1. Introduction

Let $T\subset \mathbb {R}^3$ be a domain with smooth boundary. The three-dimensional magneto hydrostatic (MHS) equations on $T$ read

(1.1)\begin{gather} J \times B = \boldsymbol{\nabla} P+ f, \quad \text{in}\ T, \end{gather}

(1.2)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} B = 0, \quad \text{in}\ T, \end{gather}

(1.3)\begin{gather}B\boldsymbol{\cdot} \hat{n} = 0, \quad \text{on}\ \partial T, \end{gather}

where $J= \boldsymbol {\nabla }\times B$ is the current, $f$ is an external force and $P$ is the pressure. The solution $B$ to (1.1)–(1.3) can be interpreted as either a stationary fluid velocity field which solves the time-independent Euler equation, or as a steady self-supporting magnetic field in a continuous medium with trivial flow velocity. The latter interpretation is robust across a variety of magnetohydrodynamic models (e.g. compressible, incompressible, non-ideal) and makes the system (1.1)–(1.3) central to the study of plasma confinement fusion.

In view of this, there is a long-standing scientific program to identify and construct MHS equilibria which are effective at confining ions during a nuclear fusion reaction. The most basic requirement for confinement is the existence of a ‘flux function’ $\psi$, whose level sets foliate the domain $T$ and which satisfies $B\boldsymbol {\cdot } \boldsymbol {\nabla } \psi = 0$. To first approximation, ions move along the integral curves of $B$ and so this condition ensures that particle trajectories are approximately constrained to the level sets of $\psi$. For this reason, it is desirable to seek equilibria with nested flux surfaces (isosurfaces of $\psi$) which foliate the plasma domain. When $T$ is the axisymmetric torus, it is natural to look for such solutions in the form of axisymmetric magnetic fields. If $(R, \varPhi , Z)$ denote the usual cylindrical coordinates on $\mathbb {R}^3$ and the centreline of the torus lies in the $Z = 0$ plane, axisymmetric solutions take the form

(1.4)\begin{equation} B_0 = \frac{1}{R^2} \left(C_0(\psi_0) Re_\varPhi + Re_\varPhi \times \boldsymbol{\nabla}\psi_0\right), \end{equation}

with flux function $\psi _0$. In order for $B_0$ to satisfy (1.1) with $P_0 = P_0(\psi _0)$, taking $f = 0$ momentarily for simplicity and taking $T$ to be the torus with inner radius $R_0 -1$ and outer radius $R_0 + 1$, say, the flux function needs to satisfy the axisymmetric Grad–Shafranov equation (Shafranov Reference Shafranov1966; Grad Reference Grad1967)

(1.5)\begin{gather} \partial_R^2 \psi_0 + \partial_Z^2\psi_0 - \frac{1}{R}\partial_R \psi_0 + R^2 P_0'(\psi_0) + C_0C_0'(\psi_0) = 0, \quad \text{in } D_0, \end{gather}

(1.6)\begin{gather}\psi_0 = \textrm{const. } \quad \text{on } \partial D_0, \end{gather}

where $D_0$ denotes the cross-section of the torus (unit disk) in the $\varPhi = 0$ half-plane centred at $R = R_0$. Conversely, if $\psi _0$ is any solutionFootnote ¹ to (1.5) with $e_\varPhi \boldsymbol {\cdot }\boldsymbol {\nabla } \psi _0 = 0$ then the vector field $B_0$ defined in (1.4) is divergence-free and satisfies (1.1). If $\psi _0$ is constant on $\partial T$, $B_0$ satisfies (1.3).

Unfortunately, these tokamak equilibria come with a slew of problems from the point of view of plasma confinement fusion (Landreman Reference Landreman2019). For example, to achieve improved confinement it is desirable for the magnetic field to ‘twist’ as it wraps around the torus and this can only be accomplished in axisymmetry with a large plasma current, $J$. Such plasma configurations are hard to control in practice. One approach to finding equilibria with better confinement properties is to consider equilibria in geometries which have the desired twist built in. This is the basic design principle behind the stellarator (Garren & Boozer Reference Garren and Boozer1991). It is still desirable for these configurations to possess a form of symmetry, which is known as quasisymmetry.

