Table Of ContentConcomitant Hamiltonian and topological structures of extended
magnetohydrodynamics
Manasvi Lingam,1,2,∗ George Miloshevich,2,† and Philip J. Morrison2,‡
1Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
2Department of Physics and Institute for Fusion Studies,
The University of Texas at Austin, Austin, TX 78712, USA
6 The paper describes the unique geometric properties of ideal magnetohydrodynamics (MHD),
1 and demonstrates how such features are inherited by extended MHD,viz. models that incorporate
0 two-fluideffects(theHalltermandelectroninertia). Thegeneralizedhelicities,andothergeometric
2 expressions for these models are presented in a topological context, emphasizing their universal
y facets. Some of the results presented include: the generalized Kelvin circulation theorems; the
a existence of two Lie-dragged 2-forms; and two concomitant helicities that can be studied via the
M Jones polynomial, which is widely utilized in Chern-Simons theory. The ensuing commonality is
traced to the existence of an underlying Hamiltonian structure for all the extended MHD models,
8 exemplified by thepresence of a uniquenoncanonical Poisson bracket,and its associated energy.
1
] I. INTRODUCTION ity has played a pivotal role in solar physics [37–40], he-
h licity injection [41, 42], reconnection [43, 44], turbulence
p
Ideal magnetohydrodynamics (MHD) is the simplest [45,46]anddynamotheory[46,47]. Weobservethatcon-
-
m model in plasma physics, and is used extensively in the currentapplicationsandusesofhelicityinfusionplasmas
s arenas of fusion, space and astrophysical plasmas; see also abound; some examples are listed in [48].
a e.g. [1] and references therein. From a mathematical
AlthoughMHDisendowedwithseveraluniqueproper-
l
p perspective,idealMHDrepresentsanaturalextensionof ties,itisalsoinapplicableinseveraldomains. Hence,sev-
. idealhydrodynamics(HD)asitisendowedwithgeomet-
s eral extensions of ideal MHD have been studied, such as
c ricpropertiesthatmimicthoseofidealHD.Muchofthis Hall MHD [49], electron MHD [50], and extended MHD
i geometric structure arises from the flux freezing condi-
s [51]. There has been much interest in Hall MHD, as it
y tion, which is intimately linked with the conservation of
possesseshelicitiesandrelaxedstatesakintothatofideal
h magnetic and cross helicities.
MHD [52, 53], and has been widely studied as a model
p It is a widely accepted maxim that topological invari-
of fast reconnection [54]. Hall MHD can be further gen-
[
antsplayakeyroleinseveralareasofphysics. InHD,the
eralizedto include the effects of electron inertia, thereby
2 fluidhelicity playsa similarrole,asit constitutes amea- resulting in extended MHD. Alternatively, a model with
v sureoftheGausslinkingnumberofvortexlines,asshown
electroninertia,butlackingtheHallterms,wasproposed
8 in the pioneering work of [2]. In MHD, an equivalent
in [55, 56] with the accompanying title of inertial MHD.
2 roleisplayedbythe magnetic helicity,whosetopological
1
properties were extensively investigated in [3]. Subse- In this paper, we propose to highlight the common-
0
quently, the topological formulations of HD and MHD, ality of all the extended MHD models through several
0
especiallytheirattendanthelicities,underwentincreasing avenues. These include the delineation of the appro-
.
2
mathematical sophistication; representative examples in priate conserved helicities and the appropriate frozen-in
0
this category include [4–19]. Fluid/magnetic helicities fluxes. Furthermore, we demonstrate that all of these
6
also emerge naturally as a consequence of the underly- models possess a virtually identical Hamiltonian struc-
1
: ingrelabellingsymmetryofHDandMHD,onaccountof ture [57, 58] – the latter refers to the existence of a
v
Noether’s theorem [20, 21]. It is worth noting that the suitable (conserved) energy and a noncanonical Poisson
i
X relativistic version of helicity [22], and its concomitant bracket. SuchPoissonbracketswerefirstconstructedfor
topological properties have been studied in [22–25]. ideal HD and MHD in [59], and are quite different in
r
a We emphasize that this topological nature has come structureas the physicalEulerianfields (suchasdensity,
under greaterexperimentalscrutiny [26–28]demonstrat- velocity, etc.) are not canonical in nature. An extended
ingthathelicityisconvertedfromlinks/knotstocoils. In discussion of these brackets, and the advantages of the
additiontotheimportanceofthesehelicitiesastopologi- Hamiltonian description of fluids/plasmas can be found
calinvariants,theyarealsoindispensableinunderstand- in [60–62].
ingtheself-organizationandrelaxationoffluids/plasmas
The outline of the paper is as follows. The common
[29–36]. In the astrophysical context, (magnetic) helic-
Hamiltonian structure of different extended MHD mod-
els is presented in Section II. In Section III, we sketch
the unifying topological aspects of the various extended
∗ [email protected] MHD models. Finally, we summarize our results in Sec-
† [email protected] tion IV, and indicate how they could play an important
‡ [email protected] role in fusion and astrophysical plasmas.
