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Plasma Physics and Controlled Fusion
Plasma Phys. Control. Fusion 66 (2024) 015002 (13pp) https://doi.org/10.1088/1361-6587/ad0a47
Analytic Hall magnetohydrodynamics
toroidal equilibria via the
energy-Casimir variational principle
A Giannis1, D A Kaltsas1,2and G N Throumoulopoulos1,∗
1Department of Physics, University of Ioannina, 45110 Ioannina, Greece
2Department of Physics, International Hellenic University, 65404 Kavala, Greece
E-mail: gthroum@uoi.gr
Received 29 July 2023, revised 16 October 2023
Accepted for publication 7 November 2023
Published 17 November 2023
Abstract
Equilibrium equations for magnetically conned, axisymmetric plasmas are derived by means
of the energy-Casimir variational principle in the context of Hall magnetohydrodynamics
(MHD). This approach stems from the noncanonical Hamiltonian structure of Hall MHD, the
simplest, quasineutral two-uid model that incorporates contributions due to ion Hall drifts. The
axisymmetric Casimir invariants are used, along with the Hamiltonian functional to apply the
energy-Casimir variational principle for axisymmetric two-uid plasmas with incompressible
ion ows. This results in a system of equations of the Grad–Shafranov–Bernoulli (GSB) type
with four free functions. Two families of analytic solutions to the GSB system are then
calculated, based on specic choices for the free functions. These solutions are subsequently
applied to Tokamak-relevant congurations using proper boundary shaping methods. The Hall
MHD model predicts a departure of the ion velocity surfaces from the magnetic surfaces which
are frozen in the electron uid. This separation of the characteristic surfaces is corroborated by
the analytic solutions calculated in this study. The equilibria constructed by these solutions
exhibit favorable characteristics for plasma connement, for example they possess closed and
nested magnetic and ow surfaces with pressure proles peaked at the plasma core. The
relevance of these solutions to laboratory and astrophysical plasmas is nally discussed, with
particular focus on systems that involve length scales on the order of the ion skin
depth.
Keywords: equilibrium, Hall-MHD model, energy-Casimir variational principle,
analytic solutions
1. Introduction
Hall magnetohydrodynamics (Hall MHD) is the simplest
MHD model that incorporates two-uid effects and is obtained
∗Author to whom any correspondence should be addressed.
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by consistently reducing the complete two-uid equations
of motion, under the assumptions of quasineutrality and
vanishing electron inertia. The non-dimensional Hall MHD
equations read (in Alfvén units) as:
E+v×B=di(J×B−∇·Pe),(1)
∂tρ=−∇·(ρv),(2)
∂tv=−∇·P+|v|2
2I+v×(∇×v) + J×B,(3)
1361-6587/24/015002+13$33.00 1 © 2023 The Author(s). Published by IOP Publishing Ltd
Plasma Phys. Control. Fusion 66 (2024) 015002 A Giannis et al
∂tB=∇×[v×B−di(J×B)],(4)
where E,Bare the electric and magnetic elds respectively, v
is the ion uid velocity, ρis the total mass density, J=∇×B
is the current density, Pis the total plasma pressure tensor and
Peis the electron pressure tensor, Iis the identity tensor, di=
c/(ωpiL)is the ion skin depth, normalized by a characteristic
length scale L,cis the speed of light, and ωpiis the ion plasma
frequency. Equation (1) is a generalized Ohm’s law resulting
from the electron momentum equation for vanishing electron
mass and innite electrical conductivity, equation (2) is the
continuity equation, equation (3) is the momentum equation,
while equation (4) is an induction equation for the magnetic
eld, resulting from combining Ohm’s law (1) with Faraday’s
law ∇×E=−∂tB. The right-hand side of equation (1) con-
tains the so-called Hall term, as well as the electron pressure
term. These terms cause a detachment of the ion uid from the
electron one, in length scales comparable or smaller than the
ion skin depth.
For the study of most laboratory and astrophysical plasma
systems, the ideal MHD model is usually being employed,
ignoring terms that scale as diin equations (1)–(4) and thus
eliminating two-uid effects. This one-uid description is an
adequate framework for the study of large time and length
scale processes i.e. dynamical phenomena with time-scales
larger than the ion cyclotron frequency and plasma struc-
tures with length scales much larger than the ion skin depth.
However, some fundamental processes cannot be described
within this framework as they originate from the two-uid
nature of the plasmas, e.g. fast magnetic reconnection, vari-
ous micro-instabilities that trigger or enhance turbulence and
transport and wave modes that are not present in a single uid
framework. On top of that, there exist laboratory plasma sys-
tems that involve processes and structures with length scales
comparable to the ion skin depth, for example magnetically
conned plasmas with current sheets, thin boundary layers,
and steep gradients, such as Tokamak plasmas with a steep
pressure gradient in the edge region and pressure pedestals
that develop in the transition to improved connement regimes
(L-H transition) [1,2], and neoclassical diffusion [3]. The
development of a theory for the pressure pedestals in high (H)
connement mode in Tokamaks in [4], which was based on
double-Beltrami Hall MHD equilibrium states, is indicative
of the relevance of Hall MHD with H-mode plasmas (see also
[5]). It has also been established that Hall effects are relevant
to the so-called tearing mode instability [6] that occurs in both
laboratory and astrophysical plasmas. To further acknowledge
the importance of the Hall MHD model in studying the mag-
netic connement of plasmas, we stress that the omission of
the Hall term in Tokamaks has been previously criticised in
[7]. As concerns astrophysical plasmas, there are many sys-
tems that can be described in the framework of Hall MHD,
with the most notable example possibly being the corrobor-
ation by in situ satellite measurements that magnetic recon-
nection in Earth’s magnetosphere is described by a two-uid
model [8]. The usefulness of Hall MHD is not limited to the
above examples; the interested reader is referred to [9] for
even more examples of astrophysical systems described by
Hall MHD, like dense molecular clouds, protostellar disks and
neutron stars, to name only a few.
