FINMHD: an adaptive finite element code for magnetic reconnection and   plasmoid chains formation in Magnetohydrodynamics

Hubert Baty

arXiv:1904.11173·astro-ph.HE·August 7, 2019

FINMHD: an adaptive finite element code for magnetic reconnection and plasmoid chains formation in Magnetohydrodynamics

Hubert Baty

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TL;DR

This paper introduces FINMHD, an adaptive finite element code designed to simulate magnetic reconnection and plasmoid formation in magnetohydrodynamics, providing insights into fast energy release in astrophysical plasmas.

Contribution

The paper presents a new adaptive finite element code specifically developed for 2D dissipative MHD simulations of magnetic reconnection and plasmoid chain formation.

Findings

01

Transition from steady-state to plasmoid-dominated reconnection with increasing Lundquist number

02

Code successfully captures formation of current sheets and magnetic islands

03

Adaptive mesh enhances resolution of small-scale structures in simulations

Abstract

Solving the problem of fast eruptive events in magnetically dominated astrophysical plasmas requires the use of particularly well adapted numerical tools. Indeed, the central mechanism based on magnetic reconnection is determined by a complex behavior with quasi-singular forming current layers enriched by their associated small scale magnetic islands called plasmoids. A new code for the solution of two dimensional dissipative magnetohydrodynamics (MHD) equations in cartesian geometry specifically developed to this end is thus presented. A current-vorticity formulation representative of an incompressible model is chosen in order to follow the formation of the current sheets and the ensuing magnetic reconnection process. A finite element discretization using triangles with quadratic basis functions on an unstructured grid is employed, and implemented via a highly adaptive characteristic-…

Figures31

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Tables17

Table 1. Table 1: Additional AAS T e X symbols

$≲$	`\lesssim`, `\la`	$≳$	`\gtrsim`, `\ga`
µm	`\micron`	−	`\sbond`
=	`\dbond`	≡	`\tbond`
$☉$	`\sun`	$\oplus$	`\earth`
⌀	`\diameter`
°	`\arcdeg`, `\degr`	□	`\sq`
′	`\arcmin`	″	`\arcsec`
$\overset{d}{.}$	`\fd`	$\overset{h}{.}$	`\fh`
$\overset{m}{.}$	`\fm`	$\overset{s}{.}$	`\fs`
$\overset{\circ}{.}$	`\fdg`	$\overset{'}{.}$	`\farcm`
$\overset{''}{.}$	`\farcs`	$\overset{p}{.}$	`\fp`
½	`\onehalf`	UBVR	`\ubvr`
⅓	`\onethird`	U B	`\ub`
⅔	`\twothirds`	B V	`\bv`
¼	`\onequarter`	V R	`\vr`
¾	`\threequarters`	U R	`\ur`

Table 2. Table 2: Text-mode accents

ò	`\‘{o}`	ō	`\={o}`	o͡o	`\t{oo}`
ó	`\’{o}`	ȯ	`\.{o}`	o̧	`\c{o}`
ô	`\^{o}`	ŏ	`\u{o}`	ọ	`\d{o}`
ö	`\"{o}`	ǒ	`\v{o}`	o̱	`\b{o}`
õ	`\~{o}`	ő	`\H{o}`

Table 3. Table 3: National symbols

œ	`\oe`	å	`\aa`	ł	`\l`
Œ	`\OE`	Å	`\AA`	Ł	`\L`
æ	`\ae`	ø	`\o`	ß	`\ss`
Æ	`\AE`	Ø	`\O`

Table 4. Table 4: Math-mode accents

$\hat{a}$	`\hat{a}`	$\dot{a}$	`\dot{a}`
$\overset{ˇ}{a}$	`\check{a}`	$\ddot{a}$	`\ddot{a}`
$\tilde{a}$	`\tilde{a}`	$\overset{˘}{a}$	`\breve{a}`
$\overset{´}{a}$	`\acute{a}`	$\bar{a}$	`\bar{a}`
$\overset{`}{a}$	`\grave{a}`	$\vec{a}$	`\vec{a}`