2
II. HAMILTONIAN STRUCTURE OF we observe that (2) and (3) can be recast into
EXTENDED MHD MODELS
∂V V2 (∇×B)×B⋆
+(∇×V)×V=−∇ h+ +
In this Section, we shall present the dynamical equa- ∂t 2 ρ
(cid:18) (cid:19)
tions of different extended MHD models, demonstrate (∇×B)2
the existence of a common Hamiltonian structure, and −d2∇ , (5)
e 2ρ2
thereby construct the associated helicities and general- " #
ized frozen-in fluxes.
∂B⋆ (∇×B)×B⋆
=∇×(V×B⋆)−d ∇×
i
A. Mathematical preliminaries ∂t ρ
(cid:18) (cid:19)
(∇×B)×(∇×V)
+d2∇× , (6)
We beginwith the equationsofextended MHD,which e ρ
(cid:20) (cid:21)
comprise of the continuity equation
where the assumption of a barotropic equation of state
∂ρ was used in simplifying the equations. The total en-
+∇·(ρV)=0, (1)
∂t thalpy h, in this scenario, is related to the pressure p
via the relation ∇h = ρ−1∇p, whilst d = c/(ω L)
i pi
the equation for the momentum density and d =c/(ω L) serve as the normalized electron and
e pe
∂V ion skin depths respectively. The quantities ωpi and ωpe
ρ +(V·∇)V =−∇p+J×B are the ion and electron plasma frequencies respectively,
∂t
(cid:18) (cid:19) −me (J·∇) J , (2) dspeeficnieeds lvaibaelω.pHλe=re,pqλnλa0nqdλ2/mε0λmaλrewtihthe c‘λh’ardgeenaontidngmtahses
e2 n ofthegivenspecies,whilstn isacharacteristicnumber
λ0
(cid:18) (cid:19)
density; for this reason, one must view d and d as nor-
i e
and the Ohm’s law malization constants expressed in terms of the fiducial
J×B−∇p +δ∇p values of the ionand electronplasma frequencies respec-
E+V×B− e i tively. The intermediate steps involved in deriving (5)
en
m ∂J 1 and (6) from (2) and (3) have been presented in [57].
= e +∇· VJ+JV− JJ . (3) Furthermore,itcanbe shownthat(5)and(6), incon-
ne2 ∂t en
(cid:20) (cid:18) (cid:19)(cid:21) junction with (1), conserve the energy:
Thevariablesρ, V andJserveasthe totalmassdensity,
centre-of-massvelocityandthecurrentrespectively. The ρV2 B2 (∇×B)2
H = d3x +ρU(ρ)+ +d2 . (7)
variable n appearing in (2) and (3) is the number den- 2 2 e 2ρ
sity,andisdefinedasn=ρ/(m +m ),withm andm ZD " #
i e i e
representingthe ionand electronmasses respectively. In
Observethattheaboveexpressiondoesdependond but
the above expressions, observe that µ J = ∇×B, the e
0 is independent of d . We observe that the last term in
i
totalpressureisrepresentedbypwhilstp andp denote
i e the above expression, proportional to d2, is absolutely
the ion and electron pressures respectively, δ = m /m e
e i necessary for energy conservation and emerges via the
is the mass ratio, and m represents the mass of the
λ lasttermontheRHSof(2). Thelatterisoftenneglected
species ‘λ’. Broadlyspeaking, the aboveset ofequations
intextbooktreatments,leadingtoerroneousconclusions;
arederivedfromthestandardtwo-fluidtheoryofplasma
see [55] for a detailed discussion of the same.
physics [1] by neglecting the displacement current, im-
posingquasineutrality,andcarryingoutasystematicex-
pansion in δ. We refer the reader to [1, 51, 63], where
B. Common Hamiltonian structure of the
a detailed, and rigorous, derivation of extended MHD
extended MHD models and associated properties
from two-fluid theory is presented (see also [57]). The
regimes of validity for extended MHD, and the specific
conditions under which certain terms can be eliminated We are now in a position to commence our analysis of
to obtain simpler models, are described in [1, 55, 63]. the different extended MHD models.
