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Additivity of relative magnetic helicity in finite volumes  

Gherardo Valori   Submitted: 2020-08-04 02:51

Relative magnetic helicity is conserved by magneto-hydrodynamic evolution even in the presence of moderate resistivity. For that reason, it is often invoked as the most relevant constraint to the dynamical evolution of plasmas in complex systems, such as solar and stellar dynamos, photospheric flux emergence, solar eruptions, and relaxation processes in laboratory plasmas. However, such studies often indirectly imply that relative magnetic helicity in a given spatial domain can be algebraically split into the helicity contributions of the composing subvolumes, i.e., that it is an additive quantity. A limited number of very specific applications have shown that this is not the case. Progress in understanding the non-additivity of relative magnetic helicity requires removal of restrictive assumptions in favour of a general formalism that can be used both in theoretical investigations as well as in numerical applications. We derive the analytical gauge-invariant expression for the partition of relative magnetic helicity between contiguous finite-volumes, without any assumptions on either the shape of the volumes and interface, or the employed gauge. he non-additivity of relative magnetic helicity in finite volumes is proven in the most general, gauge-invariant formalism, and verified numerically. More restrictive assumptions are adopted to derive known specific approximations, yielding a unified view of the additivity issue. As an example, the case of a flux rope embedded in a potential field shows that the non-additivity term in the partition equation is, in general, non-negligible. The non-additivity of relative magnetic helicity can potentially be a serious impediment to the application of relative helicity conservation as a constraint to the complex dynamics of magnetized plasmas. The relative helicity partition formula can be applied to numerical simulations to precisely quantify the effect of non-additivity on global helicity budgets of complex physical processes.

Authors: Gherardo Valori, Pascal Démoulin, Etienne Pariat, Anthony Yeates, Kostas Moraitis and Luis Linan
Projects: None

Publication Status: Accepted for publication in Astronomy & Astrophysics
Last Modified: 2020-08-05 14:12
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The 17 February 2013 sunquake in the context of the active region's magnetic field configuration  

Gherardo Valori   Submitted: 2017-09-15 00:31

Sunquakes are created by the hydrodynamic response of the lower atmosphere to a sudden deposition of energy and momentum. In this study we investigate a sunquake that occurred in NOAA active region 11675 on 17 February 2013. Observations of the corona, chromosphere and photosphere are brought together for the first time with a non-linear force-free model of the active region's magnetic field in order to probe the magnetic environment in which the sunquake was initiated. We find that the sunquake was associated with the destabilization of a flux rope and an associated M-class GOES flare. Active region 11675 was in its emergence phase at the time of the sunquake and photospheric motions caused by the emergence heavily modified the flux rope and its associated quasi-separatrix layers, eventually triggering the flux rope's instability. The flux rope was surrounded by an extended envelope of field lines rooted in a small area at the approximate position of the sunquake. We argue that the configuration of the envelope, by interacting with the expanding flux rope, created a "magnetic lens" that may have focussed energy in one particular location the photosphere, creating the necessary conditions for the initiation of the sunquake.

Authors: Lucie M. Green, Gherardo Valori, Francesco P. Zuccarello, Sergei Zharkov, Sarah Matthews, Salvo L. Guglielmino

Publication Status: Accepted
Last Modified: 2017-09-15 10:13
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Magnetic Helicity Estimations in Models and Observations of the Solar Magnetic Field. Part III: Twist Number Method  

Gherardo Valori   Submitted: 2017-03-31 03:38

We study the writhe, twist and magnetic helicity of different magnetic flux ropes, based on models of the solar coronal magnetic field structure. These include an analytical force-free Titov"D"moulin equilibrium solution, non force-free magnetohydrodynamic simulations, and nonlinear force-free magnetic field models. The geometrical boundary of the magnetic flux rope is determined by the quasi-separatrix layer and the bottom surface, and the axis curve of the flux rope is determined by its overall orientation. The twist is computed by the Berger?Prior formula that is suitable for arbitrary geometry and both force-free and non-force-free models. The magnetic helicity is estimated by the twist multiplied by the square of the axial magnetic flux. We compare the obtained values with those derived by a finite volume helicity estimation method. We find that the magnetic helicity obtained with the twist method agrees with the helicity carried by the purely current-carrying part of the field within uncertainties for most test cases. It is also found that the current-carrying part of the model field is relatively significant at the very location of the magnetic flux rope. This qualitatively explains the agreement between the magnetic helicity computed by the twist method and the helicity contributed purely by the current-carrying magnetic field.

