Small Growth Rate Research Articles

A novel method for stable thermoacoustic system design based on exceptional points - Part I: Conceptualization

In this conceptual study, we introduce the novel EPTD (Exceptional Point-based Thermoacoustic Design)-method for the design of stable thermoacoustic systems: dampen the entire thermoacoustic spectrum by shifting the EP towards smaller growth rates. For this, we first study a generic, linear parametric eigenvalue problem (LPEP) of a non-hermitian matrix featuring an EP. We derive analytical solutions for the eigenvalues and the EP as a function of arbitrary system parameters in the LPEP. Subsequently, we formulate a method for shifting the eigenvalue spectrum by relocating the EP in the complex plane. For this method, we show that not only the location of the EP in the complex plane is of relevance, but also the exceptional parameters, as they enforce the relations of the EP and the eigenvalues in the spectrum. When fixing these while shifting the EP, we demonstrate that the entire eigenvalue spectrum preserves its original shape, such that all eigenvalues only shift with the exact amount the EP was shifted. The derivation and the analysis of the relation between EP and spectrum generally holds for arbitrary system parameter combinations. Additionally, we showcase results for an exemplary configuration, i.e., a configuration with arbitrarily chosen and fixed parameters. We subsequently introduce a generic, thermoacoustic system, a Rijke tube, and study it by means of (semi-) analytical methods to confirm that the findings from the analysis of the LPEP also transfer to thermoacoustic systems. The configuration is modeled as a thermoacoustic network that allows to trace back the full system modes to their respective origins, which are either of acoustic or intrinsic thermoacoustic (ITA) nature. After identifying an EP that is formed by acoustic and ITA driven mode branches, we study its impact on the eigenfrequency spectrum and how this shift impacts the EP’s relation to the mode origins and the remaining eigenvalues. Similar to the analysis of the LPEP, the results show that not only the EP’s location in the CP is of relevance, but also its exceptional parameters: shifting the EP without consideration of the exceptional parameters leads to unsatisfactory shifts of the associated spectrum that becomes distorted. With this, we formulate the EPTD-method: shift the EP towards smaller growth rates while enforcing a well-defined relation between the mode origins and the EP by keeping the exceptional parameters constant. We show that this constraint preserves the relation between EP and mode origins when shifting the EP and finally demonstrate the practicality of the proposed method. For this, we shift the EP of the reference Rijke tube configuration towards smaller growth rates and simultaneously show that the resulting spectrum preserves its overall shape. While this paper covers the derivation and demonstration of the proposed EPTD-method based on a generic configuration, we consider a more complex combustion test rig, featuring a turbulent confined swirl flame, in part II of this study (M. Casel & A. Ghani, Combust. Flame, 2024), where we elaborate on consequences regarding the larger system parameter dimension as well as the existence of two EPs in the spectrum.Novelty and Significance statement:Thermoacoustic instabilities are actively researched for more than half a century. Despite this effort, scientifically sound strategies to design thermoacoustically stable combustors do not exist today. We present a novel method on a conceptual level that takes into account the latest discoveries in our field (e.g. ITA modes between 2014/2015 and Exceptional Points in 2018), connects them into one fast and robust numerical framework and opens the path to design the thermoacoustic stability map. This framework does not require tuning parameters or the like, which translates to an interpretable, parsimonious and computationally cheap design strategy. We demonstrate the Exceptional Point-based Thermoacoustic Design method (called EPTD-method) on two well-known experimental setups with increasing complexity and carefully perform parametric tests to unveil the key aspects of this novel design strategy. Finally, we turn the initially unstable setups into stable ones and explain why and how this was achieved.

