arXiv subjects

Wrick Sengupta

Publications and source records attributed to Wrick Sengupta.

At least 19 recordsLinked to original sources

arXiv2026

Compact quasiaxisymmetric stellarators, a near axisymmetric theory

We develop a theory of ridges in compact stellarators with quasiaxisymmetry (QA). The equilibrium with finite plasma currents and pressure is modeled by ideal magnetohydrostatics (MHS). Field lines are collimated near sharp ridges, much like X-points, making ridges attractive to divertor designs without the requirement of a rational rotational transform at the divertor. However, unlike X-points, which must cover the entire torus an integer number of times, sharp ridges are typically localized in certain parts of the flux surfaces. Motivated by recent work (Henneberg and Plunk, Phys. Rev. Research 6, L022052) on compact hybrid devices, we develop a perturbative treatment of nearly axisymmetric quasisymmetric devices by expanding in the deviation from perfect axisymmetry. As a result, we can analytically describe the key features of compact QA devices, such as the tendency for ridges to be localized on the inboard side, where the Gaussian curvature is typically negative, and the field strength is maximum. We provide comprehensive numerical evidence in support of our analytical theory.

physics.plasm-ph↗

arXiv2026

Optical analogy for stellarators: Ridges as caustics and coils as singularities

A common feature of most numerically optimized stellarator geometries is the presence of sharp ridges on outer flux surfaces, irrespective of the rotational transform. Despite their importance, an analytical theory for their existence has been lacking. In this work, we demonstrate that ridges are not artifacts but mathematical necessities. We develop such a theory for devices with quasisymmetry (QS). We demonstrate that QS exhibits close connections with the theory of geometrical optics, following Parker's ``optical analogy" (E.N. Parker, Geophys. Astrophys. Fluid Dyn, 1989). By mapping vacuum QS to the eikonal equation of geometrical optics, we derive the conditions for ridge formation, identified as field line caustics where magnetic field lines focus. Furthermore, we prove a geometric theorem for stellarator coil design: both ridges and filamentary coils must lie on the zero-determinant manifold of the magnetic gradient tensor. This topological constraint unifies the description of plasma ridges and external coils, providing a precise criterion for identifying valid coil locations and explaining the efficacy of the magnetic gradient lengthscale (J. Kappel et al., Plasma Phys. Control. Fusion, 2024) as a coil optimization parameter. We demonstrate that as the device becomes more compact, sharp ridges naturally form on the inboard side in quasiaxisymmetry. We support our analytical theory with extensive numerical evidence.

physics.plasm-ph↗

arXiv2026

Estimating coil features from an equilibrium

We present an explicit theoretical framework for constructing artificial modular coils for vacuum stellarator fields based solely on equilibrium properties, achieved through the formulation of a current potential defined on flux surfaces. Contours of constant Boozer toroidal angle can be directly interpreted as proxy coils, and so we demonstrate that key measures of coil complexity (particularly coil non-planarity) are strongly governed by local magnetic field properties. This approach shows promise as predictor for more realistic coil configurations, providing both a pathway towards deeper understanding of equilibrium-coil relationships and a potential practical proxy for coil design.

physics.plasm-ph↗

arXiv2026

Statistical equilibrium model for stellarators

In three dimensional toroidal domains without symmetry, the standard magnetohydrodynamic (MHD) equilibrium model used for magnetic confinement fusion does not generally support smooth solutions. Instead, solutions have singular plasma currents on resonant magnetic surfaces that violate the MHD assumption of length-scale separation, further leading to the non- or slow convergence of numerical approximations under refinement. In this work, we present an improved equilibrium principle derived from a statistical model for plasma fluctuations. Instead of being static, we assume that the plasma magnetic field is ergodically and rapidly fluctuating relative to the MHD time scale. By averaging the resulting force, we derive a variational equilibrium problem for the statistical mean magnetic field which depends on fluctuation variance. Then, through asymptotics, numerical simulations, and a Grad-Shafranov type argument, we show that the variational principle supports smooth solutions for specific fluctuation statistics chosen to minimally modify the standard equilibrium modeling paradigm. Physically, this model smooths singular current sheets with a length scale determined by the magnetic field fluctuations.

