Predictions Of Linear Theory Research Articles

Modeling q95 windows for the suppression of edge localized modes by resonant magnetic perturbations in the DIII-D tokamak

A toroidal asymptotic matching model of the response of a tokamak plasma to a static resonant magnetic perturbation (RMP) is used to simulate the n = 3 RMP-induced edge-localized-mode-suppression windows in q95 that are evident when the plasma current is slowly ramped in DIII-D discharge #145380. All quantities employed in the simulation are derived from experimental measurements, apart from the neutral particle data. Three cases are considered. In the first case, the natural frequencies of tearing modes resonant in the plasma are determined by the ion flows at the corresponding resonant surfaces, which is the prediction of nonlinear tearing mode theory. In the second case, the natural frequencies are determined by the local E×B velocities at the resonant surfaces. In the third case, the natural frequencies are determined by the electron flows at the resonant surfaces, which is the prediction of linear tearing mode theory. The second case gives the best agreement between the simulations and the experimental observations. The first and third cases only lead to partial agreement between the simulations and the observations. In the first case, the lack of complete agreement may be a consequence of using an inaccurate assumption for the neutral particle distribution in the pedestal. In the third case, the lack of complete agreement is probably due to the fact that the response of a tokamak plasma to an RMP is not accurately described by linear tearing mode theory.

Wave–particle interactions in a long traveling wave tube with upgraded helix

We investigate the interaction of electromagnetic waves and electron beams in a 4 m long traveling wave tube (TWT). The device is specifically designed to simulate beam-plasma experiments without appreciable noise. This TWT presents an upgraded slow wave structure (SWS) that results in more precise measurements and makes new experiments possible. We introduce a theoretical model describing wave propagation through the SWS and validated by the experimental dispersion relation, impedance, and phase and group velocities. We analyze nonlinear effects arising from the beam–wave interaction, such as the modulation of the electron beam and the wave growth and saturation process. When the beam current is low, the wave growth coefficient and saturation amplitude follow the linear theory predictions. However, for high values of current, nonlinear space charge effects become important and these parameters deviate from the linear predictions, tending to a constant value. After saturation, we also observe trapping of the beam electrons, which alters the wave amplitude along the TWT.

Amending the halo model to satisfy cosmological conservation laws

One of the most powerful tools in the arsenal of theoretical cosmologists is the halo model of large scale structure, which provides a phenomenological description of nonlinear structure in our universe. However, it is well known that there is no simple way to impose conservation laws in the halo model. This can severely impair the predictions on large scales for observables such as weak lensing or the kinematic Sunyaev-Zel'dovich effect, which should satisfy mass and momentum conservations, respectively. For example, the standard halo model overpredicts weak lensing power spectrum by $> 8\%$ on scales $>20$ degrees. To address this problem, we present an $Amended ~ Halo ~ Model$, explicitly separating the linear perturbations from $compensated$ halo profiles. This is guaranteed to respect conservation laws, as well as linear theory predictions on large scales. We then provide a simple fitting function for the compensated halo profiles, and discuss the modified predictions for 1-halo and 2-halo terms, as well as other cosmological observations such weak lensing power spectrum. Furthermore, we argue that the amended halo model provides a more accurate framework to capture physical effects that happen in the process of cosmological structure formation.

Stability of liquid film flow laden with the soluble surfactant sodium dodecyl sulphate: predictions versus experimental data

Gravity-driven liquid film flows laden with a soluble surfactant are considered, and aqueous solutions of sodium dodecyl sulphate (SDS) are taken as a case-study. Literature measurements of the critical Reynolds number for the onset of instability are set in perspective with predictions of linear stability theory. The theory is based on a Frumkin model of adsorption equilibrium, modified by the inclusion of finite compressibility of the adsorbed monolayer. Quantitative comparison between data and theory is first attempted in the limit of infinite wavelength. Though wave characteristics are satisfactorily predicted, the theoretical critical Reynolds number is an order of magnitude below measurements. This discrepancy is understood in terms of the large difference between momentum and mass diffusivities and indicates that the assumption of infinite wavelength is far more restrictive for the mass transfer than for the flow problem. Finite-wavelength effects are taken into account by numerical solution of the Orr–Sommerfeld eigenvalue problem, leading to predictions of maximum stabilization in good agreement with the measurements. Introduction of realistic values of monolayer compressibility improves further the agreement at high surfactant loadings. Finally, a strong stabilizing effect of salinity is confirmed.