A novel method for stable thermoacoustic system design based on exceptional points - Part II: Application to laminar and turbulent flame configurations

In this paper, we apply the novel EPTD (Exceptional Point-based Thermoacoustic Design)-method derived in part I of this study (M. Casel & A. Ghani, Combust. Flame, 2024) to two lab-scale combostor test rigs that exhbit thermoacoustic instabilities and demonstrate the method’s capability and practicality for stabilizing the originally unstable systems. Furthermore, we study, and refine, certain characteristics of the proposed EPTD method when applied to realistic combustor configurations. While the first configuration is laminar and unconfined, the second features a confined turbulent swirl flame. Both combustor test rigs are modeled by thermoacoustic networks incorporating measured flame responses. The models allow to trace back full system modes to their respective mode origins, which are either of intrinsic thermoacoustic (ITA) or acoustic nature, and are validated by means of experimental data. In both configurations, we identify exceptional points (EPs) that are formed by these respective mode branches. The unconfined, laminar configuration features only one EP, that we first investigate with respect to the system parameter sensitivity, and subsequently analyze its relation to the configuration’s spectrum, confirming the characteristics that were previously identified in part I of this study by means of generic (thermoacoustic) systems: not only the EP’s position in the complex plane, but also the exceptional parameters have a significant impact on the resulting thermoacoustic spectrum as they incorporate a relation between the respective mode origins and the full system modes. In contrast to the first configuration of this paper, the confined turbulent combustor allows for some further analysis of the EPTD application to realistic combustors as it features a significantly larger system parameter space as well as two EPs. Besides, it exhibits two EPs, for which we show how their respective sensitivities with respect to system parameter variations may already be utilized to identify which system parameters impact on which mode origins in the spectrum. In addition to this, we describe which of the two EPs may be chosen when applying the EPTD method. Finally, in both cases, we adopt the EPTD method derived in part I of this study for stabilizing thermoacoustic systems: Shift the EP towards smaller growth rates while constraining the exceptional parameters, i.e., shift the entire spectrum towards smaller growth rates while enforcing specific relations between the respective mode origins. Subsequently we showcase that this method in fact successfully dampens the entire spectrum of both combustion experiments and that the entire thermoacoustic spectrum shifts in growth rate with the same magnitude of the prescribed shift in EP growth rate. While the relatively small system parameter dimension of the unconfined laminar configuration does not allow for multiple system parameter combinations that realize the same EP, we identify multiple system parameter combinations that feature the same EP in the confined turbulent case. For this, we subsequently show that, as already shown for the generic configurations in part I of this study, different system parameter combinations, realizing the same EP, in fact result in the same thermoacoustic spectrum. This finally demonstrates that our proposed method is especially suited for the design of thermoacoustic systems that feature a large number of system parameters.Novelty and Significance statement:Thermoacoustic instabilities are actively researched for more than half a century. Despite this effort, scientifically sound strategies to design thermoacoustically stable combustors do not exist today. We present a novel method on a conceptual level that takes into account the latest discoveries in our field (e.g. ITA modes between 2014/2015 and Exceptional Points in 2018), connects them into one fast and robust numerical framework and opens the path to design the thermoacoustic stability map. This framework does not require tuning parameters or the like, which translates to an interpretable, parsimonious and computationally cheap design strategy. We demonstrate the Exceptional Point-based Thermoacoustic Design method (called EPTD-method) on two well-known experimental setups with increasing complexity and carefully perform parametric tests to unveil the key aspects of this novel design strategy. Finally, we turn the initially unstable setups into stable ones and explain why and how this was achieved.

Effects of Photoperiod Treatments on Stock Plants and Cutting Rooting of Three Cultivars of Ornamental Perennials

Many species of herbaceous perennials now have numerous cultivars, with growth habits and flower colors unique to each cultivar. Vegetative propagation is required so that resulting plants are genetically identical to the parent plant. Although many cultivars are selected for precocious and vigorous flowering, it is often difficult to collect adequate vegetative cuttings from such cultivars for commercial production because juvenile (vegetative) growth is preferred for high-quality cuttings. Cuttings that are reproductive (with flower buds or flowers) can have reduced or delayed rooting and increased occurrences of fungal pathogens (especially Botrytis species), resulting in lack of crop uniformity. We sought to answer the question, can growing stock plants of herbaceous perennials under defined photoperiods extend the length of the vegetative period and enhance the rooting of cuttings harvested from these stock plants? In this study, stock plants of ‘P009S’ twinspur (Diascia integerrima), ‘Furman's Red’ sage (Salvia greggii), and ‘Wild Thing’ sage (Salvia greggii) were grown under ambient, 12-hour light, 10-hour light, and 8-hour light to determine if a particular photoperiod could be used to suppress reproductive growth by promoting vegetative growth, thereby enhancing cutting rooting success. Effects of photoperiod treatments varied among the plant cultivars studied. Plants grown under 8-hour photoperiod had longer duration of vegetative growth, smaller growth rates, and lower dry weights when compared with plants grown under 12-hour or 10-hour photoperiod. Plants grown under 12-hour photoperiod had shorter duration of vegetative growth, larger growth rates, and higher dry weights when compared with plants grown under 10-hour and 8-hour photoperiods. The probability of rooting of cuttings harvested from stock plants of ‘P009S’ twinspur, ‘Furman’s Red’ sage, and ‘Wild Thing’ sage grown under 12-hour and 10-hour photoperiods was greater when compared with cuttings harvested from stock plants grown under 8 h photoperiod.