The present paper addresses the construction of analytic,
axisymmetric, Hall MHD equilibrium states for Tokamak
plasmas with shaped boundaries. The starting point of this
construction is the derivation of the Hall MHD equilib-
rium equations by exploiting the energy-Casimir principle, a
Hamiltonian variational principle that stems from the nonca-
nonical Hamiltonian structure of Hall MHD. Then the equi-
librium equations are cast in a the form of a Grad–Shafranov
system and we provide two special analytic solutions which
are used to construct Tokamak-pertinent equilibria. This work
is organized as follows: In section 2we deduce equilibrium
equations for axisymmetric two-uid plasmas with incom-
pressible ion ows in light of the Energy-Casimir variational
principle. The resulting Grad–Shafranov–Bernoulli system is
then solved analytically for two specic cases in section 3.
In section 4we apply these solutions to up-down symmetric,
International Thermonuclear Experimental Reactor (ITER)-
like congurations, while in section 5we summarize and dis-
cuss the results of the present study.
2. Energy-Casimir equilibrium variational principle
It has been established (e.g. [10,11]) that equations (2)–
(4) possess a noncanonical Hamiltonian structure [12], in the
sense that the system’s dynamics can be described by a set of
generalized Hamilton’s equations
∂tη={η,H} ,(5)
where η= (ρ,v,B)represents the dynamical variables of the
model, H=H[η]is the Hamiltonian functional, an integ-
ral over the plasma volume, and {F,G}is a noncanonical
Poisson bracket, which is bilinear and antisymmetric and also
satises the Jacobi identity [12]. The noncanonical Poisson
bracket of Hall MHD has been found in [10,11]. Such brackets
can be degenerate, i.e. there exist functionals Cthat Poisson-
commute with any other functional Fof the dynamical vari-
ables, that is
{C,F}=0,∀F.(6)
The functionals Care thus invariants of the system, called
Casimirs or Casimir invariants [12]. Regarding equilib-
rium points ηeof the Hamiltonian system, these can be
calculated by
{ηe,H} =0.(7)
However, as Csatisfy (6), the functional Hin (7) can be
replaced by a generalized Hamiltonian formed by a linear
combination of the Hamiltonian Hand the Casimirs C. Then,
nonsingular equilibrium points satisfy
δH−
i
Ci[ηe] = 0.(8)
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Plasma Phys. Control. Fusion 66 (2024) 015002 A Giannis et al
Relation (8) is an expression of the so-called Energy-Casimir
variational principle, and is essentially a sufcient condition
for equilibrium [12].
Here, we apply this principle considering a two-uid
plasma with massless electrons, homogeneous ion density and
anisotropic electron pressure. The equations of motion are
given by (1)–(4) with P=Pi+Pe. Two additional equations
are needed to close the system, one for the ion pressure Pi
and one for the electron pressure Pe. We have shown in [13]
that the electron pressure equation in the Hall MHD limit
becomes [14]
B×Pe+ (B×Pe)T=0,(9)
which is solved by gyrotropic pressure tensors of the form
Pe=Pe∥−Pe⊥
B2BB +Pe⊥I=σBB +Pe⊥I,(10)
where BB =BiBjis the tensor product of Bwith itself, Pe∥
and Pe⊥denote the electron pressures parallel and perpendic-
ular to the magnetic eld, respectively and we dened a func-
tion σ:= (Pe∥−Pe⊥)/B2which quanties the anisotropy of
the electron pressure. Evidently, for σ=0 the electron pres-
sure is isotropic.
As regards the closure equation for the ion pressure, this
is replaced by plasma incompressibility to simplify the sub-
sequent analysis. This means that the mass density is assumed
to be homogeneous (ρ=1)throughout the plasma volume and
thus the uid velocity is divergence-free ∇·v=0. The con-
straint ρ=1 can be retrieved directly from the variation, if we
introduce the ion pressure as a Lagrange multiplier in a cer-
tain manner in the energy-Casimir variational principle (8) as
follows
δH−
i
Ci+ˆd2xPiln(ρ)=0,(11)
where d2x=dRdZin the standard cylindrical coordinates
(R,ϕ,Z). Clearly, for ρ=1 the last term in (8) vanishes.
An analogous technique for introducing the pressure in the
Lagrangian description of incompressible ideal MHD has been
proposed by Newcomb in [15].
The Hamiltonian for axisymmetric Hall MHD plasmas with
electron pressure anisotropy reads
H=ˆ
S
d2xρv2
ϕ
2+B2
ϕ
2+|∇Ψ|2
2R2+ρ|∇X|2
2R2+ρUe(ρ,|B|),
(12)
where Ψand Xare the two poloidal ux functions for the mag-
netic and the velocity elds, respectively, and Sis the plasma
cross section normal to the ϕdirection with closed boundary
∂S. Note that the ion internal energy is not included in Has
it has been taken into account through the constraint intro-
duced in (11). The axisymmetric magnetic and velocity elds
are given in terms of Ψand Xby the following relations
B=RBϕ∇ϕ+∇Ψ×∇ϕ, (13)
v=Rvϕ∇ϕ+∇X×∇ϕ. (14)
In equation (12), Ueis the electron internal energy function,
which should satisfy [16]:
∂Ue
∂ρ =Pe∥
ρ2,(15)
∂Ue
∂|B|=−σ
ρ|B|.(16)
The Hall MHD Casimir invariants with axial and helical
symmetry have been previously calculated in [17] and the
recovery of the corresponding invariants in the MHD limit has
been discussed in [18]. The axisymmetric Hall MHD Casimirs
are
C1=ˆ
S
d2xR−1Bϕ+diΩF(Φ),(17)
C2=ˆ
S
d2x R−1BϕG(Ψ),(18)
C3=ˆ
S
d2xρM(Φ),(19)
C4=ˆ
S
d2xρN(Ψ),(20)
where F,G,Mand Nare free functions, i.e. arbit-
rary functions of their arguments; Ω=(∇×vp)·∇ϕ=
−1
R2∆∗X, with vp=∇X×∇ϕbeing the poloidal compon-
ent of the velocity eld and ∆∗≡R2∇·(∇/R2)is the so-
called Shafranov operator. In equations (17)–(20), Φis a gen-
eralized ux function dened as Φ = Ψ + diRvϕ. The physical
interpretation of these symmetric Casimirs has been discussed
in previous studies (e.g. [18]), however, for completeness we
mention that C1and C2are related to the generalized Hall
MHD cross-helicity and the magnetic helicity, respectively,
while C3and C4are associated with the conservation of mass,
toroidal angular momentum and magnetic ux.
