Coronal Loops Research Articles

Generic low-atmosphere signatures of swirled-anemone jets

Solar jets are collimated plasma flows moving along magnetic field lines and are accelerated at low altitude following magnetic reconnection. Several of them originate from anemone-shaped low-lying arcades, and the most impulsive ones tend to be relatively wider and display untwisting motions. We aim to establish typical behaviours and observational signatures in the low atmosphere that can occur in response to the coronal development of such impulsive jets. We analysed an observed solar jet associated with a circular flare ribbon using high-resolution observations from SST coordinated with IRIS and SDO. We related specifically identified features with those developing in a generic 3D line-tied numerical simulation of reconnection-driven jets performed with the ARMS code. We identified three features in the SST observations: the formation of a hook along the circular ribbon, the gradual widening of the jet through the apparent displacement of its kinked edge towards (and not away) from the presumed reconnection site, and the falling back of some of the jet plasma towards a footpoint offset from that of the jet itself. The 3D numerical simulation naturally accounts for these features, which were not imposed a priori. Our analyses allowed us to interpret them in the context of the 3D geometry of the asymmetric swirled-anemone loops and their sequences of reconnection with ambient coronal loops. Given the relatively simple conditions in which the observed jet occurred, together with the generic nature of the simulation that comprised minimum assumptions, we predict that the specific features that we identified and interpreted are probably typical of every impulsive jet.

Resonant absorption of standing MHD waves in coronal loops in the absence of directional symmetry: the effect of weak plasma flow

ABSTRACT The oscillation properties of standing magnetohydrodynamic waves in coronal loops has been investigated. A coronal loop is modelled as a straight cylinder with purely axial magnetic field and field aligned plasma flow. The loop model includes an inhomogeneous transitional layer that causes the wave to be resonantly damped. Our aim is to obtain the solution describing a resonantly damped standing kink wave in a flowing loop with the directional symmetry of the loop being broken due to the presence of plasma flow. In the absence of damping the standing wave is a superposition of two waves propagating in the opposite directions that have the same frequency but different wavenumbers. However, the same approach cannot be used when there is damping because the decrements of the two propagating waves are different. We suggest that in the case of relatively weak flow, as is the case in coronal loops, the effect of flow on the decrements can be neglected. Then the decrements of the two waves are the same and the solution describing the damped standing kink wave is a superposition of two resonantly damped propagating waves.

Investigating the Soft X-Ray Spectra of Solar Flare Onsets

In this study, we present the analysis of six solar flare events that occurred in 2022, using new data from the third-generation Miniature X-Ray Solar Spectrometer, also known as the Dual-zone Aperture X-ray Solar Spectrometer (DAXSS). The primary focus of this study is on the flare’s onset phase, which is characterized by elevated soft X-ray emissions even before the flare’s impulsive phase. We analyze the temporal evolution of plasma temperature, emission measure, and elemental abundance factors during the flare onset phase by fitting the DAXSS spectra with the Astrophysical Plasma Emission Code model. The model fitting results indicate that the flaring plasma is already at a high temperature (10–15 MK) during the onset period. The temperature rises during the onset phase, followed by a decrease and subsequent increase during the impulsive phase. Elemental abundance factors show a trend of falling below preflare values during the onset phase, with some recovery before the impulsive phase. During the impulsive phase, the abundance factors decrease from elevated coronal values to about photospheric values. We also analyze images from the 193 Å channel of the Atmospheric Imaging Assembly (AIA), highlighting the formation or brightening of coronal loop structures during the onset phase. Two distinct onset loop configurations are observed, which are referred to as one-loop and two-loop onsets. Both DAXSS and AIA observations indicate that the flare onset phase exhibits hot coronal plasma properties similar to those of the impulsive phase, suggesting that the onset phase may act as a preconditioning effect for some flares.

MUSE observations of small-scale heating events

Abstract Constraining the processes that drive coronal heating from observations is a difficult task due to the complexity of the solar atmosphere. As upcoming missions such as MUSE will provide coronal observations with unprecedented spatial and temporal resolution, numerical simulations are becoming increasingly realistic. Despite the availability of synthetic observations from numerical models, line-of-sight effects and the complexity of the magnetic topology in a realistic setup still complicate the prediction of signatures for specific heating processes. 3D MHD simulations have shown that a significant part of the Poynting flux injected into the solar atmosphere is carried by small-scale motions, such as vortices driven by rotational flows inside intergranular lanes. MHD waves excited by these vortices have been suggested to play an important role in the energy transfer between different atmospheric layers. Using synthetic spectroscopic data generated from a coronal loop model incorporating realistic driving by magnetoconvection, we study whether signatures of energy transport by vortices and eventual dissipation can be identified with future missions such as MUSE.

