Study Of Linear Stability Research Articles

Fast ion relaxation in ITER mediated by Alfvén instabilities

We address the critical issue for future burning plasmas of whether high-energy fusion products or auxiliary heating-beam ions will be confined for a sufficiently long time to compensate for thermal plasma energy losses. This issue can be mitigated by one of the most deleterious collective phenomena—the instability of low, sub-cyclotron frequency Alfvén eigenmodes (AEs), such as toroidicity-induced AEs and reversed-shear AEs in the ITER steady-state scenario. Using a revised quasi-linear (QL) theory applied to energetic particle (EP) relaxation in the presence of AEs, we find that the AE instabilities can affect both neutral beam ions and alpha particles, although the resulting fast ion transport is expected to be modest if classical particle slowing down is assumed. On the other hand, the QL theory predicts that the AE amplitudes will be enhanced by the background microturbulence, although this topic remains outside our scope due to the significant numerical effort required to evaluate these effects. We report our results for EP relaxation dynamics obtained utilizing several tools: (i) a comprehensive linear stability study of the sub-cyclotron Alfvénic spectrum as computed by ideal magnetohydrodynamic NOVA simulations for the AE eigenproblem, (ii) drift kinetic NOVA-C calculations for wave–particle interaction and AE growth/damping rates, and (iii) predictive QL modeling coupled with the global transport code TRANSP to assess the EP relaxation on the equilibrium timescale.

Numerical study of linear and nonlinear stability in double-diffusive Hadley-Prats flow through a horizontal porous layer with Soret and internal heat source effects

Numerical analysis is constructed to study the onset of the double-diffusion convection mechanism through an infinite parallel permeable channel with heat generation, mass flow and Soret impacts. Homogenous isotropic Darcy’s flow model is deployed to elucidate the porous features. The roll instabilities pertaining to longitudinal and transverse cases are examined through a linear and nonlinear stability analysis. The unit-less eigenvalue problem is constructed through linear and nonlinear stability assumptions and which is solved numerically using a fourth order Runge-Kutta scheme. The critical values of wave and thermal Rayleigh parameters are evaluated. Extended graphical and tabular visualization is presented to describe the onset of the convection mechanism. The results of this semi-numerical investigation may be useful in environmental, geothermal and chemical engineering processes. In addition, the critical R z value is also determined for a range of thermo-physical numbers. Significant modifications in the flow patterns are computed with mass flow parameter and vertical thermal and solutal Rayleigh number. With an increment in Soret (thermo-diffusive) parameter Sr the regime become more unstable. Based on the nonlinear stability analysis, an elevation in solutal Rayleigh number also induces earlier instability. The collective influence of the Lewis number and horizontal concentration parameter is observed to render the system more stable.

Models of resistive wall tearing mode disruptions

Disruptions are a serious issue in tokamaks. In a disruption, the thermal energy is lost by means of an instability which could be a resistive wall tearing mode (RWTM). During precursors to a disruption, the plasma edge region cools, causing the current to contract. Model sequences of contracted current equilibria are given, and their stability is calculated. A linear stability study shows that there is a maximum value of edge qa≈3 for RWTMs to occur. This also implies a minimum rational surface radius normalized to plasma radius from RWTMs to be unstable. Nonlinear simulations are performed using a similar model sequence derived from an equilibrium reconstruction. There is a striking difference in the results, depending on whether the wall is ideal or resistive. With an ideal wall, the perturbations saturate at moderate amplitude, causing a minor disruption without a thermal quench. With a resistive wall, there is a major disruption with a thermal quench, if the edge qa≤3. There is a sharp transition in nonlinear behavior at qa=3. This is consistent with the linear model and with experiments. If disruptions are caused by RWTMs, then devices with highly conducting walls, such as the International Tokamak Experimental Reactor will experience much milder, tolerable, disruptions than presently predicted.

Stability of two-dimensional potential flows using bicomplex numbers.

