensemble of ldquovirtualrdquo fields

euro

˜ B n(τ) n=1hellipN by solving along the stochastic trajectories the ldquofrozen-inrdquo equation

euro

d

dτ˜ B n(τ) = ˜ B n(τ ) sdot nablau(an(τ)τ )

euro

˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)

from τ=t0 to τ=t These ldquovirtual fieldsrdquo are shown in the movie evolving from τ=t0 to

τ=t along the stochastic trajectories stretched and rotated by the flow to the final

point (xt) In practice we do not solve the equation (S5) but use the mathematically

equivalent formula

euro

˜ B n(τ) = ˜ B n(t 0) sdot ˜ J n( ˜ a τt 0) = ˜ B n(t 0) sdot ˜ J n( ˜ a tt 0) sdot ˜ J n-1

( ˜ a tτ ) since

thematrices

euro

˜ J n( ˜ a tτ) havealreadybeencalculatedandstoredfort0leτletFinally

theldquovirtualfieldsrdquo

euro

˜ B n(t) = ˜ B n(xtt 0) n=1N at the spacetime point (xt) are

averaged in the last frame of the movie to obtain

euro

1

N˜ B n(t)

n=1

N

sum recovering the archived

magnetic field B(xt)

In Figure 3b of the Letter are plotted the relative errors

euro

RelErr(xtτ) =1

| B(xt) |B(xt) minus

1

N˜ B n(xtτ)

n=1

N

sum

for a typical point (xt) in the database as a function of times τ (called t0 in the text

figure) The small errors in this figure for large N illustrate that the stochastic flux-

freezing relation successfully recovers the magnetic field point by point This is a

very stringent test of the accuracy of the archived data and the convergence of our

numerical integration of (S3)(S4) It is important however to demonstrate that

similar convergence holds at all points in the database and not just for a particular

chosen point In Supplementary Figure 2 is plotted

euro

1

PRelErr(xptτ)p=1

P

sum averaged

over P=512 diagnostic points in the database as a function of times τ between t0 and t


1 0 | W W W N A T U R E C O M N A T U R E

RESEARCH

The errors decrease for greater N but also increase for earlier τ because larger N

values are required at earlier times to properly sample the more extended ldquocloudsrdquo of

points and also integration errors in Δτ build up going backward in time For times τ over half a turnover time the errors decrease to values lt5 as N increases This

residual error is due to the Lagrange interpolation of velocity gradients as well as to

errors from finite Δτ and N By contrast the relative error in the magnetic field using

standard flux-freezing averaged over the same P points rises rapidly going to earlier τ and is well above 100 by half an eddy turnover time

Observational Tests of Turbulent Reconnection

ThethesisofthisLetterthatturbulenceleadstoabreakdownofflux‐freezing

isintimatelyconnectedtopreviousworkonstochasticorturbulent

reconnection78InparticularRef8discussesindetailtherelationshipof

Richardsondispersionoffield‐linestoturbulentreconnectiontheoryWhile

therehasbeensomenumericalworkaimedattestingthismodel2838ultimately

thismodelneedstobecomparedwithastrophysicalobservationsThisisnot

perfectlystraightforwardbecausethemodelwasoriginallyconceivedinthe

idealizedcontextofexternallydrivenspatiallyhomogeneousturbulenceacting

onacurrentsheetwithanassociatedlarge‐scalemagneticfieldwhoseAlfveacuten

velocityisatleastasgreatastheturbulentrmsvelocityTheconvectionzonesof

starsandionizedpartsofaccretiondisks(subjecttothemagnetorotational

instability)arelikelyexamplesbutdirectdetailedobservationsareunavailable

forbothcasesWecanalsoapplythismodeltoopticallythincollisionless

plasmasassumingasaminimalconditionthatthelarge‐eddyscaleis

sufficientlylargerthananyrelevantmicroscale8Thisopensupthepossibility

oftestingthemodelagainstreconnectioneventsinthesolarcoronaSince

turbulenceinthesolarcoronaappearshighlyintermittentinspaceandtimeany

SupplementaryFig2Relativeerrorinnumericalimplementationofstochasticfluxshy

freezingaveragedoverPpointsinthedatabasePlottedaretheerrorsatfourvaluesofN

Forcomparisonisplottedtheaveragerelativeerrorusingstandardflux‐freezing(black)