Definition 1.1 (Weak quasisymmetry, Rodriguez, Helander & Bhattacharjee (Reference Rodriguez, Helander and Bhattacharjee2020))

Let $\xi$ be a non-vanishing vector field tangent to $\partial T$. We say that $\xi$ is a quasisymmetry and the field $B$ is quasisymmetric with respect to $\xi$ if

(1.7)\begin{gather} \textrm{div} \xi =0,\quad \text{in } T, \end{gather}

(1.8)\begin{gather}B \times \xi={-}\boldsymbol{\nabla} \psi,\quad \text{in } T, \end{gather}

(1.9)\begin{gather}\xi \boldsymbol{\cdot} \boldsymbol{\nabla} |B|=0, \quad \text{in } T, \end{gather}

for some function $\psi : T\to \mathbb {R}$.

The significance of the condition (1.8) is that it implies $B\boldsymbol {\cdot } \boldsymbol {\nabla } \psi = 0$ and $\xi \boldsymbol {\cdot } \boldsymbol {\nabla } \psi = 0$ and so quasisymmetric solutions possess flux functions which are symmetric with respect to $\xi$. In Rodriguez et al. (Reference Rodriguez, Helander and Bhattacharjee2020), the authors argue that (1.7)–(1.9) form sufficient conditions that ensure first-order (in gyroradius) particle confinement, hence the terminology of weak quasisymmetry. In the confinement fusion literature (Landreman Reference Landreman2019; Burby, Kallinikos & MacKay Reference Burby, Kallinikos and MacKay2020), one encounters the following alternative definition which is actually stronger than the above. It replaces (1.9) with

(1.10)\begin{equation} \xi \times J= \boldsymbol{\nabla} (B \boldsymbol{\cdot} \xi)\quad \textrm{in } T. \end{equation}

We term this set of conditions strong quasisymmetry. When $f=0$ it is this stronger form of quasisymmetry which is equivalent to other definitions in the plasma fusion literature involving Boozer angles, see § 8 of Landreman (Reference Landreman2019). If $\textrm {div} B = 0$ then (1.9) requires only that a single component of (1.10) vanish, $B\boldsymbol {\cdot } ( \xi \times J - \boldsymbol {\nabla } (B\boldsymbol {\cdot } \xi )) = 0$.Footnote ² In light of this, the additional content of strong quasisymmetry (1.7), (1.8) and (1.10) is the assumption that the other two components of $\xi \times J - \boldsymbol {\nabla } (B\boldsymbol {\cdot } \xi )$ vanish. It turns out that when there is no force and the equilibria are toroidal, strong quasisymmetry is equivalent to Definition 1.1.Footnote ³

From (1.8), if $\xi \boldsymbol {\cdot } B$ is constant on surfaces of constant $\psi$, $\xi \boldsymbol {\cdot } B = C(\psi )$ (by a result in Burby et al. (Reference Burby, Kallinikos and MacKay2020), any solution of (1.1) with $f = 0$ which satisfies (1.7)–(1.9) satisfies this condition), it follows that $B$ is of the form

(1.12)\begin{equation} B = \frac{1}{|\xi|^2} \left(C(\psi) \xi + \xi \times \boldsymbol{\nabla} \psi\right), \end{equation}

and when $f = 0$, the requirement that (1.1) holds implies that $\psi$ must satisfy the quasisymmetric Grad–Shafranov equation (introduced in Burby et al. (Reference Burby, Kallinikos and MacKay2020)) which reads

(1.13)\begin{gather} \Delta \psi - \frac{\xi \times \textrm{curl} \xi}{|\xi|^2} \boldsymbol{\cdot} \boldsymbol{\nabla} \psi + \frac{\xi \boldsymbol{\cdot} \textrm{curl} \xi}{|\xi|^2} C(\psi) + CC'(\psi) + |\xi|^2P'(\psi)=0, \quad \text{in} \ T, \end{gather}