Effect of Photoperiod on Annual Growth Cycle of Two forest Tree Species at Seedling Stage

The seedlings of Pterygota alata and Manilkara bojeriwere exposed daily to normal day length (ND), short day (SD) and long day (LD) conditions. Relative rates of extension growth were calculated under each treatment.Under ND conditions, Pterygota plants assumed dormant condition from November to March with major part of season's growth occurring in August-September. Manilkara plants under similar conditions, continued to grow though the growth rates declined progressively and were of lower order during December-January.Under LD treatment, Pterygota plants became dormant one month later and resumed growth about two months earlier than ND plants; SD conditions, however, brought about dormant growth at the same time as ND treatment but the growth was resumed as under LD plants i.e. about two months earlier than ND. Manilkara plants were induced to dormant condition both under SD and LD treatments and the period corresponded exactly with small growth rates under ND conditions.It is concluded that these two species are by and large indifferent to LD and SD conditions and that the annual growth-cycle in nature is more under the changing influence of total sum of weather conditions rather than entirely on day lengths.

Relaxation Oscillation in SEIR Epidemic Models with the Intrinsic Growth Rate

The periodic oscillation transmission of infectious diseases is widespread, deep understanding of this periodic pattern and exploring the generation mechanism, and identifying the specific factors that lead to such periodic outbreaks, which are of very importanceto predict and control the spread of infectious diseases. In this study, to further reveal the mathematical mechanism of spontaneous generation of periodic oscillation solution, we investigate a type of SEIR epidemic model with a small intrinsic growth rate. By utilizing the singular perturbation theory and center manifold theorem, we extend the relaxation oscillation of three‐dimensional SIR models to the four‐dimensional SEIR models and prove the existence of stable relaxation oscillation with a large amplitude in the model. Numerical simulations are performed to verify our theoretical results. The results presented in this study provide a new idea for the study of the intrinsic mechanism of periodic oscillation in epidemiology, enrich the dynamics of epidemic models, and deepen the understanding of the global dynamics of these models.

Effect of Cerebral Small Vessel Disease Burden on Infarct Growth Rate and Stroke Outcomes in Large Vessel Occlusion Stroke Receiving Endovascular Treatment.

This study aimed to investigate the association between cerebral small vessel disease (CSVD) burden and infarct growth rate (IGR) in patients with large vessel occlusion (LVO) stroke who underwent endovascular treatment (EVT). A retrospective analysis was conducted on a cohort of 495 patients with anterior circulation stroke who received EVT. CSVD burden was assessed using a CSVD score based on neuroimaging features. IGR was calculated from diffusion-weighted imaging (DWI) lesion volumes divided by the time from stroke onset to imaging. Clinical outcomes included stroke progression and functional outcomes at 3 months. Multivariate analyses were performed to assess the relationship between CSVD burden, IGR, and clinical outcomes. The fast IGR group had a higher proportion of high CSVD scores than the slow IGR group (24.4% vs. 0.8%, p < 0.001). High CSVD burden was significantly associated with a faster IGR (odds ratio [95% confidence interval], 26.26 [6.26-110.14], p < 0.001) after adjusting for confounding factors. High CSVD burden also independently predicted stroke progression and poor functional outcomes. This study highlights a significant relationship between CSVD burden and IGR in LVO stroke patients undergoing EVT. High CSVD burden was associated with faster infarct growth and worse clinical outcomes.