GRAMSES: a new route to general relativistic N-body simulations in cosmology. Part II. Initial conditions

We address the generation of initial conditions (ICs) for {\\sc gramses}, a code for nonlinear general relativistic (GR) N-body cosmological simulations recently introduced in ref. [1]. {\\sc gramses} adopts a constant mean curvature slicing with a minimal distortion gauge, where the linear growth rate is scale-dependent, and the standard method for realising initial particle data is not straightforwardly applicable. A new method is introduced, in which the initial positions of particles are generated from the displacement field realised for a matter power spectrum as usual, but the velocity is calculated by finite-differencing the displacement fields around the initial redshift. In this way, all the information required for setting up the initial conditions is drawn from three consecutive input matter power spectra, and additional assumptions such as scale-independence of the linear growth factor and growth rate are not needed. We implement this method in a modified {\\sc 2LPTic} code, and demonstrate that in a Newtonian setting it can reproduce the velocity field given by the default {\\sc 2LPTic} code with subpercent accuracy. We also show that the matter and velocity power spectra of the initial particle data generated for {\\sc gramses} simulations using this method agree very well with the linear-theory predictions in the particular gauge used by {\\sc gramses}. Finally, we discuss corrections to the finite difference calculation of the velocity when radiation is present, as well as additional corrections implemented in {\\sc gramses} to ensure consistency. This method can be applied in ICs generation for GR simulations in generic gauges, and simulations of cosmological models with scale-dependent linear growth rate.

Flow Interactions Between Low Aspect Ratio Hydrofoils in In-line and Staggered Arrangements.

Many species of fish gather in dense collectives or schools where there are significant flow interactions from their shed wakes. Commonly, these swimmers shed a classic reverse von Kármán wake, however, schooling eels produce a bifurcated wake topology with two vortex rings shed per oscillation cycle. To examine the schooling interactions of a hydrofoil with a bifurcated wake topology, we present tomographic particle image velocimetry (tomo PIV) measurements of the flow interactions and direct force measurements of the performance of two low-aspect-ratio hydrofoils () in an in-line and a staggered arrangement. Surprisingly, when the leader and follower are interacting in either arrangement there are only minor alterations to the flowfields beyond the superposition of the flowfields produced by the isolated leader and follower. Motivated by this finding, Garrick’s linear theory, a linear unsteady hydrofoil theory based on a potential flow assumption, was adapted to predict the lift and thrust performance of the follower. Here, the follower hydrofoil interacting with the leader’s wake is considered as the superposition of an isolated pitching foil with a time-varying cross-stream velocity derived from the wake flow measurements of the isolated leader. Linear theory predictions accurately capture the time-averaged lift force and some of the major peaks in thrust derived from the follower interacting with the leader’s wake in a staggered arrangement. The thrust peaks that are not predicted by linear theory are likely driven by spatial variations in the flowfield acting on the follower or nonlinear flow interactions; neither of which are accounted for in the simple theory. This suggests that unsteady potential flow theory that does account for spatial variations in the flowfield acting on a hydrofoil can provide a relatively simple framework to understand and model the flow interactions that occur in schooling fish. Additionally, schooling eels can derive thrust and efficiency increases of 63-80% in either a in-line or a staggered arrangement where the follower is between two branched momentum jets or with one momentum jet branch directly impinging on it, respectively.

A Numerical Investigation of Gap and Shape Effects on a 2D Plunger-Type Wave Maker

The installation of plunger-type wave makers in experimental tanks will generally include a gap between the back of the wedge and the wall of the tank. In this study, we analyze the influence of this gap on the wave making performance of the plunger using two-dimensional (2D) CFD calculations for a range of nearly linear wave conditions and compare the results with both experimental measurements and linear potential flow theory. Three wedge-shaped profiles, all with the same submerged volume, are considered. Moreover, the generated waves are compared with the predictions of linear potential flow theory. The calculations are made using the commercial ANSYS FLUENT finite-volume code with dynamic meshes to solve the Navier–Stokes equations and the volume of fluid scheme to capture the air–water interface. Furthermore, the linear potential flow solution of Wu (J Hydraul Res 26:481–493, 1988) is extended to consider an arbitrary profile and serve as a reference solution. The amplitude ratios of the generated waves predicted by the CFD calculations compare well with the predictions of linear potential flow theory for a simple wedge, indicating that viscous effects do not influence this ratio for small-amplitude motions in 2D. By contrast, significant higher harmonic components are produced by larger amplitude motions. Also, the simple wedge is found to produce the smallest spurious higher harmonic content in the far-field wave.

