10,000 – A Reason to Study Granular Heat Convection

I. Einav, P. Rognon, Y. Gan, T. Miller and D. Griffani

Particles and Grains Laboratory, School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia

Abstract. In sheared granular media, particle motion is characterized by vortex-like structures; here this is demonstrated experimentally for disks system undergoing indefinite deformation during simple shear, as often imposed by the rock masses hosting earthquake fault gouges. In traditional fluids it has been known for years that vortices represent a major factor of heat transfer enhancement via convective internal mixing, but in analyses of heat transfer through earthquake faults and base planes of landslides this has been continuously neglected. Can research proceed by neglecting heat convection by internal mixing? Our answer is astonishingly far from being yes.

Keywords: Heat transfer, convection, vortices, mixing, simple shear. PACS: 47.57.Gc, 47.55.pb, 91.32.-m, 83.50.Ax

INTRODUCTION

Heat convection by internal dispersive mixing has almost been exclusively associated with conventional fluids. On the other hand, in granular media heat transfer was attributed to conduction between and through grains and between grains and surrounding fluids, convection and conduction in the fluids, and radiation [1-3]; and bulk mass advection of the grains [4]. Significant research focused on their effective conductivity, as it was assumed to be the most important factor in granular materials [1,2]. More recently this has been studied using discrete element model (DEM) simulations [5-8].

Our group carried out a similar study with DEM, initially aiming to question the effect of grain motion during shearing on the effective conductivity. To our surprise the conductivity we measured showed dependence on particles’ heat capacity [8]. Although this appears to be a relatively moderate effect, it triggered our interest to search for a possibly more impressive phenomenon – also reported in [8] – related to heat convection due to the internal mixing of the grains. In this paper we will gradually unfold the physics of this phenomenon, its significance, and the reasons behind its overwhelming extent.

MOTIVATION

How justifiable is it to neglect heat convection in sheared granular media? To answer this question, we draw attention to two parallel streams of research: (a) heat budget of earthquakes and landslides [9-11], studied by geophysicists; and (b) particle kinematics in

granular media, a current topic of interest for the ‘granulists’ of our community [12,13].

1st Motivation – Heat Equation in Fault and Landslide Studies

In geophysics it is widely accepted that the deformation in the rock mass during earthquakes quickly localizes to produce a zone of fault gouge, of a few millimetres thickness, or even less. Such a fault gouge layer is illustrated in Fig. 1.

FIGURE 1. Schematics of Earthquake Fault Gouge. Heat production arises from shear heating, while heat transfer across the gouge was assumed to come only through conduction [9,10]; the current paper explores another mode of heat transfer, convection via internal mixing [8]. We link the latter phenomenon to the possibility of transient vortex-like motions (as shown in Fig. 2 for evidence).

Take the y-axis to be normal to the fault plane, we

use T = T (y, t) and qycond = qycond (y, t) for the temperature and normal conductive heat flux across the gouge. The following heat flow equation is commonly assumed:

shea

r he

atin

gDt

Dt

t

t

grain at time t, with Tg = T(y)

grain at time t+Dt, with Tg <  T(y+Dy)

grain at time t+Dt, with Tg >  T(y+Dy)

s y’

s y’vortex-like motion

y =  b

y =  -b

qycond

qyconv

qycond

qycond

qycondqy

conv

y

T = D0∂2T∂y2

+P (1)

where P represents the rate of heat production, and

where it is usually assumed that the heat flux follows a Fourier law of conduction qycond = −K ∂T /∂y( ) , with K being the effective conductivity. It is tacitly assumed that the thermal diffusivity D0 is not affected by solid motion normal to the fault. In an average global sense this is correct: there cannot be an overall mass flux crossing the rock boundaries. However, one cannot neglect that normal fluctuating velocities may still develop locally within the fault, such as those illustrated experimentally below.

2nd Motivation – Particle Kinematics in Sheared Granular Media

A fascinating property of dense granular flows is the spontaneous development of transient collective grain motions [8,12,13]. In this conference these motions are being confirmed for the first time experimentally in homogeneous plane simple-shear [14]. Specifically, this novel apparatus takes a stadium geometry designed to impose shear deformations as large as desired, in 2D granular disk media. Particles are allowed to circulate underneath two sprockets enforcing a stadium belt motion, as shown in Fig. 2.

