A three-dimensional reduced MHD model consistent with full MHD

Abstract

Within the context of a viscoresistive magnetohydrodynamic (MHD) model with anisotropic heat transport and cross-field mass diffusion, we introduce novel three-term representations for the magnetic field (background vacuum field, field line bending and field compression) and velocity ( E × B flow, field-aligned flow and fluid compression), which are amenable to three-dimensional treatment. Once the representations are inserted into the MHD equations, appropriate projection operators are applied to Faradays law and the Navier-Stokes equation to obtain a system of scalar equations that is closed by the continuity and energy equations. If the background vacuum field is sufficiently strong and the β is low, MHD waves are approximately separated by the three terms in the velocity representation, with each term containing a specific wave. Thus, by setting the appropriate term to zero, we eliminate fast magnetosonic waves, obtaining a reduced MHD model. We also show that the other two velocity terms do not compress the magnetic field, which allows us to set the field compression term to zero within the same reduced model. Dropping also the field-aligned flow, a further simplified model is obtained, leading to a fully consistent hierarchy of reduced and full MHD models for 3D plasma configurations. Finally, we discuss the conservation properties and derive the conditions under which the reduction approximation is valid. We also show that by using an ordering approach, reduced MHD equations similar to what we got from the ansatz approach can be obtained by means of a physics-based asymptotic expansion.

Testing of the new JOREK stellarator-capable model in the tokamak limit

“…The original hierarchy of models, as presented in Nikulsin et al. (2019), was derived from the viscoresistive MHD equations: The gradient operators parallel and perpendicular to the total magnetic field are defined as and . The usual MHD notation is followed with , , and being density, pressure, velocity and magnetic field, respectively.…”

“…Here, as in our previous paper (Nikulsin et al. 2019), we adopt an ansatz-based approach. Previously, we derived a hierarchy of models, which we intended to use for stellarator simulations.…”

“…Dropping the fast magnetosonic wave term from the velocity ansatz and the field compression term from the magnetic field ansatz, we obtain reduced MHD (Nikulsin et al. 2019). However, we later found that the projection operators used had good momentum conservation properties, but not as good energy conservation properties.…”

First predictive simulations for deuterium shattered pellet injection in ASDEX Upgrade

“…Table 1: An overview is given of the parameters used for all simulation cases investigated in this study. The following information is shown: n at is the number of atoms in the original pellet before shattering in units of 10 22 , n sh represents the number of shards (all shards have the same size as described in the main text), η 0 is the initial core resistivity in units of 10 −7 Ωm, v in is the injection velocity in units of km/s, χ ||,0 the parallel conductivity coefficient in units of 10 11 m 2 /s, B φ denotes the toroidal field strength on axis in units of T, n el gives the number of grid elements in units of 1000, N t refers to the largest toroidal mode number included in the simulation, j 0 indicates whether a source is applied to keep the current profile approximately constant in time. In the column "remark", special setups of the respective cases are mentioned, e.g., those variables are listed, which are kept constant in time.…”

General three-dimensional equilibria for gravitating ideal magnetohydrodynamics of field-aligned steady incompressible flows with an application to solar prominences

“…They used Euler potentials of the magnetic field of magnetohydrostatic equilibria to obtain solutions of the governing MHD equations for stationary flows from corresponding static MHD equilibria. In the framework of a resistive MHD model with anisotropic heat transport and cross-field mass diffusion, Nikulsin et al (2019) introduced representations for magnetic field-aligned flows which are amenable to three-dimensional treatment.…”