physics.plasm-ph↗

arXiv2026

The Geometry of Flux Surfaces with Quasi-Poloidal Symmetry

Quasi-poloidal (QP) magnetic fields have desirable properties for confining plasma: no radial drift of guiding centres (with positive implications for neoclassical transport), zero Pfirsch-Schl\"uter current, and a lower level of damping for poloidal flows. Despite their attractive properties, QP fields are not amenable to the near-axis expansion, a major theoretical tool for understanding toroidal fields. In this paper, we provide a novel framework for defining and understanding QP flux surfaces. This framework relies on a simplification that transforms the task of finding a quasi-poloidal flux surface from a 3D problem to a 2D problem. This simplification also applies to asymmetric magnetic mirrors with desirable properties. We sketch how this 2D problem can form the basis of an efficient optimisation problem for finding QP flux surfaces. We leverage this 2D problem for theoretical understanding: for instance, we identify a route to finding QP flux surfaces that are naturally flat mirrors (Velasco et al. 2023). The reduced model is qualitatively checked against numerically optimised QP equilibria. These numerical solutions only satisfy QP approximately, but we predictably find that local discrepancies with the reduced model correspond to significant local QP errors, anomalous parallel currents, and field lines deviating from geodesics.

physics.plasm-ph↗

arXiv2025

A class of high-beta, large-aspect-ratio quasiaxisymmetric Palumbo-like configurations

The space of high-beta, approximately quasiaxisymmetric, large-aspect-ratio stellarator configurations is explored using an inverse coordinate approach and a quadratic polynomial ansatz for the flux function, following the method of Palumbo, extended by Hernandes and Clemente. This approach yields a system of nonlinear ODEs that, when solved, give equilibria exhibiting positive or negative triangularity, cusps, and (in an extreme limit) current singularities. It is shown that a cubic ansatz may also be used, but that polynomials of degree four or higher will lead to overdetermination.

physics.plasm-ph↗

arXiv2025

A Grad-Shafranov model for compact quasisymmetric stellarators

A Grad-Shafranov equation (GSE) valid for compact quasisymmetric stellarators is derived by an asymptotic expansion around a vacuum field carried to first order. We obtain an equation for the existence of flux surfaces leading up to the GSE. The flux surface label must simultaneously satisfy the existence equation and the GSE, which generally leads to an overdetermined problem. We show how the overdetermined problem can be resolved within our model for a class of hybrid devices similar to that studied by Henneberg and Plunk (S. Henneberg and G. Plunk, PRR 2024). We are also able to solve the existence equation for flux surfaces analytically in the most general case by introducing a special coordinate system. This will enable us to carry out an optimization seeking to minimize the error in our GSE while obeying the flux surface existence equation, which will allow us to find solutions outside the class of hybrid devices. This will allow for a coarse-grained approximate search in the space of quasisymmetric equilibria that should be faster than a conventional stellarator optimization. Nevertheless, it would still be necessary to fine-tune the approximate solutions using conventional tools to obtain a more precise optimized equilibrium.

physics.plasm-ph↗

arXiv2024

Enhanced Collisional Losses from a Magnetic Mirror Using the Lenard-Bernstein Collision Operator

Collisions are crucial in governing particle and energy transport in plasmas confined in a magnetic mirror trap. Modern gyrokinetic codes model transport in magnetic mirrors, but some utilize approximate model collision operators. This study focuses on a Pastukhov-style method of images calculation of particle and energy confinement times using a Lenard-Bernstein model collision operator. Prior work on parallel particle and energy balances used a different Fokker-Planck plasma collision operator. The method must be extended in non-trivial ways to study the Lenard-Bernstein operator. To assess the effectiveness of our approach, we compare our results with a modern finite element solver. Our findings reveal that the particle confinement time scales like $a \exp(a^2)$ using the Lenard-Bernstein operator, in contrast to the more accurate scaling that the Coulomb collision operator would yield $a^2 \exp(a^2)$, where $a^2$ is approximately proportional to the ambipolar potential. We propose that codes solving for collisional losses in magnetic mirrors utilizing the Lenard-Bernstein or Dougherty collision operator scale their collision frequency of any electrostatically confined species. This study illuminates the collision operator's intricate role in the Pastukhov-style method of images calculation of collisional confinement.

physics.plasm-ph↗

arXiv2024

Universal non-thermal power-law distribution functions from the self-consistent evolution of collisionless electrostatic plasmas

Distribution functions of collisionless systems are known to show non-thermal power law tails. Interestingly, collisionless plasmas in various physical scenarios, (e.g., the ion population of the solar wind) feature a $v^{-5}$ tail in the velocity ($v$) distribution, whose origin has been a long-standing mystery. We show this power law tail to be a natural outcome of the self-consistent collisionless relaxation of driven electrostatic plasmas. We perform a quasilinear analysis of the perturbed Vlasov-Poisson equations to show that the coarse-grained mean distribution function (DF), $f_0$, follows a quasilinear diffusion equation with a diffusion coefficient $D(v)$ that depends on $v$ through the plasma dielectric constant. If the plasma is isotropically forced on scales much larger than the Debye length with a white noise-like electric field, then $D(v)\sim v^4$ for $\sigma<v<\omega_{\mathrm{P}}/k$, with $\sigma$ the thermal velocity, $\omega_{\mathrm{P}}$ the plasma frequency and $k$ the maximum wavenumber of the perturbation; the corresponding $f_0$, in the quasi-steady state, develops a $v^{-\left(d+2\right)}$ tail in $d$ dimensions ($v^{-5}$ tail in 3D), while the energy ($E$) distribution develops an $E^{-2}$ tail irrespective of the dimensionality of space. Any redness of the noise only alters the scaling in the high $v$ end. Non-resonant particles moving slower than the phase-velocity of the plasma waves ($\omega_{\mathrm{P}}/k$) experience a Debye-screened electric field, and significantly less (power law suppressed) acceleration than the near-resonant particles. Thus, a Maxwellian DF develops a power law tail. The Maxwellian core ($v<\sigma$) eventually also heats up, but over a much longer timescale than that over which the tail forms. We definitively show that self-consistency (ignored in test-particle treatments) is crucial for the development of the universal $v^{-5}$ tail.