The use of the complex velocity potential and the complex velocity is widely disseminated in the study of two-dimensional incompressible potential flows. The advantages of working with complex analytical functions made this representation of the flow ubiquitous in the field of theoretical aerodynamics. However, this representation is not usually employed in linear stability studies, where the representation of the velocity as real vectors is preferred by most authors, in order to allow the representation of the perturbation as the complex exponential function. Some of the classical attempts to use the complex velocity potential in stability studies suffer from formal errors. In this work, we present a framework that reconciles these two complex representations using bicomplex numbers. This framework is applied to the stability of the von Kármán vortex street and a generalized formula is found. It is shown that the classical results of the symmetric and staggered von Kármán vortex streets are just particular cases of the generalized dynamical system in bicomplex formulation.

A Numerical Analysis of Six Physics‐Dynamics Coupling Schemes for Atmospheric Models

AbstractSix strategies to couple the dynamical core with physical parameterizations in atmospheric models are analyzed from a numerical perspective. Thanks to a suitably designed theoretical framework featuring a high level of abstraction, the truncation error analysis and the linear stability study are carried out under weak assumptions. Indeed, second‐order conditions are derived which are not influenced either by the specific formulation of the governing equations, nor by the number of parameterizations, nor by the structural design and implementation details of the time‐stepping methods. The theoretical findings are verified on two idealized test beds. Particularly, a hydrostatic model in isentropic coordinates is used for vertical slice simulations of a moist airflow past an isolated mountain. Self‐convergence tests show that the sensitivity of the prognostic variables to the coupling scheme may vary. For those variables (e.g., momentum) whose evolution is mainly driven by the dry dynamics, the truncation error associated with the dynamical core dominates and hides the error due to the coupling. In contrast, the coupling error of moist variables (e.g., the precipitation rate) emerges gradually as the spatio‐temporal resolution increases. Eventually, each coupling scheme tends toward the formal order of accuracy, upon a careful treatment of the grid cell condensation. Indeed, the well‐established saturation adjustment may cap the convergence rate to first order. A prognostic formulation of the condensation and evaporation process is derived from first principles. This solution is shown effective to alleviate the convergence issues in our experiments. Potential implications for a complete forecasting system are discussed.

Linear, resistive stability studies for Wendelstein 7-X-type equilibria with external current drive

Comprehensive linear stability studies of resistive modes are presented for Wendelstein 7-X-type stellarator equilibria with electron cyclotron current drive (ECCD). The external co-current drive leads to an increase of the rotational transform and the formation of one or two ι = 1 rational flux surfaces. Using the 3D linear stability CASTOR3D code, low n*-type resistive modes (n* = 1,2...,8 with n* being the dominant toroidal Fourier harmonic contributing to the mode) are investigated. We studied the dependence of their growth rates on plasma resistivity, parallel viscosity, and shape of the rotational transform profile (especially various distances between two ι = 1 flux surfaces). Similarly to tokamak configurations, single and double tearing modes, and internal resistive kink modes are found. In addition, modes oscillating between two n*-types of the same mode family are observed. The frequencies of those modes are in the range of ≈ 10–260 Hz. Equilibria with either a large distance between the ι = 1 flux surfaces, or a single ι = 1 surface are most unstable with respect to n* = 1 resistive kink modes. The latter finding fits to the experimental observation of sawtooth-like oscillations followed by a thermal quench in W7-X discharges with ECCD.

Linear stability and saddle-node bifurcation of electromagnetically driven electrolyte flow in an annular layer

Comprehensive linear stability study of flow in an annular layer of electrolyte driven by the action of the Lorentz force is conducted following the analysis of steady axisymmetric solutions of Suslov et al. (J. Fluid Mech., vol. 828, 2017, pp. 573–600). It is shown that an experimentally observed instability in the form of anticyclonic moving vortices reported in Perez-Barrera et al. (Magnetohydrodynamics, vol. 51 (2), 2015, pp. 203–213) develops on a background of the basic flow consisting of two tori with the opposite azimuthal vorticity components. It is found that, while the background flow is driven electromagnetically, the appearance of vortices is purely due to hydrodynamic effects: shear of the flow and centrifugal inertial forcing. The current study has also revealed that the unstable two-torus basic flow has a stable single-torus counterpart, both emanating from a saddle-node bifurcation of steady states when the Lorentz force is sufficiently strong. The transition from a one-torus to two-torus flow at weaker forcing is abrupt and leads to the appearance of vortices as soon as it occurs. The ranges of layer depths and Reynolds numbers for which vortices develop on a steady background are determined. Subsequently, weakly nonlinear amplitude expansion is used to find an approximate unsteady solution beyond the saddle-node bifurcation.

