Low Mode Number Research Articles

A spectral/finite‐difference approach for narrow‐channel flow with inertia

AbstractA hybrid spectral/finite‐difference scheme is proposed to determine the inertial flow inside narrow channels. The flow field is represented spectrally in the depthwise direction, which together with the Galerkin projection lead to a system of equations that are solved using a variable step finite difference discretization. The method is particularly effective for non‐linear flow, and its validity is here demonstrated for a flow with inertia. The problem is closely related to high‐speed lubrication flow. The validity of the spectral representation is assessed by examining the convergence of the method, and comparing with the fully two‐dimensional finite‐volume solution (FLUENT), and the widely used depth‐averaging method from shallow‐water theory. It is found that a low number of modes are usually sufficient to secure convergence and accuracy. Good agreement is obtained between the low‐order description and the finite‐volume solution at low to moderate modified Reynolds number. The depth‐averaging solution is unable to predict accurately (qualitatively and quantitatively) the high‐inertia flow. The influence of inertia is examined on the flow. Copyright © 2003 John Wiley & Sons, Ltd.

Signatures of salinity variability in tropical Pacific Ocean dynamic height anomalies

The vertical variability of the salinity field, with emphasis on interannual timescales, is examined within the tropical Pacific Ocean (10°N–10°S) using a compilation of the conductivity‐temperature‐depth (CTD) casts for the period 1975–1998. Compared to the vertical dependence of temperature that exhibits a systematic maximum variability at the depth of the main thermocline, the salinity typically shows a maximum variability in the surface layers. One notable region where this rule is violated is the southwestern and central Pacific Ocean. Below the surface, salinity variability is correlated with the strong gradients in mean salinity above and below the subsurface salinity maximum. It is shown that using conventional mean T‐S curves to estimate salinity profiles from temperature observations leads to strong biases because of the large scatter around the T‐S relationships. From the surface to the bottom of the thermocline, the dispersion of the T‐S diagrams is large regardless of whether the conditions are representative of “El Niño” or “La Niña” conditions. Error introduced in computing the dynamic height anomaly (DHA) is larger than 2 dyn. cm (dynamic centimeters) throughout the tropical Pacific if salinity variability is neglected. This error represents 30% of the total variability of the sea surface height in the western Pacific, and more than 50% in the south central Pacific. In order to estimate salinity variability when direct measurements are not available, an empirical orthogonal function analysis of existing temperature and salinity profiles was conducted. It is shown that a relatively low number of dominant modes, typically less than 6, are sufficient to explain 80% or more of the total variance. The separate contributions of the temperature and salinity fields to the DHA are then examined. The salinity contribution is generally smaller than the temperature contribution, though in some instances the two oppose one another, resulting in lowered dynamic height anomalies. These results confirm that salinity variability should not be neglected in ocean analyses that attempt to infer vertical changes in density from sea level fluctuations within the tropical Pacific Ocean.

Magnetohydrodynamic Stability of Improved Confinement Plasmas in JT-60U

Progress in the understanding of magnetohydrodynamic (MHD) stability is summarized on JT-60U tokamak discharges with improved confinement such as the (hot-ion) H-mode, high-βp mode, high-βp H-mode, and reversed shear discharges. Transport barriers, which are essential for the improved confinement, play key roles in the local and global MHD stability owing to the local large pressure gradient and the related bootstrap current. Disruptive β limits of these discharges are consistent with theoretical ideal kink-ballooning stability limits with low toroidal mode numbers n. Achievable β limit is improved by broadening of the pressure profile with high plasma internal inductance, plasma shaping, and wall stabilization. Edge localized modes (ELMs) and barrier localized modes (BLMs), which are associated with edge and internal transport barriers, respectively, are analyzed carefully. Resistive interchange modes with n ≤ 3 are excited in the negative shear region in reversed shear discharges with the internal transport barrier and lead to major collapse occasionally through nonlinear coupling with a tearing mode in the positive shear region. MHD characteristics of low m/n (m: poloidal mode number) tearing modes, which are attributed to the neoclassical tearing mode, are investigated. Stabilization of tearing modes and control of sawtooth activity are demonstrated using the fundamental O-mode electron cyclotron wave injection. Resistive wall modes associated with current-driven and pressure-driven low n external kink modes are identified.

