A Fast and Accurate Method to Capture the Solar Corona/Transition Region Enthalpy Exchange
DDraft version March 5, 2019
Preprint typeset using L A TEX style emulateapj v. 12/16/11
A FAST AND ACCURATE METHOD TO CAPTURE THE SOLAR CORONA/TRANSITION REGIONENTHALPY EXCHANGE
C. D. Johnston & S. J. Bradshaw School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife, KY16 9SS, UK; [email protected] and Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA; [email protected]
Draft version March 5, 2019
ABSTRACTThe brightness of the emission from coronal loops in the solar atmosphere is strongly dependent onthe temperature and density of the confined plasma. After a release of energy, these loops undergo aheating and upflow phase, followed by a cooling and downflow cycle. Throughout, there are significantvariations in the properties of the coronal plasma. In particular, the increased coronal temperatureleads to an excess downward heat flux that the transition region is unable to radiate. This generatesan enthalpy flux from the transition region to the corona, increasing the coronal density. The enthalpyexchange is highly sensitive to the transition region resolution in numerical simulations. With a numer-ically under-resolved transition region, major errors occur in simulating the coronal density evolutionand, thus, the predicted loop emission. This Letter presents a new method that addresses the diffi-culty of obtaining the correct interaction between the corona and corona/chromosphere interface. Inthe transition region, an adaptive thermal conduction approach is used that broadens any unresolvedparts of the atmosphere. We show that this approach, referred to as TRAC, successfully removesthe influence of numerical resolution on the coronal density response to heating while maintaininghigh levels of agreement with fully resolved models. When employed with coarse spatial resolutions,typically achieved in multi-dimensional MHD codes, the peak density errors are less than 3% and thecomputation time is three orders of magnitude faster than fully resolved field-aligned models. The ad-vantages of using TRAC in field-aligned hydrodynamic and multi-dimensional magnetohydrodynamicsimulations are discussed. INTRODUCTION
One of the main difficulties encountered when usingfield-aligned hydrodynamic models to study the physicsof magnetically closed coronal loops is the need to im-plement a grid that fully resolves the steep and dynamictransition region (TR) (see e.g. Bradshaw & Cargill 2013,hereafter BC13). In a static coronal loop, the energy bal-ance in the TR is between a downward heat flux from thecorona and optically thin radiative losses to space. Defin-ing the temperature length scale as L T = T / | dT /ds | ,where s is a coordinate along the magnetic field, suchstatic loops can have L T of order 1 km in the TR. Dur-ing impulsive heating, the heat flux from the corona canbe enhanced by orders of magnitude, so that L T becomesvery small, of order 100 m and less in some cases.Properly resolving these small length scales is essen-tial in order to obtain the correct coronal density in re-sponse to heating (BC13), otherwise the downward heatflux ‘jumps’ over an under-resolved TR to the chromo-sphere, where the incoming energy is then strongly radi-ated. BC13 showed that major errors in simulating thecoronal density evolution were likely with a lack of spa-tial resolution.However, since the thermal conduction time scaleacross a grid cell scales as (∆ s ) and T − / (where ∆ s is the grid cell width), cells in the cooler lower TRmust become dramatically smaller to resolve the tem-perature gradient that supports the incoming heat flux.In addition, numerical stability requires that the mini-mum time scale across the entire grid is temporally re-solved. Thus, locally satisfying ∆ s < L T in the TRimplies long computation times for fully resolved field- aligned simulations. The