Effect of initial phase on the ablative Rayleigh–Taylor instability

The effect of initial perturbation phase on the ablative Rayleigh–Taylor instability is investigated by numerical simulations. We aim at the growth of harmonic amplitudes and the formation of spikes and bubbles in single- and two-mode coupling cases, respectively. In the two-mode coupling case, two kinds of simulations are performed: two modes with relatively small linear growth rate difference and two modes with relatively large linear growth rate difference. The initial relative phase between the original two modes has a significant effect on the growth of harmonic amplitudes, and in different initial relative phases, the structures of spikes and bubbles begin to show great differences in the nonlinear stage. Fortunately, the harmonic amplitudes are weakened at a specific initial relative phase. This has a certain enlightening significance for the stabilization of ablative Rayleigh–Taylor instability.

Synchronization and stability analysis of an exponentially diverging solution in a mathematical model of asymmetrically interacting agents.

This study deals with an existing mathematical model of asymmetrically interacting agents. We analyze the following two previously unfocused features of the model: (i) synchronization of growth rates and (ii) initial value dependence of damped oscillation. By applying the techniques of variable transformation and timescale separation, we perform the stability analysis of a diverging solution. We find that (i) all growth rates synchronize to the same value that is as small as the smallest growth rate and (ii) oscillatory dynamics appear if the initial value of the slowest-growing agent is sufficiently small. Furthermore, our analytical method proposes a way to apply stability analysis to an exponentially diverging solution, which we believe is also a contribution of this study. Although the employed model is originally proposed as a model of infectious disease, we do not discuss its biological relevance but merely focus on the technical aspects.

Growth Rate and Body Size Mapping of Male Buffaloes during the Fattening Phase

Background: Buffalo is an animal that really likes water. Generally, buffalo like to soak in muddy waters and swamps around the cage. This behavior appears because buffalo have very few sweat glands. Therefore, if one wants to develop buffalo farming, he/she should look for special habitats or existing buffalo breeding centers. Differences in growth rates are caused by physiological factors and functional demands. Growth in animal body size usually follows an exponential function, with the growth rate varying from one body size to another. An animal's body size that functions earlier will grow faster with a greater growth rate than an animal that functions later. Differences in the growth rate of animal body size are also influenced by the constituent components of these body parts. Body parts composed primarily of bone will develop earlier than those composed of muscle or fat. During fattening, the body size of male buffaloes will have a different growth rate, where this difference indicates the potential for body size. The body size of a buffalo with a high growth rate has relatively large growth potential, while those with a small growth rate have small growth potential, or the body part has stopped growing because it has reached its maximum point. The purpose of this study is to determine the body size growth rate of male buffaloes, which have high potential during fattening. Mapping the body size of male buffaloes during fattening aims to help breeders determine at what age the buffaloes start to be fattened and slaughtered for meat production purposes so that they are economically quite profitable.&#x0D; Methodology: Data was collected using a saturated sampling technique, in which the livestock taken were all buffaloes kept by the Sumber Sari Livestock Group in Kalianget village, Seririt District, Buleleng Regency, Bali, which met the requirements in terms of their health and physical condition. The data obtained were analyzed using the power model regression analysis to determine the growth rate of the body size of the buffaloes. To map the growth rate, Biplot analysis was carried out with a Promax rotation of 90, as the variable is the estimated body size of the buffaloes based on the equation of the power regression line, and the object is the age of the male buffaloes.&#x0D; Conclusion: The results showed that the fastest growth rate or the greatest potential was chest width, followed by hip width, chest depth, body length, chest circumference, and shoulder height. At the same time, the slowest part of the lowest potential was the height of the hips. The results of mapping the body size growth rate of male buffalo aged 11-74 months with biplot charts showed that their growth potential was still quite high. However, there was a tendency for male buffalo over 30 months to have a slower growth rate in body size than those under 30 months.

