3D Radiative MHD Research Articles

Effect of numerical resolution on synthetic observables of simulated coronal loops

Abstract Increasingly realistic simulations of the corona are used to predict synthetic observables for instruments onboard both existing and upcoming heliophysics space missions. Synthetic observables play an important role in constraining coronal heating theories. Choosing the spatial resolution of numerical simulations involves a trade-off between accuracy and computational cost. Since the numerical resolution not only affects the scale of structures that can be resolved, but also thermodynamic quantities such as the average coronal density, it is important to quantify the effect on synthesized observables. Using 3D radiative MHD simulations of coronal loops at three different grid spacings, from 60 km down to 12 km, we find that changes in numerical resolution lead to differences in thermodynamic quantities and stratification as well as dynamic behaviour. Higher grid resolution results in a more complex and dynamic atmosphere. The resolution affects the emission intensity as well as the velocity distribution, thereby affecting synthetic spectra derived from the simulation. The distribution of synthetic coronal loop strand sizes changes as more fine-scale structure is resolved. A number of parameters, however, seem to start to saturate from our chosen medium grid resolution on. Our study shows that while choosing a sufficiently high resolution matters when comparing forward-modelled observables with data from current and future space missions, for most purposes not much is gained by further increasing the resolution beyond a grid spacing of 24 km, which seems to be adequate for reproducing bulk loop properties and forward-modelled emission, representing a good trade-off between accuracy and computational resource.

Chromospheric Heating from Local Magnetic Growth and Ambipolar Diffusion under Nonequilibrium Conditions

The heating of the chromosphere in internetwork regions remains one of the foremost open questions in solar physics. In the present study, we tackle this old problem by using a very-high-spatial-resolution simulation of quiet-Sun conditions performed with radiative MHD numerical models and interface region imaging spectrograph (IRIS) observations. We have expanded a previously existing 3D radiative MHD numerical model of the solar atmosphere, which included self-consistently locally driven magnetic amplification in the chromosphere, by adding ambipolar diffusion and time-dependent nonequilibrium hydrogen ionization to the model. The energy of the magnetic field is dissipated in the upper chromosphere, providing a large temperature increase due to ambipolar diffusion and nonequilibrium ionization (NEQI). At the same time, we find that adding the ambipolar diffusion and NEQI in the simulation has a minor impact on the local growth of the magnetic field in the lower chromosphere and its dynamics. Our comparison between synthesized Mg ii profiles from these high-spatial-resolution models, with and without ambipolar diffusion and NEQI, and quiet-Sun and coronal hole observations from IRIS now reveal a slightly better correspondence. The intensity of profiles is increased, and the line cores are slightly broader when ambipolar diffusion and NEQI effects are included. Therefore, the Mg ii profiles are closer to those observed than in previous models, though some differences still remain.

The effects of magnetic fields on observational signatures of atmospheric escape in exoplanets: Double tail structures

ABSTRACTUsing 3D radiative MHD simulations and Lyman-α transit calculations, we investigate the effect of magnetic fields on the observational signatures of atmospheric escape in exoplanets. Using the same stellar wind, we vary the planet’s dipole field strength (Bp) from 0 to 10G. For Bp < 3G, the structure of the escaping atmosphere begins to break away from a comet-like tail following the planet (Bp = 0), as we see more absorbing material above and below the orbital plane. For Bp ≥ 3G, we find a ‘dead-zone’ around the equator, where low velocity material is trapped in the closed magnetic field lines. The dead-zone separates two polar outflows where absorbing material escapes along open field lines, leading to a double tail structure, above and below the orbital plane. We demonstrate that atmospheric escape in magnetized planets occurs through polar outflows, as opposed to the predominantly night-side escape in non-magnetized models. We find a small increase in escape rate with Bp, though this should not affect the time-scale of atmospheric loss. As the size of the dead-zone increases with Bp, so does the line centre absorption in Lyman-α, as more low-velocity neutral hydrogen covers the stellar disc during transit. For Bp < 3G the absorption in the blue wing decreases, as the escaping atmosphere is less funnelled along the line of sight by the stellar wind. In the red wing (and for Bp > 3G in the blue wing) the absorption increases caused by the growing volume of the magnetosphere. Finally we show that transits below and above the mid-disc differ caused by the asymmetry of the double tail structure.