problem is more severe in three-dimensional (3D) magnetohydrodynamic (MHD) mod-els where computational resources place significant con-straints on the achievable resolutions. Therefore, otherapproaches need to be found.One such approach has been developed by Linker etal. (2001), Lionello et al. (2009) and Miki´c et al. (2013)for use in global 3D MHD simulations. This techniquemodifies the temperature dependence of the parallel ther-mal conductivity κ (cid:107) ( T ) and radiative emissivity Λ( T ) be-low a fixed temperature value ( T c ) in the TR. It hasbeen demonstrated that such modifications artificiallybroaden the TR for T < T c without significantly chang-ing the coronal properties of the loop. While this is aparticularly attractive property the method still requiressufficiently high resolution to properly resolve the smalllength scales in the TR above T c . Furthermore, the char-acteristic temperature of the TR can change dramati-cally during the heating and cooling cycle, making it un-clear how suitable a fixed value of T c is for capturing thedynamic evolution of coronal loops that are impulsivelyheated.Another approach has been developed by Johnston etal. (2017a,b), who treated the (unresolved) TR as a dis-continuity across which energy is conserved through theimposition of a jump condition. No attempt is made toproperly resolve the TR. Instead, the method is employedwith coarse spatial resolutions that are supported by dy-namically locating the top of the unresolved transitionregion (UTR). This approach gives good agreement withfully resolved HYDRAD simulations (Bradshaw & Ma-son 2003; Bradshaw & Cargill 2006, BC13) whilst signif- a r X i v : . [ a s t r o - ph . S R ] M a r Johnston & Bradshaw icantly speeding up the computation time (e.g. Johnstonet al., submitted).This Letter proposes a new approach for modellingthe TR that incorporates the strengths of both ap-proaches. Principally, the ability to dynamically identifythe UTR to prescribe an adaptive T c and then broadenthe length scales in this unresolved region only in or-der to eliminate the need for highly resolved numericalgrids. The outcome is an extremely powerful methodthat (1) removes the influence of numerical resolution onthe coronal response to heating and (2) accurately pre-dicts the results of properly resolved field-aligned models(e.g. BC13) when employed with the coarser spatial res-olutions achieved by multi-dimensional MHD codes. NUMERICAL MODEL AND EXPERIMENTS
Numerical Model
We solve the field-aligned hydrodynamic equations us-ing the HYDRAD code (Bradshaw & Mason 2003; Brad-shaw & Cargill 2006, BC13) run in single fluid mode.HYDRAD uses adaptive regridding to ensure adequatespatial resolution, with the grid being refined such thatcell-to-cell changes in the temperature and density arekept between user defined values (taken as 5% and 10%here), where possible. The largest grid cell in all of ourcalculations has a width of 10 m (1,000 km) and eachsuccessive refinement splits the cell into two. Thus, arefinement level of RL leads to cell widths decreased by1 / RL . The maximum value of RL is limited to 14, corre-sponding to a grid cell width of 61 m in the most highlyresolved parts of the TR. Transition Region Adaptive Conduction
While it is possible to employ such high resolution gridsto properly resolve the TR in field-aligned codes, this ap-proach is unlikely to be a viable way of running multi-dimensional MHD simulations. Therefore, we have de-veloped a new method that addresses this difficulty ofobtaining adequate spatial resolution in numerical simu-lations by modelling the transition region using adaptiveconduction (TRAC) coefficients that act to broaden anyunresolved parts of the atmosphere. This treatment ofthermal conduction in the TR, referred to as the TRACmethod, relies on a dynamic capability to track resolvedand modify unresolved conductive fluxes.
