Extended-Magnetohydrodynamics in Under-Dense Plasmas C

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Extended-Magnetohydrodynamics in Under-dense Plasmas C. A. Walsh,1, a) J. P. Chittenden,1 D. W. Hill,2 and C. Ridgers2 1)Blackett Laboratory, Imperial College, London, United Kingdom, SW7 2AZ 2)York Plasma Institute, Genesis 1&2, York Science Park, University of York, Church Lane, Heslington, United Kingdom, YO10 5DQ (Dated: 18 December 2019) Extended-magnetohydrodynamics transports magnetic flux and electron energy in high-energy-density experiments, but individual transport effects remain unobserved experimentally. Two factors are responsible in defining the transport: electron temperature and electron current. Each electron energy transport term has a direct analogue in magnetic flux transport. To measure the thermally-driven transport of magnetic flux and electron energy, a simple experimental configuration is explored computationally using a laser-heated pre-magnetised under-dense plasma. Changes to the laser heating profile precipitate clear diagnostic signatures from the Nernst, cross-gradient-Nernst, anisotropic conduction and Righi-Leduc heat-flow. With a wide operating parameter range, this configuration can be used in both small and large scale facilities to bench- mark MHD and kinetic transport in collisional/semi-collisional, local/non-local and magnetised/unmagnetised regimes.

I. INTRODUCTION ICF, extended-MHD affects laboratory astrophysics experiments, such as the measurement of magnetic fields generated around laser-foil interactions8,9 and are antic- Extended-Magnetohydrodynamics (extended-MHD) is 10 a theoretical framework used to evaluate the trans- ipated to also affect 2-spot magnetic reconnection . port of energy and magnetic flux in a plasma. The While the impact of extended-MHD is widespread in electrons typically move at a higher speed than the laboratory HEDP, many of the effects are yet to be mea- ions, dominating the transport. Additional terms in- sured directly. Without experimental verification of the troduced by extended-MHD above resistive-MHD in- extended-MHD model, uncertainties remain in the design clude temperature-gradient-driven transport (such as the and analysis of ICF and laboratory astrophysics stud- Nernst term moving magnetic fields down electron tem- ies. A notable exception is the Biermann battery term, perature gradients) and electric-current-driven transport with time-dependent magnetic field generation measured around a laser-foil interaction8, in addition to systems ex- (such as the Hall term moving magnetic fields with the 11 flow of charge). For each magnetic transport term there hibiting Rayleigh-Taylor instability growth . In both of is a corresponding transport of electron energy; for exam- these examples the Nernst term significantly alters the ple, the analogue of Nernst in energy transport is thermal magnetic field distribution, but the experimental com- conduction, which moves thermal energy down tempera- plexity prohibited direct inference of a Nernst velocity ture gradients. to compare with simulations. Other key properties of Extended-MHD terms are anticipated to be important Nernst advection, such as suppression of the effect at in a wide range of high energy-density physics (HEDP) large magnetisations, also remain unverified. experiments. The Nernst effect limits the performance This paper uses simulations to investigate experi- of magnetised liner inertial fusion (MagLIF) implosions mental configurations where thermally-driven extended- by demagnetising the pre-heated fuel1. The design of MHD terms (Nernst, cross-gradient-Nernst, anisotropic laser-driven pre-magnetised inertial confinement fusion thermal conduction and Righi-Leduc heat-flow) could be (ICF) targets2,3, also requires the consideration of these measured unambiguously for the first time. An under- additional effects4. Even initially unmagnetised ICF con- dense laser-driven magnetised plasma is used, allowing

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5124144 figurations can be affected by extended-MHD phenom- for the thermally-driven transport to dominate over the ena, with self-generated fields growing through the Bier- hydrodynamic motion. Clear diagnostic signatures are mann battery process. Simulation studies have found sought for each term through simple modifications to the the Nernst (magnetic fields moving down temperature laser profile. A similar set-up has been used before without the diagnosis of magnetic field distribution to mea- gradients) and Righi-Leduc (heat-flow deflected by the 12 magnetic field) terms to change plasma properties in sure non-local heat-flow suppression , with subsequent kinetic simulations suggesting that both Nernst and non- hohlraums (increasing the temperature of the hohlraum 13 gas fill5), direct-drive ablation fronts (changing the per- locality are important . Non-local transport is outside turbation growth6) and at the compressed fusion-fuel the scope of this paper, with the plasma treated using edge (modifying the cooling process7). In addition to an MHD framework. Changes to the experimental set- up can then be used to explore different regimes, such as the transition from extended-MHD to kinetic transport for each of the terms. This paper is organised as follows. The appendix re- a)Electronic mail: [email protected] writes the traditional Braginskii extended-MHD equa- This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. 2