Table 5. Table 5: Greek and Hebrew letters (math mode)

$α$	`\alpha`	$ν$	`\nu`
$β$	`\beta`	$ξ$	`\xi`
$γ$	`\gamma`	$o$	`o`
$δ$	`\delta`	$π$	`\pi`
$ϵ$	`\epsilon`	$ρ$	`\rho`
$ζ$	`\zeta`	$σ$	`\sigma`
$η$	`\eta`	$τ$	`\tau`
$θ$	`\theta`	$υ$	`\upsilon`
$ι$	`\iota`	$ϕ$	`\phi`
$κ$	`\kappa`	$χ$	`\chi`
$λ$	`\lambda`	$ψ$	`\psi`
$μ$	`\mu`	$ω$	`\omega`
$ϝ$	`\digamma`	$ϰ$	`\varkappa`
$ε$	`\varepsilon`	$ς$	`\varsigma`
$ϑ$	`\vartheta`	$φ$	`\varphi`
$ϱ$	`\varrho`
$Γ$	`\Gamma`	$Σ$	`\Sigma`
$Δ$	`\Delta`	$Υ$	`\Upsilon`
$Θ$	`\Theta`	$Φ$	`\Phi`
$Λ$	`\Lambda`	$Ψ$	`\Psi`
$Ξ$	`\Xi`	$Ω$	`\Omega`
$Π$	`\Pi`
$ℵ$	`\aleph`	$ℶ$	`\beth`
$ℷ$	`\gimel`	$ℸ$	`\daleth`

Table 6. Table 6: Binary operators (math mode)

$\pm$	`\pm`	$\cap$	`\cap`
$\mp$	`\mp`	$\cup$	`\cup`
$∖$	`\setminus`	$⊎$	`\uplus`
$\cdot$	`\cdot`	$⊓$	`\sqcap`
$\times$	`\times`	$⊔$	`\sqcup`
$*$	`\ast`	$◁$	`\triangleleft`
$⋆$	`\star`	$▷$	`\triangleright`
$⋄$	`\diamond`	$≀$	`\wr`
$\circ$	`\circ`	$○$	`\bigcirc`
$∙$	`\bullet`	$△$	`\bigtriangleup`
$\div$	`\div`	$▽$	`\bigtriangledown`
$⊲$	`\lhd`	$⊳$	`\rhd`
$\lor$	`\vee`	$⊙$	`\odot`
$\land$	`\wedge`	$†$	`\dagger`
$\oplus$	`\oplus`	$‡$	`\ddagger`
$⊖$	`\ominus`	$∐$	`\amalg`
$\otimes$	`\otimes`	$⊴$	`\unlhd`
$⊘$	`\oslash`	$⊵$	`\unrhd`

Table 7. Table 7: AMS binary operators (math mode)

$∔$	`\dotplus`	$⋉$	`\ltimes`
$∖$	`\smallsetminus`	$⋊$	`\rtimes`
$⋒$	`\Cap`, `\doublecap`	$⋋$	`\leftthreetimes`
$⋓$	`\Cup`, `\doublecup`	$⋌$	`\rightthreetimes`
$⊼$	`\barwedge`	$⋏$	`\curlywedge`
$⊻$	`\veebar`	$⋎$	`\curlyvee`
$⩞$	`\doublebarwedge`
$⊟$	`\boxminus`	$⊝$	`\circleddash`
$⊠$	`\boxtimes`	$⊛$	`\circledast`
$⊡$	`\boxdot`	$⊚$	`\circledcirc`
$⊞$	`\boxplus`	$∙$	`\centerdot`
$⋇$	`\divideontimes`	$⊺$	`\intercal`

Table 8. Table 8: Miscellaneous symbols

†	`\dag`	§	`\S`
©	`\copyright`	‡	`\ddag`
¶	`\P`	£	`\pounds`
#	`\#`	$	`\$`
%	`\%`	&	`\&`
_	`\_`	{	`\{`
}	`\}`

Table 9. Table 9: Miscellaneous symbols (math mode)