W W W N A T U R E C O M N A T U R E | 1 1


testhastoinvolvelocalmeasurementsofthermsturbulentvelocitytheAlfveacuten

speedthelengthofthecurrentsheetandeitherthereconnectionspeedorthe

thicknessofthereconnectinglayerHerewediscusstherelationshipbetween

modelpredictionsandobservationsandgiveabriefoverviewofprospectsfora

definitiveanswer

Theoriginalworkonthistopic7assumedanisotropicforcingofturbulent

motionswithvelocityfluctuationsuLdrivenonarathersharplydefinedscale

ItwasfurtherassumedthattheAlfveacutenicMachnumberMA=uLvAisatleast

somewhatsmallerthanunityOnthelargestscalestheresultingvelocityfield

isthusonlyweaklyturbulentandcanbeapproximatedasanensembleof

interactingwavesThewaveenergycascadesanisotropicallywithfield‐parallel

wavelengthfixedbutwithperpendicularwavelengthdecreasingstepwiseuntil

itisoforderMA2Onallscalessmallerthanthistheeddiesarestrongly

nonlinearinthesensethattheircoherencetimeor``eddy‐turnovertimersquorsquois

comparabletothecorrespondingAlfveacutenwaveperiod(so‐calledcritical

balance26)Thesestronglyturbulenteddiesarethesourceofcross‐field

diffusionandtheconsequentenhancementofthereconnectionrateGivena

large‐scalefieldreversalwithatransversesizeLgtthepredictionsforthe

currentlayerwidthandreconnectioninflowspeedare78

euro

Δ ~ MA2(L)

12and

euro

vrec~ M

A

2vAL( )

12

(S6)

Here~denotesequalityuptoaprefactoroforderaboutunitythatisprobably

somewhatdependentonlarge‐scalegeometryofreconnectingfluxstructures

Theexactnatureofthesmallscalephysicsontheotherhandwillbeirrelevant

aslongasthecurrentsheetthicknessΔexceedsthemicrophysicalscalesbya

comfortablemargin

InthesituationdescribedabovethermsvelocityuisequaltouLonorderof

magnitudeThismightsuggestnaiumlvelythatinastrophysicalapplicationsthe

formulas(S6)shouldbeappliedwithuLtakentobetheturbulentrmsvelocityu

inferredfromobservationsegnon‐thermalDopplerbroadeningofline

spectra3040Howeveritshouldbeborneinmindthattheseexpressionsare

dependentontheassumedidealizedformofenergyinjectionAmorerobust

expressionfollowsfromnotingthattheturbulentvelocityatthestrong‐eddy

scaleisvT=MA2vA(denotedvtransinRef8)Ifweintroducethecorresponding

turbulentAlfveacutenicMachnumberMT=vTvA~MA2andifwespecifyasthelargest

lengthscale(integrallength)oftheturbulenceparalleltothemagneticfieldthen

wehave

euro

Δ ~ MT(L)12and

euro

vrec~ v

TL( )

12

(S7)