Employing the constrained energy-Casimir variational
principle (11) leads to the following set of Euler–Lagrange
equations
δPi:ρ=1,(21)
δρ :Pi+Pe∥
ρ+Ue=M(Φ) + N(Ψ) −v2
2,(22)
δX:di∇·∇F
R2=∇·ρ∇X
R2,(23)
δvϕ:ρvϕ=diRR−1Bϕ+diΩF′+diRρM′(Φ) .(24)
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Plasma Phys. Control. Fusion 66 (2024) 015002 A Giannis et al
δBϕ:Bϕ=F(Φ) + G(Ψ)
R2(1−σ),(25)
δψ :∇·(1−σ)∇Ψ
R2+R−1Bϕ(F′+G′)
+diΩF′(Φ) + ρN′(Ψ) + ρM′(Φ) = 0.(26)
After some manipulations the system above can be cast in a
Grad–Shafranov–Bernoulli (GSB) form
d2
iF′(Φ)R2∇·F′(Φ) ∇Φ
R2−[F(Φ) + G(Ψ)]F′(Φ)
1−σ
−M′(Φ)R2+Φ−Ψ
d2
i
=0,(27)
∇·(1−σ)∇Ψ
R2+[F(Φ) + G(Ψ)]G′(Ψ)
1−σ
+N′(Ψ)R2+Φ−Ψ
d2
i
=0,(28)
P=Pi+Pe=M(Φ) + N(Ψ) −v2
2.(29)
Relations (27) and (28) are equations of the Grad–Shafranov
type [19,20] determining the poloidal ow function Φ(it can
be deduced from (23) that Φis associated with the poloidal
ion ow) and the poloidal magnetic ux function Ψ. Finally,
relation (29) is a Bernoulli equation that determines the total
plasma pressure and is independent of the other two equations
as a result of the incompressibility assumption. Similar results
have been obtained by standard methods for reducing the gen-
eric equilibrium equations to a GSB system in [21], while as is
well known, for compressible plasmas all three equations are
coupled to each other (e.g. see [17,18,22,23]).
Henceforward we assume isotropic electron pressure, i.e.
σ=0 in order to be able to nd analytic solutions to the GS
system (27) and (28). In this case, from equations (15) and (16)
we deduce that Ueis a function of ρonly and thus the electron
pressure is constant since ρ=1. Additionally, a specic choice
for the four arbitrary functions F,G,M,Nshould be made for
the derivation of analytic solutions. More specically, we will
adopt the following general ansatz for the free functions
F(Φ) = f0+f1Φ,(30)
G(Ψ) = g0+g1Ψ,(31)
M(Φ) = m0+m1Φ + 1
2m2Φ2,(32)
N(Ψ) = n0+n1Ψ + 1
2n2Ψ2,(33)
where f0,f1,g0,g1,m0,m1,m2,n0,n1,n2are free, constant
parameters, which are determined in view of specic equi-
librium characteristics. We note that f0and g0are related to
the vacuum toroidal magnetic eld, hence a selection such
that f0+g0=0 results in a Tokamak-relevant 1/R-vacuum
eld, while a selection for which f0+g0=0 could describe
Spheromak-relevant equilibria. The rest of the parameters
f1,g1,m1,m2,n1,n2are associated with the self-consistent
elds and plasma quantities. Below we examine two notable
cases of analytic equilibria; the rst corresponds to m2=n2=
0 and the second to m2=0, n2=0.
3. Analytic solutions to the GS system
3.1. Double Beltrami equilibria
The coupled system of GS equations resulting from (27)
and (28) with (30)–(33) and m2=n2=0 can be cast in the
following matrix form
∆∗Ψ
X=W1W2
W3W4Ψ
X+A1
A2+B1
B2R2,(34)
where we dened
W1=−g2
1−1
d2
i,W2=−g1
di
+1
f1d3
i,
W3=g1
di
+1
f1d3
i,W4=1
d2
i−1
f2
1d4
i,(35)
and
A1=−g0g1−f0
d2
if1,A2=f1g0+f0
d2
if1,
B1=−n1,B2=m1
dif1
.(36)
The above matrix coefcients (35) and (36) depend solely on
the ansatz parameters (see equations (30)–(33)) and the ion
skin depth di.
One can prove that the ux functions Ψand Xin the homo-
geneous part of equation (34) correspond to elds Band v
which satisfy the following relation
∇×∇×V+b1(∇×V) + b2V=0,(37)
where
b1=−g1−1
f1d2
i
,b2=f1+g1
f1d2
i
,(38)
and Vis either the magnetic eld Bor the velocity eld v(see
for example [17,24]).
Equation (37) can be solved by appropriate superpositions
of Beltrami elds A±, i.e. by eigenvectors of the curl operator
[25,26]
∇×A±=λ±A±.(39)
These superpositions for the elds Bhand vhare found to be
Bh=1
f1di−diλ+A++1
f1di−diλ−A−,(40)
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Plasma Phys. Control. Fusion 66 (2024) 015002 A Giannis et al
vh=A++A−,(41)
where the eigenvalues λ±are given by
λ±=−b1±b2
1−4b2
2.(42)
Therefore, the homogeneous solution to (34) can be expressed
in terms of the functions Ψ±satisfying
∆∗Ψ±=−λ2
±Ψ±,(43)
which stems from (39) with A±=RA±
ϕ∇ϕ+∇Ψ±×∇ϕ.
Equation (43) can be easily solved by separation of variables
and one nds [27]:
Ψ±(R,Z)
=R
kak±J1Rλ2
±−k2cos(kZ)
+bk±Y1Rλ2
±−k2cos(kZ)+c1±R2cos(λ±Z)
+c2±cos(λ±Z) + c3±cosλ±R2+Z2,(44)
with J1(x)and Y1(x)being the rst order Bessel func-
tions of the rst and second kind respectively, and
ak±,bk±,c1±,c2±,c3±are unknown coefcients which
will be determined in view of the boundary conditions.