Identification of an adaptive cutoff temperature
The implementation of TRAC is comprised of two mainparts. The first part is to identify the maximum tempera-ture of any unresolved grid cells in the numerical domainand use the calculated value to prescribe an adaptive cut-off temperature ( T c ). This is done by using an algorithmthat is based on the method employed by Johnston et al.(2017a,b) for locating the top of the UTR in the jumpcondition method as follows.We define the temperature length scale as, L T ( T ( s )) = T | dT /ds | , (1)and the resolution in a simulation is given by the localgrid cell width, L R ( s ) = ∆ s, (2) where s is the spatial coordinate along the magnetic field.Using these definitions, the cutoff temperature is de-fined as the maximum temperature that violates the res-olution criteria of Johnston et al. (2017a,b), T c = max( T ( s )) (cid:12)(cid:12)(cid:12) L R ( s ) L T ( s ) > δ = 12 , (3)which corresponds to not having multiple grid cells acrossthe temperature length scale (i.e. unresolved temperaturegradients).An upper bound for the cutoff temperature is set as20% of the peak coronal temperature in the loop at thetime when T c is evaluated (though the results are onlyweakly dependent on the maximum temperature frac-tion), T max = 0 . T peak , (4)and a lower bound set as the temperature value of theisothermal chromosphere, T min = T chrom . (5)In this Letter the lower bound is taken as T chrom =2 × K. Employing these definitions, we dynamicallyadjust T c with the criteria that it should satisfy, T min ≤ T c ≤ T max . (6)Hence, the cutoff temperature is adaptive in identifyingthe UTR, with the value of T c used in the method chang-ing in response to coronal heating and cooling. Broadening the unresolved transition region
The second part of TRAC is to broaden the steeptemperature and density gradients in the UTR. This isachieved using the approach developed by Linker et al.(2001), Lionello et al. (2009) and Miki´c et al. (2013)Below the adaptive cutoff temperature ( T c ), the paral-lel thermal conductivity ( κ (cid:107) ) is set to a constant value, κ (cid:107) ( T ) = κ T / , ∀ T ≥ T c ; (7) κ (cid:107) ( T ) = κ (cid:107) ( T c ) , ∀ T < T c , (8)and the radiative loss rate (Λ) is modified to preserveΛ( T ) κ (cid:107) ( T ),Λ( T ) = Λ( T ) , ∀ T ≥ T c ; (9)Λ( T ) = Λ( T ) (cid:18) TT c (cid:19) / , ∀ T < T c . (10)Increasing the parallel thermal conductivity and de-creasing the radiative loss rate, at temperatures below T c , has the desired effect of broadening the tempera-ture length scales in the UTR. This helps TRAC preventthe heat flux jumping across the unresolved region whilemaintaining accuracy in the properly resolved parts ofthe atmosphere. (The broadening effect will be explainedin greater detail in forthcoming work.) Furthermore, wenote that the formulation of TRAC (1) makes no assump-tions about the spatial resolution in a simulation and (2)the parallel thermal conductivity reduces to the classicalSpitzer-Harm value when the TR is properly resolved. Fast and Accurate Method to Capture the Corona/Transition Region Enthalpy Exchange R L = R L = R L = S H R L = R L = R L = R L = S H L T RAC pu l se R L = R L = R L = L S H R L = R L = R L = R L = R L = R L = R L = T RAC S H R L = R L = R L = R L = F i g u r e . R e s u l t s f o r t h e s h e a t i n g pu l s e s i m u l a t i o n s . T h e p a n e l ss h o w t h ec o r o n a l a v e r ag e d t e m p e r a t u r e ( l e f t - h a nd c o l u m n ) a ndd e n s i t y ( ce n t r a l c o l u m n ) a s f un c t i o n s o f t i m e , a nd t h e t e m p e r a t u r e v e r s u s d e n s i t y ph a s e s p a ce p l o t( r i g h t - h a nd c o l u m n ) . T h e v a r i o u s c u r v e s r e p r e s e n t d i ff e r e n t v a l u e s o f R L , w h i c h c o n v e r g e a s R L i n c r e a s e s ( h i g h e r s p a t i a l r e s o l u t i o n i s a ss o c i a t e d w i t h l a r g e r R L ) . R o w s c o rr e s p o nd t o s i m u l a t i o n s r un w i t h t h e Sp i t ze r - H a r m ( S H ) , L i o n e ll o e t a l. ( , L ) a nd T R A C c o ndu c t i o n m e t h o d s , r e s p ec t i v e l y . T h e li n e s a r ec o l o u r - c o d e d i n a w a y t h a t r e fl ec t s t h ec o ndu c t i o n m e t h o du s e d . N o t e t h a tt h e p r o p e r l y r e s o l v e dS H s o l u t i o n ( R L = ) i s a l s o s h o w n i n t h e L nd T R A C p a n e l s . Johnston & Bradshaw