A novel linear stability analysis method for plane Couette flow considering rarefaction effects

Following the stability analysis method in classic fluid dynamics, a linear stability equation (LSE) suitable for rarefied flows is derived based on the Bhatnagar–Gross–Krook (BGK) equation. The global method and singular value decomposition method are used for modal and non-modal analysis, respectively. This approach is validated by results obtained from Navier–Stokes (NS) equations. The modal analysis shows that LSEs based on NS equations (NS-LSEs) begin to fail when the Knudsen number ($Kn$) increases past$\sim$0.01, regardless of whether a slip model is used. When$Kn\geq 0.01$, the growth rate of the least stable mode is generally underestimated by the NS-LSEs. Under a fixed wavenumber, the pattern (travelling or standing wave) of the least stable mode changes with$Kn$; when the mode presents the same pattern, the growth rate decreases almost linearly with increasing$Kn$; otherwise, rarefaction effects may not stabilize the flow. The characteristic lengths of the different modes are different, and the single-scale classic stability analysis method cannot predict multiple modes accurately, even when combined with a slip model and even for continuum flow. However, non-modal analysis shows that this error does not affect the transient growth because modes with small growth rates offer little contribution to the transient growth. In rarefied flow, as long as the Mach number ($Ma$) is large enough, transient growth will occur in some wavenumber ranges. The rarefaction effect plays a stabilizing role in transient growth. The NS-LSEs-based method always overestimates the maximum transient growth.

Assessment of Conservation Effectiveness of the Qinghai–Tibet Plateau Nature Reserves from a Human Footprint Perspective with Global Lessons

The intensity of human pressure (HP) has an important impact on the biodiversity and ecosystem services of nature reserves (NRs), and the conflict and the coordination between NRs and human activities are now key issues to solve in the construction of NR systems. This study improved and applied a human footprint (HF) model that considers population density, land use, night light, grazing intensity, and road construction as indicators of human activity to evaluate the effectiveness of NRs in the Qinghai–Tibet Plateau in mitigating HP from 2000 to 2020. The results indicated that during this period, the average HP in the national NRs of the plateau increased from 1.47646 to 1.76687, where values were generally high in the east and low in the west. The average value in wetland NRs was the largest and had the smallest growth rate, while that in desert NRs was the smallest and had the largest growth rate. From 2000 to 2020, the average HP in the core areas, buffer areas, and experimental areas of the NRs increased by 0.12969, 0.29909, and 0.44244, respectively. It is a challenge for the Chinese government to strengthen the ability of NRs to mitigate HP on the wetland reserves and experimental zones in the Qinghai–Tibet Plateau region.

Study on Small Fatigue Crack Growth Rate of SLM AlSi10Mg Alloy

In this paper, the fatigue small crack growth rate of selective laser melting (SLM) AlSi10Mg alloy is studied by in-situ SEM experiment. The experimental results show that the small crack growth rate of SLM AlSi10Mg alloy can be described by Paris law, and the crack growth rate increases significantly with the increase in temperature. In addition, the combination of finite element simulation and fracture analysis shows that the pore is one of the main factors affecting the dispersion of small crack growth rate.

Receptivity and its influence on transition prediction of a hypersonic boundary layer over a small bluntness cone

Hypersonic boundary-layer receptivity is investigated using direct numerical simulation for Mach 6 flow over a 0.79 mm nose radius, 5° half-angle cone to slow acoustic waves. Two ways of exciting the second mode are discovered: For a low-frequency forcing (region I where f &amp;lt; 370 kHz), the first mode is initially excited and subsequently evolves into the second mode, whereas for a high-frequency forcing (region II where f≥ 370 kHz), the second mode is excited through the disturbances in the entropy layer. The receptivity mechanisms corresponding to the first and second modes are consistent with those of a previously investigated large bluntness cone. The receptivity coefficient shows a dramatical decrease with the frequency for disturbances in region I while a slight increase for disturbances in region II. The results show that the correlation between the receptivity coefficient of the first mode and the frequency proposed for large bluntness cones also applies to the current case. Furthermore, with receptivity and the measured freestream noise spectra taken into account, it is found that at the measured transition location, the disturbance predicted by the traditional eN method is no longer the most amplified disturbance, since the disturbances with lower frequencies can acquire larger amplitudes due to their large receptivity coefficients despite their smaller growth rates.

