The gyrokinetic magnetohydrodynamic (MHD) equations, related to the recent paper by W. W. Lee [“Magnetohydrodynamics for collisionless plasmas from the gyrokinetic perspective,” Phys. Plasmas 23, 070705 (2016)], and their associated equilibria properties are discussed. This set of equations consists of the time-dependent gyrokinetic vorticity equation, the gyrokinetic parallel Ohm's law, and the gyrokinetic Ampere's law as well as the equations of state, which are expressed in terms of the electrostatic potential, $ϕ$, and the vector potential, $A$, and support both spatially varying perpendicular and parallel pressure gradients and the associated currents. The corresponding gyrokinetic MHD equilibria can be reached when $ϕ→0$ and A becomes constant in time, which, in turn, gives $∇·(J∥+J⊥)=0$ and the associated magnetic islands, if they exist. Examples of simple cylindrical geometry are given. These gyrokinetic MHD equations look quite different from the conventional MHD equations, and their comparisons will be an interesting topic in the future.

For the general toroidal geometry, the gyrokinetic Vlasov equation can be written as1

$∂Fα∂t+dRdt·∂Fα∂R+dv∥dt∂Fα∂v∥=0,dRdt=v∥b*+v⊥22Ωα0b̂0×∇lnB0−cB0∇Φ¯×b̂0,$
(1)

and

$dv∥dt=−v⊥22b*·∇lnB0−qαmα(b*·∇Φ¯+1c∂A¯∥∂t),$

where

$Fα=∑j=1Nαδ(R−Rαj)δ(μ−μαj)δ(v∥−v∥αj).$

$Ωα0≡qαB0/mαc$, $b*≡b+(v∥/Ωα0)b̂0×(b̂0·∇)b̂0, b=b̂0+∇×A¯/B0, Φ¯≡ϕ¯−v⊥·A⊥¯/c$, and the variables with subscript “0” represent equilibrium quantities. Using the gyrokinetic Poisson's equation of the form

$ωpi2Ωi2∇⊥2ϕ=−4πρ$
(2)

in the long wavelength limit and the Ampere's law of

$∇2A=−4πcJ,$
(3)

and by assuming that $ω2/k2vA2≪1$, where ω and k are the frequency and the wavelength of interest, respectively, and vA is the Alfvén speed, we can then proceed to obtain a simple set of gyrokinetic magnetohydrodynamic (MHD) equations as shown in Refs. 1 and 2. They are

$J⊥=cBb̂×∇⊥p⊥$
(4)

for the perpendicular current density associated for a given pressure profile, where $p⊥=p⊥i+p⊥e$, and

$ddt∇⊥2ϕ−4πvA2c2∇·(J∥+J⊥)=0$
(5)

is the vorticity equation, which is associated with the continuity equation obtained from Eqs. (1) and (2), where

$ddt≡∂∂t−cB∇ϕ×b·∇,$

together with the parallel Ohm's law of the form

$E∥≡−1c∂A∥∂t−b·∇ϕ≈−1ene∂p∥e∂x∥+ηJ∥$
(6)

obtained from the electron momentum equation of Eq. (1) by ignoring the contribution from the electron inertia in the time-dependent term, as shown earlier by Refs. 1 and 3, where $η=νme/ne2=4πν/ωpe2$ is the resistivity, ν is the collision frequency, n is the number density, and ωpe is the electron plasma frequency. The time evolution of the pressure can also be obtained from Eq. (1) as

$dp⊥dt=0,$
(7)

implying that the energy and mass convect together and

$dp∥edt=2E∥J∥,$
(8)

where $p=nT$. Here, we have a complete set of gyrokinetic MHD equations, in which quantities associated with third order velocity moments and higher are ignored. Thus, with $J∥$ given by Eq. (3), Eqs. (4)–(8) can be used for studying many interesting MHD physics. For $E∥→0$ in Eq. (6), these equations have the conservation property of

$∂∂t∫18π(|∇⊥ϕ|2+vA2c2|∇A∥|2)dx=−vA2c2∫E⊥·J⊥dx$
(9)

for a collisionless plasma, where $E⊥≈−∇⊥ϕ$. It can be regarded as Poynting's theorem for our system. Furthermore, when $ϕ→0$ in Eq. (5), we recover the MHD equilibrium as

$∇·(J∥+J⊥)=0.$
(10)

Let us now focus on Eqs. (5) and (6). Both $ϕ$ and $A∥$ as well as $J∥$ via Eq. (3) are time dependent, while $J⊥$ (as well as $A⊥$) is static. Through linearization (i.e., $d/dt≈∂/∂t$), we can simplify these equations, in the collisionless limit with no pressure gradients, as