(1.14)\begin{gather}\psi = \textrm{const. } \quad \text{on} \ \partial T. \end{gather}

Equations (1.2), (1.7) and (1.9) can be thought of as constraints relating $\psi$ to the deformation tensor of $\xi$, the symmetric two-tensor $\mathcal {L}_\xi \delta$ defined by

(1.15)\begin{equation} (\mathcal{L}_\xi\delta)(X, Y) = \boldsymbol{\nabla}_X \xi \boldsymbol{\cdot} Y + \boldsymbol{\nabla}_Y \xi \boldsymbol{\cdot} X, \end{equation}

where $\boldsymbol {\nabla }$ denotes covariant differentiation with respect to the Euclidean metric. Recall that $\xi$ generates an isometry of Euclidean space if and only if $\mathcal {L}_\xi \delta = 0$, in which case $\xi$ is called a Killing field for the metric $\delta$. Assuming that $\xi \boldsymbol {\cdot } \boldsymbol {\nabla } \psi = 0$ and $\textrm {div} \xi = 0$, from (1.12) we find

(1.16)\begin{equation} \textrm{div} B = C(\psi) (\mathcal{L}_\xi\delta)(\xi, \xi) + (\mathcal{L}_\xi \delta)(\xi, \xi\times \boldsymbol{\nabla} \psi), \end{equation}

and expanding the condition (1.9) we find

(1.17)\begin{equation} \frac{1}{2}\mathcal{L}_\xi |B|^2 = (\mathcal{L}_\xi\delta)(\xi, \xi) + \frac{2}{C(\psi)} (\mathcal{L}_\xi\delta)(\xi, \xi \times \boldsymbol{\nabla} \psi) + \frac{1}{C(\psi)^2} (\mathcal{L}_\xi\delta)(\xi\times \boldsymbol{\nabla} \psi, \xi \times \boldsymbol{\nabla} \psi), \end{equation}

see lemma C.6 of appendix C. Equation (1.17) is a complicated relationship between $\psi , C(\psi )$ and $\xi$ but notice that it holds trivially (assuming only that $\xi \boldsymbol {\cdot } \boldsymbol {\nabla } \psi = 0$) whenever when $\xi$ is a Killing field. It is well known that in Euclidean space the only Killing fields are linear combinations of translations and rotations. Therefore, up to a multiplicative constant, the only such field compatible with the geometry of the axisymmetric torus is $\xi = R e_\varPhi$ and as mentioned above, such solutions have problematic confinement properties. We have arrived at the following problem.

It is not clear that there are any smooth solutions $\psi , \xi$ to the above problem. In fact, in 1967 (long before the above notion of quasisymmetry was introduced), Grad & Rubin (Reference Grad and Rubin1958) and Grad (Reference Grad1967, Reference Grad1985) conjectured that the only smooth solutions to (1.1)–(1.3) possessing a good flux function have a Euclidean symmetry,Footnote ⁴ and this would in particular rule out any solutions of the above type. Since Grad's work, there have been some constructions of non-symmetric equilibria an infinite cylindrical domains (Salat & Kaiser Reference Salat and Kaiser1995; Kaiser & Salat Reference Kaiser and Salat1997). As these are unbounded in extent, they have limited practical appeal for the perspective of confinement. No such examples of smooth solutions have been rigorously demonstrated on toroidal domains, although there has been some work on suggestive formal near-axis expansions (Bernardin, Moses & Tataronis Reference Bernardin, Moses and Tataronis1986; Weitzner Reference Weitzner2014; Jorge, Sengupta & Landreman Reference Jorge, Sengupta and Landreman2019) and non-symmetric weak solution equilibria with pressure jumps have been rigorously constructed (Bruno & Laurence Reference Bruno and Laurence1996) which may have practical implications for the confinement fusion program (Hudson et al. Reference Hudson, Dewar, Hole and McGann2011, Reference Hudson, Dewar, Dennis, Hole, McGann, Von Nessi and Lazerson2012).Footnote ⁵

We do not address Grad's conjecture here and our goal is instead to present a robust method for constructing solutions to (1.1) with small force and which are approximately quasisymmetric with respect to a given vector field $\xi$ (sufficiently close to the axisymmetric vector field $\xi _0 = R e_\varPhi$), in the sense that (1.8) holds but that (1.9) holds up to a small error.