Excitation and evolution of coronal oscillations in self-consistent 3D radiative MHD simulations of the solar atmosphere

Context. Solar coronal loops are commonly subject to oscillations. Observations of coronal oscillations are used to infer physical properties of the coronal plasma using coronal seismology. Aims. Excitation and evolution of oscillations in coronal loops is typically studied using highly idealised models of magnetic flux tubes. In order to improve our understanding of coronal oscillations, it is necessary to consider the effect of realistic magnetic field topology and evolution. Methods. We study excitation and evolution of coronal oscillations in three-dimensional (3D) self-consistent simulations of solar atmosphere spanning from the convection zone to the solar corona using the radiation-magnetohydrodynamic (MHD) code Bifrost. We use forward-modelled extreme-ultraviolet emission and 3D tracing of magnetic field to analyse the oscillatory behaviour of individual magnetic loops. We further analyse the evolution of individual plasma velocity components along the loops using wavelet power spectra to capture changes in the oscillation periods. Results. Various types of oscillations commonly observed in the corona are present in the simulation. We detect standing oscillations in both transverse and longitudinal velocity components, including higher-order oscillation harmonics. We also show that self-consistent simulations reproduce the existence of two distinct regimes of transverse coronal oscillations: rapidly decaying oscillations triggered by impulsive events and sustained small-scale oscillations showing no observable damping. No harmonic drivers are detected at the footpoints of oscillating loops. Conclusions. Coronal loop oscillations are abundant in self-consistent 3D MHD simulations of the solar atmosphere. The dynamic evolution and variability of individual magnetic loops suggest that we need to re-evaluate our models of monolithic and static coronal loops with constant lengths in favour of more realistic models.

Connecting Atmospheric Properties and Synthetic Emission of Shock Waves Using 3D RMHD Simulations of the Quiet Sun

Abstract We analyze the evolution of shock waves in high-resolution 3D radiative MHD simulations of the quiet Sun and their synthetic emission characteristics. The simulations model the dynamics of a 12.8 × 12.8 × 15.2 Mm quiet-Sun region (including a 5.2 Mm layer of the upper convection zone and a 10 Mm atmosphere from the photosphere to corona) with an initially uniform vertical magnetic field of 10 G, naturally driven by convective flows. We synthesize the Mg II and C II spectral lines observed by the Interface Region Imaging Spectrograph (IRIS) satellite and extreme ultraviolet emission observed by the Solar Dynamics Observatory (SDO)/AIA telescope. Synthetic observations are obtained using the RH1.5D radiative transfer code and temperature response functions at both the numerical and instrumental resolutions. We found that the Doppler velocity jumps of the C II 1334.5 Å IRIS line and a relative enhancement of the emission in the 335 Å SDO/AIA channel are the best proxies for the enthalpy deposited by shock waves into the corona (with Kendall’s τ correlation coefficients of 0.59 and 0.38, respectively). The synthetic emission of the lines and the extreme ultraviolet passbands are correlated with each other during the shock-wave propagation. All studied shocks are mostly hydrodynamic (i.e., the magnetic energy carried by horizontal fields is ≤2.6% of the enthalpy for all events) and have Mach numbers >1.0–1.2 in the low corona. The study reveals the possibility of diagnosing energy transport by shock waves into the solar corona, as well as their other properties, by using IRIS and SDO/AIA sensing observations.

3D Radiative MHD Simulations of Starspots

Abstract There are no direct spatially resolved observations of spots on stars other than the Sun, and starspot properties are inferred indirectly through lightcurves and spectropolarimetric data. We present the first self-consistent 3D radiative MHD computations of starspots on G2V, K0V, and M0V stars, which will help us to better understand observations of activity, variability, and magnetic fields in late-type main-sequence stars. We used the MURaM code, which has been extensively used to compute “realistic” sunspots, for our simulations. We aim to study how fundamental starspot properties such as intensity contrast, temperature, and magnetic field strength vary with spectral type. We first simulated in 2D multiple spots of each spectral type to find out appropriate initial conditions for our 3D runs. We find that with increasing stellar effective temperature, there is an increase in the temperature difference between the umbra of the spot and its surrounding photosphere, from 350 K on the M0V star to 1400 K on the G2V star. This trend in our simulated starspots is consistent with observations. The magnetic field strengths of all the starspot umbrae are in the 3–4.5 kG range. The G2V and K0V umbrae have comparable magnetic field strengths around 3.5 kG, while the M0V umbra has a relatively higher field strength around 4 kG. We discuss the physical reasons behind both these trends. All of the three starspots develop penumbral filament-like structures with Evershed flows. The average Evershed flow speed drops from 1.32 km s−1 in the G2V penumbra to 0.6 km s−1 in the M0V penumbra.