tions into a form that is both physically intuitive and simple to implement into an MHD code. This is then FIG. 1. 2-D cylindrical simulations of the Nernst configura- summarised in section II, with each of the relevant trans- tion at 0.5ns with a 1T applied field. a) electron temperature port terms clearly formed. Section III A then outlines an with Nernst included. b) magnetic field magnitude with the experiment to measure Nernst cavitation of a magnetic Nernst effect included. c) magnetic field magnitude without field in an under-dense plasma, showing simulation re- the Nernst effect included. The magnetic field cavitation is strongly driven by Nernst advection. sults as well as synthetic diagnostics. This set-up pro- vides a baseline configuration, which is modified to explore other extended-MHD terms. By changing the laser thermal conduction perpendicular to magnetic field lines focus, the cross-gradient-Nernst twists the magnetic field, κ is seen here to be associated with the Nernst γ as shown in section III B. Anisotropic thermal conduc- ⊥ ⊥ term. Comparably, the Righi-Leduc heat-flow κ acts to tion is then demonstrated in section III C by using an ∧ move electron energy in the same direction as the cross- applied magnetic field perpendicular to the beam rather gradient-Nernst γ moves magnetic flux. than parallel. Finally, section III D takes the Nernst ex- ∧ In a similar way, the electric-current-driven magnetic periment but uses a non-circular laser spot to elucidate field advection velocity and energy advection velocity can the Righi-Leduc heat-flow in the form of a rotation of the be written: thermal profile.

j j c ⊥ c ˆ II. EXTENDED-MAGNETOHYDRODYNAMICS vjB = −(1 + δ⊥) + δ∧( × b) ene ene (5) Magnetic transport in an extended-MHD plasma is ex- j j j plored in the appendix, beginning with Braginskii’s for- c k c ⊥ c ˆ 14 vjUe = − (1 + βk) −(1 + β⊥) + β∧( × b) mulation of Ohm’s Law . The following equation is de- ene ene ene rived for the change of magnetic ﬁeld strength: (6)

In these simplified forms, a non-dimensional number ∂B α ∇P k e can be used to assess if temperature gradients or elec- = −∇× 2 2 ∇×B+∇×(vB ×B)+∇× (1) ∂t µ0e ne ene tric currents dominate the transport. Assuming that the There are only 3 terms: one for magnetic field diffu- magnetic field varies over the same length-scale as the sion, one for advection of the magnetic field at velocity temperature: vB and the Biermann Battery term as the only source of magnetic flux. The magnetic field advection velocity vB 5 c c 2 is given by: |v | γ τeTeeneµ0 γ Te Ξ = N⊥ = ⊥ ∼ ⊥ (7) j c vjB⊥ |B| me |B| ˆ c δ∧ ˆ vB = v −γ⊥∇Te −γ∧(b×∇Te)− (1+δ⊥)+ (j ×b) ene ene Where the δc factor is dropped as it is only a small (2) ⊥ correction to the collisionless current-driven magnitude i.e. the advection is based on the bulk plasma velocity, (Hall). In a regime where thermally-driven terms domi- as well as the electron temperature gradient and electric nate (Ξ ≫ 1) both the thermally-driven magnetic trans- current. The γ term is called the Nernst term, which ⊥ port and thermally-driven electron energy transport will moves magnetic field down temperature gradients. The be significant. In a regime where current-driven terms γ term is then the cross-gradient-Nernst, moving the ∧ dominate (Ξ ≪ 1) both current-driven magnetic trans- field perpendicular to both the temperature gradient and port and current-driven electron energy transport will be

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5124144 the magnetic field. Both γ and γ simply decrease with ⊥ ∧ significant. magnetisation ω τ (see figure 8). e e The appendix studies more closely the different mag- The electron temperature gradient and electric current netic field and electron energy transport terms, compar- also cause transport of electron energy. To make the ing the coefficients and outlining simply how to include physical connection between magnetic and energy trans- these terms in an extended-MHD code. port clear, the magnetic field advection velocity due to electron temperature gradients are shown alongside the heat-flow due to electron temperature gradients: III. THERMALLY-DRIVEN TRANSPORT EXPERIMENTS ˆ vN = −γ⊥∇⊥Te − γ∧b × ∇Te (3) This section outlines a set of simple experiments to q = − κ ∇ Te −κ⊥∇⊥Te − κ∧ˆb × ∇Te (4) κ k k verify the thermally-driven magnetic and electron energy where there is no field advection parallel to the field transport in equation 3 and 4. Section III A outlines the lines, as this does not change the magnetic flux. The baseline configuration for measuring the Nernst velocity, This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. 3

which is then modified in the subsequent sections in order to allow the other terms to be measurable. FIG. 2. Time-dependent synthetic proton radiographs for the The parameters used in this publication are: Nernst configuration with an initial magnetic field of 1T. The 3 Nernst velocity could be measured for the first time using this • Gas density ρ0 = 0.065kg/m technique. While the distance between the edges (λ) gives the • Gas composition = deuterium cavitation rate, there is also information within the individual edges. The background magnetic field deflects protons to- • Laser energy = 50J with wavelength of 1.055µm wards the right of this image. The distance has been re-scaled to the interaction region using the set-up magnification. • Laser spot spatial profile = Gaussian with standard deviation of 75µm