Note that since the elds (Bh,vh)satisfy (37), which
involves a double curl operator and are superpositions of two
Beltrami elds, the corresponding states are called double
Beltrami equilibrium states. Note additionally, that in the
framework of MHD, the Beltrami states are homonymous to
the well-known Taylor force-free states [27–29], which are
states with vanishing Lorentz force.
To nd the general solution to the nonhomogeneous system
of elliptic partial differential equations (34) we form a linear
combination of the homogeneous solution expressed in terms
of Ψ±and a particular solution of the inhomogeneous system.
This general solution is
Ψ(R,Z) = 1
f1di−diλ+Ψ++1
f1di−diλ−Ψ−
+κ1R2+κ2,
(45)
X(R,Z)=Ψ++ Ψ−+κ3R2+κ4,(46)
Here κj,j={1,...,4}are the coefcients of the inhomogen-
ous part of the solution and they are given by:
κ1=B1W4−B2W2
W2W3−W1W4
, κ2=A1W4−A2W2
W2W3−W1W4
,
κ3=B1W3−B2W1
W1W4−W2W3
, κ4=A2W1−A1W3
W2W3−W1W4
,
(47)
3.2. Equilibrium solutions in terms of Whittaker functions
If one selects g1=−1/(d2
if1)in the general ansatz (30)–(33),
then the system of GS equations (27) and (28) is decoupled
and the two equations read as:
∆∗Ψ + e1R2+e2Ψ + e3R2+e4=0,(48)
∆∗Φ + ˜
e1R2+˜
e2Φ + ˜
e3R2+˜
e4=0.(49)
where the coefcients ej, and ˜
ej, (j=1,...,4) are given by
e1=n2,e2=1
d4
if2
1−1
d2
i,
e3=n1,e4=−f0+g0
d2
if1
,
˜
e1=−m2
d2
if2
1
,˜
e2=e2,
˜
e3=−m1
d2
if2
1
,˜
e4=e4.(50)
We can nd analytic solutions to the homogeneous counter-
parts of the equations (48) and (49) employing the standard
method of separation of variables. These homogeneous solu-
tions depend on Rand on Zthrough Whittaker functions [30]
and trigonometric functions, respectively:
Ψh(R,Z) =
kakMνk,1
2i√e1R2cos(kZ)
+bkWνk,1
2i√e1R2cos(kZ),(51)
Φh(R,Z) =
k˜
akM˜νk,1
2i˜
e1R2cos(kZ)
+˜
bkW˜νk,1
2i˜
e1R2cos(kZ).(52)
Here Mk,m(z)and Wk,m(z)are the two independent Whittaker
functions [30], and
νk=ik2−e2
4√e1
,˜νk=ik2−˜
e2
4√˜
e1
(53)
with idenoting the imaginary unit and ak,bk,˜
ak,˜
bkare con-
stant coefcients.
In the special case where the fractions n2/n1and m2/m1
satisfy the following relation
n2
n1
=m2
m1
=f1−1/d2
if1
f0+g0
,(54)
a set of two special solutions to the inhomogeneous
equations (48) and (49) can be found by a direct similarity
reduction method [31] which is portrayed in the appendix.
These solutions are
Ψp(R,Z) = cos√e1
2R2+√e2Z
+cos√e1
2R2−√e2Z−e3
e1
,
(55)
5
Plasma Phys. Control. Fusion 66 (2024) 015002 A Giannis et al
Φp(R,Z) = cos√˜
e1
2R2+˜
e2Z
+cos√˜
e1
2R2−˜
e2Z−˜
e3
˜
e1
.
(56)
Since the partial differential equations (48) and (49) are lin-
ear, the general solutions will be constructed by superposing
the homogeneous solutions (51) and (52) with the particular
solutions (55) and (56).
4. Equilibria with shaped boundaries
In this section we examine two applications of the equilibrium
solutions of section 3, by Tokamak-relevant values for the vari-
ous physical quantities and geometric parameters determining
the plasma boundary. For simplicity, we have retained only
cosine functions of Z, thus only up-down symmetric equilib-
ria will be considered here. However, extension to up-down
asymmetric equilibria is straightforward and will be pursued
in future studies along with the construction of equilibria with
negative triangularity and additional shape control features
(e.g. [32,33]).
The solutions presented in the previous section contain a
large number of unknown parameters that ought to be specied
in view of appropriate boundary shaping conditions. For their
determination, we will approximate a D-shaped boundary of
interest with the following parametric equations
Rb(t) = 1+εcos(ϕ+sin−1δsint),
Zb(t) = εκsin t,(57)
where t∈[0,2π]is the poloidal angle, εis the inverse aspect
ratio, κis the elongation, and δ=sinαis the triangularity
[34]. In terms of these geometrical parameters, we can select
three characteristic points on the boundary with coordinates:
(1+ε,0)(outer equatorial point), (1−ε, 0)(inner equatorial
point), and (1+δε, κε)(upper point). We may also dene
three curvatures [34] in these points of interest
N1=d2Rb
dZ2
bϕ=0
=−(1+α)2
εκ2,(58)
N2=d2Rb
dZ2
bϕ=π
=(1−α)2
εκ2,(59)
N3=d2Zb
dR2
bϕ=π/2
=−κ
εcos2(α).(60)
The boundary shaping conditions that we will employ for
the determination of the unknown coefcients were rst
described in [34] and subsequently utilized in a series of papers
on Tokamak-pertinent analytic MHD equilibria [27,35–38].
These conditions are
Ψ(1+ε, 0) = 0,(61)
Ψ(1−ε, 0) = 0,(62)
Figure 1. The contours of the magnetic and the ion velocity
surfaces for the double Beltrami equilibrium. The positions of the
extra points, selected for better shaping, are also marked.