In addition to the non-trivial constraint (1.9), there are two serious difficulties in constructing solutions to (1.1)–(1.3) of the form (1.12) with given symmetry direction $\xi$. The first is that by (1.16), unlike in the axisymmetric setting, vector fields of the form (1.12) need not be divergence-free. The second difficulty is that for arbitrary $\xi$, it is not at all clear that the equations (1.13) and (1.14) admit any solutions with $\xi \boldsymbol {\cdot } \boldsymbol {\nabla } \psi = 0$, since the coefficients appearing in (1.13)–(1.14) need not be invariant under $\xi$. Both of these difficulties can be traced to the fact that $\xi$ need not be a Killing field with respect to the Euclidean metric. To circumvent these issues, inspired by Lichtenfelz, Misiolek & Preston (Reference Lichtenfelz, Misiolek and Preston2019) and Burby, Kallinikos & MacKay (Reference Burby, Kallinikos and MacKay2020), we replace the metric structure of $(\mathbb {R}^3, \delta )$ with $(\mathbb {R}^3, g)$ for a metric $g$ for which $\xi$ is a Killing field. The resulting magnetic field will not satisfy the usual MHS equations (1.1), but provided $\xi$ is sufficiently close to Killing for the Euclidean metric, the error will be small. We now explain the idea.

Let us suppose that given $\xi$, we can find a metric $g$ on $\mathbb {R}^3$ for which $\mathcal {L}_\xi g= 0$, that is, for which $\xi$ generates an isometry (we give an explicit construction of such metrics for a large class of vector fields $\xi$ after the upcoming statement of theorem 1.3). We then consider the following generalization of the ansatz (1.12), introduced in Burby et al. (Reference Burby, Kallinikos and MacKay2020):

(1.18)\begin{equation} B_g = \frac{1}{|\xi|_g^2} \left( C(\psi)\xi + \sqrt{|g|} \xi\times_g \boldsymbol{\nabla}_g \psi\right). \end{equation}

Here, $|\xi |_g, \times _g, \boldsymbol {\nabla }_g$ denote the analogues of the usual Euclidean quantities $|\xi |,\times \boldsymbol {\nabla }$ with respect to the metric $g$ (see appendix B). In lemma C.3 we use the fact that $\mathcal {L}_\xi g = 0$ to show that vector fields of this form are divergence free assuming only that $\xi \boldsymbol {\cdot } \boldsymbol {\nabla } \psi = 0$,

(1.19)\begin{equation} \textrm{div} B_g = 0 \end{equation}

and also that $\psi$ is a flux function for $B_g$,

(1.20)\begin{equation} \xi \times \boldsymbol{\nabla} \psi = B_g. \end{equation}

We emphasize the somewhat surprising fact that even though the definition (1.18) involves the metric $g$ in a non-trivial way, it is designed that way so that the identities (1.19) and (1.20) involve only Euclidean quantities. We remark that $B_g$ will not be divergence-free with respect to the $g$ metric.

We then seek $B_g$ of the form (1.18) which satisfy the MHS with respect to the metric $g$,

(1.21)\begin{equation} \textrm{curl}_g B_g \times_g B_g = \boldsymbol{\nabla}_g P. \end{equation}

This ansatz leads to the generalized Grad–Shafranov equation for $\psi$,

(1.22)\begin{gather} \textrm{div}_g\left(\sqrt{|g|} \frac{\boldsymbol{\nabla}_g \psi}{|\xi|_g^2} \right) - C(\psi) \frac{\xi}{|\xi|_g^2} \cdot_g \textrm{curl}_g \left(\frac{\xi}{|\xi|_g^2}\right) + \frac{C(\psi)C'(\psi)}{\sqrt{|g|}|\xi|_g^2}+ \frac{P'(\psi)}{\sqrt{|g|}} = 0, \quad \text{in} \ T, \end{gather}