• Laser time profile = 0.5ns with a 0.1ns linear rise FIG. 3. Synthetic proton radiographs for the Nernst configuration at 0.5ns for 1T, 5T and 10T applied magnetic fields. • Applied magnetic field magnitude B0 = 0 − 10T Nernst suppression for higher magnetisations could be mea- These parameters are realisable in a relatively small-scale sured for the first time using this technique. The background facility and give a reasonable signal for the extended- magnetic field deflects protons towards the right of this image. MHD terms considered here. There is ample room for The distance has been re-scaled to the interaction region us- moving into different regions of parameter space, thereby ing the set-up magnification and each field strength has been re-centred for comparison purposes. modifying the transport rates and providing additional tests against theory. Lower laser energies (< 1J) are also expected to give appreciable signals. The ionised electron density relative to the laser crit- of laser propagation. ical density is small ne/ncr = 0.0190, i.e. the gas is Simulations are conducted in 2-D cylindrical geome- under-dense. The low density allows for the laser to pass try (r − z). The laser and magnetic field are applied through without depositing significant amounts of en- in the direction −z, which assumes the laser is uniform ergy; laser energy coupling into the simulation domains azimuthally (in θ). shown here are between 1-5% over 1.5mm. Increasing The laser-heated plasma becomes more transparent to the coupling (e.g. using a higher density gas) would al- the laser as the temperature increases, resulting in rel- low for even lower laser energies, although this would atively uniform heating along the laser axis. Figure 1 decrease temperature uniformity along the laser axis and shows the electron temperature at 0.5ns using the base- increase laser refraction effects. line parameters described in section III and a 1T applied For the parameters chosen, thermally-driven transport field. terms dominate, i.e. Ξ is large (from equation 7). Thermal conduction transports heat radially away The extended-MHD code Gorgon7,15 is used to sim- from the laser into the cold gas. Nernst advection, which ulate the configurations, with distinct diagnostic signa- is analogous to magnetic field moving with the heat-flow, tures anticipated for each of the effects. Gorgon is a 2- reduces the field intensity in the laser-heated region. The temperature Eulerian code with laser ray tracing and ab- demagnetising effect of the Nernst term is compounded sorption by inverse Bremsstrahlung. The magnetic trans- by the increase in Nernst velocity as magnetisation de- port is treated as in equation 1, using operator splitting creases, further enhancing the demagnetisation rate. The between the advection, diffusion and generation compo- magnetic field intensity is plotted in figure 1. In the nents. The electron heat-flow is fully anisotropic, as in regime shown, near complete magnetic cavitation is ob- equation 34, using a centred-symmetric algorithm16. For served. The thermal wave corresponds with a discrete the configurations in this paper the spatial resolution is step in magnetic field intensity. The field is compressed 0.5µm. at the edge of the heat front and can resistively diffuse Synthetic proton deflectometry is used here to diag- away depending on whether or not the plasma is ini- PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5124144 nose magnetic field transport. A D3He exploding pusher tialised as cold in the simulations. is taken as the source, producing mono-energetic pro- Magnetic fields also move with the bulk plasma motion tons at 14.7MeV17. Target-normal sheath acceleration (so-called frozen-in-flow). As the laser-heated region ex- (TNSA)18,19 could also be used. The source offset is pands, the magnetic field strength decreases. A definitive taken as 6.3mm with the image plate 120mm from the measure of the Nernst effect is simpler if the Nernst ve- interaction region. The source is assumed to be infinitely locity is much larger than the hydrodynamic expansion. small, thereby reducing blurring. Electric field contribu- Figure 1 quantifies the relative impact of Nernst advec- tions to the proton images are estimated to be small. tion by also showing the magnetic field profile for a simulation without the Nernst term included. In this case, the Nernst term is clearly dominating over the hydrody- A. Nernst namic motion. For the case without Nernst, the magnetic field strength |B| = ρB0/ρ0. Significant hydrodynamic A uniform under-dense (ne ≪ ncrit) plasma is used, motion does not strictly prohibit a measurement of the with a uniform magnetic field applied along the direction Nernst velocity; a simultaneous density measurement can This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. 4