Ψ(1−δε, κε) = 0,(63)
ΨR(1−δε, κε) = 0,(64)
ΨZZ (1+ε,0) = −N1ΨR(1+ε, 0),(65)
ΨZZ (1−ε,0) = −N2ΨR(1−ε, 0),(66)
ΨRR (1−δε, κε) = −N3ΨZ(1−δε,κε),(67)
where the subscripts indicate partial derivatives with respect
to the specied variable. Analogous shaping conditions can
be considered also for Φ.
4.1. Double Beltrami ITER-relevant equilibrium
On the basis of the solution (44)–(46), we constructed an up-
down symmetric Tokamak equilibrium, with ITER-pertinent
values for the geometrical parameters: ε=0.32, κ=1.8,
δ=0.45, R0=6.2m, and (normalized) ion skin depth di=
0.03. The rest of the parameters were adjusted so that the
equilibrium quantities attain large Tokamak values and pro-
les, although further optimization is possible. The plasma
boundary was specied by equations (61)–(67), along with
some additional shaping conditions introduced to improve the
boundary representation, where it was required that Ψvan-
ishes at some additional boundary points. These conditions
result in a linear system of algebraic equations which is solved
numerically to determine the unknown coefcients ak±,bk±
(k=1,...,5) and c1±,c2±,c3±in (44) . Figure 1depicts the
nested magnetic and ion velocity surfaces in a poloidal cross-
section, as well as the extra points selected on the boundary for
improved shaping. This construction of double-Beltrami equi-
librium with shaped, Tokamak-relevant boundary is an exten-
sion of a previous work [27], concerned with exact Taylor
6
Plasma Phys. Control. Fusion 66 (2024) 015002 A Giannis et al
Figure 2. Magnetic eld proles for the double Beltrami
equilibrium. Top: the Z-component of the magnetic eld on the
plane Z=0. Bottom: the toroidal component of the magnetic eld
on the plane Z=0.
states of toroidal, axisymmetric plasmas with Tokamak geo-
metric characteristics. The present equilibrium has some novel
features due to its two-uid nature, namely it accommodates
strong plasma ows and pressure gradients with peaked pres-
sure prole and closed isobaric surfaces and can describe high-
beta equilibria. Furthermore, being a two-uid equilibrium it
contains additional information about the distinct electron and
ion uid behavior. We should note that the double-Beltrami
states have been previously employed to study high-beta toka-
mak equilibria in [5]; however, the authors constructed those
equilibria by means of numerical solutions on a domain with
a simple circular boundary. It is remarkable that this simple
analytic equilibrium model predicts a departure of the mag-
netic surfaces with the respect to the ion velocity ones, which is
similar to the departure observed in previous works concerned
with the calculation of numerical equilibria with compressible
ows [18,23]. The presented equilibrium conguration exhib-
its a positive Shafranov shift while it should be emphasised
that due to the separation of the two uids, there are two char-
acteristic axes, a magnetic and a velocity one—though their
positions differ slightly.
In gures 2and 3some proles for the magnetic and
ion velocity elds are illustrated. Both Band vhave physic-
ally acceptable proles, and their respective values are within
acceptable ranges [39,40]; the magnetic eld values are
Figure 3. Ion velocity eld proles for the double Beltrami
equilibrium. Top: the Z-component of the velocity eld on the plane
Z=0. Bottom: the toroidal component of the velocity eld on the
plane Z=0.
O(1T), while the velocity values are O(106ms−1). Moreover
we observe that the toroidal magnetic eld reverses in the core
region, while the poloidal and toroidal velocity components
have comparable magnitudes. Proles of the current density
components JZand Jϕon the mid-plane Z=0 are displayed
in gure 4. As we can see, the proles are physically accept-
able and the current density values are in the MA m−2range,
which is typical for large Tokamaks, including ITER [41,42].
The Rand Zcomponents of the electric eld, calculated by the
Ohm’s law (1) are depicted in gure 5presenting an expected
behaviour and values on the order of MV m−1.
Another equilibrium quantity that is of particular import-
ance for the scope of magnetic connement is the plasma pres-
sure. Figure 6depicts a peaked-on-axis pressure prole for
the double Beltrami equilibrium with typical values for large
Tokamaks [41,43]. Owing to the ow the isobaric surfaces
depart from the magnetic and the ow surfaces as it can be
seen in gure 7. As a result, although Pattains low values on
the boundary, it is not exactly zero because the plasma ow
does not vanish on ∂S.
4.2. ITER-relevant equilibrium in terms of the Whittaker
functions
This subsection is concerned with the construction of a second
up-down symmetric Tokamak equilibrium, on the basis of the
solutions (51)–(56). From one point of view, this equilibrium
7
Plasma Phys. Control. Fusion 66 (2024) 015002 A Giannis et al
Figure 4. Current density proles for the double Beltrami
equilibrium. Top: the Z-component of the current density on the
plane Z=0. Bottom: the toroidal component of the current density
on the plane Z=0.
Figure 5. The Rand Zcomponents of the electric eld on the Z=0
and R=1 planes respectively for the double Beltrami equilibrium.
is more general compared to the one constructed in the pre-
vious subsection since m2=0 and n2=0. The same ITER-
relevant values for the geometric parameters were used as in
Figure 6. The plasma pressure prole on the Z=0 plane for the
double Beltrami equilibrium.
Figure 7. The pressure contours in a torus cross section, along with
the magnetic and ion surfaces for the double Beltrami equilibrium.
the previous case, while for the ion skin depth we selected di=
0.02. The rest of the parameters were again adjusted so that
the equilibrium quantities approximate as much as possible
Tokamak-relevant values and proles. In this case the bound-
ary shaping conditions were imposed simultaneously on both
Ψand Φsince they are independent and as in the previous case
some additional boundary points were introduced to improve
the boundary representation. The coefcients ak,bk,˜
ak,˜
bk
where specied for k={1,2,...,11}this time. The nested
contours of the two ux functions Ψand Φare illustrated in
gure 8.
As with the double Beltrami equilibrium, the separation
of the two ux surfaces is evident. The velocity surfaces are
organised around a distinct ow axis located very close to the
magnetic one. A Shafranov shift of both surfaces can also be
observed. A discernible difference with the previous equilib-
rium is that the two boundaries for Ψand Φnow coincide as
we imposed the boundary on both Ψand Φ.