Validation Experiments
The effectiveness of the TRAC method to obtain thecorrect interaction between the corona and TR, in re-sponse to rapid heating events, is investigated by con-sidering two impulsive coronal heating events, compris-ing short and long pulses that last for a total durationof 60 s and 600 s, respectively. The temporal profileof the heating pulses is triangular with a peak value of Q H = 2 × − Jm − s − , while the spatial profile is uni-form along the loop. We release the energy in a coronalloop of total length 100 Mm, which includes a 10 Mmchromosphere attached at the base of each TR. Thus,the total energy injected into the coronal part of the loopis 4 . × Jm − (4 . × Jm − ) in the 60 s (600 s)heating pulse simulations. These heating conditions arerepresentative of reasonably powerful flares and presenta challenge to resolving the TR.For each simulation, the main assessment of the per-formance of TRAC is a comparison with the results fromtwo alternative methods that are commonly used to treatthermal conduction. Each method is applied with spa-tial resolutions that cover several orders of magnitude,ranging from those required for 3D MHD codes to runin a realistic time to those that fully resolve the TR butare achievable only in field-aligned models. The first ofthese methods is the classical Spitzer-Harm heat flux for-mulation, while the second is the approach developed byLionello et al. (2009), which artificially broadens the TRbelow a fixed specified temperature. For brevity and clar-ity we define these two conduction methods as SH andL09, respectively. We note that the L09 method uses thebroadening technique outlined in Section 2.2.2 but themodifications are applied below a fixed cutoff tempera-ture taken as T c = 250 ,
000 K (which is a typical valueused by Lionello et al. (2009) and Miki´c et al. (2013)).Using the same set of parameter values but employ-ing the three different conduction methods (SH, L09 andTRAC), we repeated each simulation for RL = [0, 1, 3,5, 7, 9, 11, 13, 14] refinement levels to create a group ofsimulations run for each conduction method.All of the simulations start from the same initial con-ditions. These are calculated using the SH parallel ther-mal conductivity and unmodified radiative loss rate. At t = 0 (the initial conditions) a small spatially uniformbackground heating term ( Q bg ) is present. This gives astarting temperature of order 1 MK. However, we notethat Q bg is switched off thereafter and the total energyin the initial conditions is negligible compared with theenergy released into the loop during the heating pulses. RESULTS
Coronal Response to Heating
Figure 1 shows the temporal evolution of the coronalaveraged temperature ( T ) and density ( n ) and the corre-sponding T versus n phase space plot (shown on a log-logscale) for the 60 s heating pulse simulation. The coronalaverages are calculated by spatially averaging over theuppermost 50% of the loop. Each conduction methodis identified with a specific colour and their results areshown in separate rows. The red curves correspond tothe SH method (first row), the purple curves the L09method (second row) and the blue curves represent the TRAC method (third row).In the panels of each method, each curve correspondsto a simulation run with a different value of maximum re-finement level. Taking the SH simulations as an example,the line styles associated with the different refinementlevels are shown in the figure legend on the tempera-ture plot, and indicate that RL increases as one movesupwards from the lowest curve (RL=0) to the highestcurve (RL=14) in the density plot. These simulationsare identical in all respects except for the value of RL.Note that the results for RL=11 and 13 are not shownin the SH panels. In addition, RL=11, 13 and 14 arenot shown in the L09 and TRAC panels, where insteadthe SH simulation run with RL=14 (solid red curves) isshown. This SH simulation (RL=14) is properly resolvedand used as a benchmark solution.The coronal response of the benchmark simulation (SHwith RL=14), to the 60 s heating pulse, follows the stan-dard evolution of an impulsively heated loop as describedin the literature (Bradshaw & Cargill 2006; Cargill et al.2012a,b, 2015; Bradshaw & Klimchuk 2015; Reale 2016).The rise in temperature is followed by the density in-creasing due to the ablation process (Antiochos & Stur-rock 1978; Klimchuk et al. 2008), then, after the time ofthe density peak, the loop cools by radiation and drainsby a downward enthalpy flux to the TR (Bradshaw &Cargill 2010a,b). The temperature cools below the start-ing value of 1 MK at around 2100s, and then continuesto decrease towards the chromospheric value of 2 × Kwhile the density is evacuated from the loop.Consistent with BC13, the coronal density evolutionin the group of SH simulations is strongly dependent onthe spatial resolution, requiring grid cell widths of atleast 488 m (RL ≥