Authors: Y. Guo, E. Pariat , G. Valori , S. Anfinogentov , F. Chen , M. Georgoulis , Y. Liu , K. Moraitis , J. K. Thalmann , S. Yang
Projects: None

Publication Status: Accepted in A&A
Last Modified: 2017-04-02 19:07
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Magnetic helicity estimations in models and observations of the solar magnetic field. Part I: Finite volume methods  

Gherardo Valori   Submitted: 2016-10-10 03:26

Magnetic helicity is a conserved quantity of ideal magneto-hydrodynamics characterized by an inverse turbulent cascade. Accordingly, it is often invoked as one of the basic physical quantities driving the generation and structuring of magnetic fields in a variety of astrophysical and laboratory plasmas. We provide here the first systematic comparison of six existing methods for the estimation of the helicity of magnetic fields known in a finite volume. All such methods are reviewed, benchmarked, and compared with each other, and specifically tested for accuracy and sensitivity to errors. To that purpose, we consider four groups of numerical tests, ranging from solutions of the three-dimensional, force-free equilibrium, to magneto-hydrodynamical numerical simulations. Almost all methods are found to produce the same value of magnetic helicity within few percent in all tests. In the more solar-relevant and realistic of the tests employed here, the simulation of an eruptive flux rope, the spread in the computed values obtained by all but one method is only 3%, indicating the reliability and mutual consistency of such methods in appropriate parameter ranges. However, methods show differences in the sensitivity to numerical resolution and to errors in the solenoidal property of the input fields. In addition to finite volume methods, we also briefly discuss a method that estimates helicity from the field lines' twist, and one that exploits the field's value at one boundary and a coronal minimal connectivity instead of a pre-defined three-dimensional magnetic-field solution.

Authors: Gherardo Valori, Etienne Pariat, Sergey Anfinogentov, Feng Chen, Manolis K. Georgoulis, Yang Guo, Yang Liu, Kostas Moraitis, Julia K. Thalmann, Shangbin Yang
Projects: None

Publication Status: Accepted in Space Science Review
Last Modified: 2016-10-12 12:24
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Accuracy of magnetic energy computations  

Gherardo Valori   Submitted: 2013-04-03 03:36

For magnetically driven events, the magnetic energy of the system is the prime energy reservoir that fuels the dynamical evolution. In the solar context, the free energy (i.e., the energy in excess of the potential field energy) is one of the main indicators used in space weather forecasts to predict the eruptivity of active regions. A trustworthy estimation of the magnetic energy is therefore needed in three-dimensional (3D) models of the solar atmosphere, e.g., in coronal fields reconstructions or numerical simulations. The expression of the energy of a system as the sum of its potential energy and its free energy (Thomson?s theorem) is strictly valid when the magnetic field is exactly solenoidal. For numerical realizations on a discrete grid, this property may be only approximately fulfilled. We show that the imperfect solenoidality induces terms in the energy that can lead to misinterpreting the amount of free energy present in a magnetic configuration. We consider a decomposition of the energy in solenoidal and nonsolenoidal parts which allows the unambiguous estimation of the nonsolenoidal contribution to the energy. We apply this decomposition to six typical cases broadly used in solar physics. We quantify to what extent the Thomson theorem is not satisfied when approximately solenoidal fields are used. The quantified errors on energy vary from negligible to significant errors, depending on the extent of the nonsolenoidal component of the field. We identify the main source of errors and analyze the implications of adding a variable amount of divergence to various solenoidal fields. Finally, we present pathological unphysical situations where the estimated free energy would appear to be negative, as found in some previous works, and we identify the source of this error to be the presence of a finite divergence. We provide a method of quantifying the effect of a finite divergence in numerical fields, together with detailed diag- nostics of its sources. We also compare the efficiency of two divergence-cleaning techniques. These results are applicable to a broad range of numerical realizations of magnetic fields.