8
Plasma Phys. Control. Fusion 66 (2024) 015002 A Giannis et al
Figure 8. The contours of the magnetic and the ion velocity surfaces for the Whittaker equilibrium. The chosen extra points on the
boundary are illustrated as well.
Figure 9. Magnetic eld proles for the Whittaker equilibrium.
Top: the Z-component of the magnetic eld on the plane Z=0.
Bottom: the toroidal component of the magnetic eld on the plane
Z=0.
In gure 9we present the proles of the Z- and ϕ-
components of the magnetic eld on the mid-plane Z=0.
Although both proles have a typical behavior and the toroidal
magnetic eld has desirable values, the poloidal magnetic
Figure 10. Ion velocity eld proles for the Whittaker equilibrium.
Top: the Z-component of the velocity eld on the plane Z=0. Right:
The toroidal component of the velocity eld on the plane Z=0.
eld Bpattains particularly small values (of the order of mT).
Regarding the velocity and the current density components,
two velocity proles are shown in gure 10, exhibiting accept-
able characteristics and two current density proles are shown
in gure 11. Although both components exhibit typical beha-
vior, the toroidal current density, associated with the poloidal
9
Plasma Phys. Control. Fusion 66 (2024) 015002 A Giannis et al
Figure 11. Current density proles for the Whittaker equilibrium. Top: the Z-component of the current density on the plane Z=0. Bottom:
the toroidal component of the current density on the plane Z=0.
Figure 12. The Rand Zcomponents of the electric eld on the
planes Z=0 and R=1 respectively for the Whittaker equilibrium.
magnetic eld, attains particularly small values. The small
toroidal current of this particular equilibrium is insufcient to
produce the rotational transform needed for effective conne-
ment, so we tried to x this inconsistency by re-scaling some
equilibrium parameters. However, this resulted in equilibria
Figure 13. The plasma pressure prole on the Z=0 plane for the
Whittaker equilibrium.
with unacceptably large values of other quantities (e.g. P), so
we intend to employ a systematic procedure to further optim-
ize the values of the free parameters for this particular class of
equilibria in future work.
Finally, we present the electric eld components and the
plasma pressure prole, calculated by means of equations (1)
and (29), respectively. The results are shown, respectively in
gures 12 and 13. The generated electric eld ranges from
103to 104V m−1, while the pressure has typical and desirable
behavior as it peaks in the plasma core and almost vanishes
on the boundary, with acceptable maximum values for large
Tokamaks. As in the previous case, the isobaric surfaces form
a distinct set of contours due to the ow contribution in the
Bernoulli equation (gure 14) although the departure from the
magnetic and ow surfaces is now less evident.
10
Plasma Phys. Control. Fusion 66 (2024) 015002 A Giannis et al
Figure 14. The pressure contours in a torus cross section, along
with the magnetic and ion surfaces for the Whittaker equilibrium.
5. Conclusions
The equilibrium GSB equations for axisymmetric, two-
uid plasmas with homogeneous ion density and aniso-
tropic electron pressure were derived using the energy-
Casimir variational principle, stemming from the nonca-
nonical Hamiltonian structure of Hall MHD. Subsequently,
electron pressure anisotropy was omitted to enable the calcu-
lation of analytic equilibrium solutions. Two families of ana-
lytic solutions to the Hall MHD GS system were derived and
employed to construct axisymmetric equilibrium states with
Tokamak-pertinent characteristics.
The construction of the rst type of equilibria (double
Beltrami states), was motivated by previous work of the
authors [44] and was effected by superposing two Beltrami
elds and a particular inhomogeneous solution. We demon-
strated that this class of equilibria possesses Tokamak-relevant
characteristics, namely nested magnetic and ow surfaces and
acceptable values and proles for the equilibrium quantities.
Nevertheless, although the pressure in the double Beltrami
equilibrium attains very low values on the boundary, it does
not vanish thereon due to the nonvanishing ow. In addition,
there is considerable evidence [45] to postulate that double
Beltrami states are essentially metastable equilibrium states,
because, as Gondal et al suggest, they tend to eventually relax
to ordinary Beltrami states. This loss of equilibrium may take
place under some circumstances, namely when certain scale
parameters become degenerate or even when the product of the
magnetic helicity with the ion helicity becomes positive [45,
46]. The abovementioned termination of the double-Beltrami
equilibrium may also give rise to a conversion of magnetic
energy to ow kinetic energy [45]. This metastability mech-
anism suggests that double-Beltrami states can be candidates
for the study of transient phenomena in laboratory plasmas but
mainly in astrophysical environments [46,47].
Successive to the double Beltrami states, we studied a
more general class of equilibria, with equilibrium solutions
expressed in terms of Whittaker functions. The Whittaker
equilibrium proved to be particularly interesting in some
aspects, mainly because the pressure prole has a desirable
behavior and optimal values. The rest of the quantities demon-
strated typical proles, with only exception the poloidal mag-
netic eld and the toroidal current density values.
For both classes of equilibria we employed a shaping
method in terms of certain conditions that concern charac-
teristic boundary points, allowing in this way the determ-
ination of the free parameters that appear in the analytic
solutions. Both equilibrium classes can describe congura-
tions with nested ux surfaces, peaked pressure proles and
isobaric surfaces which do not coincide with the other two
sets of surfaces, i.e. the magnetic and the ion ow surfaces.
Additionally, the non-parallel ow and the separation of the
characteristic surfaces that characterizes both solutions, pro-
duces large-scale poloidal electric elds that may interfere in
plasma and energy connement and transport. It is intriguing
that these analytic Hall MHD equilibria, recreate the general
characteristics of the ion uid and magnetic surface separ-
ation obtained in precedent numerical calculations [18,23].
As a general remark we should note that this departure of
the ion uid from the magnetic surfaces, which is of the
scale of the ion skin depth, is rather unimportant in terms of
macroscopic equilibrium. However, such Hall corrections can
be particularly important if these equilibria are used as ref-
erence states in the study of phenomena with scales relev-
ant to the ion drift motions, like transport, microinstabilies,
turbulence etc.