The employment of neutral approach for linear singular system stability study with additive time varying delays

The employment of neutral approach for linear singular system stability study with additive time varying delays

Comparative Study on Sixth Order Boundary Value Problems with Application to Linear Hydrodynamic Stability Problem and Benard Layer Eigenvalue Problem

Numerical estimation for higher order eigenvalue problems are promising and has accomplished significant importance, mainly due to existence of higher order derivatives and boundary conditions relating to higher order derivatives of the unknown functions. In this article, we perform a numerical study of linear hydrodynamic stability of a fluid motion caused by an erratic gravity field. We employ two methods, collocation and spectral collocation based on Bernstein and Legendre polynomials to solve the linear hydrodynamic stability problems and Benard type convection problems. In order to handle boundary conditions, our techniques state all the unknown coefficients of boundary conditions derivatives in terms of known co-efficient. The schemes have been carried out to several test problems to establish the efficiency of the two methods.

Effects of different levels of solar radiation and depth-varying vertical diffusion on the dynamics of a reaction–diffusion NPZ model

ABSTRACT Although the effects of solar radiation, diffusion and mixed layer depths on plankton populations have been studied extensively, it is not intuitive what the equilibrium states of a plankton system should be in the absence of seasonal forcing, nor what are the prime determiners of the attracting equilibrium states. In this study, the linear stability of a nutrient–phytoplankton–zooplankton (NPZ) model in reaction–diffusion form is analyzed to understand how solar radiation, diffusion and mixed layer depth influence the predicted equilibrium states. We incorporate parameters that control the response of the phytoplankton growth rate to solar radiation, control the depth of the upper mixed layer and control the strength of the diffusion. The combined effect of the constant in time parameters for these quantities is shown to significantly influence the stability properties of the NPZ reaction–diffusion model. In our parameter range, we find that diffusion and light are stronger determinants of the equilibrium state than mixed layer depth. This linear stability study gives insight into how these quantities affect the stability states of marine planktonic ecosystems and reinforces the importance of understanding light and upper mixed layer turbulence as important factors in determining the state of the plankton system.

Linear stability studies for a quasi-axisymmetric stellarator configuration including effects of parallel viscosity, plasma flow, and resistive walls

We present linear stability studies performed with the newly developed CASTOR3D code. This code is a very flexible and versatilely applicable numerical tool to study the stability properties of 3D equilibria. Ideal external kink modes of a quasi-axisymmetric (QA) equilibrium are investigated taking into account effects of parallel viscosity, plasma flow, and resistive walls. The results are compared with those obtained for an axisymmetric approximation of this equilibrium. Two major differences in the stability properties are found: (i) the QA equilibrium is vertically stable, while the axisymmetric equilibrium is vertically unstable; (ii) a mode of the QA equilibrium only oscillates when the plasma rotates with a minimum rotation frequency, or faster. This mininum rotation frequency depends on the n*-type of mode (with n* being the dominant toroidal Fourier harmonic contributing to the mode). Resistive wall structures in combination with plasma flow, however, lead to resistive wall modes and/or plasma modes for both types of equilibria.

A linear stability study of stellar rotating spheres

AbstractRecent observations of globular clusters imposed major revisions to the previous paradigm, in which they were considered to be isotropic in velocity space and non-rotating. However, the theory of collisionless spheroids with some kinematic richness has seldom been studied. We present here a first step in this direction, owing to new results regarding the linear stability of rotating Plummer spheres, with varying anisotropy in velocity space and total amount of angular momentum. We extend the well-known radial orbit instability to rotating systems, and discover a new regime of instability in fast rotating, tangentially anisotropic systems.