Horizontally and vertically polarized kink oscillations in curved solar coronal loops

Kink oscillations are frequently observed in coronal loops. This work aims to numerically clarify the influence of the loop curvature on horizontally and vertically polarized kink oscillations. Working within the framework of ideal magnetohydrodynamics (MHD), we conduct three-dimensional (3D) simulations of axial fundamental kink oscillations in curved density-enhanced loops embedded in a potential magnetic field. Both horizontal and vertical polarizations are examined, and their oscillation frequencies are compared with Wentzel-Kramers-Brillouin (WKB) expectations. We discriminate between two different density specifications. In the first (dubbed ``uniform density''), the density is axially uniform and varies continuously in the transverse direction toward a uniform ambient corona. Some further stratification is implemented in the second specification (dubbed ``stratified''), allowing us to address the effect of evanescent barriers. Examining the oscillating profiles of the initially perturbed uniform-density loops, we found that the frequencies for both polarizations deviate from the WKB expectation by $ 10<!PCT!>$. In the stratified loop however, the frequency of the horizontal polarization deviates to a larger extent ($ 25<!PCT!>$). We illustrate the lateral leakage of kink modes through wave tunneling in 3D simulations, for the first time. Despite this, in the uniform density and the stratified loops, the damping time-to-period ratios are similar and are close to the analytical predictions for straight configurations under the thin-tube-thin-boundary (TTTB) assumption. The WKB expectation for straight configurations can reasonably describe the eigenfrequency of kink oscillations only in loops without an asymmetrical cross-loop density profile perpendicular to the oscillating direction. Lateral leakage via wave tunneling is found to be less efficient than resonant absorption, meaning that the latter remains a robust damping mechanism for kink motions even when loop curvature is included.

Nonlinear Wave Damping by Kelvin–Helmholtz Instability-induced Turbulence

Magnetohydrodynamic kink waves naturally form as a consequence of perturbations to a structured medium, for example, transverse oscillations of coronal loops. Linear theory has provided many insights into the evolution of linear oscillations, and results from these models are often applied to infer information about the solar corona from observed wave periods and damping times. However, simulations show that nonlinear kink waves can host the Kelvin–Helmholtz instability (KHI), which subsequently creates turbulence in the loop, dynamics that are beyond linear models. In this paper we investigate the evolution of KHI-induced turbulence on the surface of a flux tube where a nonlinear fundamental kink mode has been excited. We control our numerical experiment so that we induce the KHI without exciting resonant absorption. We find two stages in the KHI turbulence dynamics. In the first stage, we show that the classic model of a KHI turbulent layer growing at ∝t is applicable. We adapt this model to make accurate predictions of the damping of the oscillation and turbulent heating as a consequence of the KHI dynamics. In the second stage, the now dominant turbulent motions are undergoing decay. We find that the classic model of energy decay proportional to t −2 approximately holds and provides an accurate prediction of the heating in this phase. Our results show that we can develop simple models for the turbulent evolution of a nonlinear kink wave, but the damping profiles produced are distinct from those of linear theory that are commonly used to confront theory and observations.

Generalized Coronal Loop Scaling Laws and Their Implication for Turbulence in Solar Active Region Loops