In future work we intend to extend the present study by
solving numerically the system (27)–(28) with electron pres-
sure anisotropy (σ=0) and further extend the model to incor-
porate ion-pressure-anisotropy effects and nite Larmor radius
corrections. Additionally, the analytic equilibria presented
here could be extended to accommodate additional boundary
shaping features, such as lower X-points and negative triangu-
larity while a systematic determination of the free parameters
in connection with the typical values of the physical quantities
can also be pursued.
Data availability statement
All data that support the ndings of this study are included
within the article (and any supplementary les).
Acknowledgments
We would like to thank the anonymous reviewers for their
constructive feedback and valuable insights, which helped to
improve the quality of this manuscript.
This work has received funding from the National Fusion
Program of the Hellenic Republic—General Secretariat for
Research and Innovation.
11
Plasma Phys. Control. Fusion 66 (2024) 015002 A Giannis et al
Appendix. Inhomogeneous solutions via direct
similarity reduction
The general solutions to the inhomogeneous equations (48)
and (49) split into a homogeneous and an inhomogeneous
part u(R,Z) = uh(R,Z) + up(R,Z), where up= (Φp,Ψp)is a
particular, special solution of the corresponding inhomogen-
eous equation. The homogeneous part has been calculated in
section 3in terms of Whittaker functions. On the other hand
though, a special solution up(R,Z)is hard to nd. One can
assume that up(R,Z) = u(R)and thus the particular solution
satises an ordinary differential equation of the form
u′ ′
p(R)−u′
p(R)
R+e1R2+e2up+e3R2+e4=0,(68)
where ej,j={1,...,4}are constants. The general solution
to this equation is again of the form up(R) = wh(R) + wp(R)
where wp(R)is a special inhomogeneous solution. We know
[48] that in general the inhomogeneous part wp(R)can be cal-
culated upon knowing wh(R) = c1h1(R) + c2h2(R)as follows:
wp(R) = h2(R)ˆdRe3R2+e4h1(R)
R
−h1(R)ˆdRe3R2+e4h2(R)
R,(69)
where h1(R)and h2(R)are linearly independent solutions of
the respective homogeneous equation. Here, h1(R)and h2(R)
are given in terms of Whittaker functions, and therefore one
should resort in the use of numerical methods for the compu-
tation of the above integrals. For this reason, in this study, we
employ a direct similarity reduction method [31] which solves
the system (48) and (49) in view of some constraints on the
free parameters appearing in those equations. Both equations
have the general form
∆∗u+e1R2+e2u+e3R2+e4=0.(70)
Let us consider particular solutions of the form
u=u(ξ), ξ =cRR2+cZZ,(71)
with cR,cZbeing constants. Substituting (71) into (70) we
can easily show that for cZ=±4e2
e1cRand e3/e4=e2/e1,
the function u(ξ)satises the following ordinary differential
equation
4c2
Ru′ ′ (ξ) + e1u(ξ) + e3=0,(72)
which is solved by
u(ξ) = c1cose1
4c2
R
ξ
+c2sine1
4c2
R
ξ−e3
e1
,(73)
where c1and c2are arbitrary constants and ξ=cR(R2±
4e2
e1)Z. Employing this prescription for (48) and (49) and
selecting c2=0, we obtain the solutions (55) and (56) with
the constraints (54).
ORCID iDs
A Giannis https://orcid.org/0009-0005-9065-2583
D A Kaltsas https://orcid.org/0000-0003-0076-9015
G N Throumoulopoulos https://orcid.org/0000-0003-
3336-687X
References
[1] Wagner F et al 1982 Regime of improved connement and
high beta in neutral-beam-heated divertor discharges of the
ASDEX tokamak Phys. Rev. Lett. 49 1408–12
[2] Wagner F 2007 A quarter-century of H-mode studies Plasma
Phys. Control. Fusion 49 B1
[3] Stroth U 2005 Transport in Toroidal Plasmas (Springer)
pp 213–67
[4] Guzdar P N, Mahajan S M and Yoshida Z 2005 A theory for
the pressure pedestal in high (H) mode tokamak discharges
Phys. Plasmas 12 032502
[5] Yoshida Z, Mahajan S M, Ohsaki S, Iqbal M and Shatashvili N
2001 Beltrami elds in plasmas: high-connement mode
boundary layers and high beta equilibria Phys. Plasmas
82125–31
[6] Zhang W, Ma Z W and Wang S 2017 Hall effect on tearing
mode instabilities in tokamak Phys. Plasmas 24 102510
[7] Gourdain P A 2017 The impact of the Hall term on tokamak
plasmas (arXiv:1703.00987)
[8] Berkowitz R 2019 Satellites glimpse the microphysics of
magnetic reconnection Phys. Today 72 20–23
[9] Mininni P D, Gómez D O and Mahajan S M 2003 Dynamo
action in magnetohydrodynamics and
Hall-magnetohydrodynamics Astrophys. J. 587 472
[10] Holm D D 1987 Hall magnetohydrodynamics: conservation
laws and Lyapunov stability Phys. Fluids 30 1310–22
[11] Lingam M, Morrison P J and Miloshevich G 2015 Remarkable
connections between extended magnetohydrodynamics
models Phys. Plasmas 22 072111
[12] Morrison P J 1998 Hamiltonian description of the ideal uid
Rev. Mod. Phys. 70 467–521
[13] Kaltsas D A, Throumoulopoulos G N and Morrison P J 2021
Hamiltonian kinetic-Hall magnetohydrodynamics with uid
and kinetic ions in the current and pressure coupling
schemes J. Plasma Phys. 87 835870502
[14] Hunana P, Tenerani A, Zank G P, Khomenko E,
Goldstein M L, Webb G M, Cally P S, Collados M, Velli M
and Adhikari L 2019 An introductory guide to uid models
with anisotropic temperatures. Part 1. CGL description and