astro-ph.SR↗

arXiv2024

An asymptotic Grad-Shafranov equation for quasisymmetric stellarators

A first-order model is derived for quasisymmetric stellarators where the vacuum field due to coils is dominant, but plasma-current-induced terms are not negligible and can contribute to magnetic differential equations, with $\beta$ of the order of the ratio of induced to vacuum fields. Under these assumptions, it is proven that the aspect ratio must be large and a simple expression can be obtained for the lowest-order vacuum field. The first-order correction, which involves both vacuum and current-driven fields, is governed by a Grad-Shafranov equation and the requirement that flux surfaces exist. These two equations are not always consistent, and so this model is generally overconstrained, but special solutions exist that satisfy both equations simultaneously. One family of such solutions are the first-order near-axis solutions. Thus, the first-order near-axis model is a subset of the model presented here. Several other solutions outside the scope of the near-axis model are also found. A case study comparing one such solution to a VMEC-generated solution shows good agreement.

physics.plasm-ph↗

arXiv2023

Phase-space entropy cascade and irreversibility of stochastic heating in nearly collisionless plasma turbulence

We consider a nearly collisionless plasma consisting of a species of `test particles' in 1D-1V, stirred by an externally imposed stochastic electric field. The mean effect on the particle distribution function is stochastic heating. Accompanying this heating is the generation of fine-scale structure in the distribution function, which we characterize with the collisionless (Casimir) invariant $C_2 \propto \iint dx dv \, \langle f^2 \rangle$. We find that $C_2$ is transferred from large scales to small scales in both position and velocity space via a phase-space cascade enabled by both particle streaming and nonlinear interactions between particles and the stochastic electric field. We compute the steady-state fluxes and spectrum of $C_2$ in Fourier space, with $k$ and $s$ denoting spatial and velocity wavenumbers, respectively. Whereas even the linear phase mixing alone would lead to a constant flux of $C_2$ to high $s$ (towards the collisional dissipation range) at every $k$, the nonlinearity accelerates this cascade by intertwining velocity and position space so that the flux of $C_2$ is to both high $k$ and high $s$ simultaneously. Integrating over velocity (spatial) wavenumbers, the $k$-space ($s$-space) flux of $C_2$ is constant down to a dissipation length (velocity) scale that tends to zero as the collision frequency does, even though the rate of collisional dissipation remains finite. The resulting spectrum in the inertial range is a self-similar function in the $(k,s)$ plane, with power-law asymptotics at large $k$ and $s$. We argue that stochastic heating is made irreversible by this entropy cascade and that, while collisional dissipation accessed via phase mixing occurs only at small spatial scales rather than at every scale as it would in a linear system, the cascade makes phase mixing even more effective overall in the nonlinear regime than in the linear one.

physics.plasm-ph↗

arXiv2022

Constructing the space of quasisymmetric stellarators

A simplified view of the space of optimised stellarators has the potential to guide and aid the design efforts of magnetic confinement configurations suitable for future fusion reactors. We present one such view for the class of quasisymmetric stellarators based on their approximate description near their centre (magnetic axis). The result is a space that captures existing designs and presents new ones, providing a common framework to study them. Such a simplified construction offers a basic topological approach, guided by certain theoretical and physical choices, which this paper presents in detail.

physics.plasm-ph↗

arXiv2022

Phases and phase-transitions in quasisymmetric configuration space

We explore the structure of the space of quasisymmetric configurations identifying them by their magnetic axes, described as 3D closed curves. We demonstrate that this topological perspective divides the space of all configurations into well-separated quasisymmetric phases. Each phase is characterized by the self-linking number (a topological invariant), defining different symmetry configurations (quasi-axisymmetry or quasi-helical symmetry). The phase-transition manifolds correspond to quasi-isodynamic configurations. By considering some models for closed curves (most notably torus unknots), general features associated with these phases are explored. Some general criteria are also built and leveraged to provide a simple way to describe existing quasisymmetric designs. This constitutes the first step in a program to identify quasisymmetric configurations with a reduced set of functions and parameters, to deepen understanding of configuration space, and offer an alternative approach to stellarator optimization that begins with the magnetic axis and builds outward.