FIGURE 2. Stadium Shear Device. (Above) First prototype, installed at The University of Sydney; (Below) Two snapshots of disks’ velocity colored by magnitude, taken from the same run during steady flow [14]. During steady flow, we find that the spatial mean velocity along the y-axis follows a near-linear profile, as expected from homogeneous simple shear; nevertheless, some deviation was observed from the

pure linearity, in an s-shape that we explain in [14] by drawing analogy to eddy viscosity in fluids. Figure 3 shows two snapshots during the steady flow, corresponding to the bottom images of Fig. 2, which reveal vortex-like motions of the fluctuating velocities about the mean s-shape profile.

FIGURE 3. Fluctuating Velocities Illustrating Vortex-Like Motion. These two snapshots correspond to those shown in Fig. 2. Here, it is the magnitude of the fluctuating velocities that determines their arrow color.

In light of the proven emergence of vortex-like structures during plane shear deformations we refer back to Fig. 1, where these structures are illustrated. The figure presents a sketch of how internal mixing of grain temperatures, Tg, can contribute to heat transfer via convection. Grains moving against the overall heat flux become cooler (blue) than the new local layer-mean temperature, T(y+Δy), while those moving with the flux become hotter (red). The net effect is an enhancement of the fault-normal heat transfer, with cooler gouge centre.

Accepting the idea of the vortex-motions, it is therefore clear that Eq. (1) may miss an important convective heat flux qy

conv = Nuqycond , with Nu

representing a ‘granular Nusselt number’ [8]. It is then proposed to correct previous Eq. (2) by:

T = Deff∂2T∂y2

+P , Deff = D0 1+ Nu( ) (2)

where it is shown that Deff, the effective diffusivity

now accounts for both the conductive diffusivity D0 and diffusivity by grain mixing DM = NuD0 .

MECHANISMS OF HEAT TRANSFER

How significant is this previously ignored mode of convective heat transfer through internal mixing? To put it differently, let us recap the two definitions of the granular Nusselt number

Nu =qyconv

qycond , Nu =

Dm

D0

(3)

By answering how big could be, we can illustrate how big the previously neglected heat flux qyconv is relative to the only heat flux thus far

considered, namely conduction flux qycond .

Alternatively, this will also show how big the previously neglected diffusivity by grain mixing Dm may be relative to D0 . To answer these questions, we have developed a Thermal-DEM. The method is briefly presented below, and then employed to measure both qycond and qyconv .

Thermal-DEM

The conventional DEM was developed to represent the motion and interactions of grains in granular materials, and nowadays it is likely the most frequently used research tool in our field. An extension of this method was proposed by [5,7] that enables the transfer of heat along grain contacts by a simple Fourier law of conduction (Fig. 4).

FIGURE 4. Schematics of Thermal-DEM. (a) Grain-to-grain heat flow is governed by contact area, and thus by the force-displacement contact law; (b) The heat flow into a single particle depends on the mechanical interactions with all its neighbors; (c) The overall effective conductivity is therefore a function of the contact force network [7].

Contacts between grains transmit forces and conduct heat. The contact force-displacement law determines the contact size a, as known from usual DEM. Here we employ the Hertz contact law for spherical elastic frictional grains of diameter d, mass m, Young’s modulus E, and friction coefficient µ. Thermally speaking, the particles are characterized by intrinsic conductivity ks, intrinsic heat capacity c, and intrinsic density ρ.

Simple steps may follow to turn a usual DEM into a Thermal-DEM. First, we employ a simple model for the heat flux φij from grain i to grain j with temperatures Ti and Tj at their centers [15]; this simple model proves to be reasonable for collision-dominated flows, and even more so in dense flows with sustained contacts [16,17]. Due to the constriction of the heat flux at the narrow contact, most of the temperature gradient is contained within a region of typical size a near the

contact (aij for contact between particle i and j), while the temperature outside this region is almost homogeneous. In this model the heat flux from particle i to particle j is given as [15]:

φij = −2aks Tj −Ti( ) (4)

Therefore, the effective conductance between two grains, aks, is coupled to the mechanical part of the DEM through a. In other words, the heat flux depends on the normal contact force, as illustrated by Fig. 4(a). The overall heat budget of a single grain temperature depends on the fluxes from all its contacts. For grain i, the rate of temperature change can be calculated by summing all the fluxes from neighboring particles j’s, and normalizing appropriately by the grain’s specific heat and mass:

Ti =1cmi

φijj∑ (5)

Model Parameters

Earthquakes, landslides, and general granular lubricators involve a wide range of grain properties and loading intensities [8-11]. The set of relevant dimensionless numbers describing the physical properties of the grains is then given by their typical diameter d, mass m, mass density ρ, Young’s modulus E [Pa], bulk conductivity ks [W m-1 K-1], and bulk heat capacity c [J kg-1 K-1].

