acoustics actuators addictions admsci adolescents aerobiology aerospace
agriculture agriengineering agrochemicals agronomy ai air algorithms allergies
alloys analytica analytics anatomia anesthres animals antibiotics antibodies
antioxidants applbiosci appliedchem appliedmath applmech applmicrobiol
applnano applsci aquacj architecture arm arthropoda arts asc asi astronomy
atmosphere atoms audiolres automation axioms bacteria batteries bdcc behavsci
beverages biochem bioengineering biologics biology biomass biomechanics biomed
biomedicines biomedinformatics biomimetics biomolecules biophysica biosensors
biotech birds blockchains bloods blsf brainsci breath buildings businesses
cancers carbon cardiogenetics catalysts cells ceramics challenges
chemengineering chemistry chemosensors chemproc children chips cimb civileng
cleantechnol climate clinpract clockssleep cmd coasts coatings colloids
colorants commodities complications compounds computation computers
condensedmatter conservation constrmater cosmetics covid crops cryptography
crystals csmf ctn curroncol cyber dairy data ddc dentistry dermato
dermatopathology designs devices diabetology diagnostics dietetics digital
disabilities diseases diversity dna drones dynamics earth ebj ecologies
econometrics economies education ejihpe electricity electrochem electronicmat
electronics ecm encyclopedia endocrines energies eng engproc entomology
entropy environments environsciproc epidemiologia epigenomes est fermentation
fibers fintech fire fishes fluids foods forecasting forensicsci forests
fossils foundations fractalfract fuels future futureinternet futurepharmacol
futurephys futuretransp galaxies games gases gastroent gastronomy
gastrointestdisord gels genealogy genes geographies geohazards geomatics
geosciences geotechnics geriatrics grasses gucdd hardware hazardousmatters
healthcare hearts hemato hematolrep heritage higheredu highthroughput
histories horticulturae hospitals humanities humans hydrobiology hydrogen
hydrology hygiene idr ijerph ijfs ijgi ijms ijns ijpb ijtm ijtpp ime immuno
informatics information infrastructures inorganics insects instruments
inventions iot j jal jcdd jcm jcp jcs jcto jdb jeta jfb jfmk jimaging
jintelligence jlpea jmmp jmp jmse jne jnt jof joitmc jor journalmedia jox jpm
jrfm jsan jtaer jvd jzbg kidneydial kinasesphosphatases knowledge land
languages laws life limnolrev lipidology liquids literature livers logics
logistics lubricants lymphatics machines macromol magnetism magnetochemistry
make marinedrugs materials materproc mathematics mca measurements medicina
medicines medsci membranes merits metabolites metals meteorology methane
metrology micro microarrays microbiolres micromachines microorganisms
microplastics minerals mining modelling molbank molecules mps msf mti muscles
nanoenergyadv nanomanufacturing nanomaterials ncrna ndt network neuroglia
neurolint neurosci nitrogen notspecified nri nursrep nutraceuticals nutrients
obesities oceans ohbm onco optics oral organics organoids osteology oxygen
parasites parasitologia particles pathogens pathophysiology pediatrrep pets
pharmaceuticals pharmaceutics pharmacoepidemiology pharmacy philosophies
photochem photonics phycology physchem physics physiologia plants plasma
platforms pollutants polymers polysaccharides poultry powders preprints
proceedings processes prosthesis proteomes psf psych psychiatryint
psychoactives publications quantumrep quaternary qubs radiation reactions
receptors recycling regeneration religions remotesensing reports reprodmed
resources rheumato risks robotics ruminants safety sci scipharm sclerosis
seeds sensors separations sexes signals sinusitis skins smartcities sna
societies socsci software soilsystems solar solids spectroscj sports standards
stats std stresses surfaces surgeries suschem sustainability symmetry synbio
systems targets taxonomy technologies telecom test textiles thalassrep thermo
tomography tourismhosp toxics toxins transplantology transportation traumacare
traumas tropicalmed universe urbansci uro vaccines vehicles venereology vetsci
vibration virtualworlds viruses vision waste water wem wevj wind women world
youth zoonoticdis


# A transport-like approach to turbulence-mediated magnetic reconnection

Nuno F. Loureiro<EMAIL_ADDRESS>Plasma Science and Fusion Center,
Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

###### Abstract

A transport-like framework for the study of magnetic reconnection mediated by
self-driven turbulence is proposed, based on timescale separation between the
reconnection time and the characteristic timescale of the turbulent
fluctuations which arise in the reconnection layer. We argue that the mean
fields remain on MHD scales even in collisionless cases. These observations
provide theoretical justification for an efficient computational approach to
this problem, which we discuss.

## Introduction.

Plasmas have the ability to irreversibly and non-trivially alter the topology
of the magnetic fields that thread them. This nonlinear process — known as
magnetic reconnection [1, 2] — underlies explosive events such as flares in
astrophysical bodies [3, 4], as well as a wide variety of other phenomena,
including the interaction between the magnetic fields of the Sun and the Earth
[5, 6]. Reconnection is also thought to play an important role in plasma
turbulence [e.g. 7, 8, 9, 10, 11] and in the nonlinear development of many
plasma instabilities [e.g. 12, 13, 14, 15].

As the understanding of reconnection matures, it is becoming increasingly
clear that laminar reconnection sites (current sheets) are an exception,
realizable only in systems with relatively small scale separation (low
Lundquist number, or small ratio of system size to kinetic scales). More
generally, reconnecting systems appear capable of driving their own
turbulence; and, in turn, such turbulence becomes critical in defining the
properties of the reconnecting system [16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26].

Understanding the interaction between the reconnecting field and the
turbulence that it drives — particularly in realistic three-dimensional
geometries where reconnection-driven turbulence can live up to its full
potential — is only just beginning, with progress severely limited by the
extreme computational resources required. In this Letter, we argue that a
transport-like approach — conceptually akin to how the problem of turbulent
transport in fusion devices is usually approached [e.g., 27, 28] — is
justified and both enables the derivation of interesting results as well as
yields a feasible and computationally efficient way forward in this problem.

## Mean-field decomposition.

As is standard practice in mean-field electrodynamics, we decompose fields
(such as magnetic and velocity fields) into a mean and a fluctuating part:
$\bm{Q}(\bm{r},t)=\bm{Q}_{0}(\bm{r},t)+\bm{q}(\bm{r},t)$, where
$\bm{Q}_{0}=\overline{\bm{Q}}$ denotes the background, mean value of any field
$\bm{Q}$ (with overbar representing an adequately defined spacial or temporal
average, as appropriate), and $\bm{q}$ is the fluctuating component (absent in
the laminar limit). The latter is assumed here to be driven by magnetic
reconnection; i.e., as reconnection proceeds, it triggers instabilities (of
the macroscopic sort, such as plasmoid or kink, and/or microscopic, such as
mirror [29, 30], Buneman [31], or others [e.g. 32, 33, 34]) which drive the
fluctuations in the current sheet. Away from the current sheet, such
fluctuations are assumed to be absent: there, the fields reduce to their mean
values. Thus, the reconnection geometry we consider is one where the length
$L_{\rm CS}$ (in the outflow direction) is determined by the large scale
variation of the mean (reconnecting) magnetic field $\bm{B}_{0}$. The scale of
variation of the mean field across the current sheet, $a$, on the other hand,
is dynamically set, and is to be determined; it effectively corresponds to a
critical gradient that drives the fluctuation power required to reach a
statistical steady state. We assume that $L_{\rm CS}/a\gg 1$.

Essential to a mean-field decomposition is the (time- and/or length-) scale
separation between mean and fluctuating quantities. We will argue below that
spacial separation is not, in general, guaranteed. Timescale separation,
however, is expected once a statistical steady-state is reached. Indeed, the
characteristic timescale of the fluctuations is the Alfvén time,
$\tau_{A}\equiv L_{\rm CS}/V_{A}$ (the advection time out of the sheet), or
faster; whereas the mean fields evolve on the reconnection time, $\tau_{\rm
rec}=\mathcal{R}^{-1}\tau_{A}$, with $\mathcal{R}$ the normalized reconnection
rate (a finite but small fraction of unity both in MHD and in collisionless
cases). We shall denote the typical turbulence timescale as $\tau_{\rm turb}$
(which may be equal to $\tau_{A}$ in some cases, but not necessarily).
Therefore, mean fields, $\bm{Q}_{0}(\bm{r},t)$, are defined as a time average
of the full fields over a time interval $\tau$ such that $\tau_{\rm
rec}\gg\tau\gg\tau_{\rm turb}$.