Experimental and theoretical study of two-phase flow in wide microchannels

Experimental and theoretical studies of the two-phase flow regimes in a wide microchannel with the height of 164 μm were performed. The ranges of parameters of the formation of the main flow regimes are determined: jet, bubble, churn, stratified, and annular. The pressure drop in single-phase and two-phase flows was measured. Using a flat flow model, we performed a theoretical study of the pressure drop and compared it with experimental data. Regular waves were experimentally detected in film flow regimes. The formation of waves on a liquid film was investigated, and characteristic wavelengths were measured. As for the theory, the study of linear stability of the stratified two-phase flow regime in a wide microchannel was performed. The comparison of numerical and experimental results indicated the validity of the proposed approach for modeling waves in film regime in wide microchannels.

Weakly nonlinear analysis for car-following model with consideration of cooperation and time delays

In traffic systems, cooperative driving has attracted the researchers’ attention. A lot of works attempt to understand the effects of cooperative driving behavior and/or time delays on traffic flow dynamics for specific traffic flow models. This paper is a new attempt to investigate analyses of linear stability and weak nonlinearity for the general car-following model with consideration of cooperation and time delays. We derive linear stability condition and study how the combinations of cooperation and time delays affect the stability of traffic flow. Burgers’ equation and Korteweg de Vries’ (KdV) equation for car-following model considering cooperation and time delays are derived. Their solitary wave solutions and constraint conditions are concluded. We investigate the property of cooperative optimal velocity (OV) model which estimates the combinations of cooperation and time delays about the evolution of traffic waves using both analytic and numerical methods. The results indicate that delays and cooperation are model-dependent, and cooperative behavior could inhibit the stabilization of traffic flow. Moreover, delays of sensing relative motion are easy to trigger the traffic waves; delays of sensing host vehicle are beneficial to relieve the instability effect to a certain extent.

Properties of ion temperature gradient and trapped electron modes in tokamak plasmas with inverted density profiles

We perform a numerical study of linear stability of the ion temperature gradient (ITG) mode and the trapped electron mode (TEM) in tokamak plasmas with inverted density profiles. A local gyrokinetic integral equation is applied for this study. From comprehensive parametric scans, we obtain stability diagrams for ITG modes and TEMs in terms of density and temperature gradient scale lengths. The results show that, for the inverted density profile, there exists a normalized threshold temperature gradient above which the ITG mode and the TEM are either separately or simultaneously unstable. The instability threshold of the TEM for the inverted density profile is substantially different from that for normal and flat density profiles. In addition, deviations are found on the ITG threshold from an early analytic theory in sheared slab geometry with the adiabatic electron response [T. S. Hahm and W. M. Tang, Phys. Fluids B 1, 1185 (1989)]. A possible implication of this work on particle transport in pellet fueled tokamak plasmas is discussed.

Adiabatic pumping solutions in global AdS

We construct a family of very simple stationary solutions to gravity coupled to a massless scalar field in global AdS. They involve a constantly rising source for the scalar field at the boundary and thereby we name them pumping solutions. We construct them numerically in D = 4. They are regular and, generically, have negative mass. We perform a study of linear and nonlinear stability and find both stable and unstable branches. In the latter case, solutions belonging to different sub-branches can either decay to black holes or to limiting cycles. This observation motivates the search for non-stationary exactly timeperiodic solutions which we actually construct. We clarify the role of pumping solutions in the context of quasistatic adiabatic quenches. In D = 3 the pumping solutions can be related to other previously known solutions, like magnetic or translationally-breaking backgrounds. From this we derive an analytic expression.

Revised conditions for MRI due to isorotation theorem

We re-analyze the physical conditions for Magneto-rotational Instability (MRI) within a steady axisymmetric stratified disk of plasma, in order to account for the so-called isorotation theory (the spatial profile of differential angular velocity depends on the magnetic flux surface). We develop the study of linear stability around an astrophysical background configuration, following the original derivation in [15], but implementing the isorotation condition as the orthogonality between the background magnetic field and the angular velocity gradient. We demonstrate that a dependence on the background magnetic field direction is restored in the dispersion relation and, hence, the emergence of MRI is also influenced by field orientation.