Forcomparisonwithobservationsequation(S7)shouldusuallygivea

reasonableestimateofthepropertiesofthereconnectionlayerifvTistakento

beofordertheestimatedturbulentvelocitydispersionuprimeIndeedlocally

generatedturbulenceshouldbedominatedatitslargestscalesbystrongly



RESEARCH

nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe

turbulenceisgeneratedexternallyandreachesthereconnectinglayeras

interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet

thicknessandthereconnectionspeedMorenumericalworkalongthelinesof

Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof

differenttypesofturbulencegenerationmechanismonthereconnectionrates

Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe

theoryespeciallythescalingonthevariousphysicalparametersandthe

independenceofthemicroscales

Thebestastrophysicalobservationsofmagneticreconnectioneventsboth

currentlyandinthenearfutureareinvariousregionsoftheheliospherein

particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow

explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill

generallynotapplyinthemagnetospherebutverylikelydoesholdincertain

situationsinthesolarcorona

InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates

becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe

iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures

Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis

transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos

magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof

magnitudesmallerthanthesizeofreconnectionstructures39Fora

turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis

closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable

LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi

butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi

Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe

magnetosphereandmayenhancereconnectionratesForexampleRef41

documentsaneventinthemagnetotailwhereturbulentemfappearstosupply

thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot

MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas

observedtobearoundahundredkmthickoftheorderofρiThusthe

wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe

observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion

gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta

decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed

intheframeworkofMHDTheremayneverthelessbesomesignificant

commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent

reconnection78whichdeservetobeexploredForexampleeventslikethatin

Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin

Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics

(EMHD)42

OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe

solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar

corona304043havealmostsevenordersofmagnitudeseparationinscalebetween

Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow



equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30

revealacurrentsheetthicknesstolengthratiointherange016to008while

theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler

line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note

thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness

wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation

(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge

scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured

butislikelytobesmallerthanLWeconcludefromthisthatthepredicted

currentsheetthicknessisatmostcomparabletothelowerboundfrom

observationsSubsequentobservationsintheX‐ray43suggestthattheactual

thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof

magnitudeusingUVmeasurementsThebottomlineisthatobservationsof

post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch

broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith

thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe

strongeddiesandneglectedfactorsoforderunityinthetheorymakeit

impossibletoassertmorethanaroughconsistencyThelattermaybeovercome

byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe

predictedscalingscanbetestedindependentofnumericalcoefficientsThelack

ofknowledgeofismorechallengingAreasonableestimatemaybepossible

usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe

turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly

coarse‐grainedobservations8Fortunatelythedependenceofthereconnection

ratesoniscomparativelyweak

SUPPLEMENTARY REFERENCES

31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385

(1958)

32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys

Res 77 6271-6274 (1972)

33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res

81 4019-2021 (1976)

34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the

motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)

35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems

(Springer New York 1998)

36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of

Mathematical Sciences preprint COO-3077-152 MF-92 New York University New

York 1978 httparchiveorgdetailsreconnectionofma00grad

37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma



RESEARCH

flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)

38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection

studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19

297‐314(2012)

39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania

KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace

environmentrdquoSpaceSciRev15689‐134(2010)

40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent

sheetsrdquoAstrophysJ689572‐584(2008)

41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe

presenceofguidefieldrdquoGeophysResLett39L11104(2012)

42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic

turbulencerdquoAstrophysJ615L41‐L44(2004)

43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope

ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe

2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)

44httpcdawgsfcnasagovCME_list



RESEARCH

the pressure p Also calculated and archived were the 3 components of the magnetic

vector potential a = curlminus1

b in the Coulomb gauge nablamiddota = 0 The data were stored at

every 10 DNS time-steps ie the samples are stored at time-step δt =00025

Extensive tests showed that this provides sufficient temporal resolution to allow

accurate particle tracking The total duration of the stored data is 1024 times 00025 =

256 ie about one large-eddy turnover time The energy spectra of the velocity and

magnetic fields time-averaged over this interval are plotted in Fig2b in the main text

Other statistical parameters of the archived flow are listed here

Velocity (w=u) Magnetic (w=b)

Total energy

euro

Ew = Ewint (k)dk =1

2|w |

2 Eu=77 times 10-2

Eb=85 times 10-2

Dissipation

euro

εw

= 2ν k2int Ew(k)dk εu= 11times 10

-2 εb= 22times 10

-2

Rms field component

euro

prime w = |w |23

1 2

uprime= 023 b = 024

Taylor microscale

euro

λw

= (15ν εw)1 2 prime w λu= 89times10

-2 λb= 66times 10

-2

Taylor-scale Reynolds Reλw=wprimeλwν Re u= 186 Re b= 144

Kolmogorov time scale τw=(νεw)12

u =01 b = 007

Kolmogorov length scale ηw=(ν3εw)