Assuming that time-scale separation holds, what equations to solve for the
mean fields depends on whether $a$ is larger or smaller than the kinetic
scales (the ion skin depth or the ion (sound) Larmor radius, as appropriate).
If larger, then the relevant equations are simply the well-known mean-field
equations of MHD (see, e.g., Eqs. (4.2-4.3) of Ref. [35] in the incompressible
case). If, instead, $a$ is smaller than the ion kinetic scale, a richer
description is needed. We will argue, however, that this does not, in general,
happen; i.e., that the scale of variation of the mean-fields, once a steady-
state is reached, is larger than the kinetic scales — see discussion below. As
for the fluctuating fields, one can use descriptions of varying degree of
sophistication, from MHD to first-principles (Vlasov or PIC), as required by
the specific problem at hand.

Finally, it is worth noting at the outset some key differences between the
application of the mean-field approach to magnetic reconnection versus the
kinematic dynamo, where it has been so successful [36, 37, 38]. First and
foremost, the background magnetic field is obviously not small compared to the
background flows; i.e., the kinematic approximation does not hold. Second, as
we will discuss later, the background mean fields are inhomogeneous on scales
that can be commensurate with those of the turbulent fluctuations; they are
also anisotropic. And, third, the fluctuations cannot, in general, be assumed
small — on the contrary, they must be assumed to be Alfvénic with respect to
the reconnecting (upstream) magnetic field. These constraints strongly limit
the ability to make progress on this problem (nonetheless, see Refs. [39,
40]). However, it is both possible to derive a number of useful and
interesting results using this approach, as well as devise an efficient
computational framework that casts this as a self-consistent transport
problem. We first analyze separately the MHD and the collisionless cases.

## Strong guide-field reconnection in the MHD limit.

We specialize for simplicity to guide-field reconnection in an incompressible
plasma. There is a constant, homogeneous magnetic field in the direction
perpendicular to the reconnection plane, $B_{z}\bm{\hat{z}}$. We assume
$B_{z}\gg B_{\perp,0}$ and, for concreteness, consider a reconnection geometry
where $x$ and $y$ are, respectively, the inflow and outflow directions. The
global Lundquist number (defined with the reconnecting field) is $S=L_{\rm
CS}V_{A,y}/\chi_{m}\gg S_{\rm crit}\gg 1$, where $\chi_{m}$ is the magnetic
diffusivity, and $S_{\rm crit}$ is a reference critical value required to
trigger the plasmoid instability [41, 42, 43, 44, 45] in the layer. Once the
system settles down to a statistical steady-state, we expect a spectrum of
highly anisotropic turbulent fluctuations whose outer scale in the
$x$-direction is $a$, the width of the mean current sheet; i.e., the
fluctuations define how broad the mean current sheet is.

The equation for the mean electric field is

$\displaystyle
c\bm{E}_{0}+\bm{U}_{\perp,0}\times\bm{B}_{\perp,0}+\overline{\bm{u}_{\perp}\times\bm{b}_{\perp}}=\eta
c\bm{J}_{0}.$ (1)

Away from the current sheet there are no fluctuations (and the resistive term
is negligible) so Ohm’s law yields simply $cE_{0}=U_{\rm in}B_{\rm in}$, with
$U_{\rm in}$ and $B_{\rm in}$ the magnitudes of the upstream inflow velocity
and reconnecting field (note that this would be true even if there were no
timescale separation between mean fields and fluctuations; it requires only
that the fluctuations be confined to the layer).

Seeking an approximate steady-state solution for the mean-fields implies that
the mean electric field should be spatially constant (this is nothing but the
standard Sweet-Parker argument [46, 47]). Close to the center of the layer,
where the (perpendicular) mean fields vanish (by symmetry), the electric field
must then be set by the turbulent fluctuations (assuming that those are much
larger than the resistive term, which must be the case for the problem to
differ significantly from the standard (laminar) SP configuration). Therefore,
imposing (approximate) steady-state yields an expression for the reconnection
electric field:

$\displaystyle cE_{\rm rec}\approx\text{const.}\approx U_{\rm in}B_{\rm
in}\approx\left[-\bm{\hat{z}}\cdot(\overline{\bm{u}_{\perp}\times\bm{b}_{\perp}})\right]_{X},$
(2)

where $[\cdots]_{X}$ means the expression is to be evaluated at the $X$ point
of the mean fields. This relation between upstream quantities and the
fluctuations in the layer can be directly tested in simulations, since all
quantities are directly measurable. Its validity requires only that two
hypotheses hold true: timescale separation between the evolution of the
background fields and the fluctuations, and that a (quasi) steady-state is
reached — both of which seem reasonable to assume. Although we have in mind 3D
reconnecting systems whose layers host fully developed turbulence, these two
hypotheses are also valid for the better understood (and simpler) case of 2D
MHD stochastic plasmoid chain dynamics; so, it is worthwhile checking that Eq.
(2) is consistent with that understanding. The answer is that it is. Indeed,
in a plasmoid chain, the quantity $u_{y}b_{x}$ (evaluated along $x=0$) always
has the same sign. Following Ref. [48], the expected value for $u_{y}$ is
$V_{A,y}$, and $b_{x}\sim b_{y}w_{x}/w_{y}$, where $w_{x},\,w_{y}$ are the
typical plasmoid widths in the $x$ and $y$ directions. The typical field in
the plasmoids is the upstream field, $b_{y}\sim B_{\rm in}$, and
$w_{x}/w_{y}\sim\mathcal{R}$. Therefore, $u_{y}b_{x}\sim
V_{A,y}\mathcal{R}B_{\rm in}\sim U_{\rm in}B_{\rm in}$.

More generally, since $B_{\rm in}$ is assumed to be known, (2) is an equation
for the inflow velocity $U_{\rm in}$ in terms of the fluctuations in the
layer. If one assumes that 2D plasmoid chain physics extends to 3D, then
$b_{y,\rm rms}\sim B_{\rm in}$, implying that

$\displaystyle U_{\rm in}\approx u_{x,\rm rms}.$ (3)

While perhaps somewhat obvious, this expression is a testable prediction. The
fact that it matches the analytical result found in Appendix D.7.1 of Ref.
[11] (Eq. (D108)), arrived at via a different line of reasoning, and is in
agreement with recent numerical results [22], suggests that the mean field
framework proposed here rests on solid grounds.

In itself, Eq. (3) is not a solution to the problem: neither the upstream
inflow velocity nor the outer-scale velocity fluctuations are a priori known
(indeed, determining $U_{\rm in}$, i.e., the reconnection rate, is the main
goal). To address this, imagine that one initializes a SP-type reconnection
layer (at $S\gg S_{\rm crit}$). Plasmoid turbulence will immediately ensue,
broadening the mean layer [18, 19, 20, 21, 22]; that is, the mean current
sheet thickness and outflow velocity profile will be determined by the mean
width of the fluctuations. If one now thinks of the stability of the mean
profiles, one sees that if the layer were to broaden too much, the
fluctuations would die out — such a sheet would be stable to plasmoid
formation — and a steady-state would be impossible to maintain. The
conclusion, then, is that the mean profiles have to relax and stay close to
marginal stability: that is, an effective SP sheet with $a/L_{\rm CS}\approx
S_{\rm crit}^{-1/2}$; see Fig. 1. In an approximately incompressible system,
this is also the reconnection rate. This is indeed the case in 2D MHD [48, 43,
49]. In 3D MHD there is less certainty on what $S_{\rm crit}$ is, though
simulations suggest that it may not be significantly different than the 2D
value [18, 45, 20]. If that is the case, the mean reconnection rate would be
$\approx 0.01$ — in very reasonable agreement with existing numerical
simulations [20, 21, 22]. Lastly, note that while this reconnection rate is
“fast” in the usual sense — it does not depend on resistivity — it is small
enough that time averaging of the fluctuations, as described above, is
legitimate: $\tau_{\rm turb}=a/u_{x,a}\sim a/U_{\rm in}\sim L_{\rm
CS}/V_{A,y}\ll\tau_{\rm rec}$.

Figure 1: Conceptual evolution of a turbulent reconnecting system. Left:
Aspect ratio of the mean current sheet as a function of time. Right:
Normalized turbulent e.m.f. as a function of mean current sheet aspect ratio.
The arrows indicate the time evolution of the system. In both plots, the bold
(red) line indicates a case where the initial condition is sub-critical; the
system evolves until it reaches a critical threshold for triggering an
instability (e.g., plasmoid formation), and subsequently relaxes until a
statistical steady-state is reached. The dashed (blue) line is instead the
case where the initial aspect ratio of the background is super-critical and,
therefore, strongly unstable.

## Collisionless reconnection.

Let us now consider reconnection in weakly collisional plasmas. While the
range of possible instabilities driving turbulence in the current sheet is
much greater in this case than in MHD, one might still expect tearing-driven
(and potentially kink-unstable) flux ropes (plasmoids) to be among those, as
suggested by available numerical studies [e.g., 16, 24, 25]. Unlike other
possible (intrinsically kinetic) instabilities, the structures that arise via
these instabilities are larger scale — and, as we will now conjecture, will be
on MHD scales.