IfoneadoptsthestandardAlfv´enunits,andintroduces
the dynamical variable Hall MHD: Hall MHD (HMHD) is a model that ne-
glects electron inertia, and it amounts to letting d → 0
e
∇×B in (5), (6) and (7). Alternatively, it can be viewed as
B⋆ =B+d2∇× , (4)
e ρ the model wherein the last term on the RHS of (2) is
(cid:20) (cid:21)
neglected,alongwiththelastterminthefirstlineof(3),
which is well-known from electron MHD [50] and colli- and all the terms on the second line of (3). The cur-
sionless (two-fluid based) reconnection studies [64, 65], rent formulation of barotropic Hall MHD was presented
3
in [66], andwe shallreproduce it below, as it constitutes the statement that the canonical vorticity (curl of the
a core part of our investigations: canonical momenta) flux of each species is Lie-dragged
by the corresponding velocity of that species. We shall
return to this issue in greater detail in Section III.
{F,G}HMHD =− d3x [Fρ∇·GV+FV·∇Gρ]
ZD ( Inertial MHD: Inertial MHD (IMHD) arises upon
(∇×V)
− ·(FV×GV) setting di → 0 in (6). The astute reader may wonder
ρ
why d →0 does notautomatically imply d →0 as well
i e
B
since m ≪ m . However, we emphasize that the two
− ·(FV×(∇×GB)) (8) e i
ρ parametersareindependent. Inparticular,inertialMHD
B is valid when the time scale for changes in the current is
+ ·(GV×(∇×FB)) muchshorterthan the electrongyroperiod[55]. Aprac-
ρ
) ticaluse ofinertialMHDstems fromthe factthatasim-
B plified and reduced version yields the famous Ottaviani-
−di d3x ρ ·[(∇×FB)×(∇×GB)], Porcelli reconnection model [64]. Alternatively, inertial
ZD MHD amounts to dropping the terms on the RHS of the
and we shall represent this noncanonical bracket as first line of (3). A Hamiltonian formulation for the 2D
version was presented in [56], and the full structure was
{F,G}HMHD ={F,G}MHD +{F,G}Hall, (9)
determinedin[57]. Anindependentbracketforthemodel
wasconstructedinthelatter,but[58]showedthatthein-
where (i) {F,G}MHD comprises of the first four lines
ertialMHDbracketcouldbemappedtotheHallbracket
on the RHS of (8) and is the ideal MHD bracket first
as follows:
derivedin[59],and(ii){F,G}Hall isthelasttermonthe
RHS of (8), and is characterized by the presence of the
{F,G}IMHD ≡{F,G}HMHD[∓2d ; B ], (14)
factor d . We observe that the Jacobi identity for this e ±
i
(noncanonical) bracket was first shown in [57], and an
and this indicates that the inertial MHD bracket is ex-
alternative, more detailed, versionwas presented in [58].
actly identical to the Hall MHD bracket provided that
Through the suitable use of (8), it is easy to establish B → B = B⋆ ±d ∇×V and d → ∓2d in (8). It
± e i e
that there are two helicities for Hall MHD, which serve
is evident from (14) that there exist two different trans-
as Casimir invariants of the model; Casimirs are special
formations that map the Hall MHD bracket to inertial
invariants which satisfy the property {F,C} = 0 for all
MHD.
arbitrary choices of F. The two Casimirs of interest are
We see that the new variables B , which empower us
±
totransitionbetweenthetwobrackets,arecloselyrelated
CI = d3xA·B, (10) tothetwohelicitiesofinertialMHD,whichhavetheform
ZD
C = d3x (A⋆±d V)·(B⋆±d ∇×V). (15)
I,II e e
C = d3x (A+d V)·(B+d ∇×V), (11) ZD
II i i
ZD The difference of the above two helicities leads to the
which represent the magnetic and ion canonical helici- Casimir:
ties respectively [67]. Hall MHD exhibits two frozen-in
quantities, which behave as Lie-dragged 2-forms. These C = d3xV·B⋆, (16)
III
correspond to dynamical equations of the form ZD
∂ which resembles the cross-helicity invariant of ideal
+LV (B+di∇×V)·dS=0, (12) MHD, after performing the transformation B → B⋆.