(1.23)\begin{gather}\psi = (\textrm{const.}) , \quad \text{on} \ \partial T, \end{gather}

where $|\xi |_g, \cdot _g, \textrm {curl}_g$ denote the magnitude, dot product and curl with respect to the metric $g$ (see appendix B for the definitions and appendix C for the derivation of (1.22) from (1.18) and (1.21)). Note that (1.22) and (1.23) reduces to (1.13) and (1.14) when $g=\delta$, and when $g$ is the circle-averaged metric, it agrees with the equation derived in Burby et al. (Reference Burby, Kallinikos and MacKay2020). As shown in Burby et al. (Reference Burby, Kallinikos and MacKay2020), all solutions of MHS (1.1)–(1.3) without force and non-vanishing pressure gradient must have a flux function satisfying (1.22) where $g$ is the circle-averaged metric discussed below. In light of this, the study of the generalized Grad–Shafranov equation (1.22) is of fundamental importance in the study of solutions to MHS with a generalized symmetry.

As another consequence of the fact that $\mathcal {L}_\xi g= 0$, the coefficients in (1.22) are invariant under $\xi$ and so (1.22), unlike (1.14), is consistent with the requirement $\xi \boldsymbol {\cdot }\boldsymbol {\nabla } \psi = 0$. The downside is that the equation (1.21) does not agree with (1.1) unless $g = \delta$ and so $B_g$ will not satisfy the original MHS equations. However, if we can arrange for the metric $g$ to be sufficiently close to the Euclidean metric $\delta$, then $B_g$ will satisfy the usual MHS equations $\textrm {curl} B_g \times B_g - \boldsymbol {\nabla } P = 0$ up to a small error. Our approach will be to solve the generalized Grad–Shafranov equation (1.22) by deforming an appropriate solution $\psi _0$ of the axisymmetric Grad–Shafranov equation (1.5), using the methods from Constantin, Drivas & Ginsberg (Reference Constantin, Drivas and Ginsberg2020). In particular, we seek a diffeomorphism $\gamma : D_0 \to D$ and requiring that $\psi = \psi _0\circ \gamma ^{-1}$. It turns out (see § 2) that this reduces to a system of nonlinear elliptic equations for the components of $\gamma$ which can be solved by a iteration.

In what follows, $T_0$ denotes the axisymmetric torus

(1.24)\begin{equation} T_0 = \{ (R,\varPhi, Z)\, |\, (R - R_0)^2 + Z^2 \leq a, 0\leq \varPhi \leq 2{\rm \pi}\}, \end{equation}

with thickness $0 < a \ll R_0$. Let $\xi _0 = R e_\varPhi$ be the generator of rotations in the $Z = 0$ plane. Let $D$ be any domain in the half-plane $\{\varPhi = 0\}$ sufficiently close to $D_0$. Suppose that $\xi$ is a vector field which is sufficiently close to the rotation field $\xi _0$ with the property that all the orbits of $\xi$ starting from $D$ are periodic (with possibly different period $\tau (p)$). In this case we define the toroidal domain

(1.25)\begin{equation} T = \{\varphi_s(p) \, | \, p\in D, \ s\in [0, \tau(p))\}, \end{equation}

where $\varphi _s(p)$ denotes the time-$s$ flow of $\xi$ starting from $p \in D$,

(1.26a,b)\begin{equation} \frac{\textrm{d} }{\textrm{d} s} {\varphi}_s(p) = \xi(\varphi_s(p)), \quad \varphi_0(p) = p \in D. \end{equation}

In this setting we say that the toroidal domain $T$ is swept out by $\xi$ from $D$.

Our first result is that, given a toroidal domain $T$ swept out by a vector field $\xi$ as above, sufficiently close to the axisymmetric torus $T_0$, we can find a flux function satisfying the generalized Grad–Shafranov equation (1.22). The proof is constructive and relies on deforming a known axisymmetric steady state satisfying mild conditions (H1)–(H2) stated in § 2.