Consider first the case of low plasma beta and strong guide field; and assume
that a current sheet is forming with a rate $U_{\rm dr}/L_{\rm CS}$ (which we
take to be of the same order as the inverse Alfvén time, $V_{A,y}/L_{\rm CS}$,
such that $M_{\rm dr}\equiv U_{\rm dr}/V_{A,y}\sim 1$) [50, 51]. One can then
show that the growth rate of the most unstable tearing wavelength overcomes
the current sheet formation rate when $a/L_{\rm CS}\sim M_{\rm
dr}^{-1/3}(d_{e}\rho_{s}/L_{\rm CS}^{2})^{1/3}$ [52, 53]111Assuming a Harris
sheet profile for concreteness; other choices have no qualitative impact on
the conclusion.. At this moment of time, the current sheet width is still at
MHD scales, $a>\rho_{s}$, if $L_{\rm
CS}/\rho_{s}>(\beta_{e}\,m_{i}/m_{e})^{1/2}M_{\rm dr}$; this condition is very
easy to satisfy. One arrives at a similar condition for plasma beta of order
unity (using two fluid tearing mode scalings [55, 56]); in this case,
$a>d_{i}$ requires $L_{\rm CS}/d_{i}>(m_{i}/m_{e})^{1/2}M_{\rm dr}$. Yet
another example is that of (low beta, non-relativistic) pair plasmas [57];
there, the critical aspect ratio is $a/L_{\rm CS}\sim M_{\rm
dr}^{-1/3}(d_{e}L_{\rm CS})^{-2/3}\gg(d_{e}/L_{\rm CS})$. Although we cannot
prove this, we conjecture that this conclusion is more general than the
specific examples just listed; i.e., that a current sheet forming in a large-
scale, weakly collisional plasma will disrupt due to the tearing instability
before its thickness reaches kinetic scales.

Once the plasmoid-driven turbulence sets in the layer, one expects that the
reconnection rate will converge to a (time-averaged) constant $\mathcal{R}$: a
small but finite fraction of the Alfvén rate (perhaps the canonical $0.1$
value [58, 59]). Assuming compressibility effects are negligible, this ought
to correspond to a mean current sheet whose aspect ratio is $a/L_{\rm
CS}\approx\mathcal{R}$. Now, recall that $L_{\rm CS}$ is a macroscopic,
system-size length and, thus, $\rho_{s}/L_{\rm CS}\ll\mathcal{R}\ll 1$. So,
regardless of whether the layer widens or contracts once the turbulence
develops, the thickness of the mean layer, $a$, will have to be on MHD scales
— a conclusion which, from the point of view of the mean fields, is the only
way to ensure that the reconnection rate $\mathcal{R}\approx a/L$ is
independent of kinetic scales, as is suggested by multiple numerical studies
222In agreement with these ideas, we note that Beresnyak’s simulations of 3D
Hall MHD reconnection [65] develop a turbulent current layer which broadens
well beyond the ion skin depth..

These considerations suggest that the mean field equations can still be those
of MHD in this case, and that the first-principles Vlasov (or PIC) description
is only required for computing the fluctuations in the layer. Specifically,
this implies that Eq. (1) applies both in the MHD and in the collisionless
case; and so should Eq. (3).

## Simplest possible subgrid model.

While the framework that we have discussed so far requires one to rigorously
compute the fluctuations in the layer, it may desirable, from a computational
modelling perspective, to have a subgrid model for the fluctuations, such that
only the (MHD) mean-field equations would have to be solved. The simplest such
model is one where the turbulent e.m.f. in Eq. (1) is replaced with a
turbulent resistivity. Dimensionally, this can be estimated as

$\eta_{\rm turb}\approx u_{x,a}a\approx U_{\rm in}a\approx V_{A,y}a^{2}/L_{\rm
CS},$ (4)

where we have used Eq. (3). One can simplify this further with the extra
assumption that there is a critical aspect ratio below which the current sheet
is unstable (e.g., around 0.01 in MHD):

$\eta_{\rm turb}\approx V_{A,y}(a/L_{\rm CS})^{2}_{\rm crit}L_{\rm CS}.$ (5)

This corresponds to a turbulent Lundquist number $S_{\rm turb}=(L_{\rm
CS}/a)_{\rm crit}^{2}$ which, in the MHD case, would evaluate to $S_{\rm
turb}\approx 10^{4}$, yielding the canonical plasmoid-mediated rate
$\mathcal{R}=0.01$. Naturally, a subgrid model would also prescribe a
turbulent viscosity. From dimensional analysis alone, its value would be the
same as that of $\eta_{\rm turb}$; direct numerical simulations are required
to determine the value of the turbulent Prandtl number (although note that the
numerical results of Ref. [49] hint that this may indeed be of order unity in
the MHD case).

For collisionless cases, there is not as much clarity on what the critical
aspect ratio may be; figuring that out in different regions of parameter space
is key for Eq. (5) to be usable.

## Absence of scale separation.

With the above considerations and conclusions in place, we can now inquire
about the existence, or absence, of scale separation between the mean fields
and the fluctuations. Again borrowing insight from two-dimensional plasmoid
chain dynamics, where the plasmoids’ typical width in the transverse direction
essentially defines the mean thickness of the layer, we similarly expect that
in three-dimensional systems there will be no separation between the outer
scale of the fluctuations and the scale of variation of the mean fields in the
transverse direction, $a$; indeed, as already mentioned, we expect the width
$a$ of variation of the mean fields to be defined by the fluctuations. This is
supported by existing numerical evidence [19, 20]. We expect this to be true
even in collisionless cases where kinetic-scale instabilities might be
triggered, because those are expected to coexist with kink or tearing-type
modes [29, 30] whose outer-scale in the $x$-direction will be $a$.

Consider now the out-of-plane-direction, with the background fields extending
over a region of length $L_{z}$. Let us assume the (Alfvénic) turbulence in
the sheet to be critically balanced [61], such that
$\ell_{\lambda}/\lambda\sim V_{Az}/u_{x,\lambda}$, where $\lambda\leq a$ is a
lengthscale in the $x$-direction, and $\ell_{\lambda}$ and $u_{x,\lambda}$
are, respectively, the extent of the fluctuations in the direction parallel to
the total field and the characteristic $x$-direction velocity fluctuation at
that scale. At the $x$-direction outer scale, then, we expect

$\displaystyle\frac{\ell_{a}}{a}\sim\frac{V_{Az}}{u_{x,a}}\sim\frac{V_{Az}}{U_{\rm
in}}\sim\frac{V_{Az}}{V_{Ay}}\frac{L_{\rm CS}}{a}\sim\frac{L_{z}}{a},$ (6)

where we have used Eq. (3) and assumed, in the last step, that the
reconnecting fields are critically balanced, $L_{\rm CS}/L_{z}\sim
V_{Ay}/V_{Az}$. We thus find that, at the scale of the mean current sheet,
there is no scale separation in the $z$ direction between the background and
the fluctuations (note that for moderate guide fields

Finally, one also expects $u_{x,\lambda}/\lambda\sim
u_{y,\lambda}/\xi_{\lambda}$; evaluated at the outer scale, this relation
implies $\xi_{a}/a\sim V_{Ay}/U_{\rm in}\sim L_{\rm CS}/a$ and, thus, no scale
separation between mean fields and fluctuations in the outflow direction.

These conclusions imply that spatially averaging the fluctuations, instead of
time averaging, may not be justified.

## Computational procedure.

The general description of the numerical approach to the solution of this
problem is as follows. Starting from prescribed functional forms for the
background fields, one solves the fluctuating equations until the necessary
averaged quantities reach a steady-state. This solution is then fed into the
mean-field equations, thus evolving said fields into a new state. This new
mean state becomes the new background for the fluctuations, which will
therefore adjust; and so on. The loop is repeated until the mean fields
converge.

The different steps just described are non-trivial and require some
consideration. Let us first consider the choice of the initial functional form
of the mean fields. The most efficient prescription may be to specify a
configuration that is supercritical, i.e., unstable to whatever instabilities
are going to be driving the turbulence — for example, in MHD one might
initialize a current sheet whose aspect ratio is set by the condition for
triggering the plasmoid instability [41, 44] in a forming sheet [62, 50, 63,
51]; the (simplest) collisionless counterpart of this choice is a system-size
Harris sheet [64] varying on the ion skin depth or smaller [16] (since, as
argued above, such a sheet is unstable for realistic parameter choices). The
ensuing dynamics that is captured by this numerical approach is then one of
relaxation of the mean fields to a state where the effect of the turbulence
that they drive is to balance the upstream inflow (see Fig. 1).

One question which arises is that of whether a solution found through this
procedure — while obviously an admissible solution — is, in fact, dynamically
accessible. I.e., if the mean fields were to be dynamically evolved from a
stable to an unstable configuration, would their final configuration be the
same as that which results from this procedure? In the more complex cases,
particularly when the layer is weakly collisional, there might be several
instabilities possible, with different triggering conditions (e.g., some will
depend on current gradients, others on the current itself, etc.) — and, so,
this procedure might converge to a solution that is governed by an instability
that would never be triggered if the background mean fields had been evolved
from an initially stable state. One way to address this problem is to perform
a series of runs where the initial mean fields are made progressively less
unstable (by decreasing the current gradient and/or the current itself) until
a solution is found that is independent of the initial condition.