Recent coronal loop modeling has emphasized the importance of combining both Coulomb collisions and turbulent scattering to characterize field-aligned thermal conduction, which invokes a hybrid loop model. In this work, we generalize the hybrid model by incorporating a nonuniform heating and cross section that are both formulated by a power-law function of temperature. Based on the hybrid model solutions, we construct scaling laws that relate loop-top temperature (T a ) and heating rate (H a ) to other loop parameters. It is found that the loop-top properties for turbulent loops are additionally power-law functions of the turbulent mean free path (λ T ), with the functional forms varying from situation to situation, depending on the specification of the heating and/or areal parameters. More importantly, both a sufficiently footpoint-concentrated heating and a cross-sectional expansion with height can effectively weaken (strengthen) the negative (positive) power-law dependence of T a (H a ) on λ T . The reason lies in a notable reduction of heat flux by footpoint heating and/or cross-sectional expansion in the turbulence-dominated coronal part, where turbulent scattering introduces a much weaker dependence of the conduction coefficient on temperature. In this region, therefore, the reduction of the heat flux predominately relies on a backward flattening of the temperature gradient. Through numerical modeling that incorporates more realistic conditions, this scenario is further consolidated. Our results have important implications for solar active region (AR) loops. With the factors of nonuniform heating and cross section taken into account, AR loops can bear relatively stronger turbulence while still keeping a physically reasonable temperature for nonflaring loops.

Modeling of Condensations in Coronal Loops Produced by Impulsive Heating with Variable Frequencies and Locations

We present the results of models of impulsively heated coronal loops using the 1D hydrodynamic Adaptively Refined Godunov Solver code. The impulsive heating events (which we refer to as nanoflares) are modeled by discrete pulses of energy along the loop. We explore the occurrence of cold condensations due to the effective equivalent of thermal nonequilibrium in loops with steady heating, and examine its dependence on nanoflare timing and intensity and also nanoflare location along the loop, including randomized distributions of nanoflares. We find that randomizing the distribution of nanoflares, both in time/intensity and location, diminishes the likelihood of condensation occurring as compared to distributions with regularly occurring nanoflares with the same average properties. The usual criteria that condensation is favored for heating near loop footpoints and with high cadences are more strict for randomized (as opposed to regular) nanoflare distributions, and for randomized distributions the condensations stay in the loop for a shorter amount of time. That said, condensation can sometimes occur in cases where the average values of parameters (frequency or location) are beyond the critical limits above which condensation does not occur for corresponding steady, non-randomized values of those parameters. These properties of condensation occurring due to randomized heating can be used in the future to investigate the diagnostics of coronal heating mechanisms.

Imaging a Large Coronal Loop Using Type U Solar Radio Burst Interferometry

Solar radio U-bursts are generated by electron beams traveling along closed magnetic loops in the solar corona. Low-frequency (<100 MHz) U-bursts serve as powerful diagnostic tools for studying large-sized coronal loops that extend into the middle corona. However, the positive frequency drift component (descending leg) of U-bursts has received less attention in previous studies, as the descending radio flux is weak. In this study, we utilized LOFAR interferometric solar imaging data from a U-burst that has a significant descending leg component, observed between 10 and 90 MHz on 2020 June 5th. By analyzing the radio source centroid positions, we determined the beam velocities and physical parameters of a large coronal magnetic loop that reached just about 1.3 R ⊙ in altitude. At this altitude, we found the plasma temperature to be around 1.1 MK, the plasma pressure around 0.20 mdyn, cm−2, and the minimum magnetic field strength around 0.07 G. The similarity in physical properties determined from the image suggests a symmetric loop. The average electron beam velocity on the ascending leg was found to be 0.21c, while it was 0.14c on the descending leg. This apparent deceleration is attributed to a decrease in the range of electron energies that resonate with Langmuir waves, likely due to the positive background plasma density gradient along the downward loop leg.

Persistent Upflows and Downflows at Active Region Boundaries Observed by SUTRI and AIA

Upflows and downflows at active region (AR) boundaries have been frequently observed with spectroscopic observations at extreme ultraviolet passbands. In this paper, we report the coexistence of upflows and downflows at the AR boundaries with imaging observations from the Solar Upper Transition Region Imager (SUTRI) and the Atmospheric Imaging Assembly (AIA). With their observations from 2022 September 21 to 2022 September 30, we find 17 persistent opposite flows occurring along the AR coronal loops. The upflows are prominent in the AIA 193 Å images with a velocity of 50–200 km s−1, while the downflows are best seen in the SUTRI 465 Å and AIA 131 Å images with a slower velocity of tens of kilometers per second (characteristic temperatures (log T(K)) for 193, 465, and 131 Å are 6.2, 5.7, and 5.6, respectively). We also analyze the center-to-limb variation of the velocities for both upflows and downflows. The simultaneous observations of downflows and upflows can be explained by the chromosphere–corona mass-cycling process, in which the localized chromospheric plasma is impulsively heated to coronal temperature forming a upflow and then these upflows experience radiative cooling producing a downflow with the previously heated plasma returning to the lower atmosphere. In particular, the persistent downflows seen by SUTRI provide strong evidence of the cooling process in the mass cycle. For upflows associated with open loops, part of the plasma is able to escape outward and into the heliosphere as solar wind.