As a consequence of the above theorem, we are able to produce magnetic fields with nested flux surfaces and a global symmetry that solve MHS up to a small force whose magnitude is controlled by the deviation of the symmetry from being Euclidean. These fields satisfy two of the three quasisymmetry conditions, the third holding approximately. The resulting magnetic field possesses flux surfaces which have the same topology as the axisymmetric base state.

Theorem 1.3 Suppose the hypotheses of the previous theorem hold. For any given $C\in C^{k+1,\alpha }(\mathbb {R})$ sufficiently close to $C_0$, there is a flux function $\psi \in C^{k+2,\alpha }(T)$, and a pressure $P = P(\psi ) \in C^{k+1,\alpha }(T)$ so that the magnetic field $B$ defined by (1.18) satisfies $B\boldsymbol {\cdot } \boldsymbol {\nabla }\psi = 0$, $B\times \xi = \boldsymbol {\nabla } \psi$ as well as MHS (1.1)–(1.3) with a force $f$ obeying

(1.27)\begin{equation} \|f\|_{C^{k,\alpha}(T)} \leq c \|\delta - g\|_{C^{k+2,\alpha}(T)}, \end{equation}

where $c:= c(\|\xi \|_{C^{k+2,\alpha }(T)},\| \psi \|_{C^{k+2,\alpha }(T)},\| P_0\|_{C^{k+1,\alpha }(\mathbb {R})},\| C_0\|_{C^{k+1,\alpha }(\mathbb {R})})$. Moreover, the flux surfaces of $B$ (isosurfaces of $\psi$) are diffeomorphic to the isosurfaces of $\psi _0$, and $\psi$ is a solution of the generalized Grad–Shafranov equation (1.22).

The point of the bound (1.27) is that if $\xi$ is a Killing field for the Euclidean metric $\delta$ then we can take $g = \delta$ in the above and by (1.27), the $B$ is then an exact solution of the MHS equations (1.1) with $f = 0$. In this sense, (1.27) shows that one can construct approximate solutions to MHS with symmetry direction $\xi$ with error proportional to how far $\xi$ is from being a symmetry of $\mathbb {R}^3$. We remark that the proof is quantitative in that all the small parameters can be explicitly defined in terms of the inputs $\psi _0$, $C_0$, $P_0$ and $\xi$. Let us also remark that one is not free to choose $P$ from the outset and it is instead determined in the course of the proof to enforce a certain compatibility condition, see § 2.

The above theorem is perturbative, in the sense that the resulting magnetic field will be approximately axisymmetric and have a flux function close to a given $\psi _0$ satisfying the axisymmetric Grad–Shafranov equation (1.5). As will be discussed in the upcoming section, the result follows from a theorem in Constantin et al. (Reference Constantin, Drivas and Ginsberg2020) by deforming the given solution $\psi _0$ of the axisymmetric Grad–Shafranov equation (1.5) into a solution of the generalized Grad–Shafranov equation (1.22). The same theorem from Constantin et al. (Reference Constantin, Drivas and Ginsberg2020) in fact allows one to deform a given solution $\psi _1$ to (1.22) for given $\xi = \xi _1$ into a solution $\psi _2$ to (1.22) with nearby, but different, $\xi = \xi _2$. Given a desired $\xi$, if one can produce a sequence of vector fields $\xi _0, \xi _1,\ldots , \xi _{N-1}, \xi _N = \xi$ in such a way that the resulting solutions $\psi _0, \psi _1,\ldots , \psi _{N-1}$ all satisfy the conditions (H1)–(H2) this would produce a flux function satisfying (1.22) far from axisymmetry. Note, however, that the resulting force in (1.1) could be quite large.

If one is only interested in constructing approximate equilibria, this can be achieved simply by pushing forward a given axisymmetric state by a volume-preserving diffeomorphishm. The resulting flux function need not satisfy the generalized Grad–Shafranov equation (1.22). On the other hand, the construction in theorem 1.2 does ensure that the generalized Grad–Shafranov equation is exactly satisfied. In light of the fact that Burby et al. (Reference Burby, Kallinikos and MacKay2020) show that all unforced solutions to (1.1) and (1.3) must satisfy the equation (1.22), our theorem may provide a path towards obtaining non-axisymmetric solutions without force.