FIGURE 5. Elementary Thermal-DEM Model.

(Left) Schematics of fault gouge located at depth Z, with thickness h, ambient temperature θ0, with v being the relative velocity between rock-mass plates; (Right) The Thermal-DEM model of a Representative Volume in the gouge, undergoing heat transfer from T1 to T0 over the representative thickness w < h.

The loading conditions can generally be described

by a normal stress σyy, and a shear strain rate γ . For example, the problem and corresponding Thermal-DEM model for the earthquake fault gouge problem is

Nu

November 3, 2011 17:58 Philosophical Magazine PT˙PHM

2

that allow the transition from unstable to stable flow in crustal rocks. An alter-native model to interpret the association of pseudotachylytes with mylonites isdownward propagation of seismic ruptures in the lower crust [20]. Here we developa micromechanical approach to construct a general phase diagram dependent onthe characteristic times for melting, thermal di↵usion, and loading rates, at thisstage without accounting for crystal-plastic deformations that possibly dominatein deeper faults [21].The development of melt associated with pseudotachylytes can be lead to critical

weakening factors. Such factors have been studied at the scale of laboratory tests.Spray [22] showed in experiments performed on solid granite that grain comminu-tion is precursor to friction melting. Through experiments with solidification ofmelted asperity contacts prior to bulk melting, Tsutsumi and Shimamoto [23] pro-posed that the initial peak in friction was due to welding of the asperities. Recently,Di Toro et al. [8] carried out experiments on solid rocks to measure rock friction inthe presence of melts under seismic slip conditions. Using theoretical studies withthe consideration of the rate and state friction law, fault weakening was associatedwith either flash heating involving localized asperity melts [5] and global melting[24, 25]. Finally, numerical models using the discrete element method (DEM) havebeen used to study fault localizations with a focus given to comminution [26, 27],stick-slip driven by local force chain buckling between the grains [28], and thethermal convection arising from the dispersed granular motions and temperatureswithin the fault [29]. The method of Rognon and Einav [29] is specifically rele-vant to the current paper, in that it accounts for heat transfer between grains.However, this work did not consider aspects of heat generation and subsequentmelting processes, which are critical to pseudotachylyte rocks. Here this methodis extended with added details accounting for the melting of the grain surfaces,melt viscosity, solidification of melts, and deboning of the solidified melts. In thisMeltable-DEM model, heat generation comes from intergranular viscous dampingand most importantly from intergranular friction.

2. Dimensionless numbers

We start by identifying the key parameters for idealized fault model accounting forthermal di↵usion and mechanical stick-slip dynamics. The fault under considerationis located at a depth H, with a fault core thickness h, as shown in Fig. 1(a).The mechanical boundary to the fault layer is being imposed via a spring-dashpot

Figure 1. Model schematics: (a) rock mass with a fault located at depth H, with thickness h, and ambienttemperature ✓0; (b) boundary conditions for the Meltable-DEM with the spring-dashpot system represent-ing the intact rock above the fault, d denoting the average grain diameter, �yy being the e↵ective verticalstress, qb denoting the heat flux di↵used to the environment and KD, ⌘D and vD referring to the earthcrust parameters including the e↵ective shear sti↵ness, the damping dashpot constant and the draggingvelocity, respectively; (c) grains with molten layers.

σyy!≈!ρgZ!≈!v/h!

w/d ≈ 20!

Z!

depicted in Fig. 5. In this case, as shown in this figure, the normal stress could be evaluated by the weight of a unit vertical column of height Z above the fault, while the shear strain rate can be determined by assuming, for simplicity, a uniform shear strain rate across the fault thickness h with a relative dragging velocity v.

The simulations involve a fully periodic 3D cell with Lees-Edwards boundary conditions for grain motion. When grains cross the vertical cell faces of the cells, they maintain their temperature. However, upon crossing the horizontal faces of the cell, the temperatures swap from T0 to T1, or vise versa, thus maintaining a constant temperature gradient condition in the y direction.

