Use of artificial viscosity in multidimensional fluid dynamic calculations

Wilkins, Mark L.

doi:10.1016/0021-9991(80)90161-8

Cited by 257 publications

(139 citation statements)

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“…LARE3D uses shock viscosity [33] capture the heating effect of shocks, which is implemented by adding an extra term to the energy equation. This term is expressed as the product of the rate of strain tensor, 8) and the shock tensor,…”

Section: (A) Shock Resolutionmentioning

confidence: 99%

Shock heating in numerical simulations of kink-unstable coronal loops

Bareford

Hood

2015

Phil. Trans. R. Soc. A.

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An analysis of the importance of shock heating within coronal magnetic fields has hitherto been a neglected area of study. We present new results obtained from nonlinear magnetohydrodynamic simulations of straight coronal loops. This work shows how the energy released from the magnetic field, following an ideal instability, can be converted into thermal energy, thereby heating the solar corona. Fast dissipation of magnetic energy is necessary for coronal heating and this requirement is compatible with the time scales associated with ideal instabilities. Therefore, we choose an initial loop configuration that is susceptible to the fast-growing kink, an instability that is likely to be created by convectively driven vortices, occurring where the loop field intersects the photosphere (i.e. the loop footpoints). The largescale deformation of the field caused by the kinking creates the conditions for the formation of strong current sheets and magnetic reconnection, which have previously been considered as sites of heating, under the assumption of an enhanced resistivity. However, our simulations indicate that slow mode shocks are the primary heating mechanism, since, as well as creating current sheets, magnetic reconnection also generates plasma flows that are faster than the slow magnetoacoustic wave speed.

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Section: (A) Shock Resolutionmentioning

confidence: 99%

Shock heating in numerical simulations of kink-unstable coronal loops

Bareford

Hood

2015

Phil. Trans. R. Soc. A.

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“…Viscous heating is represented by the last term of Eq. (5), which also incorporates artificial viscosity (Wilkins 1980) in order to capture the heating effect of shocks. This heating term is expressed as the product of the rate of strain tensor,…”

Section: Numerical Codementioning

confidence: 99%

Coronal heating by the partial relaxation of twisted loops

Bareford

Hood

Browning

2013

A&A

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Context. Relaxation theory offers a straightforward method for estimating the energy that is released when continual convective driving causes a magnetic field to become unstable. Thus, an upper limit to the heating caused by ensembles of nanoflaring coronal loops can be calculated and checked against the level of heating required to maintain observed coronal temperatures (T 10 6 K). Aims. We present new results obtained from nonlinear magnetohydrodynamic (MHD) simulations of idealised coronal loops. All of the initial loop configurations discussed are known to be linearly kink unstable. The purpose of this work is to determine whether or not the simulation results agree with Taylor relaxation, which will require a modified version of relaxation theory applicable to unbounded field configurations. In addition, we show for two cases how the relaxation process unfolds. Methods. A three-dimensional (3D) MHD Lagrangian-remap code is used to simulate the evolution of a line-tied cylindrical coronal loop model. This model comprises three concentric layers surrounded by a potential envelope; hence, being twisted locally, each loop configuration is distinguished by a piecewise-constant current profile, featuring three parameters. Initially, all configurations carry zero-net-current fields and are in ideally unstable equilibrium. The simulation results are compared with the predictions of helicityconserving relaxation theory. Results. For all simulations, the change in helicity is no more than 2% of the initial value; also, the numerical helicities match the analytically-determined values. Magnetic energy dissipation predominantly occurs via shock heating associated with magnetic reconnection in distributed current sheets. The energy release and final field profiles produced by the numerical simulations are in agreement with the predictions given by a new model of partial relaxation theory: the relaxed field is close to a linear force free state; however, the extent of the relaxation region is limited, while the loop undergoes some radial expansion. Conclusions. The results presented here support the use of partial relaxation theory, specifically, when calculating the heating-event distributions produced by ensembles of kink-unstable loops. The energy release increases with relaxation radius; but, once the loop has expanded by more than 50%, further expansion yields little more energy. We conclude that the relaxation methodology may be used for coronal heating studies.

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“…Evans and Hawley constrained transport (Evans & Hawley 1988) is used to ensure that the magnetic field remains divergence-free. Shock handling is achieved through Van Leer gradient limiters (Van Leer 1997), so that the code remains TVD, and Wilkins shock viscosity (Wilkins 1980) to ensure the correct entropy jump occurs across a shock. See Arber et al (2001) for more details.…”

Section: Numerical Proceduresmentioning

confidence: 99%

Coronal heating by magnetic reconnection in loops with zero net current

Hood¹,

Browning

Linden

2009

A&A

104

143

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Context. The paper is concerned with heating of the solar corona by nanoflares: a superposition of small transient events in which stored magnetic energy is dissipated by magnetic reconnection. It is proposed that heating occurs in the nonlinear phase of an ideal kink instability, where magnetic reconnection leads to relaxation to a state of minimum magnetic energy. Aims. The aim is to investigate the nonlinear aspects of magnetic relaxation on a current loop with zero net axial current. The dynamical processes leading to the establishment of a relaxed state are explored. The efficiency of loop heating is investigated. Methods. A 3D magnetohydrodynamic numerical code is used to simulate the evolution of coronal loops which are initially in ideally unstable equilibrium. The initial states have zero net current. The results are interpreted by comparison both with linear stability analysis and with helicity-conserving relaxation theory. Results. The disturbance due to the unstable mode is strongly radially confined when the loop carries zero net current. Strong current sheets are still formed in the nonlinear phase with dissipation of magnetic energy by fast reconnection. The nonlinear development consists first of reconnection in a large scale current sheet, which forms near the quasi-resonant surface of the equilibrium field. Subsequently, the current sheet extends and then fragments, leading to multiple reconnections and effective relaxation to a constant α field. Conclusions. Magnetic reconnection is triggered in the nonlinear phase of kink instability in loops with zero net current. Initially, reconnection occurs in a single current sheet, which then fragments into multiple reconnection sites, allowing almost full relaxation to the minimum energy state. The loop is heated to high temperatures throughout its volume.

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Use of artificial viscosity in multidimensional fluid dynamic calculations

Cited by 257 publications

References 1 publication

Shock heating in numerical simulations of kink-unstable coronal loops

Shock heating in numerical simulations of kink-unstable coronal loops

Coronal heating by the partial relaxation of twisted loops

Coronal heating by magnetic reconnection in loops with zero net current

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