14 u= 33times 10

-3 b= 28times 10

-3

Integral scale

euro

Lw =π

2 prime w 2

kminus1int Ew(k)dk Lu= 056 Lb= 035

Large eddy turnover time Tw=Lwwprime Tu=243 Tb=146

Cross-helicity HC=〈umiddotb〉 HC= 13times10-3

Magnetic-helicity HM=〈amiddotb〉 HM= -07times10-3

Description of the JHU Turbulence Database Cluster

The JHU Turbulence database cluster stores the output of high-resolution direct

numerical simulations (DNS) of turbulent flows in a cluster of relational databases

At present it archives two distinct datasets The first dataset is the entire 10244 space-

time history of a DNS of an isotropic turbulent flow in an incompressible fluid in

3D The second is the 10244 space-time history of a DNS of the incompressible

magneto-hydrodynamic (MHD) equations In total the database clusters stores

over 80 Terabytes of data across 8 database nodes Users of the database may write

and execute analysis programs on their host computers while the programs make

subroutine-like calls (getFunctions) requesting desired variables from the archived

datasets over the network Built-in 1st- and 2nd-order space differentiation as well as

spatial and temporal interpolations are implemented on the database

We describe the architecture of the database cluster its user interface the data

analysis functionality that it supports and provide some implementation details of the

query execution framework The database cluster makes extensive use of database

technology to partition index and query multi-Terabyte simulation data Archiving

these data in a database cluster serves several purposes ndash it preserves the

computational effort it provides for easy verification and repeatability of the results

of experiments and provides public access to high-resolution simulations

Additionally it allows for new types of experimentation that are either not possible or

difficult to execute in the traditional high-performance computing (HPC) setting










volume)
















RESEARCH








6th

and 8th



6th

and

8th























total of 221


















2+i

i=1

N

)

















































RESEARCH






euro


˜ x (at)det[nablaa

˜ x (at)] |˜ a (xt)〉

where

euro


equation

euro


euro

˜ x (at 0) = a

where

euro


euro


euro


euro


euro


and where

euro



fields

euro



euro


equation

euro

d




euro

˜ J (at 0t 0) = I




euro

˜ B (xt)at time


euro





euro

˜ x (t) if








that



euro


2dλ

K(e

Ktminus1) (S2)



euro




euro


converge to

euro


euro


ax(at)] |a(xt)





























kinematic dynamo13













RESEARCH









euro

˜ x (at) to find

euro



euro

d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )

euro

˜ a (t) = x (S3)


euro



euro

d



xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)

euro

˜ J ( ˜ a tt) = I (S4)


euro



euro

B(xt) = 〈 ˜ B (xtτ )〉









euro












euro

nablauminus1




euro

˜ a n(τ) matrices

euro


euro



euro

1

N˜ B n(xtτ)

n=1

N

sum This average







euro





euro



euro


euro

d


euro

˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)




equivalent formula

euro


( ˜ a tτ ) since

thematrices

euro



euro



euro

1

N˜ B n(t)

n=1

N




euro

RelErr(xtτ) =1


1

N˜ B n(xtτ)

n=1

N

sum








euro

1

PRelErr(xptτ)p=1

P

sum averaged




RESEARCH


































definitiveanswer















euro

Δ ~ MA2(L)

12and

euro

vrec~ M

A

2vAL( )

12

(S6)





comfortablemargin











wehave

euro

Δ ~ MT(L)12and

euro

vrec~ v

TL( )

12

(S7)







RESEARCH










































(EMHD)42


































(1958)


Res 77 6271-6274 (1972)


81 4019-2021 (1976)











RESEARCH




297‐314(2012)























volume)
















RESEARCH








6th

and 8th



6th

and

8th























total of 221


















2+i

i=1

N

)

















