Typical Times and Dimensionless Numbers

Based on the problem parameters above it is possible to detect four fundamental time scales: the shear time t γ = γ −1 , the inertial time ti = m / (dσ yy ) , the

collision time tc = m / (dE) , and a newly defined thermal time tth =mc / (dks ) . Therefore, only three independent dimensionless numbers may describe our system. Two of them have been broadly used to express constitutive laws for granular flow: the inertial number I = ti / t γ describing the shear rate, and the stiffness number κ = (tc / ti )

2 =σ yy / E reflecting the degree of deformation sustained by the grains [7,18]. We introduce a third dimensionless number, which we term the thermal number [8], defined as

τ =tthti=cks

mσ yy

d (6)

and representing the rate of heat diffusion. Another

useful way to express this thermal number is in terms of the inertial number:

τ =ρcksI!

"#

$

%&⋅ γd 2 (7)

which we will use later to illustrate the effect of the

shear strain rate for simulations where we fixed the inertial number on different values. The possible range of values these three dimensionless numbers (I, κ, and τ) can attain in practice has been compiled by Rognon and Einav [8], to which readers are referred for further details on the model. In particular, in this paper an effort is paid to specify the range of model parameters used in the following simulations to cover the problems of earthquake faults, landslides and lubrication.

Conductive Heat Flux

The numerical experiments begin with the generation of a flow by subjecting a random configuration of grains to a steady shear rate and designated normal stress (Fig. 5). After some time, the flow reaches a steady regime with no shear localization, during which the sample height, the shear stress, and the solid fraction remain constant apart from small fluctuations. The ‘‘heat’’ part of the experiment starts when the steady regime is reached; grain temperatures are initially set according to a constant gradient ∇T0 along y: Ti = yi∇T0. They are then free to evolve, but constrained to a thermal condition where the difference between the temperatures at the top and bottom boundaries remains constant to ensure a constant global temperature gradient of ∇T0. The conductive heat flux density [W m-2] may then be calculated instantaneously during the simulations by averaging the heat fluxes along all the contacts in the representative volume, Ω:

qcond = 1Ω

φcrc

c∑ (8)

Looking at the general tilt of the force chains from

the fault-normal in Fig. 6, and since contact forces control grain-to-grain contact areas a, and the contact area dictates conductance (Eq. 4), it is clear that in steady flow conditions the heat flux vector is stress-dependent and tilted from the fault-normal, and that this heat conductivity is anisotropic. In [7] we have been able to extract a robust relationship expressing these effects.

FIGURE 6. The Rheological Control Over The

Conductive Heat Flux – Force Chains. At that time, before analysing the relative significance of the conductive and convective heat fluxes, and dealing only with the conductive heat flux, we drew one unexpected observation: the component of the system’s conductive heat flux normal to the faultqycond was shown to depend on the heat capacity c. But

according to Eq. (4), local conductive heat fluxes do not depend on c. What is the reason then?

‘Freeze’ Simulation

To answer this unexpected observation, we look at an artificial experiment that involves freezing in space all the grains in a typical sample at the steady flow regime, where we let the heat flow continue towards thermal equilibrium. What we see is the development of temperature clusters, coalesced along the major carrying force chains as shown in Fig. 7. In other words, the final state can never fulfill the linear temperature gradient imposed by the thermal boundaries, such that the overall conductive heat flux depends on the heat capacity, unlike the local conduction law in Eq. (4). But the local law of advection does depend on the heat capacity c. Can this level of coupling hint towards an even bigger effect, related to convection?

FIGURE 7. ‘Freeze’ Simulation. After artificially ‘freezing’ particle positions corresponding to steady flow regime, heat flow is enabled towards thermal equilibrium. (Left) Particles cooler than the layer’s average temperature; (Right) Particles hotter than the layer’s average temperature. Lines show major force chains, over which clustering occurs.