$∂∂t∇⊥2ϕ−4πvA2c2∇·J∥=0$
(11)

and

$E∥≡−1c∂A∥∂t−b·∇ϕ=0.$
(12)

Using the ansatz of $exp (−iωt+ik·x)$ for $ϕ$ and $A∥$, the eigenmodes of the linearized equations are

$ω2=k∥2vA2,$

i.e., the usual shear Alfvén frequencies. Since these modes are weakly damped,4 which may give rise to a state that $ϕ≈0$ and $A∥≈const.$, i.e., an equilibrium state. Let us assume that Eqs. (5) and (6) can also reach the equilibrium, then, we recover Eq. (10) from Eq. (5) which, in turn, gives

$ηJ∥≈1ene∂p∥e∂x∥,$
(13)

as obtained from Eq. (6) for $E∥≈0$, where $∂/∂x∥=b·∇$.

Let us now re-examine the Ampere's law, Eq. (3), i.e.,

$∇×B=(4π/c)(J∥+J⊥).$
(14)

First, it is interesting to note that Eq. (4) can be re-written as

$∇·J⊥=cB∇⊥p⊥·(∇×b),$

where

$∇×b=1B∇×B+b×∇lnB;$

consequently, we arrive at

$∇·J⊥=cB∇⊥p⊥×b·∇lnB.$
(15)

For $∇·J⊥=0$, it means that $B$ is independent of the poloidal angle for $pi(r)≠0$ or the magnetic field is undefined in the region when $∇⊥p⊥=0$. Therefore, we need the parallel part of Eq. (14) to satisfy Eq. (10), i.e.,

$∇·J=cB∇⊥p⊥×b·∇lnB+1eη∂∂x∥(1ne∂p∥e∂x∥)=0.$
(16)

It should be mentioned here that while Eq. (4) indicates $b·∇⊥p⊥=0$, Eq. (13) indicates that $b·∇p∥e≠0.$ Note that

$∂∂x∥(1ne∂p∥e∂x∥)=1ne∂p∥e∂x∥∂∂x∥(ln∂p∥e/∂x∥ne),$

which indicates that either $p∥e$ or $(∂p∥e/∂x∥)/ne$ is constant along $x∥$, if $∇⊥p⊥=0$. Thus, Eq. (13) shows that the electron parallel current is either zero or constant along the field line in this case. However, the latter is unphysical.

Equations similar to Eqs. (5), (12), and (7) have been numerically solved as initial value problems in three dimensions by Strauss.5,6 His first set of equations for solving $ϕ$ and $A∥$ takes the form of Eqs. (11) and (12), where $J∥$ is given by the parallel part of Eq. (3). His second attempt was to solve Eqs. (5) and (12), although his $J⊥$ is different than ours as given in Eq. (4). We intend to solve our more complete equations, i.e., Eqs. (5) and (6) as initial value problems in the future and compare the results with those of Strauss6 as well as with those of the conventional MHD equations. By using the results of time-dependent problems as a guide, we also plan to use the iterative numerical procedures for solving Eqs. (10) and (14) based on Eqs. (4) and (13) for island formation in an equilibrium plasma in general geometries.

In the present paper, our immediate interest here is to obtain the time-independent solutions of these equations in simple geometry. Namely, we are looking for an $A∥$, which will modify $δB⊥$, so as to satisfy $∇·J=0$, while giving us $ϕ≈0$, for some given pressure profiles along with an initially given main $B$ field. To that end, let us first normalize the governing equations in the gyrokinetic units4 of $J¯≡J/encs, B¯≡B/(cTe/ecsρs)$ and $β≡(cs/vA)2$, where $n=ne=ni$ is the number density, cs is the ion acoustic speed, ρs is the ion gyroradius measured in terms of the electron temperature, vA is the Alfven speed and Te is the electron temperature. Thus, the Ampere's law becomes

$∇×B¯=β(J¯∥+J¯⊥),$
(17)

with the usual condition of $∇·B¯=0.$ The perpendicular current is now

$J¯⊥≈b×∇¯p⊥p0,$

where $p0≈⟨n⟩⟨Te⟩$ and $⟨⋯⟩$ denote the spatial average and $b≡B¯/B¯$. The parallel current is given by

$ν¯J¯∥=bp0∂pe∂x¯∥,$

where $ν¯≡ν/Ωe$ and Ωe is the electron cyclotron frequency. The equilibrium condition, Eq. (16), now takes the form of

$∇¯·(J¯∥+J¯⊥)≈1p0∇¯p⊥×b·∇¯B¯B¯+1ν¯p0∂∂x¯∥(∂pe∂x¯∥)=0.$
(18)