∂t
(cid:20) (cid:21) Hence, this highlights the commonality of inertial and
ideal MHD. Just as in Hall MHD, there exist two Lie-
∂ dragged 2-forms, given by
∂t +LVe B·dS=0, (13)
(cid:20) (cid:21) ∂
where LX indicates the Lie-derivative with X serving ∂t +LV± B±·dS=0, (17)
as the flow field, V = V − d ∇ × B/ρ denotes the (cid:20) (cid:21)
e i
electron velocity, whilst B·dS and (B+di∇×V)·dS where V± = V±de∇×B/ρ and B± has already been
constitutethemagneticandioncanonicalvorticityfluxes previously introduced.
respectively; here dS represents an area element. For a
discussion of the Lie derivative (and the phenomenon Extended MHD: Finally, we consider extended
of Lie dragging) in ideal HD and MHD, we refer the MHD (XMHD), whichwasfirstderivedcorrectlyin [51];
reader to [11, 68]. The two expressions are equivalent to the reader is referred to [55] where several sources that
4
use incorrect versions of this model are discussed. Ex- • Owing to this commonality, all extended MHD
tended MHD comprises of (1), (5) and (6) in their en- models are endowed with two helicities, which also
tirety, and its Hamiltonian formulation was first pre- serve as Casimir invariants. This feature is clearly
sented in [57]. However, the bracket for extended MHD inherited from the parent 2-fluid model, whose
can be mapped to Hall MHD, as in inertial MHD, as bracket exhibits similar properties [69].
follows:
• All of the extended MHD models possess two Lie-
{F,G}XMHD ≡{F,G}HMHD[d −2κ ; B ], (18) dragged 2-forms, indicating that generalizations of
i ± ±
thefrozen-fluxconditionofidealMHDcanbeeasily
indicating that the extended MHD bracket is simply re- built. Consequently, this implies that these quan-
coveredviaB→B :=B⋆+κ ∇×Vandd →d −2κ tities serve as the analogs of the magnetic field
± ± i i ±
in(8),theHallMHDbracket[58]. Weobservethatκ is in ideal MHD, enabling the generalizations of the
±
determinedviathe quadraticequationκ2−d κ−d2 =0, Cauchy formula for the latter.
i e
implyingtheexistenceoftwosuchsolutions(κ andκ ).
+ −
As a result, it is worth emphasizing that there are two • The above property makes it possible to construct
possible mappings from the Hall MHD bracket to ex- a unified (Lagrangianvariable) actionprinciple for
tended MHD in (18). Upon taking the limits d → 0 these models, by building the constraints into the
i
and d → 0, and mapping to the original variables, it model a priori, akin to the ideal MHD action [70].
e
is straightforwardto verify that one recoversthe inertial Byemployingthereductionprocedure,wecanalso
and Hall MHD brackets respectively. We also note that derive the noncanonical bracket (8) in a rigorous
the above definition of B reduces to the inertial MHD manner. Wenotethatboththeseaspectshavebeen
±
definition for B when d →0. successfully tackled in [71].
± i
Extended MHD is also endowed with two helicities,
given by • The unified action principle delineated in [71] is
complementary to the approach espoused in [63],
where an alternative (Eulerian-Lagrangian vari-
C = d3x (A⋆+κ V)·(B⋆+κ ∇×V), (19)
I,II ± ± able) action principle was studied.
ZD
and one can verify the existence of two Lie-dragged 2-
forms, which are governed via
III. GEOMETRIC AND TOPOLOGICAL
PROPERTIES OF THE EXTENDED MHD
∂∂t +LV± B±·dS=0, (20) MODELS
(cid:20) (cid:21)
Hitherto,ouranalyseshavenotconsideredtheexplicit
where V =V−κ ∇×B/ρ and B =B⋆+κ ∇×V.
± ∓ ± ± consequences of the commonalities described in Section
We note that V and B duly reduce to their Hall
± ± II, and have, instead, focused primarily on highlighting
and inertial MHD counterparts upon taking d →0 and
e them. Now, we shall present a few applications of our
d →0 respectively.
i (unified) Hamiltonian formulation, and highlight its ad-
vantages. Henceforth, we shall adopt a coordinate inde-
From the preceding analysis,it is possible to draw the
pendent language wherever possible, as it simplifies and
following conclusions:
generalizes our discussion.