be used to infer that the magnetic field is not frozen into generally using a simple comparison of characteristic the flow. velocities. The Nernst velocity (from equation 20) is c Experimentally, the key measurements to constrain the −γ⊥τe∇Te/me. Unlike the near-instantaneous Nernst Nernst effect involve measuring the field cavitation rate velocity, the bulk plasma has inertia and takes time to be and the decrease in cavitation with a larger applied field. accelerated to a significant velocity by the pressure im- The field cavitation is diagnosed synthetically using pro- balance. From simple hydrodynamics this velocity can ton radiography with protons traversing the system per- be taken as ∆t∇P/ρ. By using the fact that the density pendicular to the initial magnetic field axis. Figure 2 is initially uniform (overestimating the impact of hydro- shows proton deflectometry evolution with time. An av- dynamics) the relative magnitude of hydrodynamics to erage over the length of z in figure 1 is used. While Nernst advection can be estimated: all protons are deflected by the background magnetic field, the protons passing through the cavity are deflected v ∆t Zm 1 less. This creates distinct regions with higher and lower Hydro = e (9) c proton counts at the cavity edge. The separation dis- |vN⊥| τe mi γ⊥ tance of these regions, λ, indicates the cavitation region Zm /m varies little between the choice of gas, al- size, which increases with time. The Nernst velocity can e i though deuterium gives larger Nernst velocities than hy- then be inferred through measurement of ∂λ/∂t. Struc- drogen. γc is larger (factor of 10) for high-Z material ture in the proton image at each of the cavitation edges ⊥ (see figure 8) and decreases with magnetisation; the more could then provide more information on the compressed magnetised the plasma, the more hydrodynamic motion field profile, although this measurement requires higher will play a role (until magnetic pressure becomes impor- resolution. Further information can be extracted from tant). ∆t/τ is a measure of the experiment collisionality, the proton radiographs by using a grid and assuming e i.e how many electron-ion collisions a typical electron un- small proton deflections20; cylindrical symmetry allows dergoes in the time since the laser was switched on ∆t. the path-integrated magnetic field to be calculated. Therefore, the more collisional (and as a consequence the Combining proton radiographs with Thomson scatter- more MHD-like) the experiment, the more hydrodynamic ing and interferometry diagnostics can be used to con- motion will also play a role in de-magnetising the laser- strain the electron temperature and density; experimen- heated plasma. tal and theoretical Nernst advection rates can then be Further variations in the experiment can be used to compared without the need to estimate plasma proper- test theoretical predictions. Increasing the laser energy ties from simulations. increases the temperature gradients, which is expected The separation distance on the proton radiograph, λ, to increase the cavitation rate. Using a higher Z plasma can be related to the cavitation radius r by assuming a c can increase the Nernst coefficient. In the regime shown uniform completely cavitated cylinder (i.e. fully demag- here, increasing the initial density increases the Nernst netised from initial magnetic field strength B to 0T ) 0 cavitation rate, primarily because the plasma magnetisa- probed by protons of initial velocity v undergoing only p tion lowers. small deflections, with the distance from the cylinder to The transition from MHD-like transport to kinetic-like image plate as D: transport could be probed by decreasing the laser spot size while keeping the intensity constant, thereby lower- eB02rcD ing the spatial scale relative to the mean-free path. The λ = (8) experiment times should be kept constant to maintain mpvp the same balance of Nernst to hydrodynamics (equation For an increased background magnetic field, the above 9). While the cavitation rate will increase due to larger relation predicts that the offset distance increases for a temperature gradients, the proton images only measure given cavitation radius. Synthetic proton radiographs at the path-integrated change in field strength. A proton PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5124144 0.5ns for varying magnetic field strengths are shown in image scaled in size with the laser radius is expected, figure 3. Instead of λ increasing, the offset distance de- with any differences due to non-local effects. creases due to Nernst suppression by magnetisation. The proton deposition structure at each cavity edge is more apparent for larger applied magnetic fields, allowing for B. Cross-Gradient-Nernst a finer resolution of the magnetic field profile. In par- ticular, an experimental measurement of the non-local The cross-gradient-Nernst is a magnetic transport ve- ’pre-Nernst’ may be possible21 in these regions. If the locity acting perpendicular to both the temperature gra- gas outside of the heat front remains cold and unionised, dient and the magnetic field (equation 21). Figure 8 then the magnetic field can diffuse away, removing the demonstrates that the cross-gradient-Nernst velocity is small-scale structure within the cavity edges in the pro- large whenever the Nernst velocity is large. However, this ton radiographs. does not necessarily mean that the cross-gradient-Nernst An assessment of whether Nernst or hydrodynamic ad- significantly alters the magnetic field profile. In the pre- vection demagnetises the plasma can be explored more vious Nernst configuration, the cross-gradient-Nernst al- This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. 5

ters the field profile indistinguishably, as the plasma profile is effectively 1-D (not changing in θ or along the field FIG. 4. Electron temperature and axial/azimuthal mag- lines). netic field components for the cross-gradient-Nernst config- In this section the laser focus is changed such that the uration. Along the labelled magnetic field line the cross- beam decreases with intensity along its path, giving a gradient-Nernst moves out of the page at low Z and is zero at high Z, resulting in a twisted field component. hotter plasma further along the cylindrical axis (high z) and cooler plasma occupying a larger radius at low z. One standard deviation from the centre of the laser spot, the rays are at an angle of 3.6 degrees to the axis. The FIG. 5. Synthetic proton image at 0.5ns for the cross- z electron temperature profile at 0.5ns is shown in figure gradient-Nernst configuration using a proton source at + . The images have been re-scaled to the interaction region us- 4, with labels for the Nernst and cross-gradient-Nernst ing the magnification. directions. As before, the Nernst effect moves magnetic field radially outwards. The cross-gradient-Nernst, however, acts independently of the cavitated field by probing with pro- perpendicular to the 2-D simulation plane, moving the tons along the laser axis. Figure 5 shows the synthetic field azimuthally. The equation for the magnetic field proton radiograph with source situated at +z. The rel- advection velocity vN∧ affects the magnetic field profile ative path-integrated strengths of the positive/negative can be expanded from the form in equation 1 to: Bθ could be determined by looking at the outer/inner portions of the radiographs. ∂B +(vN∧ ·∇)B = −B(∇·vN∧)+(B·∇)vN∧ (10) ∂t N∧ C. Anisotropic Thermal Conduction