11) for convergence. These resolutionrequirements are slightly weaker in the L09 simulations,with convergence seen for minimum grid cell widths of7.81 km or finer (RL ≥ T c = 250 ,
000 K).It is therefore striking that the TRAC solutions areonly weakly dependent on the spatial resolution, whilemaintaining high levels of accuracy throughout the fullcoronal response range. Grid cell widths of 125 km(RL=3, medium dashed curves) are sufficient to observeconvergence to the properly resolved SH solution whichemployed TR grid cell widths of 61 m. The TRAC solu-tion computed with RL=3 correctly captures the inter-action between the corona and chromosphere during theablation phase, up to the time of peak density where theerror is less than 3% and the subsequent decay phase.Even the density oscillations (that are characteristic ofthe short heating pulse imposed e.g Reale (2016)) andthe global cooling of the loop down to 2 × K, areboth correctly captured. This demonstrates that adap-tively broadening unresolved parts of the TR removes theinfluence of numerical resolution on the coronal densityresponse to impulsive heating.Figure 2 shows the results for the 600 s heating pulsesimulations. Releasing the energy for an extended periodof time but with the same peak heating rate has two mainconsequences. Firstly, the density oscillations associated
Fast and Accurate Method to Capture the Corona/Transition Region Enthalpy Exchange R L = R L = R L = S H R L = R L = R L = R L = S H L T RAC pu l se R L = R L = R L = L S H R L = R L = R L = R L = R L = R L = R L = T RAC S H R L = R L = R L = R L = F i g u r e . R e s u l t s f o r t h e s h e a t i n g pu l s e s i m u l a t i o n s . N o t a t i o n i s t h e s a m e a s F i g u r e . Johnston & Bradshaw
SHL09TRAC t= 75 st = 150 st = 300 st = 350 s t = 1000 st = 2000 st = 2500 st = 3000 s
SH(RL=14)TRAC(RL=3) t = 150 st = 350 st = 2000 st = 3000 s
SH(RL=14)TRAC(RL=3)
Figure 3.
Results for the 600 s heating pulse simulations. Starting from the top left, the upper four panels show time ordered snapshotsof the temperature, density, pressure and velocity as functions of position along the loop for times during the ablation and decay phases.The lower four panels show an enlargement about the transition region of a subset of these snapshots. The various solid curves representthe properly resolved SH solution (RL=14) at different times (with the lines colour-coded in a way that reflects the temporal evolution)and the dashed blue curves correspond to the TRAC solution that is computed with RL=3 at these times. Animations of the 60 s and600 s heating pulse simulations can be viewed in the online version of this article. These compare the full time evolution of the
T, n, p and v profiles of the SH simulations run with RL=14 and the TRAC simulations run with RL=3. Fast and Accurate Method to Capture the Corona/Transition Region Enthalpy Exchange ≥ ≥
9) were necessary for the solutions toconverge in the SH and L09 simulations, respectively.On the other hand, the TRAC solutions once againconverge at RL=3 (125 km grid cells) with a peak den-sity error of less than 2%, indicating that this treatmentof the TR gives rise to a coronal response that can ac-curately follow the complete coronal heating and coolingcycle with coarse spatial resolutions, irrespective of theheating duration and resulting density regime.In particular, TRAC eliminates the need for the veryshort time steps that are imposed by a highly resolvednumerical grid. This enables the method to accuratelycapture the corona/TR enthalpy exchange significantlyfaster than fully resolved models. A typical speed-upis between two and three orders of magnitude. For ex-ample, the computation time of the properly resolvedSH simulation (RL=14) is 3.5 days for the 600 s heat-ing pulse, while the run time for the TRAC simulationcomputed with RL=3 is only 6 minutes.
