Scale Fluctuations Research Articles

Hypoplastic modeling post-failure behavior of sandy slope with porosity spatial variability

Recent probabilistic studies have thoroughly examined how uncertainties in soil shear strength affect slope stability, while the impact of spatial variability in soil porosity on post-failure behaviors of slopes has received less attention. Overlooking the influence of porosity may lead to inaccurate assessments of slope post-failure behaviors. This study presents a random smoothed particle hydrodynamics (RSPH) framework for probabilistic analysis of slope failure with large deformation using a critical state-based hypoplastic constitutive model. The results indicate that the proposed RSPH method provides reasonable estimates for slope post-failure behaviors. Increases in both the coefficient of variation (COV) and the scale of fluctuation (SOF) of porosity give rise to underestimations of the run-out distance and sliding volume. Additionally, slope failure modes are affected by increasing uncertainty of porosity, which may contribute to the occurrence of deep-seated failures.

Revisiting the role of cosmic-ray driven Alfven waves in pre-existing magnetohydrodynamic turbulence. I. Turbulent damping rates and feedback on background fluctuations

Alfv\'en waves (AWs) excited by the cosmic-ray (CR) streaming instability (CRSI) are a fundamental ingredient for CR confinement. The effectiveness of such self-confinement relies on a balance between the CRSI growth rate and the damping mechanisms acting on quasi-parallel AWs excited by CRs. One relevant mechanism is called turbulent damping, in which an AW packet injected in pre-existing turbulence undergoes a cascade process due to its nonlinear interaction with fluctuations of the background. The turbulent damping of an AW packet in pre-existing magnetohydrodynamic (MHD) turbulence is re-examined, revised, and extended to include the most recent theories of MHD turbulence that account for dynamic alignment and reconnection-mediated regimes. The case in which the role of feedback of CR-driven AWs on pre-existing turbulence is important is also discussed. The Els\"asser formalism is employed. Particular attention is given to the role of a nonlinearity parameter $ that estimates the strength of the nonlinear interaction between CR-driven AW packets and the background fluctuations. We point out the difference between $ and the parameter $ that instead describes the intrinsic strength of nonlinear interactions between pre-existing fluctuations. Turbulent damping rates of quasi-parallel AW packets and cosmic-ray feedback (CRF) are derived within this formalism. When the strength of the nonlinear interaction is properly taken into account, we find that (i) the turbulent damping rate of quasi-parallel AWs in sub-Alfv\'enic turbulence depends on the background-fluctuation amplitude to the third power, and hence is strongly suppressed; (ii) the dependence on the AW's wavelength (and thus on the CR gyro-radius from which it is excited) is different from what has been previously obtained; and (iii) when dynamic alignment of cascading fluctuations and the possibility of a reconnection-mediated range is included in the picture, the turbulent damping rate exhibits novel regimes and breaks. Finally, a criterion for CRF is derived and a simple phenomenological model of CR-modified scaling of background fluctuations is provided.

A deep learning-based surrogate model for probabilistic analysis of high-speed railway tunnel crown settlement in spatially variable soil considering construction process

In reality, many high-speed railway tunnels are built by sequential excavation method in spatially variable soil. This study introduces an efficient deep learning-based solution for the probabilistic analysis of tunnel crown settlements considering the spatial variability of surrounding soil. The two-dimensional convolutional neural network (2D-CNN) is employed to uncover the implicit correlation between the soil elastic modulus random field and tunnel crown settlements. The correlation coefficient (R) of the surrogate model is greater than 0.9664. Meanwhile, the mean relative error of the predictions remains within 2.38%, under various scales of fluctuation (SOFs). An additional 10,000 samples were produced to assess the practical performance of the trained model in probabilistic analysis. The relative errors of predictions generally fall within 5%, with a minimum confidence interval of 98.15%. These results demonstrate that the developed 2D-CNN model can effectively substitute the traditional method in predicting tunnel crown settlements considering soil spatial variability.