collisionless uid hierarchy J. Plasma Phys. 85 205850602
[15] Newcomb W A 1962 Lagrangian and hamiltonian methods in
magnetohydrodynamics Nucl. Fusion 2451
[16] Morrison P J 1982 Poisson brackets for uids and plasmas AIP
Conf. Proc. 88 13–46
[17] Kaltsas D A, Throumoulopoulos G N and Morrison P J 2018
Helically symmetric extended magnetohydrodynamics:
Hamiltonian formulation and equilibrium variational
principles J. Plasma Phys. 84 745840301
[18] Kaltsas D A, Throumoulopoulos G N and Morrison P J 2017
Translationally symmetric extended MHD via Hamiltonian
12
Plasma Phys. Control. Fusion 66 (2024) 015002 A Giannis et al
reduction: energy-Casimir equilibria Phys. Plasmas
24 092504
[19] Grad H and Rubin H 1958 Hydromagnetic equilibria and
force-free elds p iNIS-XU–021 (available at: https://
citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&
doi=b464a18baf52ebbf9b9a1e5a3f526f87f79afa52)
[20] Shafranov V D 1966 Plasma equilibrium in a magnetic eld
Rev. Plasma Phys. 2103
[21] Throumoulopoulos G N and Tasso H 2006 On Hall
magnetohydrodynamics equilibria Phys. Plasmas
13 102504
[22] Hameiri E 2013 Ertel’s vorticity theorem and new ux
surfaces in multi-uid plasmas Phys. Plasmas 20 092503
[23] Guazzotto L and Betti R 2015 Two-uid equilibrium with
ow: FLOW2 Phys. Plasmas 22 092503
[24] Mahajan S M and Yoshida Z 1998 Double curl beltrami ow:
diamagnetic structures Phys. Rev. Lett. 81 4863–6
[25] Hudson S R, Hole M J and Dewar R L 2007 Eigenvalue
problems for Beltrami elds arising in a three-dimensional
toroidal magnetohydrodynamic equilibrium problem Phys.
Plasmas 14 052505
[26] Buniy R V and Kephart T W 2014 Generalized helicity and
Beltrami elds Ann. Phys., NY 344 179–93
[27] Cerfon A J and O’Neil M 2014 Exact axisymmetric Taylor
states for shaped plasmas Phys. Plasmas 21 064501
[28] Marsh G E 1996 Force Free Magnetic Fields: Solutions,
Topology and Applications (World Scientic Publishing
Company)
[29] Wiegelmann T and Sakurai T 2012 Solar force-free magnetic
elds Living Rev. Sol. Phys. 95
[30] Whittaker E T and Watson G N 1996 A Course of Modern
Analysis (Cambridge Mathematical Library) 4th edn
(Cambridge University Press)
[31] Kaltsas D A and Throumoulopoulos G N 2016 Exact solutions
of the Grad–Shafranov equation via similarity reduction
and applications to magnetically conned plasmas Phys.
Lett. A380 3373–7
[32] Farengo R 2020 Extended Solov’ev type equilibria for rotating
plasmas with positive and negative triangularity Phys.
Plasmas 27 122502
[33] Farengo R 2020 Quasi-analytical equilibria with toroidal
rotation and shape control Plasma Phys. Control. Fusion
62 092001
[34] Cerfon A J and Freidberg J P 2010 “One size ts all” analytic
solutions to the Grad–Shafranov equation Phys. Plasmas
17 032502
[35] Throumoulopoulos G N and Tasso H 2012 International
thermonuclear experimental reactor-like extended
Solovév equilibria with parallel ow Phys. Plasmas
19 014504
[36] Kaltsas D A and Throumoulopoulos G N 2014 Generalized
solovev equilibrium with sheared ow of arbitrary direction
and stability consideration Phys. Plasmas 21 084502
[37] Evangelias A and Throumoulopoulos G N 2016 Axisymmetric
equilibria with pressure anisotropy and plasma ow Plasma
Phys. Control. Fusion 58 045022
[38] Kaltsas D A, Kuiroukidis A and Throumoulopoulos G N 2019
A tokamak pertinent analytic equilibrium with plasma ow
of arbitrary direction Phys. Plasmas 26 124501
[39] Guazzotto L and Freidberg J P 2007 A family of analytic
equilibrium solutions for the Grad–Shafranov equation
Phys. Plasmas 14 112508
[40] Ernst D R et al 1998 Notched velocity proles and the radial
electric eld in high ion temperature plasmas in the
Tokamak Fusion Test Reactor Phys. Plasmas 5665–81
[41] Aymar R, Barabaschi P and Shimomura Y 2002 The ITER
design Plasma Phys. Control. Fusion 44 519
[42] Sips A C C (for the Steady State Operation and the Transport
Physics topical groups of the International Tokamak
Physics Activity) 2005 Advanced scenarios for ITER
operation Plasma Phys. Control. Fusion 47 A19
[43] Sips A C C, Giruzzi G, Ide S, Kessel C, Luce T C, Snipes J A
and Stober J K (The Integrated Operation Scenario Topical
Group of the ITPA) 2014 Progress in preparing scenarios
for operation of the International Thermonuclear
Experimental Reactor Phys. Plasmas 22 021804
[44] Giannis A, Kaltsas D A and Throumoulopoulos G N 2021
Three-dimensional Beltrami states for toroidal, shaped
plasmas 19th European Fusion Theory Conf. (Consorzio
RFX, Padova) (available at: https://indi.to/KGm33)
[45] Gondal S M, Iqbal M, Ullah S, Asghar M and Khosa A H 2019
Double Beltrami states and loss of equilibrium in electron,
positron and ion plasmas J. Plasma Phys. 85 905850306
[46] Ohsaki S, Shatashvili N L, Yoshida Z and Mahajan S M 2002
Energy transformation mechanism in the solar atmosphere
associated with magnetouid coupling: explosive and
eruptive events Astrophys. J. 570 395
[47] Kagan D and Mahajan S M 2010 Application of double
Beltrami states to solar eruptions Mon. Not. R. Astron. Soc.
406 1140–5
[48] Abell M L and Braselton J P 2019 Introductory Differential
Equations 5th edn (Academic)
13
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