Nonlinear Evolution of a Steep, Focusing Wave Group in Deep Water Simulated With oceanwave3d

Abstract Steep, focusing waves can experience fast and local nonlinear evolution of the spectrum due to wave–wave interactions resulting in energy transfer to both higher and lower wavenumber components. The shape and kinematics of a steep wave may, thus, differ substantially from the predictions of linear theory. We have investigated the role of nonlinear interactions on group shape for a steep, narrow-banded, directionally spread wave group focusing in deep water using the fully nonlinear potential flow solver, oceanwave3d. Exact second-order correction of the initial conditions has been implemented together with a novel third-order approximate correction based on a Stokes-type formulation for surface elevation combined with a scaling argument for the third-order velocity potential. Four-phase separation reveals that the third-order scheme provides a good estimate for the third-order superharmonics. A quantitative assessment of numerical error has also been performed for the spatial and temporal discretization, including energy conservation, a reversibility check, and validation against previous simulations performed with a higher-order spectral (HOS) code. The initially narrow-banded amplitude spectrum exhibits the formation of “sidelobes” at angles of approximately ±35deg to the spectral peak during the simulated extreme wave event, occurring in approximately ten wave periods, with a preferential energy transfer to high-wavenumber components. The directional energy transfer is attributed to resonant third-order interactions with a discussion of the engineering implications.

A nonlinear model for a hanging cantilevered pipe discharging fluid with a partially-confined external flow

In this paper, a nonlinear theoretical model is developed for the dynamics of a flexible cantilevered pipe that is simultaneously subjected to internal and partially-confined external axial flows. The pipe under consideration discharges fluid downwards, which accumulates in a relatively large tank, and then flows upwards through a generally shorter annular region surrounding the pipe. Thus, the internal and external flows are interdependent and in opposite directions. A practical application of this system may be found in solution mining processes for brine production, and in the subsequent usage of the salt-mined caverns for hydrocarbon storage. The equation of motion is derived using the extended Hamilton’s principle to third-order accuracy with a separate derivation of the fluid-related forces associated with the internal and external flows. The equation is discretized using Galerkin’s scheme and solved via the pseudo-arclength continuation method and a direct time integration technique. Two pipes of different dimensions and materials are considered in this study; the stability of these pipes is investigated with increasing flow velocity. Also, the influence of varying the length and tightness of the annular region on the dynamical behaviour of the pipes is explored theoretically. The predictions of the proposed model are compared to experimental observations from the literature for systems with the same parameters as those considered in this paper, as well as to predictions of an earlier linear theory. The results obtained are in excellent qualitative and good quantitative agreement with the experimental observations. Furthermore, this model predicts the frequencies of oscillation more accurately than linear theory.

A plane strain analysis of the elastostatic fields near the notch-tip of a Blatz-Ko material

An asymptotic plane strain analysis is carried out in order to study the notch problem in a homogeneous isotropic compressible hyperelastic material whose behavior is governed by the Blatz-Ko constitutive law. The work is realized within the framework of the fully nonlinear fracture mechanics at large strain. Close to the notch tip, a stress-free state is assumed in the notch edges, while at infinity (sufficiently far to the notch tip) fracture modes I and II are considered. The results emphasizes many discrepancies with linear theory predictions like the emergence of a logarithmic singularity and show that solution depends on the notch initial opening angle.

The Effective Field Theory of Dark Energy Diagnostic of Linear Horndeski Theories After GW170817 and GRB170817A

We summarize the effective field theory of dark energy construction to explore observable predictions of linear Horndeski theories. We review the diagnostic of these theories on the correlation of the large-scale structure phenomenological functions: the effective Newton constant, the light deflection parameter, and the growth function of matter perturbations. We take this opportunity to discuss the evolution of the bounds the propagation speed of gravitational waves has undergone and use the most restrictive one to update the diagnostic.