Lastly, we remark on the different mesh sizes required for the solution of the
mean fields and the fluctuations. Whereas resolution of the full spectrum of
fluctuations is required in the layer (but only there), the only requirement
for the computation of the mean fields is to resolve the macroscopic, MHD-
scale lengths $a$, $L_{\rm CS}$ and $L_{z}$.

## Conclusions.

In this Letter, we describe a framework for approaching reconnection as a
transport-type problem, in which the evolution of the mean fields is self-
consistently determined by the turbulence that they drive in the reconnection
layer. The validity of this method relies on the timescale separation between
the fluctuations and the mean fields — the former being Alfvénic or faster;
the latter evolving on the reconnection timescale. It is argued that mean
fields should remain on MHD scales even in the collisionless limit, a
conclusion which has powerful implications for numerical modelling. We
describe a computational method to implement this approach whose efficiency
gains over the brute force approach that has so far been used should be
significant enough to enable three-dimensional reconnection simulations with
asymptotically large scale separation between the system size and the micro-
scales.

###### Acknowledgements.

The author thanks Bill Daughton, Alex Schekochihin Adam Stanier and Dmitri
Uzdensky for useful discussions. He is also thankful to the Center for
Nonlinear Studies of the Los Alamos National Laboratory, where parts of this
work were carried out. Work supported by DOE award no. DE-SC0022012 and NSF-
DOE Partnership in Basic Plasma Science and Engineering Award No. PHY-2010136.