Predicting fatigue crack growth through the small and long crack regimes for a military transport aircraft loading spectrum using FASTRAN

Predicting fatigue crack growth across the small and long crack regimes in airframe components is a challenging task. Current fracture mechanics methods require, amongst other things, high fidelity small crack growth rate data and a thorough understanding of the different behaviours of small and long cracks. This paper presents a case study where a fatigue crack growth prediction of Aluminium Alloy 7075-T7351 coupons tested to a military transport aircraft loading spectrum, from nucleation to failure, is attempted using the most recent version of the linear-elastic fracture mechanics-based analysis tool FASTRAN v5.76. The military transport aircraft loading spectrum was applied to different coupon configurations to show the effectiveness of FASTRAN for a range of crack growth scenarios. Crack growth rate data for the prediction in the near-threshold region was experimentally obtained from small cracks using quantitative fractography and this data is shown to be a determining factor in the accuracy of the prediction. This paper highlights the challenges faced by engineers in predicting fatigue crack growth under variable amplitude loading from nucleation to failure and demonstrates the ability of FASTRAN to accurately predict load history effects. Areas for continued improvement are identified with the aim of accurate and reliable fatigue crack growth prediction in a range of structures, conditions, and scenarios.

Evaluation of the influences of the stress ratio, temperatures, and local microstructure on small fatigue crack propagation behavior of the FGH96 superalloy

Here, the influences of the stress ratios, temperatures, and local microstructure on the propagation behavior of small fatigue crack for FGH96 superalloy are quantitatively investigated. Following an increase in the stress ratio and temperature, the small fatigue crack growth rates increased. The influences of the local microstructure on the restrained ability for small fatigue crack propagation are analyzed, and the critical stress range of small fatigue crack propagation for the FGH96 superalloy at different stress ratios is proposed based on an amendatory-three-dimensional Kitagawa–Takahashi diagram. These results can aid in the fatigue damage tolerance design of FGH96 superalloy components.

Multimode theory of electron hole transverse instability

We present Vlasov–Poisson three-dimensional linear stability analysis of an initially planar electron hole structure, solving for the distribution function by integration along unperturbed orbits. The non-sinusoidal potential perturbation shape (parallel to$B$) is expanded in eigenfunctions of the adiabatic Poisson operator, generalizing the prior assumption of a rigid shift of the equilibrium. We show that the shiftmode is then modified by a second discrete mode plus an integral over a continuum of wave-like modes. A rigorous treatment shows that the continuum can be approximated effectively by a single mode that satisfies the external wave dispersion relation, thus making the perturbation a weighted sum of three modes. We find numerically the solution for the complex instability frequency, and the corresponding three mode amplitudes determining the perturbation eigenmode. This multimode analysis refines the accuracy of the prior single-mode results, giving slightly higher growth rates at most parameters, as expected from the extra mode shape freedom. Oscillating modes near stability boundaries have larger mode distortions which help explain particle-in-cell simulations that observe instability up to${\sim }20$ % beyond the prior shiftmode thresholds, and narrowing of the perturbation. At high magnetic field, the multimode analysis predicts a reduction of the already small growth rate.

Effect of grain structure on fatigue crack propagation behavior of Al-Cu-Li alloys

Recrystallization behavior during optimized heat treatments provides a potential to obtain desirable grain structure, which significantly improves the mechanical properties of aluminum alloys. The influence of grain structures on fatigue crack propagation (FCP) behaviors of Al-Cu-Li alloy with hot-rolled (HR) and cold-rolled (CR) was investigated. Subgrain boundaries have a significant impact on small crack growth rates, which is reflected in the pronounced fluctuation of fatigue crack growth of HR specimens after solution treatment. Moreover, the specific cellular structure within grains can improve the deformation capacity of alloys due to their accommodation of plastic deformation, which contributes to the lower fatigue crack growth rates and higher threshold values in HR specimens. The intragranular deflection also decelerates the FCP rate and occurs in these regions of large grain without subgrain boundaries. Recrystallization occurs in the CR specimens, resulting in small anisotropy on the fatigue resistance for the different orientations in the Paris stage due to the recrystallization texture. Fatigue cracks can be deflected and tend to propagate along the grain boundaries when it goes into the grain with a relatively low Schmidt factor value.