Multiple Spiral Arms in Protoplanetary Disks: Linear Theory

Abstract Recent observations of protoplanetary disks, as well as simulations of planet–disk interaction, have suggested that a single planet may excite multiple spiral arms in the disk, in contrast to the previous expectations based on linear theory (predicting a one-armed density wave). We reassess the origin of multiple arms in the framework of linear theory by solving for the global two-dimensional response of a nonbarotropic disk to an orbiting planet. We show that the formation of a secondary arm in the inner disk, at about half of the orbital radius of the planet, is a robust prediction of linear theory. This arm becomes stronger than the primary spiral at several tenths of the orbital radius of the planet. Several additional, weaker spiral arms may also form in the inner disk. On the contrary, a secondary spiral arm is unlikely to form in the outer disk. Our linear calculations, fully accounting for the global behavior of both the phases and amplitudes of perturbations, generally support the recently proposed WKB phase argument for the secondary arm origin (as caused by the intricacy of constructive interference of the azimuthal harmonics of the perturbation at different radii). We provide analytical arguments showing that the process of a single spiral wake splitting up into multiple arms is a generic linear outcome of wave propagation in differentially rotating disks. It is not unique to planet-driven waves and also occurs in linear calculations of spiral wakes freely propagating with no external torques. These results are relevant for understanding formation of multiple rings and gaps in protoplanetary disks.

Transition in a separation bubble under tonal and broadband acoustic excitation

Transition and flow development in a separation bubble formed on an airfoil are studied experimentally. The effects of tonal and broadband acoustic excitation are considered since such acoustic emissions commonly result from airfoil self-noise and can influence flow development via a feedback loop. This interdependence is decoupled, and the problem is studied in a controlled manner through the use of an external acoustic source. The flow field is assessed using time-resolved, two-component particle image velocimetry, the results of which show that, for equivalent energy input levels, tonal and broadband excitation can produce equivalent changes in the mean separation bubble topology. These changes in topology result from the influence of excitation on transition and the subsequent development of coherent structures in the bubble. Both tonal and broadband excitation lead to earlier shear layer roll-up and thus reduce the bubble size and advance mean reattachment upstream, while the shed vortices tend to persist farther downstream of mean reattachment in the case of tonal excitation. Through a cross-examination of linear stability theory (LST) predictions and measured disturbance characteristics, nonlinear disturbance interactions are shown to play a crucial role in the transition process, leading to significantly different disturbance development for the tonal and broadband excited flows. Specifically, tonal excitation results in transition being dominated by the excited mode, which grows in strong accordance with linear theory and damps the growth of all other disturbances. On the other hand, disturbance amplitudes are more moderate for the natural and broadband excited flows, and so all unstable disturbances initially grow in accordance with LST. For all cases, a rapid redistribution of perturbation energy to a broad range of frequencies follows, with the phenomenon occurring earliest for the broadband excitation case. By taking nonlinear effects into consideration, important ramifications are made clear in regards to comparing LST predictions and experimental or numerical results, thus explaining the trends reported in recent investigations. These findings offer new insights into the influence of tonal and broadband noise emissions, resulting from airfoil self-noise or otherwise, on transition and flow development within a separation bubble.

Polar alignment of a protoplanetary disc around an eccentric binary – II. Effect of binary and disc parameters

In a recent paper Martin & Lubow showed that a circumbinary disc around an eccentric binary can undergo damped nodal oscillations that lead to the polar (perpendicular) alignment of the disc relative to the binary orbit. The disc angular momentum vector aligns to the eccentricity vector of the binary. We explore the robustness of this mechanism for a low mass disc (0.001 of the binary mass) and its dependence on system parameters by means of hydrodynamic disc simulations. We describe how the evolution depends upon the disc viscosity, temperature, size, binary mass ratio, orbital eccentricity and inclination. We compare results with predictions of linear theory. We show that polar alignment of a low mass disc may occur over a wide range of binary-disc parameters. We discuss the application of our results to the formation of planetary systems around eccentric binary stars.