## References

  * Biskamp [2000] D. Biskamp,  _Magnetic Reconnection in Plasmas_ (Cambridge University Press, 2000).
  * Yamada  _et al._ [2010] M. Yamada, R. Kulsrud, and H. Ji, Magnetic reconnection, Reviews of Modern Physics 82, 603 (2010).
  * Pettersen [1989] B. R. Pettersen, A Review of Stellar Flares and Their Characteristics, Solar Physics 121, 299 (1989).
  * Lyubarsky [2021] Y. Lyubarsky, Emission mechanisms of fast radio bursts, Universe 7, 10.3390/universe7030056 (2021).
  * Sonnerup  _et al._ [1981] B. U. O. Sonnerup, G. Paschmann, I. Papamastorakis, N. Sckopke, G. Haerendel, S. J. Bame, J. R. Asbridge, J. T. Gosling, and C. T. Russell, Evidence for magnetic field reconnection at the earth’s magnetopause, J. Geophys. Res. 86, 10049 (1981).
  * Burch and Phan [2016] J. L. Burch and T. D. Phan, Magnetic reconnection at the dayside magnetopause: Advances with mms, Geophysical Research Letters 43, 8327 (2016).
  * Matthaeus and Lamkin [1986] W. H. Matthaeus and S. L. Lamkin, Turbulent magnetic reconnection, Physics of Fluids 29, 2513 (1986).
  * Franci  _et al._ [2017] L. Franci, S. S. Cerri, F. Califano, S. Landi, E. Papini, A. Verdini, L. Matteini, F. Jenko, and P. Hellinger, Magnetic reconnection as a driver for a sub-ion-scale cascade in plasma turbulence, The Astrophysical Journal Letters 850, L16 (2017).
  * Loureiro and Boldyrev [2020] N. F. Loureiro and S. Boldyrev, Nonlinear Reconnection in Magnetized Turbulence, Astrophys. J. 890, 55 (2020), arXiv:1907.09610 [physics.plasm-ph] .
  * Zhou  _et al._ [2020] M. Zhou, N. F. Loureiro, and D. A. Uzdensky, Multi-scale dynamics of magnetic flux tubes and inverse magnetic energy transfer, Journal of Plasma Physics 86, 535860401 (2020), arXiv:2001.07291 [astro-ph.HE] .
  * Schekochihin [2022] A. A. Schekochihin, MHD turbulence: a biased review, Journal of Plasma Physics 88, 155880501 (2022).
  * Wesson [1990] J. Wesson, Sawtooth reconnection, Nuclear Fusion 30, 2545 (1990).
  * Nykyri and Otto [2001] K. Nykyri and A. Otto, Plasma transport at the magnetospheric boundary due to reconnection in Kelvin-Helmholtz vortices, Geophysical Research Letters 28, 3565 (2001).
  * Leonard [2014] A. W. Leonard, Edge-localized-modes in tokamaks), Physics of Plasmas 21, 090501 (2014).
  * Inchingolo  _et al._ [2018] G. Inchingolo, T. Grismayer, N. F. Loureiro, R. A. Fonseca, and L. O. Silva, Fully Kinetic Large-scale Simulations of the Collisionless Magnetorotational Instability, Astrophys. J. 859, 149 (2018), arXiv:1801.08657 [astro-ph.HE] .
  * Daughton  _et al._ [2011] W. Daughton, V. Roytershteyn, H. Karimabadi, L. Yin, B. J. Albright, B. Bergen, and K. J. Bowers, Role of electron physics in the development of turbulent magnetic reconnection in collisionless plasmas, Nature Physics 7, 539 (2011).
  * Lapenta and Bettarini [2011] G. Lapenta and L. Bettarini, Spontaneous transition to a fast 3D turbulent reconnection regime, EPL (Europhysics Letters) 93, 65001 (2011), arXiv:1102.4791 [astro-ph.SR] .
  * Oishi  _et al._ [2015] J. S. Oishi, M.-M. Mac Low, D. C. Collins, and M. Tamura, Self-generated Turbulence in Magnetic Reconnection, The Astrophysical Journal Letters 806, L12 (2015), arXiv:1505.04653 [astro-ph.SR] .
  * Huang and Bhattacharjee [2016] Y.-M. Huang and A. Bhattacharjee, Turbulent Magnetohydrodynamic Reconnection Mediated by the Plasmoid Instability, Astrophys. J. 818, 20 (2016), arXiv:1512.01520 [physics.plasm-ph] .
  * Beresnyak [2017] A. Beresnyak, Three-dimensional Spontaneous Magnetic Reconnection, Astrophys. J. 834, 47 (2017), arXiv:1301.7424 [astro-ph.SR] .
  * Kowal  _et al._ [2017] G. Kowal, D. A. Falceta-Gonçalves, A. Lazarian, and E. T. Vishniac, Statistics of Reconnection-driven Turbulence, Astrophys. J. 838, 91 (2017), arXiv:1611.03914 [astro-ph.GA] .
  * Yang  _et al._ [2020] L. Yang, H. Li, F. Guo, X. Li, S. Li, J. He, L. Zhang, and X. Feng, Fast Magnetic Reconnection with Turbulence in High Lundquist Number Limit, The Astrophysical Journal Letters 901, L22 (2020), arXiv:2009.06253 [astro-ph.SR] .
  * Kowal  _et al._ [2020] G. Kowal, D. A. Falceta-Gonçalves, A. Lazarian, and E. T. Vishniac, Kelvin-Helmholtz versus Tearing Instability: What Drives Turbulence in Stochastic Reconnection?, Astrophys. J. 892, 50 (2020), arXiv:1909.09179 [astro-ph.HE] .
  * Guo  _et al._ [2021] F. Guo, X. Li, W. Daughton, H. Li, P. Kilian, Y.-H. Liu, Q. Zhang, and H. Zhang, Magnetic Energy Release, Plasma Dynamics, and Particle Acceleration in Relativistic Turbulent Magnetic Reconnection, Astrophys. J. 919, 111 (2021), arXiv:2008.02743 [astro-ph.HE] .
  * Zhang  _et al._ [2021] Q. Zhang, F. Guo, W. Daughton, H. Li, and X. Li, Efficient nonthermal ion and electron acceleration enabled by the flux-rope kink instability in 3d nonrelativistic magnetic reconnection, Phys. Rev. Lett. 127, 185101 (2021).
  * Beg  _et al._ [2022] R. Beg, A. J. B. Russell, and G. Hornig, Evolution, Structure, and Topology of Self-generated Turbulent Reconnection Layers, Astrophys. J. 940, 94 (2022), arXiv:2209.04492 [astro-ph.SR] .
  * Barnes  _et al._ [2010] M. Barnes, I. G. Abel, W. Dorland, T. Görler, G. W. Hammett, and F. Jenko, Direct multiscale coupling of a transport code to gyrokinetic turbulence codes, Physics of Plasmas 17, 056109 (2010), https://doi.org/10.1063/1.3323082 .
  * Abel  _et al._ [2013] I. G. Abel, G. G. Plunk, E. Wang, M. Barnes, S. C. Cowley, W. Dorland, and A. A. Schekochihin, Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows, Reports on Progress in Physics 76, 116201 (2013).
  * Alt and Kunz [2019] A. Alt and M. W. Kunz, Onset of magnetic reconnection in a collisionless, high-$\beta$ plasma, Journal of Plasma Physics 85, 764850101 (2019), arXiv:1810.04135 [astro-ph.HE] .
  * Winarto and Kunz [2022] H. W. Winarto and M. W. Kunz, Triggering tearing in a forming current sheet with the mirror instability, Journal of Plasma Physics 88, 905880210 (2022), arXiv:2112.04018 [astro-ph.HE] .
  * Che  _et al._ [2010] H. Che, J. F. Drake, M. Swisdak, and P. H. Yoon, Electron holes and heating in the reconnection dissipation region, Geophys. Res. Lett. 37, L11105 (2010), arXiv:1001.3203 [physics.plasm-ph] .
  * Carter  _et al._ [2002] T. A. Carter, M. Yamada, H. Ji, R. M. Kulsrud, and F. Trintchouk, Experimental study of lower-hybrid drift turbulence in a reconnecting current sheet, Physics of Plasmas 9, 3272 (2002).
  * Che  _et al._ [2011] H. Che, J. F. Drake, and M. Swisdak, A current filamentation mechanism for breaking magnetic field lines during reconnection, Nature 474, 184 (2011).
  * Li  _et al._ [2022] X. Li, R. Wang, Q. Lu, C. T. Russell, S. Lu, I. J. Cohen, R. E. Ergun, and S. Wang, Three-dimensional network of filamentary currents and super-thermal electrons during magnetotail magnetic reconnection, Nature Communications 13, 3241 (2022).
  * Biskamp [2003] D. Biskamp,  _Magnetohydrodynamic Turbulence_ (Cambridge University Press, Cambridge, UK, 2003).
  * Hughes [2018] D. W. Hughes, Mean field electrodynamics: triumphs and tribulations, Journal of Plasma Physics 84, 735840407 (2018), arXiv:1804.02877 [astro-ph.SR] .
  * Rincon [2019] F. Rincon, Dynamo theories, Journal of Plasma Physics 85, 205850401 (2019), arXiv:1903.07829 [physics.plasm-ph] .
  * Moffatt and Dormy [2019] K. Moffatt and E. Dormy,  _Self-Exciting Fluid Dynamos_, Cambridge Texts in Applied Mathematics (Cambridge University Press, 2019).
  * Higashimori  _et al._ [2013] K. Higashimori, N. Yokoi, and M. Hoshino, Explosive turbulent magnetic reconnection, Phys. Rev. Lett. 110, 255001 (2013).
  * Widmer  _et al._ [2019] F. Widmer, J. Büchner, and N. Yokoi, Analysis of fast turbulent reconnection with self-consistent determination of turbulence timescale, Physics of Plasmas 26, 102112 (2019), https://doi.org/10.1063/1.5109020 .
  * Loureiro  _et al._ [2007] N. F. Loureiro, A. A. Schekochihin, and S. C. Cowley, Instability of current sheets and formation of plasmoid chains, Physics of Plasmas 14, 100703 (2007), arXiv:astro-ph/0703631 [astro-ph] .
  * Bhattacharjee  _et al._ [2009] A. Bhattacharjee, Y.-M. Huang, H. Yang, and B. Rogers, Fast reconnection in high-Lundquist-number plasmas due to the plasmoid Instability, Physics of Plasmas 16, 112102 (2009), arXiv:0906.5599 [physics.plasm-ph] .
  * Huang and Bhattacharjee [2010] Y.-M. Huang and A. Bhattacharjee, Scaling laws of resistive magnetohydrodynamic reconnection in the high-Lundquist-number, plasmoid-unstable regime, Physics of Plasmas 17, 062104 (2010), arXiv:1003.5951 [physics.plasm-ph] .
  * Loureiro  _et al._ [2013] N. F. Loureiro, A. A. Schekochihin, and D. A. Uzdensky, Plasmoid and Kelvin-Helmholtz instabilities in Sweet-Parker current sheets, Phys. Rev. E 87, 013102 (2013), arXiv:1208.0966 [physics.plasm-ph] .
  * Loureiro and Uzdensky [2016] N. F. Loureiro and D. A. Uzdensky, Magnetic reconnection: from the Sweet-Parker model to stochastic plasmoid chains, Plasma Physics and Controlled Fusion 58, 014021 (2016), arXiv:1507.07756 [physics.plasm-ph] .
  * Sweet [1958] P. A. Sweet, The Neutral Point Theory of Solar Flares, in  _Electromagnetic Phenomena in Cosmical Physics_ , Vol. 6, edited by B. Lehnert (1958) p. 123.
  * Parker [1957] E. N. Parker, Sweet’s Mechanism for Merging Magnetic Fields in Conducting Fluids, J. Geophys. Res. 62, 509 (1957).
  * Uzdensky  _et al._ [2010] D. A. Uzdensky, N. F. Loureiro, and A. A. Schekochihin, Fast Magnetic Reconnection in the Plasmoid-Dominated Regime, Phys. Rev. Lett. 105, 235002 (2010), arXiv:1008.3330 [astro-ph.SR] .
  * Loureiro  _et al._ [2012] N. F. Loureiro, R. Samtaney, A. A. Schekochihin, and D. A. Uzdensky, Magnetic reconnection and stochastic plasmoid chains in high-Lundquist-number plasmas, Physics of Plasmas 19, 042303 (2012), arXiv:1108.4040 [astro-ph.SR] .
  * Uzdensky and Loureiro [2016] D. A. Uzdensky and N. F. Loureiro, Magnetic Reconnection Onset via Disruption of a Forming Current Sheet by the Tearing Instability, Phys. Rev. Lett. 116, 105003 (2016), arXiv:1411.4295 [astro-ph.SR] .
  * Tolman  _et al._ [2018] E. A. Tolman, N. F. Loureiro, and D. A. Uzdensky, Development of tearing instability in a current sheet forming by sheared incompressible flow, Journal of Plasma Physics 84, 905840115 (2018), arXiv:1711.09042 [physics.plasm-ph] .
  * Del Sarto  _et al._ [2016] D. Del Sarto, F. Pucci, A. Tenerani, and M. Velli, “Ideal” tearing and the transition to fast reconnection in the weakly collisional MHD and EMHD regimes, Journal of Geophysical Research (Space Physics) 121, 1857 (2016), arXiv:1511.00035 [physics.plasm-ph] .
  * Mallet [2020] A. Mallet, The onset of electron-only reconnection, Journal of Plasma Physics 86, 905860301 (2020), arXiv:1912.02286 [physics.plasm-ph] .
  * Note [1] Assuming a Harris sheet profile for concreteness; other choices have no qualitative impact on the conclusion.
  * Fitzpatrick and Porcelli [2004] R. Fitzpatrick and F. Porcelli, Collisionless magnetic reconnection with arbitrary guide field, Physics of Plasmas 11, 4713 (2004).
  * Fitzpatrick and Porcelli [2007] R. Fitzpatrick and F. Porcelli, Erratum: Collisionless magnetic reconnection with arbitrary guide-field [Phys. Plasmas 11, 4713 (2004)], Physics of Plasmas 14, 049902 (2007).
  * Loureiro and Boldyrev [2018] N. F. Loureiro and S. Boldyrev, Turbulence in Magnetized Pair Plasmas, The Astrophysical Journal Letters 866, L14 (2018), arXiv:1805.09224 [physics.plasm-ph] .
  * Birn  _et al._ [2001] J. Birn, J. F. Drake, M. A. Shay, B. N. Rogers, R. E. Denton, M. Hesse, M. Kuznetsova, Z. W. Ma, A. Bhattacharjee, A. Otto, and P. L. Pritchett, Geospace Environmental Modeling (GEM) magnetic reconnection challenge, J. Geophys. Res. 106, 3715 (2001).
  * Cassak  _et al._ [2017] P. A. Cassak, Y.-H. Liu, and M. Shay, A review of the 0.1 reconnection rate problem, Journal of Plasma Physics 83, 715830501 (2017).
  * Note [2] In agreement with these ideas, we note that Beresnyak’s simulations of 3D Hall MHD reconnection [65] develop a turbulent current layer which broadens well beyond the ion skin depth.
  * Goldreich and Sridhar [1995] P. Goldreich and S. Sridhar, Toward a Theory of Interstellar Turbulence. II. Strong Alfvenic Turbulence, Astrophys. J. 438, 763 (1995).
  * Pucci and Velli [2013] F. Pucci and M. Velli, Reconnection of quasi-singular current sheets: The “ideal” tearing mode, The Astrophysical Journal Letters 780, L19 (2013).
  * Tenerani  _et al._ [2016] A. Tenerani, M. Velli, F. Pucci, S. Landi, and A. F. Rappazzo, ‘ideally’ unstable current sheets and the triggering of fast magnetic reconnection, Journal of Plasma Physics 82, 535820501 (2016).
  * Harris [1962] E. G. Harris, On a plasma sheath separating regions of oppositely directed magnetic field, Il Nuovo Cimento (1955-1965) 23, 115 (1962).
  * Beresnyak [2018] A. Beresnyak, The rate of three-dimensional hall-mhd reconnection, Journal of Physics: Conference Series 1031, 012001 (2018).