Climate change fluctuations can increase population abundance and range size

AbstractClimate change threatens many species by a poleward/upward movement of their thermal niche. While we know that faster movement has stronger impacts, little is known on how fluctuations of niche movement affect population outcomes. Environmental fluctuations often affect populations negatively, but theory and experiments have revealed some positive effects. We study how fluctuations around the average speed of the niche impact a species' persistence, abundance and realized niche width under climate change. We find that the outcome depends on how fluctuations manifest and what the relative time scale of population growth and climate fluctuations are. When populations are close to extinction with the average speed, fluctuations around this average accelerate population decline. However, populations not yet close to extinction can increase in abundance and/or realized niche width from such fluctuations. Long‐lived species increase more when their niche size remains constant, short‐lived species increase more when their niche size varies.

Geotechnical correlation field-informed and data-driven prediction of spatially varying geotechnical properties

Geotechnical measurements are often limited, leading to the use of interpolation techniques for interpreting spatial variations in geotechnical properties from sparse geo-data. Traditional geostatistical methods suffer from significant computational complexity. On the other hand, data-driven approaches often lack integration with geotechnical domain knowledge, potentially oversimplifying or complicating predictions related to the spatial variability of geotechnical properties. This study introduces a novel framework that combines geotechnical knowledge with data-driven methods to model inherent soil spatial variability incorporating Geotechnical Correlation Field (GCF) that reflects domain knowledge. The GCF, influenced by Autocorrelation Function (ACF) types and Scale of Fluctuation (SOF), provides a flexible basis for accurately representing spatially varying geotechnical properties. Using a large synthetic database comprising known ACF types and SOFs, we constructed a series of specialized neural networks. These networks identify random field parameters at different sites based on sparse data, and the estimated parameters can be directly used to calculate GCFs for a given site. The performance of the proposed method is validated using a set of synthetic data and a real case history in New Zealand. The results demonstrate the model can accurately predict random field parameters for irregularly spaced geo-data, even with limited information. Significantly, the GCFs offer improved physical interpretations and enhance the performance of subsurface modeling. The computational complexity of this method is independent of the number of soil cells, making it highly efficient and scalable.

Random Responses of Shield Tunnel to New Tunnel Undercrossing Considering Spatial Variability of Soil Elastic Modulus

This paper investigates the effect of spatial variability of soil elastic modulus on the longitudinal responses of the existing shield tunnel to the new tunnel undercrossing using a random two-stage analysis method (RTSAM). The Timoshenko–Winkler-based deterministic method considering longitudinal variation in the subgrade reaction coefficient and the random field of the soil elastic modulus discretized by the Karhunen–Loeve expansion method are combined to establish the RTSAM. Then, the proposed RTSAM is applied to carry out a random analysis based on an actual engineering case. Results show that the increases in the scale of fluctuation and the coefficient of variation of the soil elastic modulus lead to higher variabilities of tunnel responses. A decreasing pillar depth and mean value of the soil elastic modulus and an increasing skew angle strengthen the effect of the spatial variability of the soil elastic modulus on tunnel responses. The variabilities of tunnel responses under the random field of the soil elastic modulus are overestimated by the Euler–Bernoulli beam model. The results of this study provide references for the uncertainty analysis of the new tunneling-induced responses of the existing tunnel under the random field of soil properties.

Anomalous criticality coexists with giant cluster in the uniform forest model

We show by extensive simulations that the whole supercritical phase of the three-dimensional uniform forest model simultaneously exhibits an infinite tree and a rich variety of critical phenomena. Besides typical scalings like algebraically decaying correlation, power-law distribution of cluster sizes, and divergent correlation length, a number of anomalous behaviors emerge. The fractal dimensions for off-giant trees take different values when being measured by linear system size or gyration radius. The giant-tree size displays two-length scaling fluctuations, instead of following the central-limit theorem.

