692 research outputs found
Three discontinuous Galerkin schemes for the anisotropic heat conduction equation on non-aligned grids
We present and discuss three discontinuous Galerkin (dG) discretizations for
the anisotropic heat conduction equation on non-aligned cylindrical grids. Our
most favourable scheme relies on a self-adjoint local dG (LDG) discretization
of the elliptic operator. It conserves the energy exactly and converges with
arbitrary order. The pollution by numerical perpendicular heat fluxes degrades
with superconvergence rates. We compare this scheme with aligned schemes that
are based on the flux-coordinate independent approach for the discretization of
parallel derivatives. Here, the dG method provides the necessary interpolation.
The first aligned discretization can be used in an explicit time-integrator.
However, the scheme violates conservation of energy and shows up stagnating
convergence rates for very high resolutions. We overcome this partly by using
the adjoint of the parallel derivative operator to construct a second
self-adjoint aligned scheme. This scheme preserves energy, but reveals
unphysical oscillations in the numerical tests, which result in a decreased
order of convergence. Both aligned schemes exhibit low numerical heat fluxes
into the perpendicular direction. We build our argumentation on various
numerical experiments on all three schemes for a general axisymmetric magnetic
field, which is closed by a comparison to the aligned finite difference (FD)
schemes of References [1,2
The collisional drift wave instability in steep density gradient regimes
The collisional drift wave instability in a straight magnetic field
configuration is studied within a full-F gyro-fluid model, which relaxes the
Oberbeck-Boussinesq (OB) approximation. Accordingly, we focus our study on
steep background density gradients. In this regime we report on corrections by
factors of order one to the eigenvalue analysis of former OB approximated
approaches as well as on spatially localised eigenfunctions, that contrast
strongly with their OB approximated equivalent. Remarkably, non-modal phenomena
arise for large density inhomogeneities and for all collisionalities. As a
result, we find initial decay and non-modal growth of the free energy and
radially localised and sheared growth patterns. The latter non-modal effect
sustains even in the nonlinear regime in the form of radially localised
turbulence or zonal flow amplitudes.Comment: accepted at Nuclear Fusio
Non-Oberbeck-Boussinesq zonal flow generation
Novel mechanisms for zonal flow (ZF) generation for both large relative
density fluctuations and background density gradients are presented. In this
non-Oberbeck-Boussinesq (NOB) regime ZFs are driven by the Favre stress, the
large fluctuation extension of the Reynolds stress, and by background density
gradient and radial particle flux dominated terms. Simulations of a nonlinear
full-F gyro-fluid model confirm the predicted mechanism for radial ZF
propagation and show the significance of the NOB ZF terms for either large
relative density fluctuation levels or steep background density gradients
Unified transport scaling laws for plasma blobs and depletions
We study the dynamics of seeded plasma blobs and depletions in an (effective)
gravitational field. For incompressible flows the radial center of mass
velocity of blobs and depletions is proportional to the square root of their
initial cross-field size and amplitude. If the flows are compressible, this
scaling holds only for ratios of amplitude to size larger than a critical
value. Otherwise, the maximum blob and depletion velocity depends linearly on
the initial amplitude and is independent of size. In both cases the
acceleration of blobs and depletions depends on their initial amplitude
relative to the background plasma density, is proportional to gravity and
independent of their cross-field size. Due to their reduced inertia plasma
depletions accelerate more quickly than the corresponding blobs. These scaling
laws are derived from the invariants of the governing drift-fluid equations and
agree excellently with numerical simulations over five orders of magnitude. We
suggest an empirical model that unifies and correctly captures the radial
acceleration and maximum velocities of both blobs and depletions
Beyond the Oberbeck-Boussinesq and long wavelength approximation
We present the first simulations of a reduced magnetized plasma model that
incorporates both arbitrary wavelength polarization and non-Oberbeck-Boussinesq
effects. Significant influence of these two effects on the density, electric
potential and ExB vorticity and non-linear dynamics of blobs are reported.
Arbitrary wavelength polarization implicates so-called gyro-amplification that
compared to a long wavelength approximation leads to highly amplified
small-scale ExB vorticity fluctuations. These strongly increase the coherence
and lifetime of blobs and alter the motion of the blobs through a faster
blob-disintegration. Non-Oberbeck-Boussinesq effects incorporate plasma
inertia, which substantially decreases the growth rate and linear acceleration
of high amplitude blobs, while the maximum blob velocity is not affected.
Finally, we generalize and numerically verify unified scaling laws for blob
velocity, acceleration and growth rate that include both ion temperature and
arbitrary blob amplitude dependence
Angular momentum and rotational energy of mean flows in toroidal magnetic fields
We derive the balance equation for the Favre averaged angular momentum in toroidal not necessarily axisymmetric magnetic field equilibria. We find that the components of angular momentum are given by the covariant poloidal and toroidal components of E x B and parallel flow velocities and we separately identify all relevant stress tensors, torques and source terms for each of these components. Our results feature the Favre stress generalisations of previously found Reynolds stresses like the diamagnetic or parallel E x B stress, as well as the density gradient drive term. Further, we identify the magnetic shear as a source of poloidal E x B angular momentum and discuss the mirror and the Lorentz force. Here, we find that the geodesic transfer term, the Stringer-Winsor spin-up term and the ion-orbit loss term are all part of the Lorentz force and are in fact one and the same term. Discussing the relation to angular velocity we build the inertia tensor with the help of the first fundamental form of a flux-surface. In turn, the inertia tensor is used to construct a flux-surface averaged rotational energy for E x B surface flows of the plasma. The evolution of this rotational energy features a correction of previous results due to the inertia tensor. In particular, this correction suggests that density sources on the high-field side contribute much more to zonal flow energy generation than on the low field side. Our derivation is based on a full-F, electromagnetic, gyro-kinetic model in a long-wavelength limit. The results can be applied to gyro-kinetic as well as gyro-fluid theories and can also be compared to drift-kinetic and drift-fluid models. Simplified cases for the magnetic field geometry including the axisymmetric purely toroidal and purely poloidal magnetic fields are discussed, as are the angular momentum balance of the electromagnetic fields, the ion-orbit loss mechanism and the parallel acceleration
Beyond the Oberbeck-Boussinesq and long wavelength approximation
We present the first simulations of a reduced magnetized plasma model that incorporates both arbitrary wavelength polarization and non-Oberbeck-Boussinesq effects. Significant influence of these two effects on the density, electric potential and E x B vorticity and non-linear dynamics of interchange blobs are reported. Arbitrary wavelength polarization implicates so-called gyro-amplification that compared to a long wavelength approximation leads to highly amplified small-scale E x B vorticity fluctuations. These strongly increase the coherence and lifetime of blobs and alter the motion of the blobs through a slower blob-disintegration. Non-Oberbeck-Boussinesq effects incorporate plasma inertia, which substantially decreases the growth rate and linear acceleration of high amplitude blobs, while the maximum blob velocity is not affected. Finally, we generalize and numerically verify unified scaling laws for blob velocity, acceleration and growth rate that include both ion temperature and arbitrary blob amplitude dependence
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