Realistic 3D MHD modeling of self-organized magnetic structuring of the solar corona

AbstractThe dynamics of solar magnetoconvection spans a wide range of spatial and temporal scales and extends from the interior to the corona. Using 3D radiative MHD simulations, we investigate the complex interactions that drive various phenomena observed on the solar surface, in the low atmosphere, and in the corona. We present results of our recent simulations of coronal dynamics driven by underlying magnetoconvection and atmospheric processes, using the 3D radiative MHD code StellarBox (Wray et al. 2018). In particular, we focus on the evolution of thermodynamic properties and energy exchange across the different layers from the solar interior to the corona.

The Origin of Deep Acoustic Sources Associated with Solar Magnetic Structures

Abstract It is generally accepted that solar acoustic (p) modes are excited by near-surface turbulent motions, in particular by downdrafts and interacting vortices in intergranular lanes. Recent analysis of Solar Dynamics Observatory data by Zhao et al. (2015) revealed fast-moving waves around sunspots, which are consistent with magnetoacoustic waves excited approximately 5 Mm beneath the sunspot. We analyzed 3D radiative MHD simulations of solar magnetoconvection with a self-organized pore-like magnetic structure, and identified more than 600 individual acoustic events both inside and outside this structure. By performing a case-by-case study, we found that acoustic sources surrounding the magnetic structure are associated with downdrafts. Their depth correlates with downdraft speed and magnetic field strength. The sources often can be transported into deeper layers by downdrafts. The wave front shape, in the case of a strong or inclined downdraft, can be stretched along the downdraft. Inside the magnetic structure, excitation of acoustic waves is driven by converging flows. Frequently, strong converging plasma streams hit the structure boundaries, causing compressions in its interior that excite acoustic waves. Analysis of the depth distribution of acoustic events shows the strongest concentration at 0.2–1 Mm beneath the surface for the outside sources and mostly below 1 Mm inside the magnetic region, that is, deeper than their counterparts outside the magnetic region.

3D radiation MHD simulations of gas and dust in protoplanetary disks

Abstract3D global radiation MHD simulations of gas and dust in protoplanetary disks allow us to understand the dynamical and thermal evolution of protoplanetary disks. At the same time, recent observations in the mm-dust emission by the Atacama Large Millimeter Array (ALMA) allow us to resolve structures at scales of the disk scale height.From our recent simulation results by Flock et al. (2015) and Flock et al. (2017) we are able to directly compare for the first time detailed observational constraints from high-resolution observations by ALMA with the gas and dust dynamics obtain in 3D state-of-art simulations of protoplanetary disks. Especially measurements of the dust scale height obtained from the disk around the young system HL Tau allow us to compare for different gas disk instability models. Further we use Monte Carlo radiation transfer models of the dusty disk to compare our results of the dust scale height in 3D radiation HD and MHD simulations. Our findings are that magnetized models fit perfectly the observational constraints, showing a strongly settled disk, while hydrodynamical turbulence leads to a dust uplifting which is larger than expected. These results open a new window to compare future multi-wavelength observations to simulations.

HIGH SPATIAL RESOLUTION Fe xii OBSERVATIONS OF SOLAR ACTIVE REGIONS

ABSTRACT We use UV spectral observations of active regions with the Interface Region Imaging Spectrograph (IRIS) to investigate the properties of the coronal Fe xii 1349.4 Å emission at unprecedented high spatial resolution (∼0.33″). We find that by using appropriate observational strategies (i.e., long exposures, lossless compression), Fe xii emission can be studied with IRIS at high spatial and spectral resolution, at least for high-density plasma (e.g., post-flare loops and active region moss). We find that upper transition region (TR; moss) Fe xii emission shows very small average Doppler redshifts ( ∼ 3 km s−1) as well as modest non-thermal velocities (with an average of ∼24 km s−1 and the peak of the distribution at ∼15 km s−1). The observed distribution of Doppler shifts appears to be compatible with advanced three-dimensional radiative MHD simulations in which impulsive heating is concentrated at the TR footpoints of a hot corona. While the non-thermal broadening of Fe xii 1349.4 Å peaks at similar values as lower resolution simultaneous Hinode Extreme Ultraviolet Imaging Spectrometer (EIS) measurements of Fe xii 195 Å, IRIS observations show a previously undetected tail of increased non-thermal broadening that might be suggestive of the presence of subarcsecond heating events. We find that IRIS and EIS non-thermal line broadening measurements are affected by instrumental effects that can only be removed through careful analysis. Our results also reveal an unexplained discrepancy between observed 195.1/1349.4 Å Fe xii intensity ratios and those predicted by the CHIANTI atomic database.