• There exists a clear hierarchy of models starting
from extended MHD. Upon neglecting the Hall
terms via d →0, we arrive at inertial MHD. Sim- A. Generalized circulation and helicity
i
conservation theorems
ilarly, neglecting electron inertia via d → 0 leads
e
toHallMHD, andneglecting bothofthem concur-
rently yields ideal MHD. Firstly, we begin by noting that one can define a gen-
eralized vector potential from B :=B⋆+κ ∇×V via
± ±
• This hierarchy is best encapsulated by (18) which B =∇×A :=∇×(A⋆+κ V). Aftersomeextensive
± ± ±
demonstrates the application of the above lim- algebra, it is possible to show that
its leads to the emergence of inertial, Hall and
idealMHDbracketsfromtheoverarchingextended ∂A± =∇A ·V −V ·∇A +∇ψ , (21)
MHD noncanonical bracket. ∂t ± ± ± ± ±
• The commonality between all the extended MHD where V =V−κ ∇×B/ρ was defined earlier, and
± ∓
models has been highlighted through the existence
of a common bracket, whose basic structure takes d2 J2 J·V
ψ :=κ h − κ + e h −φ+κ d2 −d2 . (22)
the form of (8). ± ∓ e ± d i ∓ e 2ρ e ρ
i
(cid:16) (cid:17)
5
In(22),notethath istheenthalpyofspeciesλ, φisthe K := Tr(K )representthe generalizedhelicities of
electrostatic potentλial and J is the current. It is more ide±alMHVD±,andT±rdenotesthe(ad-invariant)innerprod-
R
intuitive to rewrite (21) as uct. We shall drop this notation (Tr) henceforth, but it
is implicitly present whenever we deal with helicity-like
∂
∂t +LV± A± =dψ±, (23) qtouaynietlidties. We find that (23) can be duly manipulated
(cid:20) (cid:21)
Awh±e.reSAim±iliasrtlhye, 1w-efocramnaisnstorcoidautecdewthiteh2t-hfoercmomBp±on=endtAs±of, ∂∂Kt± +LV±K± =dψ±∧dA± =d(ψ±dA±), (28)
whose evolution is determined by applying the exterior
derivative ‘d’ to (23). We use the the fact that d2 = and by invoking Stokes’ theorem, we end up with
0, along with the commutative property of the exterior
derivative and the Lie derivative [72], thereby leading us d
K = d(ψ dA )= ψ dA =0,
to the relations dt ± ± ± ± ±
ZV±(t) ZV±(t) Z∂V±(t)
(29)
∂
∂t +LV± B± =0, (24) as long as the generalized vorticity vanishes on the
(cid:20) (cid:21) boundary. It is evident that (29) constitutes another
proof for helicity conservation, thereby complementing
andthisisidenticalto(20). Inotherwords,inour(new)
the earlier (coordinate dependent) results presented in
notation,B ≡B ·dS. Hence,itispossibletoundertake
± ± [57, 58, 71]. It was shown in [20, 21] - see also [74, 75]
a consistency check, and verify that (24) leads to
forassociatedtreatments-thatmagneticorfluidhelicity
∂B conservationwasanaturalconsequenceofNoether’sthe-
± =∇×(V±×B±), (25) orem on account of the (Lagrangian)particle relabelling
∂t
symmetry of the ideal HD and MHD actions. By apply-
uponusing∇·B =0andnotingthatthevectordensity ingasimilarproceduretotheextendedMHDaction[71],
±
B is dualto the 2-formB [68]. We canalso introduce the invariance of the helicities of extended MHD can be
± ±
the 3-form K = A ∧dA , which we shall return to established accordingly.
± ± ±
shortly hereafter.
From fluid mechanics, the conservation of circulation
hasbeenknownsincethe19thcentury. Itisnowstraight- B. Topological aspects of the generalized helicities
forward to show that one can derive a generalized circu- of extended MHD
lation theorem.
d Now, we shall take a greater look at the topological
A ·dl
± ramifications of K and (29), viz. the generalized helic-
dt ±
ZL±(t) (cid:12)t=t0 ities and their conservation properties respectively.
(cid:12)
= ddtZL±(t)A±(t)(cid:12)(cid:12)(cid:12)t=t0 = ddtZL±(t0)Φ∗V±,tA±(t)(cid:12)t=t0 1-fLoermt su,sabpepgrionprbiayterleycaclolninsgtruthctaetdAfr+omanAd±A,−whseerrveethaes
= ddt A±(t)(cid:12)(cid:12)(cid:12)+(t−t0)LV±A±+O (t−t0(cid:12)(cid:12)(cid:12))2 Ilaftotnerewleatssddeefi→ned0taonwdadrdis→th0e,bweegihnanviengalorfeaSdeyctiinodnicIIaItAed.