EXXONMOBIL AWARDS CONTRACT TO TECHNIP

This paper presents a numerical study on vortex-induced vibration (VIV) of a vertical riser subject to uniform and linearly sheared currents. The model vertical riser tested at the MARINTEK by ExxonMobil is considered. The predicted numerical results are in good agreement with the experimental data. It is found that the dominant mode numbers, the maximum root mean square amplitudes, the dominant frequencies and the fatigue damage indices increase with the flow velocity. A standing wave response is observed for the single-mode in-line (IL) and cross-flow (CF) vibrations. Dual resonance is found to occur at most of the locations along the riser. At some locations along the riser, a third harmonic frequency component is observed in the CF response and a frequency component at the CF response frequency is found in the IL response apart from the frequency component at twice the CF response frequency. The majority of the vortex shedding shows a clear 2S pattern, whereas a 2P mode is observed near the position where the maximum vibration amplitude appears. The higher IL fatigue damage in the present study emphasises the importance of the IL fatigue damage especially in the design of low flow velocity or low mode number applications.

Methods to describe barotropic vortices by global fields and vortex characteristics

Abstract. Results from an experimental study of vortices in a rotating shear layer are presented. The data are in the form of maps of the instantaneous horizontal velocity field obtained by a particle tracking technique. Two fundamentally different methods to analyse time series of these velocity fields are presented and compared. One technique is the empirical orthogonal function (EOF) analysis, and the other method describes the flow field in terms of a few individual localised vortices. The flows discussed here are time-dependent two-vortex flows, which could either be described as a global mode 2 or as a collection of four unequal vortices. The results show that, while EOF analysis is a very powerful tool to detect fairly regular travelling modes or stationary features, it cannot detect local dynamics. The vortex identification technique is very good at detecting local structures and events but cannot put them into the context of a global flow structure. The comparison of the techniques shows indications that the time-dependence found in the system for low mode numbers could arise from an interaction of the large scale, global-mode flow with a local mechanism of vortex generation and shedding at a solid boundary.

Effect of Magnetohydrodynamic Perturbations on the Orbit Lossof Alpha Particles in Tokamak Plasma

We investigate the orbit loss of alpha particles under helicalmagnetic perturbation in a tokamak. The results show thatlow-frequency and low-mode number magnetic perturbation can causestochastic loss of alpha particles. This effect is significant forthose particles close to the boundary between the transit zone andthe trapped zone. The particle loss is sensitive to the phase of themagnetic perturbation, indicating the modulation of the particle losswith respect to magnetic perturbation. It is also found that theprecession of the particle banana orbit can even further enhance theparticle loss.

Excitation of axisymmetric Alfvénic modes in Ohmic tokamak discharges

Magnetohydrodynamic (MHD)mode activity in the Alfvén frequency range has been detected in the absence of energetic ions during discharges in several conventional tokamaks and spherical tokamaks, including the Tokamak Fusion Test Reactor (TFTR) and the Mega-Amp Spherical Tokamak (MAST). In TFTR the dominant toroidal mode number n was found to be zero; this is also the case in MAST discharges for which mode number information is available. The observed properties of these modes are shown to be consistent with global Alfvén eigenmodes (GAEs). Although they appear to have little or no effect on plasma performance in present-day devices, the fact that they are frequently observed in MAST Ohmic discharges suggests that they could be used as a diagnostic of plasma equilibrium parameters. In principle, they couldalso provide the basis for a plasma heating scheme. A possible mechanism for the excitation of the Alfvén eigenmodes in the absence of fast ions is suggestedby two-fluid simulations of various tokamaks, in which high-frequency mode activity is found to be correlated with relatively long-timescale MHD events in the plasma, such as internal reconnection events (IREs) or edge localized modes (ELMs). A simple analytical model describing the excitation of Alfvénicmodes by either IREs or ELMs is proposed. The coupling of low- and high-frequency MHD is predicted to be strongest for radially-extended modes: this is consistent with the low mode numbers of the activity observed in TFTR and MAST.