$ℵ$	`\aleph`	$'$	`\prime`
$ℏ$	`\hbar`	$\emptyset$	`\emptyset`
$ı$	`\imath`	$\nabla$	`\nabla`
$ȷ$	`\jmath`	$\sqrt$	`\surd`
$ℓ$	`\ell`	$⊤$	`\top`
$℘$	`\wp`	$⊥$	`\bot`
$ℜ$	`\Re`	$∥$	`\\|`
$ℑ$	`\Im`	$∠$	`\angle`
$\partial$	`\partial`	$△$	`\triangle`
$\infty$	`\infty`	$\$	`\backslash`
$□$	`\Box`	$◇$	`\Diamond`
$\forall$	`\forall`	$♯$	`\sharp`
$\exists$	`\exists`	$♣$	`\clubsuit`
$\neg$	`\neg`	$♢$	`\diamondsuit`
$♭$	`\flat`	$♡$	`\heartsuit`
$♮$	`\natural`	$♠$	`\spadesuit`
$℧$	`\mho`

Table 10. Table 10: AMS miscellaneous symbols (math mode)

$ℏ$	`\hbar`	$‵$	`\backprime`
$ℏ$	`\hslash`	$\emptyset$	`\varnothing`
$△$	`\vartriangle`	$▲$	`\blacktriangle`
$▽$	`\triangledown`	$▼$	`\blacktriangledown`
$□$	`\square`	$■$	`\blacksquare`
$◆$	`\lozenge`	$◆$	`\blacklozenge`
$Ⓢ$	`\circledS`	$★$	`\bigstar`
$∠$	`\angle`	$∢$	`\sphericalangle`
$∡$	`\measuredangle`
$∄$	`\nexists`	$∁$	`\complement`
$℧$	`\mho`	$ð$	`\eth`
$Ⅎ$	`\Finv`	$╱$	`\diagup`
$⅁$	`\Game`	$╲$	`\diagdown`
$𝕜$	`\Bbbk`	$↾$	`\restriction`

Table 11. Table 11: Arrows (math mode)

$\leftarrow$	`\leftarrow`	$⟵$	`\longleftarrow`
$\Leftarrow$	`\Leftarrow`	$⟸$	`\Longleftarrow`
$\to$	`\rightarrow`	$⟶$	`\longrightarrow`
$\Rightarrow$	`\Rightarrow`	$⟹$	`\Longrightarrow`
$\leftrightarrow$	`\leftrightarrow`	$⟷$	`\longleftrightarrow`
$\Leftrightarrow$	`\Leftrightarrow`	$⟺$	`\Longleftrightarrow`
$\mapsto$	`\mapsto`	$⟼$	`\longmapsto`
$↩$	`\hookleftarrow`	$↪$	`\hookrightarrow`
$↼$	`\leftharpoonup`	$⇀$	`\rightharpoonup`
$↽$	`\leftharpoondown`	$⇁$	`\rightharpoondown`
$⇌$	`\rightleftharpoons`	$↝$	`\leadsto`
$↑$	`\uparrow`	$⇕$	`\Updownarrow`
$⇑$	`\Uparrow`	$↗$	`\nearrow`
$↓$	`\downarrow`	$↘$	`\searrow`
$⇓$	`\Downarrow`	$↙$	`\swarrow`
$↕$	`\updownarrow`	$↖$	`\nwarrow`

Table 12. Table 12: AMS arrows (math mode)