ZL±(t0) (cid:12)t=t0 that the vector potential A follows from A . Yet, it
= ∂∂At± +LV±A± = d(cid:0)ψ± =0,(cid:1)(cid:12)(cid:12)(cid:12)(26) itsenimdepdorMtaHnDt thoavree,congontizoenet,habtutaltlwoothsuerchve1r-sf±oiornmss.ofItexis-
ZL±(t0) (cid:12)t=t0 ZL±(t0)
(cid:12) wellknownthatthegeneralexpressionforahelicity-type
where Φ∗V±,t denotes the(cid:12)(cid:12)pullback with vector field V± quantityisgivenbyH = MP∧dP, whereMis acom-
parametrized by t [73]. The integration is carried over pact3-manifoldandP isa1-form. Wehavedroppedthe
the contour L±(t), and the above statement indicates inner product operator(TRr) as noted earlier. Hence, one
that the generalized vorticity flux is frozen-in for a fluid can duly construct two helicity-like quantities by setting
movingwithvelocityV± –ageneralizationofthefamous P = A± and the corresponding (generalized) helicities
frozen-fluxconditionofidealMHD.Thiscanbeexplicitly are given by K .
±
worked out, as shown below Wehavereiteratedthe abovestepsbecausethecrucial
aspect of our work is that these generalized 1-forms, 2-
d d
B ·dS = B (t) forms and helicities can be seen as the exact analogues
± ±
dt dt
ZS±(t) (cid:12)t=t0 ZS±(t) (cid:12)t=t0 of the vector potential/velocity, magnetic field/vorticity,
= ∂∂Bt± +(cid:12)(cid:12)(cid:12)LV±B± =0. (cid:12)(cid:12)(cid:12) (27) aanredinmtahgenerteimc/aflrukiadblheepliocistiytiornesopfecetxipvleolyit.inAgseaverreysuklnt,owwne
ZS±(t0) (cid:12)(cid:12)t=t0 topological property of ideal HD or MHD by generaliz-
The3-formsassociatedwithext(cid:12)endedMHDweredefined ingittoextendedMHD viathe variabletransformations
(cid:12)
earlier via K := A ∧ dA , and we emphasize that introduced here, and in [58].
± ± ±
6
For instance, consider the description of the fluid he- richness of the helicity/Chern-Simons correspondence,it
licity in terms of thin vortex filaments, which are repre- hasn’t been sufficiently exploited from a knot-theoretic
sented collectively by an oriented knot (or link) in M. perspective – the mathematical works by [11, 91, 92] on
The expression for the fluid helicity is given by theJonesandHOMFLYPTpolynomialsinHDandMHD
constitute the only such examples of this specific line of
H = νi2Lki+2 νiνjLkij, (30) enquiry. Although [91, 92] utilized the formal equiva-
i ij lencebetweenthefluid(ormagnetic)helicityandAbelian
X X
Chern-Simons theory, there have been prior studies in
where ν denotes the vortex circulation, whilst Lk and
i i
high energy physics and topologicalhydrodynamics that
Lk are the self-linking and Gauss linking numbers re-
ij
werecognizantofthisconcept(seee.g. [11,86]). Itisalso
spectively [6, 76]. Moreover, we observe that Lk =
i
straightforward to apply this framework to non-Abelian
Wr + Tw , implying that the self-linking number can
i i
magnetofluid models, as briefly stated in [87].
be decomposed into its writhing and twisting numbers;
the latter duo are topologically relevant in their own Thus, we are free to import the results of [11, 91, 92]
right [6, 13, 14, 77]. The decomposition of helicity into in the context of the generalized helicities. In particu-
its variouscomponents hasalso been verifiedempirically lar, following the mathematical reasoning delineated in
through a series of ingenious experiments [26–28], and [91], we are free to compute the Jones polynomial for a
numerical simulations in dynamos [78]. If we replace given configuration of the generalized helicity (of which
the vortex filaments, circulation, etc. by the general- therearetwoinall). Theproofreliesontheconstruction
ized counterparts (corresponding to B ), we find that of the skein relations by means of the Kauffman bracket
±
the generalized helicities can be decomposed in a man- polynomial, and then introducing orientation to obtain
ner exactly identical to (30). the skein relations of the corresponding Jones polyno-
For all its elegance and utility, the linking number is mial. Letusinterpretthe resultsfromthe precedingdis-
beset by anumber of limitations. The foremostamongst cussion for the (simpler) case of Hall MHD. One of the
them is that it cannot distinguish between certain topo- Jonespolynomialswouldarisefromthemagnetichelicity,
logicalconfigurations,suchastheWhiteheadlinkandthe whilst the other arises from the canonical helicity. The
Borromeanrings [79]. The conventionalmeans of distin- difference of these two helicities is the sum of the cross
guishing between such configurations is via the Massey and fluid helicities. Hence, the associated Jones polyno-
product[80]andits generalizations[81], orotherhigher- mial, arisingfrom this remainder,wouldencapsulate the
order invariants [82–84]. As per the correspondence be- topological properties of the fluid and cross helicities.