Overview of new high gain target design for a laser fusion power plant

We have developed a new direct-drive target design that has a predicted energy gain of 127 using a 1.3 MJ KrF laser, and a gain of 155 using 3.1 MJ. The DT fuel is surrounded by an ablator consisting of a low density CH foam filled with frozen DT. The ablator is then surrounded by a thin CH coating and a very thin high-Z overcoat. The energy gain of 127–155 is possible through the use of (1) direct-drive laser-target coupling; (2) controlled levels of radiative preheating that keeps the DT fuel on a low isentrope; (3) a short 1/4 μm laser wavelength for maximum absorption and rocket efficiencies; (4) reduction of the laser beam focal spot size during the implosion (zooming) so that the focal spot size better matches the imploding target size; and (5) ISI optical smoothing to minimize the laser nonuniformities at both high and low mode numbers. In addition to its high energy gain, this target design has several other attractive features: a low target fabrication cost; the potential for a modest-size 300 MWe power plant; the physical strength to withstand the acceleration into the chamber; and a high infrared albedo to better protect the target from preheating during the injection into the chamber.

Resistive instabilities in reversed shear discharges and wall stabilization on JT-60U

Resistive instabilities and wall stabilization of ideal kink modes with low toroidal mode number n are investigated in JT-60U reversed shear discharges. Localized burst-like MHD activity with n⩽3 is observed in the negative shear region with large pressure gradient near the internal transport barrier; this activity is identified as the resistive interchange mode.The burst-like MHD activity is benign with respect to the internal transport barrier and no clear degradation of plasma stored energy by the burst-like MHD activity is observed. The resistive interchange mode, however, results in a major collapse through non-linear coupling with a tearing mode in the positive shear region. A resistive wall mode (RWM) followed by a major collapse is observed in wall stabilized high β discharges exceeding the ideal stability limit with the wall at infinity (βN>βNno-wall).No clear continuous slowing down of the plasma toroidal rotation is observed before the growth of an RWM.

Observation of Resistive Wall Modes in JT-60U.

Resistive wall modes (RWM) associated with ideal magnetohydrodynamic current-driven (βN 2.4) kink modes with low toroidal mode number n (n = 1) have been identified in JT-60U. The pressure-driven RWM occurs at the plasma toroidal rotation of about 1% the Alfven speed without clear continuous slowing down of the plasma toroidal rotation. Occurrence of n = 1 RWMs result in thermal quench accompanied by higher n (n ≥ 2) modes. In the case of current-driven (˜ zero β) RWMs, a thermal quench occurs only at the peripheral region just after the RWM. In contrast, a thermal quench occurs in the whole plasma region following a drastic increase in the growth rate of the RWM from the order of τw-1 (τw is the resistive diffusion time of the wall) to larger than 102 τw-1in the case of pressure-driven (high β) RWMs.

Equilibrium properties of spherical torus plasmas in NSTX

Research in NSTX has been conducted to establish spherical torus plasmas to be used for high β, auxiliary heated experiments. This device has a major radius R0 = 0.86 m and a midplane halfwidth of 0.7 m. It has been operated with toroidal magnetic field B0 ⩽ 0.3 T and Ip ⩽ 1.0 MA. The evolution of the plasma equilibrium is analysed between discharges with an automated version of the EFIT code. Limiter, double null and lower single null diverted configurations have been sustained for several energy confinement times. The plasma stored energy reached 92 kJ (βt = 17.8%) withneutral beam heating. A plasma elongation in the range 1.6 ⩽ κ ⩽ 2.0 and a triangularity in the range 0.25 ⩽ δ ⩽ 0.45 have been sustained, with values of κ = 2.6 and δ = 0.6 being reached transiently. The reconstructed magnetic signals are fitted to the corresponding measured values with low errors. Aspects of the plasma boundary, pressure and safety factor profiles are supported by measurements from non-magnetic diagnostics. Plasma densities have reached 0.8 and 1.2 times the Greenwald limit in deuterium and helium plasmas, respectively, with no clear limit encountered. Instabilities including sawteeth and reconnection events, characterized by Mirnov oscillations, and a perturbation of the Ip, κ and li evolutions, have been observed. A low q limit was observed and is imposed by a low toroidal mode number kink instability.