RESEARCH






euro


˜ x (at)det[nablaa

˜ x (at)] |˜ a (xt)〉

where

euro


equation

euro


euro

˜ x (at 0) = a

where

euro


euro


euro


euro


euro


and where

euro



fields

euro



euro


equation

euro

d




euro

˜ J (at 0t 0) = I




euro

˜ B (xt)at time


euro





euro

˜ x (t) if








that



euro


2dλ

K(e

Ktminus1) (S2)



euro




euro


converge to

euro


euro


ax(at)] |a(xt)





























kinematic dynamo13













RESEARCH









euro

˜ x (at) to find

euro



euro

d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )

euro

˜ a (t) = x (S3)


euro



euro

d



xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)

euro

˜ J ( ˜ a tt) = I (S4)


euro



euro

B(xt) = 〈 ˜ B (xtτ )〉









euro












euro

nablauminus1




euro

˜ a n(τ) matrices

euro


euro



euro

1

N˜ B n(xtτ)

n=1

N

sum This average







euro





euro



euro


euro

d


euro

˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)




equivalent formula

euro


( ˜ a tτ ) since

thematrices

euro



euro



euro

1

N˜ B n(t)

n=1

N




euro

RelErr(xtτ) =1


1

N˜ B n(xtτ)

n=1

N

sum








euro

1

PRelErr(xptτ)p=1

P

sum averaged




RESEARCH


































definitiveanswer















euro

Δ ~ MA2(L)

12and

euro

vrec~ M

A

2vAL( )

12

(S6)





comfortablemargin











wehave

euro

Δ ~ MT(L)12and

euro

vrec~ v

TL( )

12

(S7)







RESEARCH










































(EMHD)42


































(1958)


Res 77 6271-6274 (1972)


81 4019-2021 (1976)











RESEARCH




297‐314(2012)
















RESEARCH








6th

and 8th



6th

and

8th























total of 221


















2+i

i=1

N

)

















































RESEARCH






euro


˜ x (at)det[nablaa

˜ x (at)] |˜ a (xt)〉

where

euro


equation

euro


euro

˜ x (at 0) = a

where

euro


euro


euro


euro


euro


and where

euro



fields

euro



euro


equation

euro

d




euro

˜ J (at 0t 0) = I




euro

˜ B (xt)at time


euro





euro

˜ x (t) if








that



euro


2dλ

K(e

Ktminus1) (S2)



euro




euro


converge to

euro


euro


ax(at)] |a(xt)





























kinematic dynamo13













RESEARCH









euro

˜ x (at) to find

euro



euro

d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )

euro

˜ a (t) = x (S3)


euro



euro

d



xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)

euro

˜ J ( ˜ a tt) = I (S4)


euro



euro

B(xt) = 〈 ˜ B (xtτ )〉









euro












euro

nablauminus1




euro

˜ a n(τ) matrices

euro


euro



euro

1

N˜ B n(xtτ)

n=1

N

sum This average







euro





euro



euro


euro

d


euro

˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)




equivalent formula

euro


( ˜ a tτ ) since

thematrices

euro



euro



euro

1

N˜ B n(t)

n=1

N




euro

RelErr(xtτ) =1


1

N˜ B n(xtτ)

n=1

N

sum








euro

1

PRelErr(xptτ)p=1

P

sum averaged




RESEARCH


































definitiveanswer















euro

Δ ~ MA2(L)

12and

euro

vrec~ M

A

2vAL( )

12

(S6)





comfortablemargin











wehave

euro

Δ ~ MT(L)12and

euro

vrec~ v

TL( )

12

(S7)







RESEARCH










































(EMHD)42


































(1958)


Res 77 6271-6274 (1972)


81 4019-2021 (1976)











RESEARCH




297‐314(2012)

















2+i

i=1

N

)

















