In the cylindrical geometry, if we have

$J¯⊥=1p0∂p⊥(r¯)∂r¯eθ,$

Eq. (18) gives us

$∂δB¯∂θ=B¯0ν¯[∂∂x¯∥(∂pe∂x¯∥)]/(∂p⊥∂r¯)$
(19)

for $B¯=B¯0ẑ+δB¯.$ Thus, $δB¯$ is now a function of θ, which gives the pre-requisite for the existence of magnetic islands in the presence of both perpendicular and parallel pressure gradients in the equilibrium state. Furthermore, the existence of $δB¯$ gives rise to a new $b=B¯/B¯≈ẑ+δB¯/B¯0$, which, in turn, gives a new $J¯⊥$. To obtain a more self-consistent magnetic configuration, it then calls for an iterative procedure to calculate the new magnetic field until it converges. However, for a very low-β plasma, the iterative procedures between Eqs. (17) and (18) are not necessary, and Eq. (18) alone can determine the magnetic configuration. For

$(∂pe/∂x¯∥)/p0=A sin (2πr/a) cos mθ sin (2πx∥/L∥)$
(20)

and

$p⊥(r)p0=12−tanh[(r−r0)/w]2,$
(21)

the magnetic islands at $x∥=L∥/4$ with m = 2 for Eq. (20), as well as a = 69 and $r0=36$ for w = 5 and w = 15, respectively, in Eq. (21), are shown in Fig. 1, which are produced by the spatially dependent equilibrium parallel electron current associated with Eq. (20) as given by Fig. 2. The associated perpendicular pressure profiles are given by Fig. 3, where $r=x2+y2$. If we also include the finite Larmor radius effects as given by Eq. (21) of Ref. 1, Eq. (19) becomes

$∂δB¯∂θ=B¯0ν¯[∂∂x¯∥(∂pe∂x¯∥)]/[(∂p⊥∂r¯)(1−14TiTe1p⊥∂2p⊥∂r¯2)],$
(22)

where $p⊥≈p⊥i$ and Ti is the ion temperature. However, the extra term in Eq. (22) has no appreciable effect for the present parameters, which are similar to the ones used in the recent paper by Lee and White.7 Nevertheless, it is interesting to note that a small variation of the perpendicular pressure profile can significantly changes the magnetic island formation in gyrokinetic MHD equilibrium, i.e., a sharper gradient seems to produce less stochastic regions.

FIG. 1.

Magnetic islands produced by the equilibrium parallel pressure gradient for (a) w = 5 and (b) w = 15 in Eq. (21).

FIG. 1.

Magnetic islands produced by the equilibrium parallel pressure gradient for (a) w = 5 and (b) w = 15 in Eq. (21).

Close modal
FIG. 2.

Parallel current produced by the equilibrium parallel pressure gradient.

FIG. 2.

Parallel current produced by the equilibrium parallel pressure gradient.

Close modal
FIG. 3.

Radial pressure profiles for different w's, as defined in Eq. (21).

FIG. 3.

Radial pressure profiles for different w's, as defined in Eq. (21).

Close modal

In this paper, we have shown the connection between the fully electromagnetic gyrokinetics and the corresponding gyrokinetic MHD, which is quite different from the conventional MHD. Further studies on the differences between the two are needed. Moreover, the present paper has shown that the gyrokinetic MHD in its equilibrium state can support magnetic islands through spatially varying parallel pressure gradient, which should also be a subject of interest for the fusion community in the future. For example, the PIES code8 solves the similar equations as ours, i.e., Eqs. (4), (10), and (14), along with $∇·B=0$, and it uses an iterative scheme to calculate the magnetic field $B$ and pressure by first evaluating $J⊥$ for a given $B$ and a given pressure. It then uses $∇·J=0$ to calculate $J∥$, which, in turn, gives a new $B$ via the Ampere's law and a new pressure. As we have mentioned earlier, what we propose here is also an iterative scheme for $B$, which for a given $p⊥$ and a given $p∥$, it solves for $J⊥$ and $J∥$, through $∇·J=0$ and the Ampere law, it then calculates for a new $B$. Therefore, future collaborations using such a code with some code modifications for tokamak/stellarator research are possible and would be fruitful.

This work, which was partially supported by U.S. DoE Grant No. DE-AC02-09CH11466, represents one of the theoretical foundations for our 2017 SciDAC proposal to the Office of Fusion Science, U.S. Department of Energy—entitled “First Principles Based Transport and Equilibrium Module for Whole Device Modeling and Optimization.” Two of us (W.W.L. and S.R.H.) would like to thank Dr. E. A. Startsev, Dr. W. X. Wang, and Dr. S. Ethier at Princeton Plasma Physics Laboratory (PPPL) for the collaboration of that proposal. One of us (W.W.L.) would also like to thank Dr. Donald Monticlello for very useful discussions on the PIES code.

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