Authors: G. Valori, P. Demoulin, E. Pariat, S. Masson
Projects: None

Publication Status: in press
Last Modified: 2013-04-03 12:09
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Comparing Values of the Relative Magnetic Helicity in Finite Volumes  

Gherardo Valori   Submitted: 2012-02-13 08:59

Relative magnetic helicity, as a conserved quantity of idealmagnetohydrodynamics,has been highlighted as an important quantity to study inplasma physics. Due to its non-local nature, its estimation is notstraightforwardin both observational and numerical data. In the present study wederive expressionsfor the practical computation of the gauge-independent relative magnetichelicity in three-dimensional finite domains. The derived expressionsare easyto implement and rapid to compute. They are derived in Cartesiancoordinates,but can be easily written in other coordinate systems. We apply ourmethod to anumerical model of a force-free equilibrium containing a flux rope,and comparethe results with those obtained employing known half-space equations.We findthat our method requires a much smaller volume than half-spaceexpressionsto derive the full helicity content. Additionally, we prove thatvalues of relativemagnetic helicity of different magnetic fields can be compared witheach otherin the same sense as free-energy values can. Therefore, relativemagnetic helicitycan be meaningfully and directly compared between different datasets,such asthose from different active regions, but also the same dataset atdifferent times.Typical applications of our formulae include the helicity computationin threedimensionalmodels of the solar atmosphere, e.g. coronal-field reconstructions byforce-free extrapolation and discretized magnetic fields of numericalsimulations.

Authors: Valori, G., Demoulin, P., Pariat E.

Publication Status: Solar Phys (accepted)
Last Modified: 2012-02-17 05:38
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Testing magnetofrictional extrapolation with the Titov-D?moulin model of solar active regions  

Gherardo Valori   Submitted: 2010-05-04 01:58

We examine the nonlinear magnetofrictional extrapolation scheme using the solar active region model by Titov and D?moulin as test field. This model consists of an arched, line-tied current channel held in force-free equilibrium by the potential field of a bipolar flux distribution in the bottom boundary. A modified version, having a parabolic current density profile, is employed here. We find that the equilibrium is reconstructed with very high accuracy in a representative range of parameter space, using only the vector field in the bottom boundary as input. Structural features formed in the interface between the flux rope and the surrounding arcade-''hyperbolic flux tube'' and ''bald patch separatrix surface''-are reliably reproduced, as are the flux rope twist and the energy and helicity of the configuration. This demonstrates that force-free fields containing these basic structural elements of solar active regions can be obtained by extrapolation. The influence of the chosen initial condition on the accuracy of reconstruction is also addressed, confirming that the initial field that best matches the external potential field of the model quite naturally leads to the best reconstruction. Extrapolating the magnetogram of a Titov-D?moulin equilibrium in the unstable range of parameter space yields a sequence of two opposing evolutionary phases which clearly indicate the unstable nature of the configuration: a partial buildup of the flux rope with rising free energy is followed by destruction of the rope, losing most of the free energy.

Authors: G. Valori, B. Kliem, T. Török, V. S. Titov
Projects: None

Publication Status: A&A (accepted)
Last Modified: 2010-05-04 12:44
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Abstracts by Author
Additivity of relative magnetic helicity in finite volumes
The 17 February 2013 sunquake in the context of the active region's magnetic field configuration
Magnetic Helicity Estimations in Models and Observations of the Solar Magnetic Field. Part III: Twist Number Method
Magnetic helicity estimations in models and observations of the solar magnetic field. Part I: Finite volume methods
Accuracy of magnetic energy computations
Comparing Values of the Relative Magnetic Helicity in Finite Volumes
Testing magnetofrictional extrapolation with the Titov-D?moulin model of solar active regions

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