Relative Yield of Thermal and Nonthermal Emission during Weak Flares Observed by STIX during 2021 September 20–25

The disparate nature of the thermal–nonthermal energy partition during flares, particularly during weak flares, is still an open issue. Following the Neupert effect, quantifying the relative yield of X-ray emission in different energy bands can enable the inferring of the underlying energy release mechanism. During 2021 September 20–25, the Solar Orbiter mission—being closer to the Sun (∼0.6 au) and having a moderate separation angle (<40°) from the Sun–Earth line—offered a unique opportunity to analyze multiwavelength emission from ∼200 (mostly weak) flares, commonly observed by the Spectrometer Telescope for Imaging X-rays (STIX), STEREO-A, GOES, and the Solar Dynamics Observatory. Associating the quotient (q f ) of hard X-ray fluence (12–20 keV) and soft X-ray flux (4–10 keV) with the peak soft X-ray flux enabled us to identify strongly nonthermal flares. A multiwavelength investigation of spectral and imaging-mode observations of the 20 strongly nonthermal weak flares reveals an inverse relationship of q f with the emission measure (and density), and a positive relationship with the flare plasma temperature. This indicates that the plasma in tenuous loops attains higher temperatures compared to that in the denser loops, in response to nonthermal energy deposition. This is in agreement with the plasma parameters of the coronal loops, as derived by applying the one-dimensional Palermo–Harvard hydrodynamical code to coronal loop plasma with different initial coronal loop base pressures when subjected to similar heating inputs. Our investigation, therefore, indicates that the plasma parameters of the flaring loop in the initial phase have a decisive role in the thermal–nonthermal energy partitioning.

Observational characteristics of oscillations and waves in and around sunspots. Difficulties in observing and interpreting

This paper summarizes the body of work that we have done over the years on the oscillation processes in sunspots, including their umbra, penumbra, and close vicinity. The study analyzes a number of aspects that impede adequate determining of some characteristics of propagating oscillations and lead to misinterpretation. Using running penumbral waves as an example, we show that their horizontal propagation with decreasing frequency is delusive. The effect is due to different oscillations propagating along magnetic field lines with gradually increasing inclination. This also applies to the three-minute oscillations in the sunspot umbral chromosphere. The change in the inclination of the strips in the half-tone space-time diagrams, which are employed to determine the oscillation propagation velocities along coronal loops, is caused by the projection effect as opposed to real changes in the velocity. We propose to use flare modulation of the natural oscillations of the medium to eliminate the uncertainties that arise while measuring the phase differences between signals of the same parameters, which is employed for estimating wave propagation velocities in the solar atmosphere.

Наблюдательные характеристики колебательно-волновых процессов в пятне и его окрестностях. Сложности наблюдений и интерпретации

This paper summarizes the body of work that we have done over the years on the oscillation processes in sunspots, including their umbra, penumbra, and close vicinity. The study analyzes a number of aspects that impede adequate determining of some characteristics of propagating oscillations and lead to misinterpretation. Using running penumbral waves as an example, we show that their horizontal propagation with decreasing frequency is delusive. The effect is due to different oscillations propagating along magnetic field lines with gradually increasing inclination. This also applies to the three-minute oscillations in the sunspot umbral chromosphere. The change in the inclination of the strips in the half-tone space-time diagrams, which are employed to determine the oscillation propagation velocities along coronal loops, is caused by the projection effect as opposed to real changes in the velocity. We propose to use flare modulation of the natural oscillations of the medium to eliminate the uncertainties that arise while measuring the phase differences between signals of the same parameters, which is employed for estimating wave propagation velocities in the solar atmosphere.