Global Evolution
The main question that needs to be addressed in under-standing these results is why are the TRAC simulationsso successful in describing the coronal response to heat-ing with such large grid cell widths? A comprehensiveanswer requires a detailed assessment of how the TRACmodifications to the parallel thermal conductivity andradiative loss rate affect the local energy balance andsubsequent dynamics in the TR, coupled together withthe resulting global evolution of the loop. Here we con-centrate just on the latter. The former will be presentedin future work.Figure 3 shows the time evolution of the global temper-ature, density, pressure, and velocity profiles for the 600s heating pulse simulations. Each solid curve representsa different snapshot from the properly resolved SH sim-ulation (RL=14) and the dashed blue curves imposed ontop are the corresponding snapshots from the TRAC sim-ulation computed with 125 km grid cells (RL=3). (Thecomparison with an under-resolved SH solution (e.g. SHrun with RL=3) has been previously discussed in BC13and Johnston et al. (2017a) and will not be consideredfurther.)First we focus on the evolution of the pressure. Ofparticular importance is the formation of the pressuregradients in the TR. Adaptively broadening the temper-ature and density profiles in the unresolved parts of theTR prevents the downward heat flux jumping across theunresolved region. This ensures that the incoming en-ergy goes into increasing the gas pressure locally, ratherthan being lost due to artificially high radiative losses (e.g. BC13; Johnston et al. 2017a,b). The scale of thebroadening is relatively small but Figure 3 shows thatthe resulting TRAC approximations of the SH pressuregradients are remarkably good. These pressure gradientsare then responsible for driving the flows.To that end it is clear the TRAC solution correctlycaptures the global velocity evolution of the properly re-solved SH simulation, during both the ablation and decayphases in response to the 600 s heating pulses. (The 60 sheating pulse simulations show the same fundamentalproperties). This ensures that TRAC captures the massand energy exchange that takes place between the chro-mosphere, TR and corona correctly, enabling the methodto maintain high levels of accuracy in the coronal temper-ature and density evolution throughout the full heatingand upflow followed by cooling and downflow cycle. DISCUSSION
The difficulty of obtaining adequate spatial resolutionin numerical models of the outer solar atmosphere hasbeen a long-standing problem. BC13 demonstrated thatlack of adequate spatial resolution during impulsive heat-ing events led to coronal densities that are erroneouslysmall. Johnston et al. (submitted) then went on to showthat one consequence of these artificially low coronal den-sities is to (artificially) suppress thermal non-equilibrium(TNE) in coronal loops. Thus, under-resolving the TRin numerical simulations has very significant implicationsfor (1) the resultant loop dynamics and (2) any compar-isons between model predictions and observations.Several different approaches have been proposed in or-der to side-step the need for highly resolved numericalgrids and the commensurate very short time steps thatare required for numerical stability, yet no fully satisfac-tory solution is available to date. For example, Lionelloet al. (2009) artificially broaden the TR below a fixedcutoff temperature ( T c ), while Johnston et al. (2017a,b)dynamically locate the top of an unresolved TR and im-pose a jump condition that is derived from an integratedform of energy conservation. However, when employedin simulations with coarse spatial resolutions, the latterapproach suffers from overestimating the coronal densityresponse to heating while the former, in contrast, under-estimates the density.This Letter has presented a new approach, referred toas the TRAC method, that, for the first time, has suc-cessfully removed the influence of under-resolving the TRon the coronal density response to heating while retain-ing remarkable levels of accuracy compared with fullyresolved models. The new method combines the basicideas from the approaches developed previously by Li-onello et al. (2009) and Johnston et al. (2017a,b).We have considered only impulsive heating eventswhere the spatial distribution of the energy depositionis uniform along the loop. However, the TRAC method-ology is designed to deal with steep transition regionswhenever they arise, independent of the nature of theheating. A detailed investigation of different forms ofheating (e.g. footpoint heating) will be presented in fu-ture work.TRAC accurately captures the coronal response ofproperly