Impact of pedestal density gradient and collisionality on ELM dynamics

BOUT++ turbulence simulations are conducted to capture the underlying physics of small ELM characteristics achieved by increasing separatrix density via controlling strike points from vertical to horizontal divertor plates for three EAST discharges. BOUT++ linear simulations show that the most unstable modes change from high-n ideal ballooning modes to intermediate-n peeling–ballooning modes and eventually to peeling–ballooning stable plasmas in the pedestal. Nonlinear simulations show that the fluctuation is saturated at a high level for the lowest separatrix density. The ELM size decreases with increasing separatrix density, until the fraction of this energy lost during the ELM crash becomes less than 1% of the pedestal stored energy, leading to small ELMs. Simulations indicate that small ELMs can be triggered either by the marginally peeling–ballooning instability near the peak pressure gradient position inside the pedestal or by a local instability in the pedestal foot with a larger separatrix density gradient. The pedestal collisionality scan for type-I ELMs with steep pedestal density gradient shows that both linear growth rate and ELM size decrease with increasing collisionality. On the contrary, the pedestal collisionality and pedestal density width scan with a weak pedestal density gradient indicate small ELMs can either be triggered by a high-n ballooning mode or by a low-n peeling mode in a low collisionality region 0.04–0.1. The simulations indicate the weaker the linear unstable modes near marginal stability with small linear growth rate, the lower nonlinearly saturated fluctuation intensity and the smaller turbulence spreading from the linear unstable zone to stable zone in the nonlinear saturation phase, leading to small ELMs.

Effect of pedestal density on the formation of small edge localized modes

Recent experiments have demonstrated that a high separatrix density and a large ratio of separatrix density to pedestal top density are two crucial conditions for achieving high confinement operation with small edge localized modes (ELMs). In order to identify the underlying physics of this phenomenon, a series of equilibria with different separatrix and pedestal top densities are constructed, and their peeling–ballooning (P–B) instabilities are analyzed through simulation. It is found that there is a threshold value of pedestal top density which comes from competition between ion inertia and diamagnetic effect, and ELM energy loss can be minimized at the threshold value for a fixed separatrix density. When the pedestal top density is smaller than the threshold value, the ion inertial effect induced by the density profile has a significant influence on the growth of ELMs, resulting in an increased linear growth rate and more ELM energy loss by trigging low-n modes (n being the toroidal mode number) in the nonlinear phase. When the pedestal top density is larger than the threshold value, the diamagnetic effect is the main factor determining the mode spectrum, which moves to the high-n region with a larger growth rate and the nonlinear ELM energy loss increases. However, for a fixed pedestal top density, a larger separatrix density leads to a wider mode spectrum with a smaller growth rate; thus ELM energy loss is reduced. The results of this research provide a new mechanism, namely that the P–B mode is possibly transferred to a resistive ballooning mode, to interpret the experimental findings during high pedestal density operation.

Hyperbolic Coxeter groups and minimal growth rates in dimensions four and five

For small n , the known compact hyperbolic n -orbifolds of minimal volume are intimately related to Coxeter groups of smallest rank. For n=2 and 3 , these Coxeter groups are given by the triangle group [7,3] and the tetrahedral group [3,5,3] , and they are also distinguished by the fact that they have minimal growth rate among all cocompact hyperbolic Coxeter groups in \operatorname{Isom}\mathbb H^n , respectively. In this work, we consider the cocompact Coxeter simplex group G_4 with Coxeter symbol [5,3,3,3] in \operatorname{Isom}\mathbb H^4 and the cocompact Coxeter prism group G_5 based on [5,3,3,3,3] in \operatorname{Isom}\mathbb H^5 . Both groups are arithmetic and related to the fundamental group of the minimal volume arithmetic compact hyperbolic n -orbifold for n=4 and 5 , respectively. Here, we prove that the group G_n is distinguished by having smallest growth rate among all Coxeter groups acting cocompactly on \mathbb H^n for n=4 and 5 , respectively. The proof is based on combinatorial properties of compact hyperbolic Coxeter polyhedra, some partial classification results and certain monotonicity properties of growth rates of the associated Coxeter groups.