# Text-dependent Speaker Verification (TdSV) Challenge 2024: Challenge
Evaluation Plan

Hossein Zeinali, Kong Aik Lee, Jahangir Alam, Lukaš Burget

(Version 1.1, April 20, 2024)

##  1 Introduction

This document outlines the Text-dependent Speaker Verification (TdSV)
Challenge 2024, which centers on analyzing and exploring novel approaches for
text-dependent speaker verification. The primary goal of this challenge is to
motive participants to develop single yet competitive systems, conduct
thorough analyses, and explore innovative concepts such as multi-task
learning, self-supervised learning, few-shot learning, and others, for text-
dependent speaker verification.

Building upon the achievements of the Short-duration Speaker Verification
(SdSV) Challenge 2020 and 2021, the TdSV Challenge 2024 focuses exclusively on
text-dependent verification in two distinct scenarios. The first scenario
involves conventional TdSV, while the second track entails speaker enrollment
using user-defined passphrases. The evaluation dataset utilized for the
challenge is derived from the second version of the versatile DeepMine dataset
[1, 2]. For this challenge, Parts 1 and 3 of the dataset are employed. The
subsequent section provides a comprehensive description of both tasks.

##  2 Tasks Description

###  2.1 Task 1 - Text-Dependent Speaker Verification

Task 1 of the TdSV Challenge 2024 involves speaker verification in the
conventional text-dependent mode: determining whether a specific phrase was
spoken by the target speaker, given a test segment of speech, and the target
speaker’s enrollment data. Unlike text-independent speaker verification, Task
1 considers the lexical content of the utterance, making it a twofold
verification task verifying both the speaker and the phrase.

In Task 1, each trial includes a test segment, a model identifier indicating
three enrollment utterances, and a phrase ID uttered in the utterances. The
system processes each trial independently, producing a log-likelihood ratio
(LLR) score that combines speaker and phrase verification scores.

Enrollment and test phrases, except for target-wrong trials where some tests
were selected from free-text utterances, are drawn from a fixed set of ten
phrases, comprising five Persian and five English phrases, respectively. The
in-domain training data includes utterances from 1620 speakers, some of whom
only have Persian phrases. Model enrollment is conducted in a phrase and
language-dependent manner, using three utterances for each model. With the
same set of target speakers, Task 1 offers a solid basis for analyzing the
language factor in text-dependent speaker recognition.

####  2.1.1 Trial types

Given the ground truth, there are four trial types in a TdSV task [3]. The
first is Target-Correct (TC), where the target speaker utters the correct
passphrase. The second is Target-Wrong (TW), where the target speaker utters
an incorrect passphrase. Similarly, Imposter-Correct (IC) and Imposter-Wrong
(IW) refer to cases where the impostor utters the correct or incorrect
passphrase. The system should accept TC trials as target trials and reject the
other three types as non-target (impostor) trials. Note that the main
difference between text-dependent and text-independent speaker verification is
the consideration of TW trials as impostor trials, while both TC and TW are
considered target trials in text-independent mode. There are no cross-language
and cross-gender trials in Task 1 of the challenge.

####  2.1.2 Training condition

The training condition is defined as the amount of data/resources used to
build a Speaker Recognition (SR) system. We adopted a fixed training condition
where the system should only be trained using a designated set. The fixed
training set consists of the following:

  * •

VoxCeleb 1&2

  * •

LibriSpeech

  * •

Mozilla Common Voice Farsi

  * •

DeepMine (Task 1 Train Partition)

The use of other public or private speech data for training is prohibited,
while the use of non-speech data for data augmentation purposes is permitted.
The in-domain DeepMine training data can be utilized for any purpose,
including neural network training, LDA or PLDA model training, and score
normalization. Regarding the possibility of using pre-trained models, it
should be noted that the use of any public pre-trained model is allowed in
this challenge.

We provide a separate development set for the challenge. Teams can use the
development set for parameter tuning and system evaluation before submitting
them to the leaderboard to minimize the number of submissions. Any other
training, such as fusion training on the development set, is not permitted.
Additionally, any usage of Task 2 in-domain data and its development set data
for this task is not allowed.

####  2.1.3 Enrollment Condition

Enrollment is conducted using three utterances of a specific phrase for each
model. We opted for three utterances for model enrollment, as it is commonly
practiced. It’s important to note that using enrollment utterances from other
models is prohibited, such as for calculating score normalization parameters
(i.e., trials are to be processed independently).

####  2.1.4 Test Condition

Each evaluation trial includes a test utterance and a target model. As
mentioned earlier, there are four trial types in the evaluation, with only TC
considered as the target, and the rest classified as imposture. Among the
trial types, IW trials can be easily rejected by a simple verification system.
Consequently, there are no IW trials in the trial list.

Please be aware that there is a limit of 10 submissions per team.
Additionally, teams can submit at most two score files per day on the
evaluation platform. During the last five days of the challenge deadline, the
leaderboard will be hidden.

###  2.2 Task 2 - Text-dependent Speaker Verification Using User-defined
Passphrases

Task 2 of the TdSV challenge revisits the problem of speaker verification in a
text-dependent manner. This means that with three repetitions of a phrase as
enrollment data and a test file, we automatically determine whether the target
speaker has uttered the test speech or not. We also need to determine if the
uttered phrase in the test speech is the same as the one uttered by the
speaker in the enrollment speech. This is the same task addressed by Task 1 of
the challenge, with the difference that here we assume the passphrase has been
chosen by the user, and we have no training data for it. Additionally, we do
not know the specific phrase chosen by the user in terms of content.

To examine the impact of using additional utterances from the speaker, which
are not constrained by any content limitations, for enrollment, in this task,
besides the three repetitions of the passphrase chosen by the speaker, there
are also other speech files available from the same speaker, in which the
uttered phrase is not specified. Teams are allowed to use these utterances in
any way they choose to build the speaker model. It is recommended that a
comparison be made in the final report between using these utterances and the
scenario where these utterances are not used.

Each trial in this task consists of a test speech segment accompanied by a
model identifier, indicating three repetitions of the passphrase, as well as
several additional free-text enrollment utterances. The system is required to
process each trial independently and generate a log-likelihood ratio (LLR) for
each of them. The trial types here are the same as in Task 1, meaning TC
trials should be accepted, while TW and IC trials should be rejected.

To simulate user-defined passphrases for this task, four out of the 10
available text-dependent phrases in the dataset are exclusively used in the
test set, while the remaining six phrases are retained in the training data.
Therefore, the in-domain training data for this task comprises text-
independent Persian utterances from 1620 speakers, along with utterances from
the aforementioned six phrases. This data can be utilized for various
purposes, such as LDA/PLDA, score normalization, training data for neural
networks, etc.

####  2.2.1 Training Condition

The training data conditions in this task are similar to Task 1, except that
here, specific in-domain data has been included, which is exclusive to this
task and should not be utilized for Task 1. Additionally, it is prohibited to
use any Task 1 in-domain data and its development set data for this task.

####  2.2.2 Enrollment Condition

As mentioned before, the enrollment data in Task 2 consists of three
repetition of the user’s passphrase plus one to several variable-length free-
text utterances. Teams can use either both phrase dependent and phrase
independent utterances for model creation or one of them. Since each
enrollment utterance is a complete recording without trimming to a specific
duration, the overall duration might not be exactly uniform. Note that using
the enrollment utterances from the other models is forbidden, for example, for
calculating score normalization parameters.

####  2.2.3 Test Condition

Each evaluation trial comprises a test utterance and a target model. Similar
to Task 1, the content of a test utterance should correspond to the user’s
passphrase; however, here, we assume that the phonetic content of the
passphrase is unknown.

Similarly to Task 1, there is a development set (a smaller evaluation set)
with keys that enable teams to monitor performance improvements before
submitting scores to the evaluation platform. Additionally, there is a limit
of 10 submissions per team, and submitting two score files per day is allowed.
During the last five days leading up to the challenge deadline, the
leaderboard will be hidden.

###  2.3 Performance Measurement for Both Tasks

The main metric for the challenge is normalized minimum Detection Cost
Function (DCF) as defined is SRE08. This detection cost function is defined as
a weighted sum of miss and false alarm error probabilities:

$C_{Det}=C_{Miss}\times P_{Miss\>|\>Target}\times
P_{Target}+C_{FalseAlarm}\times
P_{FalseAlarm\>|\>NonTarget}\times(1-P_{Target})\>,$

where $C_{Miss}=10$, $C_{FalseAlarm}=1$ and $P_{Target}=0.01$. Based on the
parameters, the normalized DCF ($DCF_{norm}$) will be DCF divide by 0.1 as the
best cost that could be obtained without processing the input data. In
addition to $minDCF_{norm}^{0.01}$, the Equal Error Rate (EER) will be
reported.