physics.plasm-ph↗

arXiv2021

Weakly Quasisymmetric Near-Axis Solutions to all Orders

We show that the equations satisfied by weakly quasisymmetric magnetic fields can be solved to arbitrarily high order in powers of the distance from the magnetic axis. This demonstration does not consider force balance. The existence of solutions requires an appropriate choice of parameters, most notably the toroidal current or rotational transform profiles. We do not prove that the expansion converges (it is likely divergent but asymptotic), and thus the demonstration here should not be taken as definitive proof of the existence of global solutions. Instead, we provide a systematic construction of solutions to arbitrarily high order.

physics.plasm-ph↗

arXiv2021

Generalized Boozer coordinates: a natural coordinate system for quasisymmetry

We prove the existence of a straight-field-line coordinate system we call generalized Boozer coordinates. This coordinate system exists for magnetic fields with nested toroidal flux surfaces} provided $ \oint\mathrm{d}l/B\:(\mathbf{j}\cdot\nabla\psi)=0$, where symbols have their usual meaning, and the integral is taken along closed magnetic field lines. All quasisymmetric fields, regardless of their associated form of equilibria, must satisfy this condition. This coordinate system presents itself as a convenient form in which to describe general quasisymmetric configurations and their properties. Insight can be gained analytically into the difference between strong and weak forms of quasisymmetry, as well as axisymmetry, and the interaction of quasisymmetry with different forms of equilibria.

physics.plasm-ph↗

arXiv2021

Steady plasma flows in a periodic non-symmetric domain

Steady plasma flows have been studied almost exclusively in systems with continuous symmetry or in open domains. In the absence of continuous symmetry, the lack of a conserved quantity makes the study of flows intrinsically challenging. In a toroidal domain, the requirement of double-periodicity for physical quantities adds to the complications. In particular, the magnetohydrodynamics (MHD) model of plasma steady-state with the flow in a non-symmetric toroidal domain allows the development of singularities when the rotational transform of the magnetic field is rational, much like the equilibrium MHD model. In this work, we show that steady flows can still be maintained provided the rotational transform is close to rational and the magnetic shear is weak. We extend the techniques developed in carrying out perturbation methods to all orders for static MHD equilibrium by Weitzner (Physics of Plasmas 21, 022515 (2014)) to MHD equilibrium with flows. We construct perturbative MHD equilibrium in a doubly-periodic domain with nearly parallel flows by systematically eliminating magnetic resonances order by order. We then utilize an additional symmetry of the flow problem, first discussed by E. Hameiri in (J. Math. Phys. \textbf{22}, 2080 (1981) Sec. III), to obtain a generalized Grad-Shafranov equation for a class of non-symmetric three-dimensional MHD equilibrium with flows both parallel and perpendicular to the magnetic field. For this class of flows, we are able to obtain non-symmetric generalizations of integrals of motion, such as Bernoulli's function and angular momentum. Finally, we obtain the generalized Hamada conditions, which are consistency conditions necessary to suppress singular currents in such a system when the magnetic field lines are closed. We do not attempt to address the question of neoclassical damping of flows.

physics.plasm-ph↗

arXiv2020

Vacuum magnetic fields with exact quasisymmetry near a flux surface. Part 1: Solutions near an axisymmetric surface

While several results have pointed to the existence of exactly quasisymmetric fields on a surface (Garren & Boozer 1991a,b; Plunk & Helander 2018), we have obtained the first such solutions using a vacuum surface expansion formalism. We obtain a single nonlinear parabolic PDE for a function $η$ such the field strength satisfies $B = B(η)$. Closed-form solutions are obtained in cylindrical, slab, and isodynamic geometries. Numerical solutions of the full nonlinear equations in general axisymmetric toroidal geometry are obtained, resulting in a class of quasi-helical local vacuum equilibria near an axisymmetric surface. The analytic models provide additional insight into general features of the nonlinear solutions, such as localization of the surface perturbations on the inboard side.

physics.plasm-ph↗

arXiv2020

Exact non-symmetric closed line vacuum magnetic fields in a topological torus

Non-symmetric vacuum magnetic fields with closed magnetic field lines are of interest in the construction of stellarator equilibria. Beyond the result of D.Lortz (ZAMP \textbf{21}, 196 (1970)), few results are available. This work presents a closed-form expression for a class of vacuum magnetic fields in a topological torus with closed field lines. We explicitly obtain the invariants of such a field. We finally show that a three-dimensional low beta magnetohydrodynamic (MHD) equilibrium may be constructed in a topological torus starting with these closed line vacuum magnetic fields.

physics.plasm-ph↗