$⇠$	`\dashleftarrow`	$⇢$	`\dashrightarrow`
$⇇$	`\leftleftarrows`	$⇉$	`\rightrightarrows`
$⇆$	`\leftrightarrows`	$⇄$	`\rightleftarrows`
$⇚$	`\Lleftarrow`	$⇛$	`\Rrightarrow`
$↞$	`\twoheadleftarrow`	$↠$	`\twoheadrightarrow`
$↢$	`\leftarrowtail`	$↣$	`\rightarrowtail`
$↫$	`\looparrowleft`	$↬$	`\looparrowright`
$⇋$	`\leftrightharpoons`	$⇌$	`\rightleftharpoons`
$↶$	`\curvearrowleft`	$↷$	`\curvearrowright`
$↺$	`\circlearrowleft`	$↻$	`\circlearrowright`
$↰$	`\Lsh`	$↱$	`\Rsh`
$⇈$	`\upuparrows`	$⇊$	`\downdownarrows`
$↿$	`\upharpoonleft`	$↾$	`\upharpoonright`
$⇃$	`\downharpoonleft`	$⇂$	`\downharpoonright`
$⊸$	`\multimap`	$↝$	`\rightsquigarrow`
$↭$	`\leftrightsquigarrow`
$↚$	`\nleftarrow`	$↛$	`\nrightarrow`
$⇍$	`\nLeftarrow`	$⇏$	`\nRightarrow`
$↮$	`\nleftrightarrow`	$⇎$	`\nLeftrightarrow`

Table 13. Table 13: Relations (math mode)

$\leq$	`\leq`	$\geq$	`\geq`
$≺$	`\prec`	$≻$	`\succ`
$⪯$	`\preceq`	$⪰$	`\succeq`
$≪$	`\ll`	$≫$	`\gg`
$\subset$	`\subset`	$\supset$	`\supset`
$\subseteq$	`\subseteq`	$\supseteq$	`\supseteq`
$⊏$	`\sqsubset`	$⊐$	`\sqsupset`
$⊑$	`\sqsubseteq`	$⊒$	`\sqsupseteq`
$\in$	`\in`	$∋$	`\ni`
$⊢$	`\vdash`	$⊣$	`\dashv`
$⌣$	`\smile`	$∣$	`\mid`
$⌢$	`\frown`	$∥$	`\parallel`
$\neq$	`\neq`	$⟂$	`\perp`
$\equiv$	`\equiv`	$≅$	`\cong`
$\sim$	`\sim`	$⋈$	`\bowtie`
$≃$	`\simeq`	$\propto$	`\propto`
$≍$	`\asymp`	$⊧$	`\models`
$\approx$	`\approx`	$≐$	`\doteq`
		$⨝$	`\Join`

Table 14. Table 14: AMS binary relations (math mode)

$≦$	`\leqq`	$≧$	`\geqq`
$⩽$	`\leqslant`	$⩾$	`\geqslant`
$⪕$	`\eqslantless`	$⪖$	`\eqslantgtr`
$≲$	`\lesssim`	$≳$	`\gtrsim`
$⪅$	`\lessapprox`	$⪆$	`\gtrapprox`
$≊$	`\approxeq`	$≂$	`\eqsim`
$⋖$	`\lessdot`	$⋗$	`\gtrdot`
$⋘$	`\lll`, `\llless`	$⋙$	`\ggg`, `\gggtr`
$≶$	`\lessgtr`	$≷$	`\gtrless`
$⋚$	`\lesseqgtr`	$⋛$	`\gtreqless`
$⪋$	`\lesseqqgtr`	$⪌$	`\gtreqqless`
$≑$	`\doteqdot`, `\Doteq`	$≖$	`\eqcirc`
$≓$	`\risingdotseq`	$≗$	`\circeq`
$≒$	`\fallingdotseq`	$≜$	`\triangleq`
$∽$	`\backsim`	$\sim$	`\thicksim`
$≌$	`\backsimeq`	$\approx$	`\thickapprox`
$⫅$	`\subseteqq`	$⫆$	`\supseteqq`
$⋐$	`\Subset`	$⋑$	`\Supset`
$⊏$	`\sqsubset`	$⊐$	`\sqsupset`
$≼$	`\preccurlyeq`	$≽$	`\succcurlyeq`
$⋞$	`\curlyeqprec`	$⋟$	`\curlyeqsucc`
$≾$	`\precsim`	$≿$	`\succsim`
$⪷$	`\precapprox`	$⪸$	`\succapprox`
$⊲$	`\vartriangleleft`	$⊳$	`\vartriangleright`
$⊴$	`\trianglelefteq`	$⊵$	`\trianglerighteq`
$⊨$	`\vDash`	$⊩$	`\Vdash`
$⊪$	`\Vvdash`
$⌣$	`\smallsmile`	$∣$	`\shortmid`
$⌢$	`\smallfrown`	$∥$	`\shortparallel`
$≏$	`\bumpeq`	$≬$	`\between`
$≎$	`\Bumpeq`	$⋔$	`\pitchfork`
$\propto$	`\varpropto`	$϶$	`\backepsilon`
$◀$	`\blacktriangleleft`	$▶$	`\blacktriangleright`
$∴$	`\therefore`	$∵$	`\because`