###  2.4 Data Description

The primary dataset for the challenge is the DeepMine dataset, collected
through crowdsourcing [4]. Participants in the data collection project
installed an Android application and recorded phrases within the application.
A comprehensive description of the project and dataset can be found in [2, 1].
Below are the BibTeX sources for citing the database for easy reference:

    Ψ@inproceedings{deepmine2018odyssey,
    Ψ    title={{DeepMine} Speech Processing Database: Text-Dependent and
    Ψ    Independent Speaker Verification and Speech Recognition in
    Ψ    {Persian and English}.},
    Ψ    author={Zeinali, Hossein and Sameti, Hossein and Stafylakis, Themos},
    Ψ    year=2018,
    Ψ    booktitle={Proc. Odyssey 2018 The Speaker and Language Recognition
    Ψ    Workshop},
    Ψ    pages={386--392},
    Ψ}
    Ψ
    Ψ@inproceedings{deepmine2019asru,
    Ψ    title={A Multi Purpose and Large Scale Speech Corpus in {Persian and
    Ψ    English} for Speaker and Speech Recognition: the {DeepMine} Database},
    Ψ    author={Zeinali, Hossein and Burget, Lukas and Cernocky, Jan},
    Ψ    year=2019,
    Ψ    booktitle={Proc. ASRU 2019 The 2019 IEEE Automatic Speech Recognition
    Ψ    and Understanding Workshop},
    Ψ}
    Ψ

The database was recorded in real-life environments in Iran, incorporating
various types of noise during collection. The primary language is Persian
(Farsi), with most participants also contributing to the English portion. Part
1 of the dataset includes five Persian and five English phrases used in both
tasks of the challenge. Table 1 displays the English phrases and
transliterations of the Persian phrases. The phoneme transcription of the
phrases are shown in Table 2, and participants are free to use them as needed.
Part 3 of the dataset contains text-independent phrases in Persian and is
utilized in the training data, as well as TW trials in the evaluation set.

Table 1: Phrases in Task1 of the challenge.

Id Phrase 01 sedaye man neshandahandeye hoviyyate man ast. 02 sedaye har kas
monhaser be fard ast. 03 hoviyyate man ra ba sedaye man tayid kon. 04 sedaye
man ramze obure man ast. 05 baniadam azaye yekdigarand. 06 My voice is my
password. 07 OK Google. 08 Artificial intelligence is for real. 09 Actions
speak louder than words. 10 There is no such thing as a free lunch.

There are 6 vowels in Persian and the following map can be used roughly to map
them with English vowels. You can also find another mapping for doing this.
Also, you can do a speech analysis to map them.

  * •

Three short vowels:

    1. 1.

A like ”AE” in ”rat - R AE T”

    2. 2.

E like ”EH” in ”red - R EH D”

    3. 3.

O like ”AO” in ”short - SH AO R T” but it is pronounced shorter than in
English.

  * •

Three long vowels:

    1. 1.

AA like ”AA” in ”God - G AA D”

    2. 2.

I like ”IY” in ”seed - S IY D”

    3. 3.

U something like ”UW” in ”too - T UW”

Table 2: Transcriptions of the challenge phrases. The character ”-” is used as
a word-separator.

Id Phrase 01 S E D AA Y E - M A N - N E SH AA N D A H A N D E Y E - H O V I Y
Y A T E - M A N - AH A S T 02 S E D AA Y E - H A R - K A S - M O N H A S E R -
B E - F A R D - AH A S T 03 H O V I Y Y A T E - M A N - R AA - B AA - S E D AA
Y E - M A N - T A AH Y I D - K O N 04 S E D AA Y E - M A N - R A M Z E - AH O
B U R E - M A N - AH A S T 05 B A N I AH AA D A M - AH A AH Z AA Y E - Y E K D
I G A R A N D 06 M AY - V OY S - IH Z - M AY - P AE S W ER D 07 OW K EY - G UW
G AH L 08 AA R T AH F IH SH AH L - IH N T EH L AH JH AH N S - IH Z - F AO R -
R IY L 09 AE K SH AH N Z - S P IY K - L AW D ER - DH AE N - W ER D Z 10 DH EH
R - IH Z - N OW - S AH CH - TH IH NG - AE Z - EY - F R IY - L AH N CH

###  2.5 Data Organization

Data will be provided in separate zip files for each task. Each zip file will
include the following components: the in-domain DeepMine training data,
enrollment data (for development and evaluation), the model definition file,
the trial file (for development and evaluation), and the test utterances. When
extracting the files into a directory, the directory structure should be as
follows:

    Ψ<base directory>/
    Ψ  README.txt
    Ψ  docs/
    Ψ    dev_model_enrollment.txt
    Ψ    dev_trials.txt
    Ψ    eval_model_enrollment.txt
    Ψ    eval_trials.txt
    Ψ    train_labels.txt
    Ψ  wav/
    Ψ    enrollment/
    Ψ      enr_000001.wav
    Ψ      enr_000002.wav
    Ψ      ...
    Ψ    evaluation/
    Ψ      evl_000001.wav
    Ψ      evl_000002.wav
    Ψ      ...
    Ψ    train/
    Ψ      trn_000001.wav
    Ψ      trn_000002.wav
    Ψ      ...
    Ψ

Please note that the zip files for Task 1 and Task 2 should be extracted into
separate directories due to partial overlap in their contents.

###  2.6 Format of Model Enrollment File

####  2.6.1 Task 1 Enrollment File

The enrollment file for Task 1 is a space-separated six-column text file named
model_enrollment.txt, located in the docs directory. The file begins with a
header line. Each line afterward represents a record. The first column
indicates the model ID, the second column shows the phrase ID corresponding to
the uttered phrase, the third column indicates the speaker’s gender and the
remaining three columns display the enrollment file IDs. There is only one
space between two records in each line. The format of the enrollment file is
as follows:

    Ψmodel-id<SPACE>phrase-id<SPACE>gender<SPACE>enroll-file-id1<SPACE>enroll-file-id2
    Ψ<SPACE>enroll-file-id3<NEWLINE>
    Ψ

where, model-id represents the model identifier, phrase-id denotes the phrase
identifier, gender shows the speaker’s gender, and enroll-file-ids indicate
the identifiers for the enrollment utterances.

For example:

    Ψmodel-id phrase-id gender enroll-file-id1 enroll-file-id2 enroll-file-id3
    Ψeval_model_000001 05 m enr_036529 enr_036530 enr_036531
    Ψeval_model_000002 08 f enr_010048 enr_010049 enr_010050
    Ψeval_model_000003 01 f enr_042226 enr_042227 enr_042228
    Ψeval_model_000004 09 f enr_021046 enr_021047 enr_021048
    Ψeval_model_000005 04 f enr_000253 enr_000254 enr_000255
    Ψ

####  2.6.2 Task 2 Enrollment File

The enrollment file for Task 2 is a space-separated text file named
model_enrollment.txt, located in the docs directory. The file begins with a
header line. Each subsequent line contains at least eight columns. The first
column indicates the model ID and the second column displays the speaker’s
gender. The following three columns represent enrollment files containing the
speaker’s passphrase. The remaining columns contain file identifiers for free
text enrollment utterances. The number of columns in each line may vary
depending on the number of enrollment files. There is only one space between
two columns in each line. The format of the enrollment file is as follows:

    Ψmodel-id<SPACE>gender<SPACE>enroll-file-id1<SPACE>enroll-file-id2
    Ψ<SPACE>enroll-file-id3<SPACE>enroll-file-id4<SPACE>enroll-file-id5...<NEWLINE>
    Ψ

In this format, model-id represents the model identifier, gender indicates the
speaker’s gender, and enroll-file-ids denote the identifiers for the
enrollment utterances.

For example:

    Ψmodel-id gender enroll-file-ids ...
    Ψeval_model_000001 f enr_014718 enr_014719 enr_014720 enr_014721 enr_014722
        enr_014723 enr_014687
    Ψeval_model_000002 f enr_039732 enr_039733 enr_039734 enr_039735 enr_015670
        enr_017664 enr_017668 enr_039736 enr_039737
    Ψeval_model_000003 m enr_052051 enr_052052 enr_052053 enr_006366 enr_006382
        enr_031688 enr_052054 enr_052055
    Ψeval_model_000004 f enr_045576 enr_045577 enr_045578 enr_045579 enr_041021
        enr_045580
    Ψeval_model_000005 f enr_013768 enr_013769 enr_013770 enr_013771 enr_013772
        enr_013773 enr_013774 enr_013775 enr_013776 enr_013777
    Ψ

###  2.7 Format of Trial File

The trial file for both tasks is a space-separated two-column text file named
trials.txt, located in the docs directory. The file begins with a header line.
Each subsequent line consists of two columns. The first column indicates the
model ID and the second column indicates the evaluation file ID. There is only
one space between two columns in each line. The format of the trial file is as
follows:

    Ψmodel-id<SPACE>evaluation-file-id<NEWLINE>
    Ψ

where model-id represents the model identifier, and evaluation-file-id denotes
the test utterance identifier.