Radiative power and x-ray spectrum numerical estimations for wire array Z-pinches

Magnetically driven plasma implosion is studied numerically with the use of a 3D radiative MHD model. We consider a Z-pinch formed by an array of thin tungsten wires. In our calculations we take into account a time-extended plasma production due to a material evaporation by an individual wire caused by the heating electric current. Using a detailed model of the pinch kernel, a soft x-ray radiation intensity is analyzed numerically with resolution of temporal, spatial, angular and spectral radiation characteristics. The results are represented for the conditions pertinent to experiments with cylindrical tungsten wire arrays at Angara-5-1 facility (TRINITI, RF).

NUMERICAL SIMULATIONS OF CORONAL HEATING THROUGH FOOTPOINT BRAIDING

Advanced 3D radiative MHD simulations now reproduce many properties of the outer solar atmosphere. When including a domain from the convection zone into the corona, a hot chromosphere and corona are self-consistently maintained. Here we study two realistic models, with different simulated area, magnetic field strength and topology, and numerical resolution. These are compared in order to characterize the heating in the 3D-MHD simulations which self-consistently maintains the structure of the atmosphere. We analyze the heating at both large and small scales and find that heating is episodic and highly structured in space, but occurs along loop shaped structures, and moves along with the magnetic field. On large scales we find that the heating per particle is maximal near the transition region and that widely distributed opposite-polarity field in the photosphere leads to a greater heating scale height in the corona. On smaller scales, heating is concentrated in current sheets, the thicknesses of which are set by the numerical resolution. Some current sheets fragment in time, this process occurring more readily in the higher-resolution model leading to spatially highly intermittent heating. The large scale heating structures are found to fade in less than about five minutes, while the smaller, local, heating shows time scales of the order of 2 minutes in one model and 1 minutes in the other, higher-resolution, model.

REALISTIC MODELING OF LOCAL DYNAMO PROCESSES ON THE SUN

Magnetic fields are usually observed in the quiet Sun as small-scale elements that cover the entire solar surface (the `salt and pepper' patterns in line-of-sight magnetograms). By using 3D radiative MHD numerical simulations we find that these fields result from a local dynamo action in the top layers of the convection zone, where extremely weak 'seed' magnetic fields (e.g., from a $10^{-6}$ G) can locally grow above the mean equipartition field, to a stronger than 2000~G field localized in magnetic structures. Our results reveal that the magnetic flux is predominantly generated in regions of small-scale helical downflows. We find that the local dynamo action takes place mostly in a shallow, about 500~km deep, subsurface layer, from which the generated field is transported into the deeper layers by convective downdrafts. We demonstrate that the observed dominance of vertical magnetic fields at the photosphere and horizontal fields above the photosphere can be explained by small-scale magnetic loops produced by the dynamo. Such small-scale loops play an important role in the structure and dynamics of the solar atmosphere and that their detection in observations is critical for understanding the local dynamo action on the Sun.

Comparison of solar photospheric bright points between Sunrise observations and MHD simulations