Convective Heat Flux

The overall convective heat flux is simply related to the average of the individual advective heat fluxes. The heat advection of a particle i is simply given by the product of its heat, micTi, and its velocity vi :

φi =micTi

vi (9) The spatial average of these local advective fluxes

within a granular system of representative volume Ω gives the global system’s convective heat flux:

qconv = 1Ω

micTivi

i∑ (10)

In the current system there is no net mass flux

along y, which is why the heat convection cannot be associated with some advection along the mean flow

velocity. Instead, the convection comes directly from both the velocity and temperature fluctuations [8]. For that purpose, we denote the fluctuation parts of the temperatures and velocities from the corresponding mean field by Ti and vi . In the current simulations, the temperature and velocity of a given grain can then be expressed by Ti =∇T0yi + Ti and vy,i = 0+ vi , such that:

qconv = mcΩ

Ti vii∑ (11)

Therefore, the convective heat flux measured from

the current simulations depends only on the fluctuating components of the temperatures and velocities.

10,000

How significant is the contribution of the fault-normal component of the convective heat flux? To answer this question Figure 8 compares the heat flux by convection to that of conduction, for various thermal numbers, by altering the shear rates but fixing the inertial number.

For extreme earthquake conditions we find Nusselt numbers as high as 10,000. In this case, convection within the fault gouge simply ‘eats’ the conductive flux thus far considered. In our view, this astonishing finding of such a high Nusselt number is a reason for further studies.

FIGURE 8. 10,000. For high thermal numbers τ the convective heat flux can be up to 10,000 times the conductive heat flux . See more details in Rognon & Einav [8].

ELEMENTS OF DIFFUSIVITY

A 10,000 factor was certainly higher than what we originally expected before embarking on the study of

105!104!103!102!10!1!10-1!10-1!

103 !

102!

10 !

102! 1 !

10-1!

10-2!

10-3!

10-4!

10-5!

τ =ρcksI!

"#

$

%&⋅ γd 2

Nu = 10,000!

qyconv

qycond

heat convection using the Thermal-DEM. Needless to say that while we were very happy to stumble on this finding with the Thermal-DEM, we became a bit nervous of the full validity of this Nu=10,000 in reality. Therefore, we then moved on to look for a further validation by developing an alternative method to the DEM.

Towards this aim we recently developed and used [19] a numerical method, which we termed the tracer method as described below (see schematics in Fig. 9). The system comprises a periodic array of rotational vortices. We analysed the transfer properties obtained for various vortex sizes R, rotational frequencies f, and vortex lifetimes l, for cortices centred in one spot before randomly moving to another.

FIGURE 9. The Tracer Method. Motion of tracers is comprised of Brownian motion superposed to advection driven by a rotational vortex of radius R, rotation frequency f, and a lifetime l.

In systems where both diffusion and advection

occurs simultaneously, the local evolution of temperature, satisfies the advection-diffusion equation (i.e., corresponding to Eq. (1) without heat production, but with local advection). Assuming an incompressible flow and molecular/conductive diffusivity that is homogeneous, this equation reads:

T = D0∇

2T − v ⋅∇T (12) It is of course possible to solve the above equation by employing continuum mechanics computational tools, but these are known to be delicate for such problems. The tracer method, however, helps us solve this equation with calm. Here each tracer is advected by the rotational vortex motion (represented by the 2nd term on the r.h.s. of Eq. 12), superposed by a random velocity with a designated norm v0 and an orientation that changes randomly after each time step dt. The

norm of the random velocity is set to achieve the desired intrinsic diffusivity, D0, using v0 = 2 D0 / dt (and thus describing the 1st term in the r.h.s. of Eq. 12). The effective diffusivity of the system, Deff, is then measured in terms of the mean square displacement of the tracers' trajectories.

Simulations With the Tracer Method

In the absence of vortices the transfer is solely governed by conduction (Deff = D0), which corresponds to Nu = 0.   For such systems, the results from the method were validated against closed form analytic solutions. In the following we study how the effective system’s diffusivity depends on the four system parameters (D0, R, f, and l).

All simulations comprise 10,000 tracers initially randomly displaced in a square, two-dimensional domain of size 2R, with R being the radius of the vortex. The domain is periodic in all directions; tracers that leave the domain through one boundary re-enter it through the opposite boundary. In order to illustrate the mixing, we further employ constant temperature/colour boundaries consistent with the Thermal-DEM simulations before. See the bottom images in the typical example of Fig. 10.

FIGURE 10. Example of Rapidly Rotating

Vortices with Short Transient Lifetime (f = 1 s-1, l = 0.5s, D0 = 10-6 m2/s). (Above) Flow velocity field; (Below) Scalar field illustrating tracers’ positions at different times.