resolved field-aligned models (e.g. BC13) whenemployed with spatial resolutions that were up to threeorders of magnitude coarser. For example, with TRAC, Johnston & Bradshaw grid cell widths of order 100 km were sufficient to observeconvergence to the fully resolved field-aligned model,which required grid cell widths of order 100 m. Thepeak density errors were less than 3% and the relaxationof the TR resolution culminated in computation timesthat were three orders of magnitude faster (e.g. 6 minuterun times instead of 3.5 days).The advantages of this new approach are multiple. Forfield-aligned hydrodynamic simulations of the coronal re-sponse to heating (see e.g. Reale 2014, for a review), theshort computation time means that (1) simulations ofcoronal heating events can be run quickly, permittingextensive surveys of large parameter spaces (e.g. as doneby Froment et al. (2018) to study the occurrence of TNEin coronal loops) to be completed significantly faster at afraction of the computational cost and (2) simulations ofmultiple loop strands (thousands or more) that comprisean entire active region (e.g. experiments seeking to rec-oncile heating models with the Hi-C observational data),can be performed with relative ease and high accuracy forthe coronal emission. However, full numerical resolutionis still required to deduce the details of the emission inthe TR at temperatures below the adaptive cutoff tem-perature.In 3D MHD codes, the method can be included with-out the need for high spatial resolution in the TR and acorresponding extended computation time, “freeing up”resources to resolve better the current sheets responsiblefor the heating. Indeed, our results suggest that highlevels of accuracy can be obtained with grid cell widthsof order 100 km, which is achievable in current 3D MHDsimulations.This research has received funding from the EuropeanResearch Council (ERC) under the European Union’sHorizon 2020 research and innovation program (grantagreement No 647214). S.J.B. is grateful to the National Science Foundation for supporting this work throughCAREER award AGS-1450230. C.D.J. acknowledgessupport from the International Space Science Institute(ISSI), Bern, Switzerland to the International Team 401“Observed Multi-Scale Variability of Coronal Loops as aProbe of Coronal Heating”. We also thank Dr Z. Miki´cfor providing important clarifications on the implemen-tation of the L09 method.REFERENCESgrid cell widths of order 100 km were sufficient to observeconvergence to the fully resolved field-aligned model,which required grid cell widths of order 100 m. Thepeak density errors were less than 3% and the relaxationof the TR resolution culminated in computation timesthat were three orders of magnitude faster (e.g. 6 minuterun times instead of 3.5 days).The advantages of this new approach are multiple. Forfield-aligned hydrodynamic simulations of the coronal re-sponse to heating (see e.g. Reale 2014, for a review), theshort computation time means that (1) simulations ofcoronal heating events can be run quickly, permittingextensive surveys of large parameter spaces (e.g. as doneby Froment et al. (2018) to study the occurrence of TNEin coronal loops) to be completed significantly faster at afraction of the computational cost and (2) simulations ofmultiple loop strands (thousands or more) that comprisean entire active region (e.g. experiments seeking to rec-oncile heating models with the Hi-C observational data),can be performed with relative ease and high accuracy forthe coronal emission. However, full numerical resolutionis still required to deduce the details of the emission inthe TR at temperatures below the adaptive cutoff tem-perature.In 3D MHD codes, the method can be included with-out the need for high spatial resolution in the TR and acorresponding extended computation time, “freeing up”resources to resolve better the current sheets responsiblefor the heating. Indeed, our results suggest that highlevels of accuracy can be obtained with grid cell widthsof order 100 km, which is achievable in current 3D MHDsimulations.This research has received funding from the EuropeanResearch Council (ERC) under the European Union’sHorizon 2020 research and innovation program (grantagreement No 647214). S.J.B. is grateful to the National Science Foundation for supporting this work throughCAREER award AGS-1450230. C.D.J. acknowledgessupport from the International Space Science Institute(ISSI), Bern, Switzerland to the International Team 401“Observed Multi-Scale Variability of Coronal Loops as aProbe of Coronal Heating”. We also thank Dr Z. Miki´cfor providing important clarifications on the implemen-tation of the L09 method.REFERENCES