Bright points (BPs) in the solar photosphere are radiative signatures of magnetic elements described by slender flux tubes located in the darker intergranular lanes. They contribute to the ultraviolet (UV) flux variations over the solar cycle and hence may influence the Earth's climate. Here we combine high-resolution UV and spectro-polarimetric observations of BPs by the SUNRISE observatory with 3D radiation MHD simulations. Full spectral line syntheses are performed with the MHD data and a careful degradation is applied to take into account all relevant instrumental effects of the observations. It is demonstrated that the MHD simulations reproduce the measured distributions of intensity at multiple wavelengths, line-of-sight velocity, spectral line width, and polarization degree rather well. Furthermore, the properties of observed BPs are compared with synthetic ones. These match also relatively well, except that the observations display a tail of large and strongly polarized BPs not found in the simulations. The higher spatial resolution of the simulations has a significant effect, leading to smaller and more numerous BPs. The observation that most BPs are weakly polarized is explained mainly by the spatial degradation, the stray light contamination, and the temperature sensitivity of the Fe I line at 5250.2 \AA{}. The Stokes $V$ asymmetries of the BPs increase with the distance to their center in both observations and simulations, consistent with the classical picture of a production of the asymmetry in the canopy. This is the first time that this has been found also in the internetwork. Almost vertical kilo-Gauss fields are found for 98 % of the synthetic BPs. At the continuum formation height, the simulated BPs are on average 190 K hotter than the mean quiet Sun, their mean BP field strength is 1750 G, supporting the flux-tube paradigm to describe BPs.

A model for the formation of the active region corona driven by magnetic flux emergence

We present the first model that couples the formation of the corona of a solar active region to a model of the emergence of a sunspot pair. This allows us to study when, where, and why active region loops form, and how they evolve. We use a 3D radiation MHD simulation of the emergence of an active region through the upper convection zone and the photosphere as a lower boundary for a 3D MHD coronal model. The latter accounts for the braiding of the magnetic fieldlines, which induces currents in the corona heating up the plasma. We synthesize the coronal emission for a direct comparison to observations. Starting with a basically field-free atmosphere we follow the filling of the corona with magnetic field and plasma. Numerous individually identifiable hot coronal loops form, and reach temperatures well above 1 MK with densities comparable to observations. The footpoints of these loops are found where small patches of magnetic flux concentrations move into the sunspots. The loop formation is triggered by an increase of upwards-directed Poynting flux at their footpoints in the photosphere. In the synthesized EUV emission these loops develop within a few minutes. The first EUV loop appears as a thin tube, then rises and expands significantly in the horizontal direction. Later, the spatially inhomogeneous heat input leads to a fragmented system of multiple loops or strands in a growing envelope.

A DETAILED COMPARISON BETWEEN THE OBSERVED AND SYNTHESIZED PROPERTIES OF A SIMULATED TYPE II SPICULE

We performed a 3D radiative MHD simulation of the solar atmosphere. This simulation shows a jet-like feature that shows similarities to the type II spicules observed for the first time with Hinode. Rapid Blueshifted Events (RBEs) on the solar disk are associated with these spicules. Observational results suggest they may contribute significantly in supplying the corona with hot plasma. We perform a detailed comparison of the properties of the simulated jet with those of type II spicules (observed with Hinode) and RBEs (with ground-based instruments). We analyze variety of synthetic emission and absorption lines from the simulations including chromospheric Ca II and Ha to TR and coronal temperatures (10E4 to several 10E6K). We compare their synthetic intensities, line profiles, Doppler shifts, line widths and asymmetries with observations from Hinode/SOT and EIS, SOHO/SUMER, SST and SDO/AIA. Many properties of the synthetic observables resemble the observations, and we describe in detail the physical processes that lead to these observables. Detailed analysis of the synthetic observables provides insight into how observations should be analyzed to derive information about physical variables in such a dynamic event. For example, we find that LOS superposition in the optically thin atmosphere requires the combination of Doppler shifts and spectral line asymmetry to determine the velocity in the jet. Other properties differ from the observations, especially in the chromospheric lines. The mass density of the part of the spicule with a chromospheric temperature is too low to produce significant opacity in chromospheric lines. These and other discrepancies are described in detail, and we discuss which mechanisms and physical processes may need to be included in the MHD simulations to mimic the thermodynamic processes of the chromosphere and corona, in particular to reproduce type II spicules.

INVESTIGATING THE RELIABILITY OF CORONAL EMISSION MEASURE DISTRIBUTION DIAGNOSTICS USING THREE-DIMENSIONAL RADIATIVE MAGNETOHYDRODYNAMIC SIMULATIONS