RESEARCH






euro


˜ x (at)det[nablaa

˜ x (at)] |˜ a (xt)〉

where

euro


equation

euro


euro

˜ x (at 0) = a

where

euro


euro


euro


euro


euro


and where

euro



fields

euro



euro


equation

euro

d




euro

˜ J (at 0t 0) = I




euro

˜ B (xt)at time


euro





euro

˜ x (t) if








that



euro


2dλ

K(e

Ktminus1) (S2)



euro




euro


converge to

euro


euro


ax(at)] |a(xt)





























kinematic dynamo13













RESEARCH









euro

˜ x (at) to find

euro



euro

d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )

euro

˜ a (t) = x (S3)


euro



euro

d



xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)

euro

˜ J ( ˜ a tt) = I (S4)


euro



euro

B(xt) = 〈 ˜ B (xtτ )〉









euro












euro

nablauminus1




euro

˜ a n(τ) matrices

euro


euro



euro

1

N˜ B n(xtτ)

n=1

N

sum This average







euro





euro



euro


euro

d


euro

˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)




equivalent formula

euro


( ˜ a tτ ) since

thematrices

euro



euro



euro

1

N˜ B n(t)

n=1

N




euro

RelErr(xtτ) =1


1

N˜ B n(xtτ)

n=1

N

sum








euro

1

PRelErr(xptτ)p=1

P

sum averaged




RESEARCH


































definitiveanswer















euro

Δ ~ MA2(L)

12and

euro

vrec~ M

A

2vAL( )

12

(S6)





comfortablemargin











wehave

euro

Δ ~ MT(L)12and

euro

vrec~ v

TL( )

12

(S7)







RESEARCH










































(EMHD)42


































(1958)


Res 77 6271-6274 (1972)


81 4019-2021 (1976)











RESEARCH




297‐314(2012)
















RESEARCH






euro


˜ x (at)det[nablaa

˜ x (at)] |˜ a (xt)〉

where

euro


equation

euro


euro

˜ x (at 0) = a

where

euro


euro


euro


euro


euro


and where

euro



fields

euro



euro


equation

euro

d




euro

˜ J (at 0t 0) = I




euro

˜ B (xt)at time


euro





euro

˜ x (t) if








that



euro


2dλ

K(e

Ktminus1) (S2)



euro




euro


converge to

euro


euro


ax(at)] |a(xt)





























kinematic dynamo13













RESEARCH









euro

˜ x (at) to find

euro



euro

d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )

euro

˜ a (t) = x (S3)


euro



euro

d



xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)

euro

˜ J ( ˜ a tt) = I (S4)


euro



euro

B(xt) = 〈 ˜ B (xtτ )〉









euro












euro

nablauminus1




euro

˜ a n(τ) matrices

euro


euro



euro

1

N˜ B n(xtτ)

n=1

N

sum This average







euro





euro



euro


euro

d


euro

˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)




equivalent formula

euro


( ˜ a tτ ) since

thematrices

euro



euro



euro

1

N˜ B n(t)

n=1

N




euro

RelErr(xtτ) =1


1

N˜ B n(xtτ)

n=1

N

sum








euro

1

PRelErr(xptτ)p=1

P

sum averaged




RESEARCH


































definitiveanswer















euro

Δ ~ MA2(L)

12and

euro

vrec~ M

A

2vAL( )

12

(S6)





comfortablemargin











wehave

euro

Δ ~ MT(L)12and

euro

vrec~ v

TL( )

12

(S7)







RESEARCH










































(EMHD)42


































(1958)


Res 77 6271-6274 (1972)


81 4019-2021 (1976)











RESEARCH




297‐314(2012)
















euro


2dλ

K(e

Ktminus1) (S2)



euro




euro


converge to

euro


euro


ax(at)] |a(xt)





























kinematic dynamo13













RESEARCH









euro

˜ x (at) to find

euro



euro

d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )

euro

˜ a (t) = x (S3)


euro



euro

d



xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)

euro

˜ J ( ˜ a tt) = I (S4)


euro



euro

B(xt) = 〈 ˜ B (xtτ )〉









euro












euro

nablauminus1




euro

˜ a n(τ) matrices

euro


euro



euro

1

N˜ B n(xtτ)

n=1

N

sum This average







euro





euro



euro


euro

d


euro

˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)




equivalent formula

euro


( ˜ a tτ ) since

thematrices

euro



euro



euro

1

N˜ B n(t)

n=1

N




euro

RelErr(xtτ) =1


1

N˜ B n(xtτ)

n=1

N

sum








euro

1

PRelErr(xptτ)p=1

P

sum averaged




RESEARCH


































definitiveanswer















euro

Δ ~ MA2(L)