We now describe how to produce a base state $\psi _0$ and metric $g$ which are suitable inputs for theorem 1.2. In appendix A, we provide an example of a base state $\psi _0$ satisfying (H1)–(H2) living on a large aspect ratio torus. This is obtained as a perturbation of an explicit profile on an ‘infinite aspect ratio’ torus. It should be stressed that the conditions (H1)–(H2) are not very stringent and should hold for a wide class of axisymmetric solutions that possess simple nested flux surfaces (which could, for example, be numerically obtained). Next, we describe two large classes of vector fields $\xi$ and metrics $g$ satisfying the hypotheses of our theorem.

We conclude with some remarks about achieving exact quasisymmetry. By construction, the magnetic field $B$ from the previous theorem will satisfy (1.8) but will only approximately satisfy the property (1.9) of quasisymmetry. Thus, our fields confine particles to zeroth but not first order in the guiding centre approximation (Rodriguez et al. Reference Rodriguez, Helander and Bhattacharjee2020). The error from being an exact weak quasisymmetry can be easily quantified; for a vector field $B_g$ of the form (1.18), assuming that $g$ is such that $\mathcal {L}_\xi g = 0$ the condition (1.9) reads

(1.29)\begin{align} \xi \boldsymbol{\cdot}\boldsymbol{\nabla} |B_g| &= \frac{C^2(\psi)}{2|\xi|_g^4|B_g|} \left[ (\mathcal{L}_\xi \delta) (\xi ,\xi) +2 C^{{-}1}(\psi) (\mathcal{L}_\xi \delta) (\xi ,\boldsymbol{\nabla}_g^\perp \psi ) \right.\nonumber\\ &\quad\left.+\,C^{{-}2}(\psi) (\mathcal{L}_\xi \delta) (\boldsymbol{\nabla}_g^\perp \psi ,\boldsymbol{\nabla}_g^\perp \psi ) \right], \end{align}

see lemma C.6. Since (1.29) involves the Euclidean deformation tensor alone, it is controlled by the deviation of $\xi$ from being a Euclidean isometry and our solution will have $\xi \boldsymbol {\cdot }\boldsymbol {\nabla } |B_g|$ small. The error from being a strong quasisymmetry is also quantifiably small.

It is worth remarking that there are additional freedoms in our construction that could, in principle, be used to further constrain the constructed solution. Specifically, in our theorem, we treat $\xi$ as a fixed vector field sufficiently close to $\xi _0$ and we made the somewhat arbitrary choice that the map $\gamma$ should be volume preserving. The results in Constantin et al. (Reference Constantin, Drivas and Ginsberg2020) actually allow one to construct the map $\gamma$ so that $\det \boldsymbol {\nabla } \gamma := \rho$ is any given function, sufficiently close to one; in fact by iterating that result, one can additionally achieve that $\det \boldsymbol {\nabla } \gamma = X(\phi , \eta , \partial \phi , \partial \eta ,\partial \partial _s\phi ,\partial \partial _s\eta )$ for a suitable nonlinearity $X$ sufficiently close to one when $\phi , \eta = 0$. Using this freedom, it is possible to show that, under some (possibly restrictive and undesirable) assumptions on the field $\xi$, the Jacobian $\rho$ can be used to achieve exact quasisymmetry on a slice of the torus (namely on the cross-section $D$). Ensuring this property holds seems of little practical interest for ion confinement in a stellarator, since particles starting on the slice will immediately leave. In contrast, Garren and Boozer Garren & Boozer (Reference Garren and Boozer1991) and Plunk & Helander (Reference Plunk and Helander2018) show that exact quasisymmetry is possible to achieve on one flux surface while maintaining the MHS force balance, which is of greater relevance to confinement in a stellarator. It is unknown whether or not quasisymmetry can be achieved in a volume. We leave open the question of whether or not, using our approach, a carefully designed field $\xi$ (perhaps constructed dynamically alongside the solution) can be used to ensure quasisymmetry on a flux surface or a volume.