Table 15. Table 15: AMS negated relations (math mode)

$≮$	`\nless`	$≯$	`\ngtr`
$≰$	`\nleq`	$≱$	`\ngeq`
$⩽̸$	`\nleqslant`	$⩾̸$	`\ngeqslant`
$≦̸$	`\nleqq`	$≧̸$	`\ngeqq`
$⪇$	`\lneq`	$⪈$	`\gneq`
$≨$	`\lneqq`	$≩$	`\gneqq`
$≨$	`\lvertneqq`	$≩$	`\gvertneqq`
$⋦$	`\lnsim`	$⋧$	`\gnsim`
$⪉$	`\lnapprox`	$⪊$	`\gnapprox`
$⊀$	`\nprec`	$⊁$	`\nsucc`
$⋠$	`\npreceq`	$⋡$	`\nsucceq`
$⪵$	`\precneqq`	$⪶$	`\succneqq`
$⋨$	`\precnsim`	$⋩$	`\succnsim`
$⪹$	`\precnapprox`	$⪺$	`\succnapprox`
$≁$	`\nsim`	$≇$	`\ncong`
$∤$	`\nshortmid`	$∦$	`\nshortparallel`
$∤$	`\nmid`	$∦$	`\nparallel`
$⊬$	`\nvdash`	$⊭$	`\nvDash`
$⊮$	`\nVdash`	$⊯$	`\nVDash`
$⋪$	`\ntriangleleft`	$⋫$	`\ntriangleright`
$⋬$	`\ntrianglelefteq`	$⋭$	`\ntrianglerighteq`
$⊈$	`\nsubseteq`	$⊉$	`\nsupseteq`
$⫅̸$	`\nsubseteqq`	$⫆̸$	`\nsupseteqq`
$⊊$	`\subsetneq`	$⊋$	`\supsetneq`
$⊊$	`\varsubsetneq`	$⊋$	`\varsupsetneq`
$⫋$	`\subsetneqq`	$⫌$	`\supsetneqq`
$⫋$	`\varsubsetneqq`	$⫌$	`\varsupsetneqq`

Table 16. Table 16: Variable-sized symbols (math mode)

$\begin{matrix} \sum & \sum & \sum & ⋂ & ⋂ & \bigcap \\ \prod & \prod & \prod & ⋃ & ⋃ & \bigcup \\ ∐ & ∐ & \coprod & ⨆ & ⨆ & \bigsqcup \\ \int & \int & \int & ⋁ & ⋁ & \bigvee \\ \oint & \oint & \oint & ⋀ & ⋀ & \bigwedge \\ ⨀ & ⨀ & \bigodot & ⨂ & ⨂ & \bigotimes \\ ⨁ & ⨁ & \bigoplus & ⨄ & ⨄ & \biguplus \end{matrix}$

Table 17. Table 17: Delimiters (math mode)

$($	`(`	$)$	`)`
$[$	`[`	$]$	`]`
${$	`\{`	$}$	`\}`
$⌊$	`\lfloor`	$⌋$	`\rfloor`
$⌈$	`\lceil`	$⌉$	`\rceil`
$⟨$	`\langle`	$⟩$	`\rangle`
$/$	`/`	$\$	`\backslash`
$\|$	`\vert`	$∥$	`\Vert`
$↑$	`\uparrow`	$⇑$	`\Uparrow`
$↓$	`\downarrow`	$⇓$	`\Downarrow`
$↕$	`\updownarrow`	$⇕$	`\Updownarrow`
$⌜$	`\ulcorner`	$⌝$	`\urcorner`
$⌞$	`\llcorner`	$⌟$	`\lrcorner`

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