The term on the left-hand side is now the convective A configuration is now explored to assess the transi- derivative, i.e. how the magnetic field intensity changes tion from unmagnetised thermal conduction to highly- moving with the flow. The first term on the right-hand magnetised conduction. The same conditions as the side is then the field being compressed/rarefied by con- Nernst configuration are used (uniform under-dense verging/diverging flows. Both the convective and com- laser-heated plasma with a background magnetic field), pressive terms are zero for the cross-gradient-Nernst in but with the magnetic field applied perpendicular to laser these 2-D simulations, as both the field and velocities are propagation. In Cartesian geometry, the laser is along uniform around θ. −z and the magnetic field is in the x direction. The The final term in equation 10 represents the twisting of radial thermal conduction wave from the laser core is the magnetic field and is non-zero here. To demonstrate anisotropic, with thermal conduction in y suppressed due this, a magnetic field line is considered in isolation, pass- to magnetisation and heat-flow in x uninhibited. ing between the heated (low z in figure 4) and unheated Figure 6 shows 2-D x − y slices of the electron temper- regions (large z). At large z, the cross-gradient-Nernst ature profile at 0.5ns for various applied field strengths. velocity acting on this field line is zero, as there is no tem- The laser-heated region is hotter for the 5T case (340eV perature gradient. At low z, cross-gradient-Nernst acts compared with 260eV for 1T), as thermal cooling along to advect the field line azimuthally. Between these two re- y is reduced. The peak magnetisation at this time is only gions there is necessarily a twist in the field line, creating ωeτe ≈ 1 for the 1T case, corresponding to thermal con- a θ component. In reality the dependence of the cross- ductivity suppression κ⊥/κk ≈ 1/3. For the 5T case the gradient-Nernst velocity on magnetisation in the twisted peak magnetisation is ωeτe > 10, with κ⊥/κk ≈ 0.01. region results in both a field component into and out of While magnetisation does not directly change the ther- the page at different regions. mal conductivity along field lines (κ ), the hotter core PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5124144 k Figure 4 shows the Bz and Bθ magnetic field compo- plasma results in greater heat-flows along x for higher nents at 0.5ns. The Nernst advection still occurs, low- magnetisations, increasing the heating radius. At 0.5ns ering the axial field strength in the laser-heated region. the aspect ratio of the outer heat front is only 1.04 for A Bθ field develops, up to a peak of 0.8T. While this is 1T, increasing to 1.51 for 5T. small compared to the 5T applied field, the twisting oc- The anisotropic temperature profile can be diagnosed curs in the same region depleted of axial field by regular through self-emission (if the density remains relatively Nernst advection, resulting in a peak twisting angle of 15 unperturbed) or by Thomson scattering. The relative degrees. extent of the heated region along y and x (along with The cross-gradient-Nernst velocity changes sign when a measurement of the peak temperature) will help con- the magnetic field is applied in the opposite direction, strain the analytic form for thermal conductivity mag- but the resulting Bθ component is independent of the netisation. initial applied magnetic field. As with the Nernst configuration, a clear diagnostic While Nernst and cross-gradient-Nernst are significant signature here is aided by lack of hydrodynamic motion. in the same configuration, it is possible to measure Bθ A relation similar to 9 can be derived by comparing the This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. 6

FIG. 6. Electron temperature profiles at 0.5ns with laser FIG. 7. Electron temperature profile in x − y with laser and propagating into the page and a magnetic field applied along magnetic field (3T) applied out of the plane. A square laser x. As the applied magnetic field is increased, the heat-flow spot is used to enforce a temperature variation around θ, anisotropy increases. which allows the Righi-Leduc heat-flow to modify the temperature distribution. The standard deviation of the square laser spot is shown with a dashed white line. hydrodynamic speed to the thermal conduction speed (rearranging perpendicular thermal conduction into the c propagating out of the page. The square laser spot edges form ∂Ue/∂t = ∇ · (Uevκ⊥), where vκ⊥ = κ⊥τe∇Te/me): are aligned with x and y, with ray powers set to a Gaus- sian of the maximum between x and y. The Righi-Leduc v ∆t Zm 1 heat-flow rotates the electron energy profile. The rotated Hydro = e (11) c profile could be measured using self-emission or Thom- |vκ⊥| τe mi κ⊥ son scattering. Reversal of the applied magnetic field This is a variant on the Péclet number, relating advec- direction swaps the rotation direction. tive and diffusive effects. It is no surprise that equations To further understand the rotation, an analytic form 9 and 11 give near-identical constraints on the experi- for the change in electron energy due to Righi-Leduc can ment for hydrodynamics to be of secondary significance, be derived. Assuming magnetic field direction is constant as the thermal conduction and Nernst are inherently re- (e.g. ˆb = (0, 0, 1) here) and with Z = constant, the rate lated. In fact, it is important to note that the Nernst of change of electron energy density is given by22:

effect will play a significant role in this configuration. If c Nernst advection is larger than simulations anticipate, ∂Ue ∂κ∧ neTeτe = ˆb · ∇Te × ∇ωeτe (12) ∂t κ∧ ∂ω τ m the plasma will become less magnetised, allowing larger h i e e e perpendicular thermal conductivities. This could be er- For initially uniform density and field, frozen-in-flow roneously interpreted as a different dependence of κc on ⊥ gives |B|/n = constant. Therefore, if the mag- ω τ . Therefore, further proton measurements in this e e e netic field is frozen into the flow (i.e. ignoring Nernst configuration are valuable to avoid such misunderstand- and cross-gradient-Nernst), this relation is zero (as ings. The Nernst configuration in section III A is pre- ω τ ∼ T 3/2|B|/n ). In reality, the movement of mag- ferred for diagnosing magnetic transport, as it avoids the e e e netic field relative to the bulk plasma allows gradi- complications of anisotropy by keeping the plasma uni- ents in ω τ . These magnetisation gradients then re- form around the laser axis. e e duce/enhance the Righi-Leduc heat-flow and result in Kinetic effects will also play a role in this configura- accumulation/rarefaction of electron energy. The re- tion. In the Nernst experiment heat-flow can be kept sponse of Righi-Leduc heat-flow to the magnetisation de- in the MHD regime through magnetisation (as measured pends on the sign of ∂κc /∂ω τ , with ∂κc /∂ω τ > 0 for under similar experimental conditions12). Here the heat- ∧ e e ∧ e e ω τ < 0.1 and ∂κc /∂ω τ < 0 for ω τ > 0.1. In the flow along the field lines will be dominated by electrons e e ∼ ∧ e e e e ∼ highly magnetised plasma simulated here, Righi-Leduc with a mean-free-path on the order of the temperature heat-flow reduces when it enters more magnetised re- scale length, requiring kinetic modelling. For a more ap- gions. propriate MHD comparison, the laser radius can be in- Assuming uniform n (i.e. looking at time-scales over creased while keeping the intensity constant (to give the e which the thermal evolution dominates over the hydro- same ∆t/τ for equation 11). e dynamic motion):

c 2 ∂Ue ˆ ∂κ∧ neTeτe e D. Righi-Leduc = b · ∇Te × ∇ |B| 2 (13) PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5124144 ∂t κ∧ ∂ω τ m h i e e e Righi-Leduc heat-flow is in the same direction as the For a square laser pulse, the temperature initially cross-gradient-Nernst velocity; for an applied magnetic peaks in θ along y = x and y = −x. This results field in the same direction as the laser propagation, it acts in Nernst demagnetisation of those regions, increasing around the azimuthal direction. In the cross-gradient- the Righi-Leduc heat-flow. With the prevailing heat-flow Nernst configuration (section III B) no azimuthal vari- travelling clockwise, the upstream heat-flow from the de- ations in plasma/magnetic field profile is required, as magnetised region is increasing, while the downstream the magnetic field can twist. The heat-flow, however, heat-flow is slowing down. This rotates the temperature only changes the electron temperature profile for varia- profile clockwise, and the process continues. tions in θ. Therefore, a non-circular laser spot is required The interplay between Righi-Leduc and Nernst have to bring about a clear diagnostic signature from Righi- been studied theoretically and computationally for a sim- Leduc. Here a square spot is considered. ilar regime23, although the cross-gradient-Nernst was not Figure 7 shows a 2-D x−y slice of the electron temper- included. While figure 7 predominantly shows rotation ature profile at 0.5ns with the laser and magnetic field of the temperature profile, higher applied magnetic field This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. 7