Linear stability study of the resistive ballooning and ion temperature gradient modes using a hybrid formalism in L-mode edge-like plasmas

Motivated by several recent works suggesting the importance of resistive ballooning mode (RBM) in L-mode edge plasmas, a modeling study is presented on the linear stability of the RBM and its interaction with the ion temperature gradient mode (ITG). A hybrid code is developed and utilized for the study, which solves the gyrokinetic and fluid equations for ions and electrons, respectively. It is shown that the RBM stability has a substantial difference, depending on whether it is driven by the density gradient, electron temperature gradient (ηe), or ion temperature gradient (ηi). The interaction with the ITG also occurs quite differently, depending on the driving force type. The strongest interaction is observed between the density-gradient driven RBM and the ITG, with the RBM being completely stabilized or smoothly transited to the ITG when ηi increases. In contrast, the ηe driven RBM is not connected to the ITG, with a significant difference in their real frequencies. In the relatively large ηi and ηe regime, the ITG and the ηe or ηi driven RBMs can co-exist, with their relative role varying, mainly depending on electron temperature and the ratio between ηe and ηi.

CASTOR3D: linear stability studies for 2D and 3D tokamak equilibria

The CASTOR3D code, which is currently under development, is able to perform linear stability studies for 2D and 3D, ideal and resistive tokamak equilibria in the presence of ideal and resistive wall structures and coils. For these computations ideal equilibria represented by concentric nested flux surfaces serve as input (e.g. computed with the NEMEC code). Solving an extended eigenvalue problem, the CASTOR3D code takes simultaneously plasma inertia and wall resistivity into account. The code is a hybrid of the CASTOR_3DW stability code and the STARWALL code. The former is an extended version of the CASTOR and CASTOR_FLOW code, respectively. The latter is a linear 3D code computing the growth rates of resistive wall modes in the presence of multiply-connected wall structures. The CASTOR_3DW code, and some parts of the STARWALL code have been reformulated in a general 3D flux coordinate representation that allows to choose between various types of flux coordinates. Furthermore, the implemented many-valued current potentials in the STARWALL part allow a correct treatment of the m = 0, n = 0 perturbation. In this paper, we outline the theoretical concept, and present some numerical results which illustrate the present status of the code and demonstrate its numerous application possibilities.

Compressibility effects on the structural evolution of transitional high-speed planar wakes

The compressibility effects on the structural evolution of the transitional high-speed planar wake are studied. The relative Mach number ($Ma_{r}$) of the laminar base flow modifies two fundamental features of planar wake transition: (i) the characteristic length scale defined by the most unstable linear mode; and (ii) the domain of influence of the structures within the staggered two-dimensional vortex array. Linear stability results reveal a reduced growth (approximately 30 % reduction up to$Ma_{r}=2.0$) and a quasilinear increase of the wavelength of the most unstable, two-dimensional instability mode (approximately 20 % longer over the same$Ma_{r}$range) with increasing$Ma$. The primary wavelength defines the length scale imposed on the emerging transitional structures; naturally, a longer wavelength results in rollers with a greater streamwise separation and hence also larger circulation. A reduction of the growth rate and an increase of the principal wavelength results in a greater ellipticity of the emerging rollers. Compressibility effects also modify the domain of influence of the transitional structures through an increased cross-wake and inhibited streamwise communication as characteristic paths between rollers are deflected due to local$Ma$ gradients. The reduced streamwise domain of influence impedes roller pairing and, for a sufficiently large relative$Ma$, pairing is completely suppressed. Thus, we observe an increased two-dimensionality with increasing Mach number: directly contrasting the increasing three-dimensional effects in high-speed mixing layers. Temporally evolving direct numerical simulations conducted at$Ma=0.8$and 2.0, for Reynolds numbers up to 3000, support the physical insight gained from linear stability and vortex dynamics studies.