12and

euro

vrec~ M

A

2vAL( )

12

(S6)





comfortablemargin











wehave

euro

Δ ~ MT(L)12and

euro

vrec~ v

TL( )

12

(S7)







RESEARCH










































(EMHD)42


































(1958)


Res 77 6271-6274 (1972)


81 4019-2021 (1976)











RESEARCH




297‐314(2012)
















RESEARCH









euro

˜ x (at) to find

euro



euro

d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )

euro

˜ a (t) = x (S3)


euro



euro

d



xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)

euro

˜ J ( ˜ a tt) = I (S4)


euro



euro

B(xt) = 〈 ˜ B (xtτ )〉









euro












euro

nablauminus1




euro

˜ a n(τ) matrices

euro


euro



euro

1

N˜ B n(xtτ)

n=1

N

sum This average







euro





euro



euro


euro

d


euro

˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)




equivalent formula

euro


( ˜ a tτ ) since

thematrices

euro



euro



euro

1

N˜ B n(t)

n=1

N




euro

RelErr(xtτ) =1


1

N˜ B n(xtτ)

n=1

N

sum








euro

1

PRelErr(xptτ)p=1

P

sum averaged




RESEARCH


































definitiveanswer















euro

Δ ~ MA2(L)

12and

euro

vrec~ M

A

2vAL( )

12

(S6)





comfortablemargin











wehave

euro

Δ ~ MT(L)12and

euro

vrec~ v

TL( )

12

(S7)







RESEARCH










































(EMHD)42


































(1958)


Res 77 6271-6274 (1972)


81 4019-2021 (1976)











RESEARCH




297‐314(2012)



















euro





euro



euro


euro

d


euro

˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)




equivalent formula

euro


( ˜ a tτ ) since

thematrices

euro



euro



euro

1

N˜ B n(t)

n=1

N




euro

RelErr(xtτ) =1


1

N˜ B n(xtτ)

n=1

N

sum








euro

1

PRelErr(xptτ)p=1

P

sum averaged




RESEARCH


































definitiveanswer















euro

Δ ~ MA2(L)

12and

euro

vrec~ M

A

2vAL( )

12

(S6)





comfortablemargin











wehave

euro

Δ ~ MT(L)12and

euro

vrec~ v

TL( )

12

(S7)







RESEARCH










































(EMHD)42


































(1958)


Res 77 6271-6274 (1972)


81 4019-2021 (1976)











RESEARCH




297‐314(2012)
















RESEARCH


































definitiveanswer















euro

Δ ~ MA2(L)

12and

euro

vrec~ M

A

2vAL( )

12

(S6)





comfortablemargin











wehave

euro

Δ ~ MT(L)12and

euro

vrec~ v

TL( )

12

(S7)







RESEARCH










































(EMHD)42


































(1958)


Res 77 6271-6274 (1972)


81 4019-2021 (1976)











RESEARCH




297‐314(2012)




















definitiveanswer















euro

Δ ~ MA2(L)

12and

euro

vrec~ M

A

2vAL( )

12

(S6)





comfortablemargin











wehave

euro

Δ ~ MT(L)12and

euro

vrec~ v

TL( )

12

(S7)







RESEARCH










































(EMHD)42


































(1958)


Res 77 6271-6274 (1972)


81 4019-2021 (1976)











RESEARCH




297‐314(2012)
















RESEARCH










































(EMHD)42


































(1958)


Res 77 6271-6274 (1972)


81 4019-2021 (1976)











RESEARCH




297‐314(2012)











































(1958)


Res 77 6271-6274 (1972)


81 4019-2021 (1976)











RESEARCH




297‐314(2012)
















RESEARCH




297‐314(2012)














SUPPLEMENTARY INFORMATION - Nature · SUPPLEMENTARY INFORMATION ... were integrated using a...

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