Magnetoconductance fluctuations and weak localization in quantum dots: Reliability of the semiclassical approach

We utilize a semiclassical (SC) approach to calculate the conductance and weak-localization (WL) corrections in a triangular billiard of a given shape in the presence of nonzero magnetic field. The semiclassical conductance is given as a sum of all classical trajectories between the leads, each of them carrying the quantum-mechanical phase. The present SC approach is numerically exact (i.e., free from any approximations), explicitly includes diffractive effects in the leads, and is valid for arbitrary (low) mode numbers in the leads. We find however that the symmetry of the SC conductance/reflectance with respect to the direction of magnetic field or direction of the current is not satisfied, as well as that the WL corrections for the conductance and reflectance are inconsistent with each other; the SC approach does not satisfy the current conservation requirements and does not reproduce the corresponding exact quantum-mechanical results. The reason for that is traced to the topological difference in the sets of classical trajectories between the leads for different current or magnetic field directions which determine the conductance in the SC approximation. Our findings raise a question as to what extent one can rely on numerous predictions for statistical properties of the conductance oscillations of ballistic cavities including the WL line shapes and fractal conductance which were essentially based on the standard SC approach.

Electrostatic fluctuations in a direct current magnetron sputtering plasma

Electrostatic fluctuations (up to 500 kHz) have been studied in a planar dc magnetron device using Langmuir probes measuring the floating potential and the ion saturation current. Fluctuation levels as high as 30% have been found inside the magnetic trap. A two-point spectral analysis has shown that the fluctuations are due to coherent modes with a low azimuthal mode number. The modes are present only when the discharge power and the neutral gas pressure are above a threshold. Their frequency spacing decreases when the neutral gas pressure is raised, so that increasing the pressure leads to a more turbulent state. The modes have been interpreted as unstable E×B/density gradient modes.

Dependence of edge stability on plasma shape and local pressure gradients in the DIII-D and JT-60U tokamaks

The dependence of edge stability on plasma shape and local pressuregradients P' in the DIII-D and JT-60U tokamaks is studied. Thestronger plasma shaping in DIII-D allows the edge region of DIII-Ddischarges with type I (giant) ELMs to have access to the secondregion of stability for ideal ballooning modes and a larger edge pedestalpressure gradient P' than JT-60U type I ELM discharges. These JT-60U discharges are near the ballooning mode first regime stability limit. The DIII-D results support an ideal stability based working model of type I ELMs as low tointermediate toroidal mode number, n, MHD modes. The results from astability analysis of JT-60U type I ELM discharges indicate that thepredictions of this model are also consistent with JT-60U edgestability observations.

A LOW-DIMENSIONAL APPROACH FOR LINEAR AND NONLINEAR HEAT CONDUCTION IN PERIODIC DOMAINS

A low-order spectral method is used to solve steady-state linear and nonlinear heat conduction problems with periodic boundary conditions and periodic geometry. The study consists of first mapping the complex geometry into a rectangular domain. The Galerkin projection method is applied to solve the mapped equations. It is found that a low number of modes usually are sufficient to capture an accurate solution. Good agreement is obtained between the low-order description and existing formulations. Both the finite element method (FEM) and boundary element methods (BEM) are used for comparison.