Offshore monopile in the southern North Sea: part II, simulated hydrodynamics and loading

This paper considers two methods for determining the local wave particle kinematics and hydrodynamic forces on an idealised wind turbine monopile in the southern North Sea using sea-state data from Teesside Offshore Wind Farm. An assessment of local flow hydrodynamics is important with regard to safe access of personnel from a crew transfer vessel to a monopile. The hydrodynamic behaviour was calculated using an analytical solution from linear diffraction theory and numerical predictions using OpenFoam, with both slip and no-slip cylinder boundary conditions. Provided the underlying sea state is unidirectional, it was found that close agreement is obtained between analytical and numerical spectra derived from the time series of local free surface elevation, water particle velocity components and in-line wave force loading on the monopile. Less satisfactory agreement is achieved with sea states possessing a bimodal spectrum, which suggests that bimodal spectra may not be unidirectional.

Evaluation of video-based linear depth inversion performance and applications using altimeters and hydrographic surveys in a wide range of environmental conditions

The performance of a linear depth inversion algorithm, cBathy, applied to coastal video imagery was assessed using observations of water depth from vessel-based hydrographic surveys and in-situ altimeters for a wide range of wave conditions (0.3 < significant wave height < 4.3 m) on a sandy Atlantic Ocean beach near Duck, North Carolina. Comparisons of video-based cBathy bathymetry with surveyed bathymetry were similar to previous studies (root mean square error (RMSE) = 0.75 m, bias = −0.26 m). However, the cross-shore locations of the surfzone sandbar in video-derived bathymetry were biased onshore 18–40 m relative to the survey when offshore wave heights exceeded 1.2 m or were greater than half of the bar crest depth, and broke over the sandbar. The onshore bias was 3–4 m when wave heights were less than 0.8 m and were not breaking over the sandbar. Comparisons of video-derived seafloor elevations with in-situ altimeter data at three locations onshore of, near, and offshore of the surfzone sandbar over ∼1 year provide the first assessment of the cBathy technique over a wide range of wave conditions. In the outer surf zone, video-derived results were consistent with long-term patterns of bathymetric change (r2 = 0.64, RMSE = 0.26 m, bias = −0.01 m), particularly when wave heights were less than 1.2 m (r2 = 0.83). However, during storms when wave heights exceeded 3 m, video-based cBathy over-estimated the depth by up to 2 m. Near the sandbar, the sign of depth errors depended on the location relative to wave breaking, with video-based depths overestimated (underestimated) offshore (onshore) of wave breaking in the surfzone. Wave speeds estimated by video-based cBathy at the initiation of wave breaking often were twice the speeds predicted by linear theory, and up to three times faster than linear theory during storms. Estimated wave speeds were half as fast as linear theory predictions at the termination of wave breaking shoreward of the sandbar. These results suggest that video-based cBathy should not be used to track the migration of the surfzone sandbar using data when waves are breaking over the bar nor to quantify morphological evolution during storms. However, these results show that during low energy conditions, cBathy estimates could be used to quantify seasonal patterns of seafloor evolution.

Parametric Decay Instability and Dissipation of Low-frequency Alfvén Waves in Low-beta Turbulent Plasmas

Abstract Evolution of the parametric decay instability (PDI) of a circularly polarized Alfvén wave in a turbulent low-beta plasma background is investigated using 3D hybrid simulations. It is shown that the turbulence reduces the growth rate of PDI as compared to the linear theory predictions, but PDI can still exist. Interestingly, the damping rate of the ion acoustic mode (as the product of PDI) is also reduced as compared to the linear Vlasov predictions. Nonetheless, significant heating of ions in the direction parallel to the background magnetic field is observed due to resonant Landau damping of the ion acoustic waves. In low-beta turbulent plasmas, PDI can provide an important channel for energy dissipation of low-frequency Alfvén waves at a scale much larger than the ion kinetic scales, different from the traditional turbulence dissipation models.