strengths result in instability growth, which is in agree- non-local transport regime, increasing the relevance of ment with conditions for the onset of the magnetothermal the measurements to laser-plasma interaction regimes in instability23. ICF drive5,6,26 and MagLIF pre-heat1. A further inves- tigation of this transition will be the subject of a further publication comparing kinetic and MHD results. IV. CONCLUSION The configuration outlined could also be modified to study the current-driven transport terms (equations 5 A magnetised under-dense platform has been outlined and 6). By comparing the Nernst velocity and Hall ve- for the measurement of extended-MHD effects. The di- locity (equation 7) it is clear that a larger magnetic field verse operating range, low laser energies and relatively would be suitable, which both suppresses the Nernst and symmetric behaviour makes the base configuration suit- increases the electron current. While recent advances in magnetic field generation would allow for access to this able for measuring the Nernst velocity and its suppres- 27–29 sion at larger magnetisations for the first time. Magnetic regime , an initially uniform magnetic field does not field cavitation down the laser axis grows with time and result in an electric current. Hydrodynamic motion is can be diagnosed using proton deflectometry. In a regime required to first perturb the magnetic field distribution, where the Nernst velocity is much larger than the hydro- increasing the experiment complexity. While the laser dynamic motion, the proton measurement alone can be drive will naturally change the field distribution to cause used to estimate the effect. Otherwise, a simultaneous an electric current, the further perturbation of this field measurement of plasma density and temperature can be by the current-driven transport terms will have to be used to give a Nernst estimate independent of simula- disentangled from the hydrodynamic motion by a com- tions. parison with the density profile. The laser spatial profile was then modified to show the effect of other extended-MHD terms. By allowing an APPENDIX angle between the heating beam and the applied magnetic field, the cross-gradient-Nernst can twist the field, allowing diagnosis independently of the regular Nernst A. Magnetic Transport by proton probing down the laser axis. Anisotropic thermal conduction can then be demon- The transport of magnetic flux in a plasma is typically 14 strated by using a magnetic field perpendicular to the written as an induction equation of the form : heating beam, with heat flowing faster along the field lines. By modifying the applied field strength, the ∂B j × B α · j transition between unmagnetised and highly-magnetised = ∇ × (v × B) − ∇ × − ∇ × ∂t n e n2e2 regimes can be investigated. e e β · ∇T Finally, Righi-Leduc heat-flow was shown to be im- e ∇Pe +∇ × + ∇ × (14) portant when using a square laser spatial profile, with e nee azimuthal heat-flows rotating the electron energy profile. Where the terms on the right-hand side represent bulk Applying the magnetic field in the opposite direction then fluid advection, the Hall term, resistivity, thermally- reverses the rotation direction. This regime could also be driven transport and Biermann battery generation. The used to observe the magnetothermal instability for the tensor transport coefficients α and β are those defined by first time23. Laser plasma instabilities could hinder the measure- Epperlein and Haines30, with components parallel and ment of extended-MHD terms. The regime in this paper perpendicular to the magnetic field. is estimated to be a factor of 20 above the ponderomotive While equation 14 fully describes the transport of filamentation threshold24. Filamentation could create lo- magnetic fields in an extended-MHD plasma, the conse- PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5124144 cally hot regions where Nernst transport is high. The lo- quences of the equation are not immediately clear. Here cal spikes in temperature could slow the cavitation rate, the equation is re-written into a physically intuitive form, as field can be trapped between two hot regions. The where each term acts to either advect, diffuse or generate strong axial gradients will also increase the twisting of magnetic field. This form is also simpler to implement field by the cross-gradient-Nernst term. Ponderomotive into a code, with clearly defined stability criteria. filamentation could be mitigated by lowering the laser The thermally-driven magnetic transport term is ex- wavelength24, while changes to the laser intensity and panded as: resultant plasma temperature must ensure that Nernst transport still dominates over hydrodynamic expansion. ∂B ˆ ˆ ˆ ˆ ˆ Magnetisation of under-dense plasmas is also expected to = ∇×(βkb(b·∇Te)+β⊥b×(∇Te ×b)+β∧b×∇Te) ∂t increase thermal filamentation25, but in all regimes tried, β the simulations remained unaffected. (15) By changing the laser spot radius and keeping the in- Where ˆb is the magnetic field unit vector. This equa- tensity constant, the experiment could be scaled into the tion can be rearranged using the vector triple product This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. 8

[ˆb(ˆb · ∇Te) = ∇Te − ˆb × (∇Te × ˆb)] and the fact that ∇ × βk∇Te is zero (βk is constant for a given ionisation) FIG. 8. The dependence of the γ magnetic transport coeffi- 4 to give : cients on ωeτe for Z = 1 (solid line) and Z = ∞ (dashed). γ∧, the cross-gradient-Nernst transport coefficient, is approx- imately equal to or larger than the regular Nernst coefficient, 30 ∂B βV βk − β⊥ γ⊥ for all magnetisations. Data from . = ∇× − ∇Te×B− (ˆb×∇Te)×B ∂t β e |B| e |B| (16) These terms are now in the common advection velocity

form ∇ × (v × B). Therefore, the thermally-driven mag- |vN⊥| γ⊥ netic transport can be completely described as an advec- = (22) |vN∧| γ∧ tion of the magnetic field with velocity vN = vN⊥ +vN∧. Using a tensor transport coefficient γ further simplifies Figure 8 plots the Nernst and cross-gradient-Nernst co- the equations, where: efficients against magnetisation for Z = 1 and ∞, showing that the cross-gradient-Nernst coefficient is similar in magnitude at low magnetisations (≈ 30% lower for c β∧ Z = 1) and is larger at high magnetisations30. This sug- γ⊥ = (17) ωeτe gests that in all cases where the Nernst velocity is large, the cross-gradient-Nernst velocity should also be evalu- ated. Another advantage of the γ formulation is the sim- c βk − β⊥ γ∧ = (18) ple monotonically-decreasing behaviour of the two coef- ωeτe ficients with magnetisation. The resistive magnetic transport term in equation 14 m is typically expanded as: γc = γ e (19) τe

The superscript c represents the dimensionless form ∂B αk ˆ ˆ α⊥ ˆ ˆ α∧ ˆ = −∇× 2 2 b(b·j)+ 2 2 b×(j×b)− 2 2 b×j of the coeﬃcient, which is only dependent on the Hall ∂t α e ne e ne e ne parameter (ωeτe) and the plasma ionisation state Z. The (23) magnetic ﬁeld advection velocities are then: Using the same methodology as for the thermally- driven terms, this can be re-arranged into the form of

βV magnetic field advection velocities. In this case, how- vN⊥ = − ∇Te = −γ⊥∇Te (20) ever, the parallel term contains an additional resistive e |B| 2 2 component, as ∇ × (αk/µ0e ne)j 6= 0: which is called the Nernst velocity, and: ∂B α = − ∇ × k ∇ × B βk − β⊥ ∂t µ e2n2 v = − (ˆb × ∇T ) = −γ (ˆb × ∇T ) (21) α 0 e N∧ e |B| e ∧ e α⊥ − α α + ∇ × k j × ˆb − ∧ j × B 4 2 2 2 2 which is the cross-gradient-Nernst velocity . The e ne |B| e ne |B| ˆ (24) Nernst velocity can equally be written as vN⊥ = −γ⊥b × (∇T × ˆb) (i.e. removing the component parallel with e where the first term on the right is diffusive. The lat-