Internal kink stability of large aspect ratio high-β tokamaks

Theoretically, high-β (β∼1) tokamaks offer a large fusion power efficiency advantage over low-β devices. However, if high-β tokamaks are inherently unstable then such devices will never be realized. In particular, kink modes are thought to be the most serious obstacle to high-β operations. High-β tokamaks are characterized by a very large Shafranov shift with a thin “boundary layer” on the outboard side of the device and a large “core” region of vertical flux surfaces comprising most of the central volume. In this paper, the energy principle is used to compute the magnetohydrodynamic internal kink stability of such devices in the large aspect ratio limit with a low toroidal mode number. A class of internal kink mode similar to the usual low-β kink is present; the stability against these modes is computed. A set of parameters describing a kink stable high-β equilibrium is given. Stability is shown to be dependent on the shape of the plasma boundary.

Modification of high mode pedestal instabilities in the DIII-D tokamak

The amplitude and frequency of modes driven in the edge region of tokamak high mode (H-mode) discharges [type I edge-localized modes (ELMs)] are shown to depend on the discharge shape. The measured pressure gradient threshold for instability and its scaling with discharge shape are compared with predictions from ideal magnetohydrodynamic theory for low toroidal mode number (n) instabilities driven by pressure gradient and current density and good agreement is found. Reductions in mode amplitude are observed in discharge shapes with either high squareness or low triangularity where the stability threshold in the edge pressure gradient is predicted to be reduced and the most unstable mode is expected to have higher values of n. The importance of access to the ballooning mode second stability regime is demonstrated through the changes in the ELM character that occur when second regime access is not available. An edge stability model is presented that predicts that there is a threshold value of n for second regime access and that the most unstable mode has n near this threshold.

Capillary Effects on Surface Waves

▪ Abstract We concentrate on the rich effects that surface tension has on free and forced surface waves for linear, nonlinear, and especially strongly nonlinear waves close to or at breaking or their limiting form. These effects are discussed in the context of standing gravity and gravity-capillary waves, Faraday waves, and parasitic capillary waves. Focus is primarily on post-1989 research. Regarding standing waves, new waveforms and the large effect that small capillarity can have are considered. Faraday waves are discussed principally with regard to viscous effects, hysteresis, and limit cycles; nonlinear waveforms of low mode numbers; contact-line effects and surfactants; breaking and subharmonics; and drop ejection. Pattern formation and chaotic and nonlinear dynamics of Faraday waves are mentioned only briefly. Gravity and gravity-capillary wave generation of parasitic capillaries and dissipation are considered at length. We conclude with our view on the direction of future research in these areas.

Global Alfvén eigenmode stability in thermonuclear tokamak plasmas

Relying on the good agreement observed between the gyrokinetic PENN model and the low toroidal mode number n damping measurements from JET, the stability of Alfvén eigenmodes (AEs) is predicted for reactor relevant conditions. Full non-local wave-particle power transfers are computed between global wavefields and alpha particles in an ITER reference equilibrium, showing that low-n eigenmodes (n ≅ 2) are strongly damped and intermediate-n eigenmodes (n ≅ 12) with a global radial extension are stable with a damping rate γ/ω ≅ 0.02. Even though an excitation of alpha particle driven instabilities remains in principle possible in reactors, this study suggests that realistic operation scenarios exist where all the AEs of global character are stable.

Ideal interchange instabilities in stellarators with a magnetic hill

The prediction of the Mercier (Suydam) criterion for localized interchange modes in stellarators shows stability in cases in which global modes are unstable. One case is non-resonant pressure driven instabilities with low mode numbers which become unstable even if the mode resonant surface does not exist inside the plasma column. The other case is interchange instabilities when the pressure gradient vanishes at the mode resonant surface. If the pressure becomes flat in a narrow region around the mode resonant surface, high mode number instabilities are suppressed and the beta limit at the particular resonant surface increases. The radial mode structure at nearly marginal beta also changes significantly. The properties of the non-resonant mode and the transition from a resonant to a non-resonant mode are clarified with a cylindrical plasma model for a low shear stellarator with a magnetic hill.