Solar Type J Radio Bursts and the Associated Coronal Loop

The solar type J radio burst is a variant of type III bursts, which are a probe for understanding solar energetic electrons and local electron density. This study investigates a type J burst event on 2017 September 9. We have combined the data from the extreme-ultraviolet (EUV) imaging and the EUV Imaging Spectrometer (EIS) to analyze the event. Within 4 minutes several type J bursts with similar morphology occur. Two of them, with clear fundamental and second harmonic bands, are studied in detail. We find a delay of 2 ± 0.5 s between their different harmonic bands. During type J bursts, only one coronal loop brightens significantly at its northern footpoint, in correlation with the continuous injection of erupting jets into the loop. The EUV intensity of the brightening footpoint is correlated with the radio flux at 245 and 410 MHz, with correlation coefficients of 0.2 and 0.4, respectively. These observations suggest that the type J bursts should originate from this coronal loop. By analyzing the electron number density distribution along the coronal loop diagnosed from the EIS data and the time evolution of the plasma frequency calculated from the type J burst, we determine that the velocities of the energetic electrons exciting the two type Js are 0.10 ± 0.02c and 0.12 ± 0.02c. Our results confirm previous studies on type J bursts.

Anomalous Emission from Li- and Na-like Ions in the Corona Heated via Alfvén Waves

The solar ultraviolet intensities of spectral lines originating from Li- and Na-like ions have been observed to surpass the expectations derived from plasmas with coronal approximation. The violation of the coronal approximation can be partially attributed to nonequilibrium ionization (NEI) due to dynamic processes occurring in the vicinity of the transition region. To investigate the impact of these dynamics in the Alfvén wave-heated coronal loop, a set of equations governing NEI for multiple ion species was solved numerically in conjunction with 1.5-dimensional magnetohydrodynamic equations. Following the injection of Alfvén waves from the photosphere, the system undergoes a time evolution characterized by phases of evaporation, condensation, and quasi-steady states. During the evaporation phase, the ionization fractions of Li- and Na-like ions were observed to increase when compared to the fractions in ionization equilibrium, which led to an enhancement in the intensity of up to 1.6. This over-fractionation of Li- and Na-like ions was found to be induced by the evaporation process. While collisions between shocks and the transition region temporarily led to deviations from ionization equilibrium, on average over time, these deviations were negligible. Conversely, under-fractions of the ionization fraction led to a reduction in intensity down to 0.9 during the condensation phase and the quasi-steady state. Given the dependency of the over/under-fractionation on mass circulations between the chromosphere and the corona, these observations will serve as valuable benchmarks to validate not only Alfvén wave models but also other existing mechanisms on coronal heating.

Damped kink motions in a system of two solar coronal tubes with elliptic cross sections

This study is motivated by observations of coordinated transverse displacements in neighboring solar active region loops, addressing specifically how the behavior of kink motions in straight two-tube equilibria is impacted by tube interactions and tube cross-sectional shapes. We worked with linear, ideal, pressureless magnetohydrodynamics. Axially standing kink motions were examined as an initial value problem for transversely structured equilibria involving two identical, field-aligned, density-enhanced tubes with elliptic cross sections (elliptic tubes). Continuously nonuniform layers were implemented around both tube boundaries. We numerically followed the system response to external velocity drivers, largely focusing on the quasi-mode stage of internal flows to derive the pertinent periods and damping times. The periods and damping times that we derive for two-circular-tube setups justify the available modal results found with the T-matrix approach. Regardless of cross-sectional shapes, our nonuniform layers feature the development of small-scale shears and energy accumulation around resonances, indicative of resonant absorption and phase mixing. As with two-circular-tube systems, our configurational symmetries still make it possible to classify lower-order kink motions by the polarization and symmetric properties of the internal flows; hence, such motions are labeled as $S_x$ and $A_x$. However, the periods and damping times for two-elliptic-tube setups further depend on cross-sectional aspect ratios, with $A_x$ motions occasionally damped less rapidly than $S_x$ motions. We find uncertainties up to $ 20&lt;!PCT!&gt;$ ($ 50&lt;!PCT!&gt;$) for the axial time (the inhomogeneity lengthscale) if the periods (damping times) computed for two-elliptic-tube setups are seismologically inverted with canonical theories for isolated circular tubes. The effects of loop interactions and cross-sectional shapes need to be considered when the periods, and in particular the damping times, are seismologically exploited for coordinated transverse displacements in adjacent coronal loops.