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5124144 the magnetic field). Note the change in subscripts from ter two terms become collisional current-driven transport β to γ, which is to make clear that the Nernst veloc- velocities vjB = vjB⊥ + vjB∧. A new tensor transport ity (⊥) acts perpendicular to the magnetic field (in the coefficient δ is also used here to simplify the equations: ˆb × (∇Te × ˆb) direction), while the cross-gradient-Nernst (∧) acts perpendicular to both the driving term (in this αc case ∇Te) and the magnetic field. A clear benefit of re- c ∧ δ⊥ = (25) writing the transport coefficients is the simple compari- ωeτe son of term magnitudes. As only the advection velocity component perpendicular to the magnetic field changes αc − αc ∂B c ⊥ k the field ( = ∇ × (vN × B) = 0 for vN parallel to δ = (26) ∂t vN ∧ ω τ B), both terms are only significant when the temperature e e gradient has a component perpendicular to the magnetic field. The ratio of the Nernst and cross-gradient-Nernst 1 δ = δc (27) velocity magnitudes is simply: ene This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. 9

The first term on the right-hand side represents ther- FIG. 9. The dependence of the δ magnetic transport coeffi- mal conduction, then electro-thermal terms and finally cients on ωeτe for Z = 1 (solid line) and Z = ∞ (dashed). the heat-flow associated with the flow of charge. The c Note the similarity with the γ coefficients in figure 8. Data thermal diffusion component can be expanded as: from30.

q = −κ ˆb(ˆb.∇T ) − κ ˆb × (∇T × ˆb) − κ ˆb × ∇T (34) The magnetic field advection velocities are then: κ k e ⊥ e V e The first term represents the heat-flow along magnetic α∧ field lines, which remains unchanged by magnetisation. vjB⊥ = − 2 2 j = −δ⊥j (28) e ne |B| The second term is the heat-flow perpendicular to the field and decreases monotonically with magnetisation. The final term is the Righi-Leduc heat-flow, which rep- α⊥ − αk v = j × ˆb = δ∧j × ˆb (29) resents heat-flow deflected by the magnetic field into the jB∧ e2n2 |B| e direction perpendicular to both the magnetic field and The collisionless Hall term from equation 14 can also the temperature gradient. be manipulated into a velocity: By using simplified operators the connection between thermally-driven electron energy transport and magnetic ˆ ˆ j transport can be made clear. Here ∇kTe = b(b.∇Te) and vHall = − (30) ∇ T = ˆb × (∇T × ˆb) are used. The magnetic field en ⊥ e e e advection velocity due to temperature gradients can be i.e. the magnetic field moves with the bulk electron re-written alongside the flow of electron energy due to motion rather than the ion population. The resistive temperature gradients, as in equations 3 and 4. advection velocities from equations 28 and 29 are clearly The current-driven heat-flows are best understood by related to the Hall term, and will be referred to as the using the velocity of the electron population relative to collisonal corrections to the Hall term. Figure 9 compares the ions, −j/ene. The change in electron energy due to c c the magnitude of δ⊥ and δ∧ for Z = 1 and ∞, with both the current (last two terms in equation 33) can then be monotonically decreasing for increasing magnetisation. written: Overall, the magnetic transport equation can be re- written as in equation 2. In this rewritten form, each term is simple to imple- ∂Ue + vj · ∇Ue = −Ue∇ · vj (35) ment in an MHD code, where advection and diffusion op- ∂t j erations are commonplace. The stability limits are also clear, with the CFL condition for the advection in 1-D31: j j j j ˆ ˆ ˆ ˆ ˆ vj = − − βkb(b · ) − β⊥b × ( × b) − β∧b × ∆x ene ene ene ene ∆tadvection ≤ (31) (36) |v | B i.e. the electron energy moves with the electron pop- and the Von Neumann stability limit for the diffusion: ulation relative to the ions, with collisional corrections. The comparison with the magnetic transport becomes even more clear by using j = ˆb·(ˆb·j) and j = ˆb×(j×ˆb). 2 2 2 k ⊥ ∆x µ0e ne ∆tdiffusion ≤ (32) The current-driven velocities transporting the magnetic 2α k field and electron energies are then written as in equa- It is interesting to note that the remaining transport tions 5 and 6. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5124144 coefficients in the re-written induction equation have a simple dependence: increasing with Z, maximum at zero magnetisation and monotonically decreasing for increas- ACKNOWLEDGEMENTS ing ωeτe (apart from the αk term, which is constant). The results reported in this paper were obtained using the Imperial College High Performance Computer Cx1. B. Thermal Transport This work was supported by NNSA under DOE Coop- erative Agreement DE-NA0003764 and has been carried The electron heat-flow equation is14: out within the framework of the EUROfusion Consortium as well as receiving funding from the Euratom research and training programme 2014-2018 under grant agree- ∂U U U e = −∇ · q = −∇ · − κ · ∇T − e β · j − e j ment No. 633053 (project reference CfP-AWP17-IFE- e e ∂t q ene ene CCFE-01). The views and opinions expressed herein do (33) not necessarily reflect those of the European Commission. This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. 10

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