tween ideal MHD (or HD) and the different variants of Quite intriguingly, the Chern-Simons forms are odd-
extended MHD established earlier, we may be able to dimensional differential forms [93], implying that the
construct the equivalent (higher-order) topological in- Chern-Simons action (31) is meaningful only for odd di-
variants for the latter class of models. It is at this junc- mensions, given that it is proportional to the integral
ture that we introduce the remarkable insight provided of the Chern-Simons form. In turn, owing to its iden-
by Witten[85]betweentopologicalquantumfieldtheory tification with the generalized helicities, the latter ac-
(TQFT) and knot theory. In particular, Witten demon- quire this distinct mathematical structure only in odd
strated that the Jones polynomial, a staple of knot the- dimensions. Ipso facto, this may imply that helicities
ory,couldbenaturallyinterpretedintermsoftheChern- (magnetic, fluid or generalized) of this form will natu-
Simons action of (2+1) Yang-Mills theory. The Chern- rally emerge in non-relativistic (3D) theories, but not,
Simons action for a non-Abelian field theory is given by perforce, in the case of relativistic theories, as they are
intrinsically four-dimensional in nature. In particular,
2
S = P ∧dP + P ∧P ∧P , (31) we note that relativistic MHD possesses a cross helicity
3
ZM(cid:18) (cid:19) akintoits3Dcounterpart,butthe4Dversionofthecon-
up to constant factors. Now, suppose that the underly- ventional(3D)magnetichelicityhasproventobeelusive
inggaugegroupisAbelian,andthischoiceeliminatesthe fromaHamiltonianperspective[94],althoughithasbeen
secondtermonthe RHSofthe aboveexpression. Conse- derived through other avenues [22, 23, 25].
quently,weareledtothe strikingresultthatthehelicity It must be recognized that knot polynomials are not
is an Abelian Chern-Simons action [86, 87]. As a result, the only means of distinguishing between different topo-
one can employ the versatile mathematical formulations logical configurations. Thus, one can easily utilize more
ofChern-Simonstheory(a3-dimensionalTQFT)[88–90] powerful mathematical formalisms to study ideal and
intherealmofplasmaandfluidmodels,therebyopening extended MHD, examples of which include Khovanov
up a potentially rich and diverse line of future research, and Heegaard Floer homologies, and possibly contact
as these methods are more sophisticated than standard topology onaccountof its relevance in Legendrianknots
paradigm of computing the linking number(s); for in- [95, 96]. In the theory of contact structures, one deals
stance,the Jonespolynomialis capableof distinguishing with a plane field ξ on a manifold M, which can be lo-
between the Whitehead link and the Borromean rings callyrepresentedasthekernelofa1-formα(the contact
(which have an identical linking number of zero, as pre- form). A necessary condition for the plane field to be
viously mentioned). Despite the inherent mathematical a contact structure is that α ∧ dα is non-zero. If we
7
identify α with A , it is evident that K := A ∧dA Quite evidently, a host of avenues open up for future
± ± ± ±
must be non-zero – as a result, a potential connection analyses. The first, and possibly, the most significant
between the generalized helicities (constructed from the is the derivation of reduced extended MHD models that
integrals of K ) and contact geometry arises. We also retain the Hamiltonian properties of the parent model.
±
note that the relationshipbetween contact topology and Such models are likely to be of considerable relevance
hydrodynamics has already been probed in the context in reconnection studies, thereby furthering the basic ap-
of Beltrami fields by [97]. proach adopted in [64, 65, 99, 100]. For this reason, it
At this stage, we observe that K = 0 also leads to is equally important to conduct a detailed examination
±
several interesting results that arise from the Frobenius of their stability via Hamiltonian methods, analogous to
theorem;seefore.g. Theorem2.2.26(pg. 93)of[73]. The the extensive study of idealMHD by [101]. We alsonote
condition K = 0 is equivalent to the associated plane the possibility of using extended MHD models to study
±
fieldξ =kerαbeingclosedundertheLiebracket. Mathe- dynamos and jets [102], as well as helicity injection [31],
matically,the latter amountsto the followingstatement: the last of which appears to be a completely unexplored
if v and v are sections of ξ, their Lie bracket [v ,v ] arena. Although these models are endowed with the ion
1 2 1 2
mustalsobeasectionofξ. Ifaplanefieldisclosedunder and electron skin depths, the absence of the correspond-
the Lie bracket, the Frobenius theorem implies that ξ is ing Larmor radii is evident. To rectify this limitation, it
foliated (simply covered) by surfaces (tangent to ξ) [98]. is feasible to use the gyromap[103, 104] in the extended
Given that the Frobenius theorem has important ramifi- MHDcontext,todevelopagyroviscoustheoryanalogous
cationsforintegrability,andtheevidentconnectionswith to the one formulated by Braginskii.