A new hybrid code (CHIEF) implementing the inertial electron fluid equation without approximation

We present a new hybrid algorithm implemented in the code CHIEF ( Code Hybrid with Inertial Electron Fluid) for simulations of electron–ion plasmas. The algorithm treats the ions kinetically, modeled by the Particle-in-Cell (PiC) method, and electrons as an inertial fluid, modeled by electron fluid equations without any of the approximations used in most of the other hybrid codes with an inertial electron fluid. This kind of code is appropriate to model a large variety of quasineutral plasma phenomena where the electron inertia and/or ion kinetic effects are relevant. We present here the governing equations of the model, how these are discretized and implemented numerically, as well as six test problems to validate our numerical approach. Our chosen test problems, where the electron inertia and ion kinetic effects play the essential role, are: 0) Excitation of parallel eigenmodes to check numerical convergence and stability, 1) parallel (to a background magnetic field) propagating electromagnetic waves, 2) perpendicular propagating electrostatic waves (ion Bernstein modes), 3) ion beam right-hand instability (resonant and non-resonant), 4) ion Landau damping, 5) ion firehose instability, and 6) 2D oblique ion firehose instability. Our results reproduce successfully the predictions of linear and non-linear theory for all these problems, validating our code. All properties of this hybrid code make it ideal to study multi-scale phenomena between electron and ion scales such as collisionless shocks, magnetic reconnection and kinetic plasma turbulence in the dissipation range above the electron scales.

Prograde, retrograde, and oscillatory modes in rotating Rayleigh–Bénard convection

Rotating Rayleigh–Bénard convection is typified by a variety of regimes with very distinct flow morphologies that originate from several instability mechanisms. Here we present results from direct numerical simulations of three representative set-ups: first, a fluid with Prandtl number $Pr=6.4$, corresponding to water, in a cylinder with a diameter-to-height aspect ratio of $\unicode[STIX]{x1D6E4}=2$; second, a fluid with $Pr=0.8$, corresponding to $\text{SF}_{6}$ or air, confined in a slender cylinder with $\unicode[STIX]{x1D6E4}=0.5$; and third, the main focus of this paper, a fluid with $Pr=0.025$, corresponding to a liquid metal, in a cylinder with $\unicode[STIX]{x1D6E4}=1.87$. The obtained flow fields are analysed using the sparsity-promoting variant of the dynamic mode decomposition (DMD). By means of this technique, we extract the coherent structures that govern the dynamics of the flow, as well as their associated frequencies. In addition, we follow the temporal evolution of single modes and present a criterion to identify their direction of travel, i.e. whether they are precessing prograde or retrograde. We show that for moderate $Pr$ a few dynamic modes suffice to accurately describe the flow. For large aspect ratios, these are wall-localised waves that travel retrograde along the periphery of the cylinder. Their DMD frequencies agree with the predictions of linear stability theory. With increasing Rayleigh number $Ra$, the interior gradually fills with columnar vortices, and eventually a regular pattern of convective Taylor columns prevails. For small aspect ratios and close enough to onset, the dominant flow structures are body modes that can precess either prograde or retrograde. For $Pr=0.8$, DMD additionally unveiled the existence of so far unobserved low-amplitude oscillatory modes. Furthermore, we elucidate the multi-modal character of oscillatory convection in low-$Pr$ fluids. Generally, more dynamic modes must be retained to accurately approximate the flow. Close to onset, the flow is purely oscillatory and the DMD reveals that these high-frequency modes are a superposition of oscillatory columns and cylinder-scale inertial waves. We find that there are coexisting prograde and retrograde modes, as well as quasi-axisymmetric torsional modes. For higher $Ra$, the flow also becomes unstable to wall modes. These low-frequency modes can both coexist with the oscillatory modes, and also couple to them. However, the typical flow feature of rotating convection at moderate $Pr$, the quasi-steady Taylor vortices, is entirely absent in low-$Pr$ flows.

Reconstructing the metric of the local Universe from number counts observations

Number counts observations available with new surveys such as the Euclid mission will be an important source of information about the metric of the Universe. We compute the low red-shift expansion for the energy density and the density contrast using an exact spherically symmetric solution in presence of a cosmological constant. At low red-shift the expansion is more precise than linear perturbation theory prediction. We then use the local expansion to reconstruct the metric from the monopole of the density contrast. We test the inversion method using numerical calculations and find a good agreement within the regime of validity of the red-shift expansion. The method could be applied to observational data to reconstruct the metric of the local Universe with a level of precision higher than the one achievable using perturbation theory.