Particle trapping and acceleration in turbulent post-flare coronal loops

ABSTRACT We present a study of energetic-electron trapping and acceleration in the Kelvin–Helmholtz-induced magnetohydrodynamic (MHD) turbulence of post-flare loops in the solar corona. Using the particle-tracing capabilities of MPI-AMRVAC 3.0, we evolve ensembles of test electrons (i.e. without feedback to the underlying MHD) inside the turbulent looptop, using the guiding-centre approximation. With the MHD looptop model of Ruan et al., we investigate the relation between turbulence and particle trapping inside the looptop structure, showing that better-developed turbulent cascades result in more efficient trapping primarily due to mirror effects. We then quantify the electron acceleration in the time-evolving MHD turbulence, and find that ideal-MHD processes inside the looptop can produce non-thermal particle spectra from an initial Maxwellian distribution. Electrons in this turbulence are preferentially accelerated by mirror effects in the direction perpendicular to the local magnetic field while remaining confined within small regions of space between magnetic islands. Assuming dominance of Bremsstrahlung radiation mechanisms, we employ the resulting information from accelerated electrons (combined with the MHD background) to construct HXR spectra of the post-flare loop that include non-thermal-particle contributions. Our results pave the way to constructing more realistic simulations of radiative coronal structure for comparison with current and future observations.

Multithermal apparent damping of slow waves due to strands with a Gaussian temperature distribution

Context. Slow waves in solar coronal loops are strongly damped, but the current theory of damping by thermal conduction cannot explain some observational features. Aims. We investigated the propagation of slow waves in a coronal loop built up from strands of different temperatures. Methods. We considered the loop to have a multithermal, Gaussian temperature distribution. The different propagation speeds in different strands led to a multithermal apparent damping of the wave, similar to observational phase mixing. We used an analytical model to predict the damping length and propagation speed for the slow waves, including in imaging with filter telescopes. Results. We compared the damping length due to this multithermal apparent damping with damping due to thermal conduction and found that the multithermal apparent damping is more important for shorter period slow waves. We quantified the influence of instrument filters on the wave’s propagation speed and damping. This allowed us to compare our analytical theory to forward models of numerical simulations. Conclusions. We find that our analytical model matches the numerical simulations very well. Moreover, we offer an outlook for using the slow wave properties to infer the loop’s thermal properties.

Non-reflective Propagation of Kink Waves in Magnetic-Flux Tubes in the Solar Atmosphere

We study the non-reflective propagation of kink waves in inhomogeneous magnetic-flux tubes. We use the thin-tube and zero-beta plasma approximations. The wave equation with the variable velocity is reduced to the Euler–Poisson–Darboux equation. This equation contains one dimensionless parameter. There are two infinite sequences of this parameter, one monotonically increasing and the other monotonically decreasing, when exact analytical solutions for the Euler–Poisson–Darboux equation can be obtained. For the monotonically increasing sequences the Euler–Poisson–Darboux equation becomes the equation describing spherically symmetric waves in multi-dimensional spaces. The general results are applied to kink-wave propagation in coronal magnetic loops. We consider a coronal magnetic loop of a half-circular shape. We find that for a fixed loop height there is a one-parametric family of dependences of the loop cross-sectional radius on the coordinate along the loop corresponding to the non-reflective kink-wave propagation.

Influence of Ionization on the Polytropic Index of the Solar Atmosphere within Local Thermodynamic Equilibrium Approximation

An initial theoretical attempt to explain the observed decrease of the polytropic/adiabatic index γ in the solar corona has been accomplished. The chemical reactions of the ionization–recombination processes in local thermodynamic equilibrium (LTE) of a solar plasma cocktail containing heavy elements are found to cause 1.1 < γ ≤ 5/3 in the quiet solar atmosphere. It is also shown that the quiet solar atmosphere is in LTE, justifying this theoretical study. This result is obtained by numerically solving the Saha equation and subsequently using a newly derived equation for calculation of the polytropic index from thermodynamic partial derivatives of the enthalpy and pressure with respect to density and temperature. In addition, a comparison measured from spectroscopic observations of propagating slow magnetohydrodynamic waves in coronal loops shows that LTE ionization accounts for a very small part of the observed decrease of γ, meaning that the solar plasma in the active region is not in LTE as expected. However, the observed dependency of higher polytropic index at higher temperatures is confirmed by the current theoretical approach. It is concluded that in order to account for the polytropic index decrease in the active regions of the solar corona, it is necessary for kinetic non-LTE ionization calculations to be performed.