For example:

    Ψmodel-id evaluation-file-id
    Ψeval_model_000001 evl_000122
    Ψeval_model_000001 evl_000186
    Ψeval_model_000001 evl_000548
    Ψeval_model_000001 evl_001205
    Ψeval_model_000001 evl_001416
    Ψeval_model_000001 evl_002123
    Ψeval_model_000001 evl_002129
    Ψeval_model_000001 evl_002230
    Ψ

##  3 In-domain Training Set

As previously mentioned, the in-domain data for both tasks comprises
utterances from 1620 speakers. All training utterances are stored in the
wav/train directory. The file train_labels.txt in the docs directory is a
space-separated text file containing information for each utterance. Each line
in this file consists of three columns: the first column indicates the train-
file-id, the second column denotes the speaker-id, and the last column shows
the phrase-id. For training utterances that contain free-text, the phrase-id
is labeled as “FT”. There is a header line at the beginning of the file. The
format of the train label file is as follows:

    Ψtrain-file-id<SPACE>speaker-id<SPACE>phrase-id<NEWLINE>
    Ψ

where train-file-id represents the train utterance identifier, speaker-id
indicates the speaker label, and phrase-id is the identifier of the phrase for
each utterance.

For example:

    Ψtrain-file-id speaker-id phrase-id
    Ψtrn_000001 spk_000001 03
    Ψtrn_000002 spk_000001 03
    Ψtrn_000003 spk_000001 03
    Ψtrn_000004 spk_000001 03
    Ψtrn_000005 spk_000001 03
    Ψtrn_000006 spk_000001 03
    Ψtrn_000007 spk_000001 03
    Ψtrn_000008 spk_000001 03
    Ψ

##  4 Evaluation Rules and Requirements

The overall rules are quite similar to NIST SREs. Firstly, participants must
adhere to the data restrictions, where only a fixed training condition is
permitted. Participants are also obligated to process the test data according
to the following rules and upload the results to the challenge website for
evaluation. These rules are:

  * •

Participants agree to make at least one valid submission for one of the tasks.

  * •

Participants agree to process each trial independently. That is, each decision
for a trial is to be based solely upon the specified test segment and target
speaker enrollment data. The use of information about other test segments
and/or other target speaker data is not allowed.

  * •

Participants agree not to probe the enrollment or test segments via
manual/human means, such as listening to the data or producing a manual
transcript of the speech.

  * •

Participants are allowed to use any automatically derived information for
training, development, enrollment, or test segments.

  * •

Participants are not allowed to use the development set for any training
purposes.

  * •

Participants may make multiple submissions for each task. Based on the
leaderboard results, participants should select the best-performing systems,
write a proper system description based on the results, and send it to
challenge organizers via email<EMAIL_ADDRESS>
##  5 Baseline

There is a common ECAPA-TDNN baseline for both tasks.

##  6 Prizes

There will be three cash prizes for each task. Winners will be selected based
on the results of the evaluation dataset and other qualitative factors such as
the novelty and contribution of the proposed system. Moreover, the results
achieved by a single system or fusions of fewer systems rather than a mega
fusion are more interesting. In addition to the cash prize, each winner will
receive a certificate for their achievement. The cash prizes for each task are
as follows:

  * •

Rank 1: 2000 USD

  * •

Rank 2: 1000 USD

  * •

Rank 3: 500  USD

##  7 Evaluation Protocol

###  7.1 Challenge Website

The challenge has a GitHub page (https://tdsvc.github.io/). All information
and updates about the challenge will be posted on the website. System
descriptions submitted by the participants will also be available on the
challenge website.

###  7.2 Registration

In summary, to register for this challenge, you first need to have a username
in the Codabench system. Then, access the Google form available at the
following address, complete the dataset license agreement file, sign it, and
send it to us along with completing the form. After reviewing your
information, your registration will be completed, and access information to
the challenge dataset will be sent to you. You can find more registration
details in the challenge on the website.

https://forms.gle/q2kXMuvomkpG73Gj9

###  7.3 Leaderboard Platform

As mentioned earlier, there is an online leaderboard for each task, and
participants can submit a maximum of two score files per day during the
evaluation period. There is also a limit of 10 submissions in total for each
task. The leaderboard displays the performance of the systems on the final
evaluation set. Please note that 5 days before the challenge deadline, the
public leaderboard will be hidden. Teams will be ranked based on the best-
performing submitted system.

The challenge leaderboard platforms are available at:

  * •

Task 1: https://www.codabench.org/competitions/2760/

  * •

Task 2: https://www.codabench.org/competitions/2761/

###  7.4 System Output Format

The results should be submitted in a single ZIP file containing a single text
file named answer.txt. The text file must be located at the root of the ZIP
file, and the ZIP file should not contain any folders or other unwanted files.

The answer.txt file is a one-column text file. Each line of the file indicates
an LLR score (a float number) for the corresponding trial. The order of scores
must match the order of trials in the trial file, and all trials must be
scored. Any inconsistency will result in an error during the evaluation of the
system. Please note that while the trial file includes a header line, this
header should not be included in the answer.txt file.

For example:

    Ψ-6.1284
    Ψ-97.8528
    Ψ-16.8025
    Ψ-44.3276
    Ψ4.4121
    Ψ-61.0123
    Ψ-42.9890
    Ψ

If the answer.txt file is not found inside the ZIP file or is not correctly
formatted as specified above, your submission will likely fail and will NOT be
scored. Additionally, improper formatting may lead to the wrong interpretation
of your results and therefore to an incorrect score.

###  7.5 Data License Agreement

The evaluation data for this challenge is a subset of the DeepMine dataset. To
access this dataset, participants must sign a data license agreement specific
to the TdSV Challenge 2024. This license permits participants to use the data
for the challenge and subsequent paper publications. Any other use of the data
is strictly prohibited. The license agreement file can be found on the
challenge website.

###  7.6 System Description

Each participant is required to submit a comprehensive system description,
which will be made available online on the challenge website. Additionally, we
strongly encourage participants to submit papers to the special session
related to the challenge in SLT 2024. These papers will undergo standard
review procedures and should adhere to proper formatting and demonstrate
sufficient novelty for acceptance.

The system description must be at least 2 pages long and include the following
information about the submitted systems. You can use the INTERSPEECH or SLT
paper template for writing the description.

  * •

A thorough description of the system components, including front-end and back-
end modules, along with their configurations.

  * •

A detailed explanation of the data partitions used to train the various
models.

  * •

Performance metrics of the submitted systems on both the development set and
the evaluation set, as reported on the leaderboard website.

A BibTeX source for citing the evaluation plan is provided below for easy
reference.

    Ψ@techreport{tdsvc2024plan,
    Ψ    title={Text-dependent Speaker Verification (TdSV) Challenge 2024:
    Ψ        Challenge Evaluation Plan.},
    Ψ    author={Zeinali, Hossein and Lee, Kong Aik and Alam, Jahangir and
    Ψ    Burget, Luka\v{s}},
    Ψ    institution={arXiv preprint arXiv:1xxx.0xxxx},
    Ψ    year=2024,
    Ψ}
    Ψ

##  8 Planned Evaluation Schedule

Release of Evaluation Plan: Apr 10, 2024 Release of Train, Dev, and Eval Sets:
Apr 10, 2024 Evaluation Platform Open: Apr 20, 2024 Challenge Deadline: Jun
10, 2024 System Description Deadline: Jun 17, 2024 SLT Paper Submission
Deadline: Jun 20, 2024 TdSV Challenge 2024 Special Session at SLT: Dec 02 –
05, 2024

## References

  * [1] H. Zeinali, H. Sameti, T. Stafylakis, DeepMine speech processing database: Text-dependent and independent speaker verification and speech recognition in Persian and English., in: Proc. Odyssey 2018 The Speaker and Language Recognition Workshop, 2018, pp. 386–392.
  * [2] H. Zeinali, L. Burget, J. Cernocky, A multi purpose and large scale speech corpus in Persian and English for speaker and speech recognition: the DeepMine database, in: Proc. ASRU 2019 The 2019 IEEE Automatic Speech Recognition and Understanding Workshop, 2019.
  * [3] A. Larcher, K. A. Lee, B. Ma, H. Li, Text-dependent speaker verification: Classifiers, databases and RSR2015, Speech Communication 60 (2014) 56–77.
  * [4] K. A. Lee, A. Larcher, G. Wang, P. Kenny, N. Brümmer, D. v. Leeuwen, H. Aronowitz, M. Kockmann, C. Vaquero, B. Ma, et al., The RedDots data collection for speaker recognition, in: Sixteenth Annual Conference of the International Speech Communication Association, 2015.