the generalized helicities via K , we shall defer further
± From the unified Hamiltonian structure of these mod-
investigations to future publications.
els, we demonstrated that they possess a common class
Apart from the topological properties of helicity, as
of Casimir invariants - the generalized helicities. Moti-
seeninisolation,onecanalsoprobeits relationshipwith
vated by these helicities, we sought the generalizations
energy. Forinstance,aclassicresultbyMoffatt[4]estab-
of the vorticity (or magnetic field), and thereby estab-
lished a relation between the minimum magnetic energy
lished the existence of two Lie-dragged 2-forms. Thus,
E , the flux Φ and the volume V of a magnetic flux
min the whole enterprise demonstrated that the topological
tube as follows: properties of these models are a natural consequence of
E =mΦ2V−1/3, (32) theirHamiltonianstructure. We believethatthis isavi-
min tal, but rather unrecognized, fact that merits further at-
where m depends on the specific properties of the knot, tention. By constructingthese helicities and 2-forms,we
and it is a topological invariant; see also [5, 8, 17] for derived properties such as the generalization of Kelvin’s
similar results. When dealing with extended MHD, the circulation theorem in a geometric setting. Moreover,
magneticcomponentoftheenergydensitymustbetrans- we also showed that these helicities can be viewed as
formed from B2 to B·B⋆. As a result, it is natural to Abelian Chern-Simons theories, and that the methodol-
ask whether one generalize the result (32) to extended ogyintroducedbyWitten, forgaininginsightsintotopo-
MHD,andweintendtopursuethislineofenquiryinour logical quantum field theory, could be employed here.
subsequent works. Consequently, we concluded that the Jones polynomials
The applications we have outlined thus far barely maybeusedtocharacterizedifferent(generalizedvortic-
scratch the surface. There are many other results from ity) configurations,serving as a more powerful tool than
HDandMHDthatcanbeimportedtoextendedMHDin- the standard Gauss linking number used to characterize
volving helicity. For instance, one such example is helic- fluid or magnetic helicity. By introducing such topolog-
ity injection. This phenomenon has been widely studied ical methods for characterizing helicity, their relevance
inthe solarcontext[41, 42]asit hasimportantramifica- in the domains of astrophysicsand fusion is self-evident.
tions,butthere havebeenno studies dealingwithgener- One such application, of paramount importance, is to
alized helicity injection. We shall leave such subjects for deploy these topologicalmethods in gaininga better un-
later investigations – it is our present goal to highlight derstanding of solar magnetic fields [105].
the correspondence with HD/MHD, thereby paving the In summary, we have used the noncanonical Hamil-
way for conducting in-depth researchin these areas. tonian formulation of extended MHD models to arrive
at their common mathematical structure, which mani-
fests itself via the existence of generalized helicities and
IV. DISCUSSION AND CONCLUSION Lie-dragged 2-forms. These helicities, which are topo-
logical invariants, can be further studied through a host
In this paper, we have emphasized and exploited the of techniques, including the Jones polynomial [11, 91].
inherent mathematical power of the unified Hamiltonian Froma conceptualpoint-of-view,our results areelegant,
structure of several extended MHD models. This enter- as they exemplify the spirit of unification common to
prise was rendered possible owing to the work of [57], most physical theories. On the other hand, we also be-
and the unified Hamiltonian (and its underlying action lieve that the results presented herein possess manifold
principle) structure was established in [58, 71]. concreteapplications,especially since the helicities serve
8
bothasimportanttopologicalinvariants,andcrucialme- ACKNOWLEDGMENTS
diators of relaxationand self-organization,reconnection,
turbulence, and magnetic field generation (dynamos) in ML was supported by the NSF Grant No. AGS-
fusion and astrophysicalplasmas. 1338944 and the DOE Grant No. DE-AC02-09CH-
11466. PJMwassupportedbyDOEOfficeofFusionEn-
ergy Sciences, under DE-FG02-04ER-54742. ML wishes
to acknowledge Eric d’Avignon, Lee Gunderson, Stuart
Hudson, Timothy Magee, Swadesh Mahajan and Taliya
Sahihi for their perceptive insights and remarks.
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