Next, Fig. 11. collates results from a large

parametric study, in terms of two dimensionless diffusivities, diffusivity ratio Deff / D0, and the Péclet number

Pe = R2 f /D0 (13) When the Péclet number is less than one, there is no transfer enhancement, Nu = 0 (Deff = D0). For higher Péclet numbers, the vortices enhance transfers;

If crossing, reappear at the bottom!

If cr

ossi

ng, r

eapp

ear i

n th

e ri

ght! If crossing, reappear in the left!

If crossing, reappear at the bottom!

the Nusselt number increases as a power law of the Péclet number, Deff/D0 = Pen. The value of the power n depends on the vortex lifetime. This power ranges from a lower bound of 0.5 for stationary vortices of lifetime infinity, to an upper bound of 1 for vortices of lifetimes shorter than half a rotation. This difference reflects internal mechanisms involving the coupling of diffusion and advection.

FIGURE 11. Effective Diffusivity in Terms of

Péclet Number Using the Tracer Method. According to this method, and considering earthquake faults where the lifetime l is characteristically short, we find Nu = 10,000, which is consistent with the separate observation in Fig. 8.

BACK TO FAULTS

According to the experimental observations in Fig. 3, the idea of employing rotational vortices to idealise fault gouge motion is sensible. It may also be reasonable to assume rotational frequencies at the order of fault shear rates, and to consider very short lifetimes in the gouge layer, given the transient nature observed during the experiments. Therefore, to a first order the findings from the tracer method may be applicable to understand diffusion within earthquake fault gouges using Deff/D0 = Pe, i.e. the effective diffusivity may be calculated by Deff = R2f, which is independent on the conductive diffusivity D0.

Next, note that the number of grains across the Stadium Shear experiments in Fig. 2 is around 20 to 25, which is close to the number of median grain sizes across faults [10]. It is therefore reasonable to expect vortices at the order of the fault thickness R ~ h/2, as seen in the experimental Fig. 3. Similarly, with reference to those experiments, the rotational frequency may take the order of the fault’s nominal shear strain rate ( γ = vf/h = fault’s seismic velocity divided by its thickness). With this first order mapping

in mind, the effective diffusivity scales with γd 2 , which has been suggested by Eq. (7) for constant I, as enforced for the Thermal-DEM simulations. Also, to a first order, the mapping of the tracer method results to earthquake faults gives the following Nusselt number:

Nu =max 0 , vf h4D0

−1"

#$

%

&' (14)

Literature in earthquakes research (e.g., see [10,11] and references therein) motivates seismic slip velocities vf up to 0.1 to 2 m/s, and ultracataclastic fault thicknesses anywhere from 1 to 10 mm (where all the shearing occurs). Putting these together, we find Nusselt numbers much larger than zero, which may approach, again, 10,000.

CONCLUSIONS

This paper reviewed some of latest work of our group in the area of heat transfer through granular media. The most striking result was the finding of extraordinarily high convective heat fluxes, even in the absence of bulk mass advection. It was demonstrated that this phenomenon originates from the fluctuating nature of the internal temperatures and velocities, the latter being typified experimentally by vortex-like motions. Two entirely different methods have been used, the Thermal-DEM and the Tracer Method, both suggesting agreeable granular Nusselt numbers that in earthquake fault gouges may be as high as 10,000. The implications of this finding cannot be overstated. For example, it can be shown by solving Eq. (2) with Nu = 10,000 that the maximum temperatures within earthquake faults might be half those frequently predicted in geophysics using Eq. (1), for the same amount of heat produced. This, in return, can carry further implications on our understanding of phase transition processes, e.g., melting and chemical reactions. Our recently developed Thermal-DEM model with melting features [11] has illustrated how competing comminution, melting and solidification explain the reoccurrence of earthquakes. That being said, in such Meltable-DEM simulations granular heat convection is being considered automatically, as part of the simulations. However, applying continuum methods without including high Nusselt numbers could miss finding such reoccurrences, and would instead predict slower motions along viscous melts.

ACKNOWLEDGMENTS

The authors would like to thanks continuous discussions with B Marks, and B Metzger. IE and PR

Pe = R2 f / D0!

Def

f / D

0!

Nu = 10,000!

would like to thank the financial support by the Australian Research Council (ARC), through grant DP1096958, which helped us initiating the work in this area. IE and YG acknowledge current support by ARC through DP120104926, DP130101291, and DP130101639.

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