Determining the temperature distribution of coronal plasmas can provide stringent constraints on coronal heating. Current observations with the Extreme ultraviolet Imaging Spectrograph onboard Hinode and the Atmospheric Imaging Assembly onboard the Solar Dynamics Observatory provide diagnostics of the emission measure distribution (EMD) of the coronal plasma. Here we test the reliability of temperature diagnostics using 3D radiative MHD simulations. We produce synthetic observables from the models, and apply the Monte Carlo Markov chain EMD diagnostic. By comparing the derived EMDs with the "true" distributions from the model we assess the limitations of the diagnostics, as a function of the plasma parameters and of the signal-to-noise of the data. We find that EMDs derived from EIS synthetic data reproduce some general characteristics of the true distributions, but usually show differences from the true EMDs that are much larger than the estimated uncertainties suggest, especially when structures with significantly different density overlap along the line-of-sight. When using AIA synthetic data the derived EMDs reproduce the true EMDs much less accurately, especially for broad EMDs. The differences between the two instruments are due to the: (1) smaller number of constraints provided by AIA data, (2) broad temperature response function of the AIA channels which provide looser constraints to the temperature distribution. Our results suggest that EMDs derived from current observatories may often show significant discrepancies from the true EMDs, rendering their interpretation fraught with uncertainty. These inherent limitations to the method should be carefully considered when using these distributions to constrain coronal heating.

DYNAMICS OF MAGNETIZED VORTEX TUBES IN THE SOLAR CHROMOSPHERE

We use 3D radiative MHD simulations to investigate the formation and dynamics of small-scale (less than 0.5 Mm in diameter) vortex tubes spontaneously generated by turbulent convection in quiet-Sun regions with initially weak mean magnetic fields. The results show that the vortex tubes penetrate into the chromosphere and substantially affect the structure and dynamics of the solar atmosphere. The vortex tubes are mostly concentrated in intergranular lanes and are characterized by strong (near sonic) downflows and swirling motions that capture and twist magnetic field lines, forming magnetic flux tubes that expand with height and which attain magnetic field strengths ranging from 200 G in the chromosphere to more than 1 kG in the photosphere. We investigate in detail the physical properties of these vortex tubes, including thermodynamic properties, flow dynamics, and kinetic and current helicities, and conclude that magnetized vortex tubes provide an important path for energy and momentum transfer from the convection zone into the chromosphere.

FORWARD MODELING OF EMISSION INSOLAR DYNAMICS OBSERVATORY/ATMOSPHERIC IMAGING ASSEMBLY PASSBANDS FROM DYNAMIC THREE-DIMENSIONAL SIMULATIONS

It is typically assumed that emission in the passbands of the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO) is dominated by single or several strong lines from ions that under equilibrium conditions are formed in a narrow range of temperatures. However, most SDO/AIA channels also contain contributions from lines of ions that have formation temperatures that are significantly different from the "dominant" ion(s). We investigate the importance of these lines by forward modeling the emission in the SDO/AIA channels with 3D radiative MHD simulations of a model that spans the upper layer of the convection zone to the low corona. The model is highly dynamic. In addition, we pump a steadily increasing magnetic flux into the corona, in order to increase the coronal temperature through the dissipation of magnetic stresses. As a consequence, the model covers different ranges of coronal temperatures as time progresses. The model covers coronal temperatures that are representative of plasma conditions in coronal holes and quiet sun. The 131, 171, and 304 \AA{} AIA passbands are found to be least influenced by the so-called "non-dominant" ions, and the emission observed in these channels comes mostly from plasma at temperatures near the formation temperature of the dominant ion(s). On the other hand, the other channels are strongly influenced by the non-dominant ions, and therefore significant emission in these channels comes from plasma at temperatures that are different from the "canonical" values. We have also studied the influence of non-dominant ions on the AIA passbands when different element abundances are assumed (photospheric and coronal), and when the effects of the electron density on the contribution function are taken into account.

Decay of a simulated mixed-polarity magnetic field in the solar surface layers

Magnetic flux is continuously being removed and replenished on the solar surface. To understand the removal process we carried out 3D radiative MHD simulations of the evolution of patches of photospheric magnetic field with equal amounts of positive and negative flux. We find that the flux is removed at a rate corresponding to an effective turbulent diffusivity, of 100--340 km^2/s, depending on the boundary conditions. For average unsigned flux densities above about 70 Gauss, the percentage of surface magnetic energy coming from different field strengths is almost invariant. The overall process is then one where magnetic elements are advected by the horizontal granular motions and occasionally come into contact with opposite-polarity elements. These reconnect above the photosphere on a comparatively short time scale after which the U loops produced rapidly escape through the upper surface while the downward retraction of inverse-U loops is significantly slower, because of the higher inertia and lower plasma beta in the deeper layers.