Multiple scattering, radiative transfer, and parameter regimes

We have established the radiative transfer regime in a four-parameter space spanned by amplitude and scale of fluctuations, convergence rate, and spatial range. The derivation of radiative transfer equations from wave equations offers a better understanding of the underlying physics in radiative transfer theory. One approach is based on tools borrowed from quantum field theory (QFT) and employs the Dyson and Bethe–Salpeter equations as the foundation. This has allowed us to investigate the small scale fluctuation regime, kl = O ( ε ) . We also investigated this regime using the scale-separated asymptotics approach and found that the conditions on the parameters are the same as in the QFT approach if the convergence rate of the Dyson series is O ( ε 2 ) . Furthermore, we found that there is a multitude of regimes where radiative transfer manifests. We hence infer that for a fixed spatial range, increase in the scale size of fluctuations leads to decrease in the upper bound of variance of permittivity fluctuations along with diminished convergence rate of the Wigner series. Also, for a fixed scale size and convergence rate, the spatial range where radiative transport manifests decreases with increase in strength of fluctuations.

Uncertainty Analysis of Post-Failure Behavior in Landslides Based on SPH Method and Generalized Geotechnical Random Field Theory

In slope stability analysis, the inherent heterogeneity and spatial variability of soil significantly influence the aftermath of landslides, including critical aspects like runout distance, influence distance, and volume. This research integrates Smoothed Particle Hydrodynamics (SPH) with random field theory to precisely model the large deformation events in slopes with anisotropic shear parameters, while leveraging Graphics Processing Unit (GPU) parallel computing to expedite the generation of random field samples and SPH simulations. This approach meticulously examines how spatial heterogeneity of soil affects slope instability, simulating soil parameters across varied geological settings. It considers the fluctuation scale of anisotropy, the cross-correlation of cohesion and the angle of internal friction, along with their coefficient of variation (CV), to elucidate their impact on landslide magnitude and severity. This study furnishes a robust tool for a holistic assessment of slope impacts and landslide volumes. Additionally, by delineating the computational efficiency disparities between GPUs and Central processing units (CPUs), it underscores the pronounced efficiency benefits of GPU computing. The insights garnered offer fresh perspectives and methodologies in slope stability analysis and disaster risk evaluation, contributing to the finer prediction and management of this prevalent and grave geological hazard.

Effective Spatial Degrees of Freedom of Natural Temperature Variability as a Function of Frequency

Abstract A fundamental statistic of climate variability is its spatiotemporal correlation function. Its complex structure can be concisely summarized by a frequency-dependent measure of the effective spatial degrees of freedom (ESDOF). Here we present, for the first time, frequency-dependent ESDOF estimates of global natural surface temperature variability from purely instrumental measurements, using the HadCRUT4 dataset (1850–2014). The approach is based on a newly developed method for estimating the frequency-dependent spatial correlation function from gappy data fields. Results reveal a multicomponent structure of the spatial correlation function, including a large-amplitude short-distance component (with weak time scale dependence) and a small-amplitude long-distance component (with increasing relative amplitude toward the longer time scales). Two frequency-dependent ESDOF measures are applied, each responding mainly to either of the two components. Both measures exhibit a significant ESDOF reduction from monthly to multidecadal time scales, implying an increase of the effective spatial scale of natural surface temperature fluctuations. Moreover, it is found that a good approximation to the global number of equally spaced samples needed to estimate the variance of global mean temperature is given, at any frequency, by the greater one of the two ESDOF measures, decreasing from ∼130 at monthly to ∼30 at multidecadal time scales. Finally, the multicomponent structure of the correlation function together with the detected ESDOF scaling properties indicate that the ESDOF reduction toward the longer time scales cannot be explained simply by diffusion acting on stochastically driven anomalies, as it might be suggested from simple stochastic-diffusive energy balance models.

Effects of subgrade spatial variability on critical strains and effectiveness of geogrid reinforcement in flexible pavement

The objective of this study is to assess the impact of spatial variability in the subgrade layer on the critical response of pavements and the effectiveness of geogrid reinforcement, employing the random field finite difference analysis (RFFDA). A comprehensive parametric study was conducted to examine the influence of two crucial factors: the coefficient of variation (COVE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ ext{COV}}}_{E}$$\\end{document}) and scale of fluctuation (SOF) of the subgrade modulus. Further investigation was conducted to uncover the statistical and mechanical mechanisms underlying the impact of subgrade spatial variability with emphasis on the critical strain distributions and their correlation with both the overall modulus and the local spatial variability of the key influence zone. Furthermore, this study explored the influence of subgrade spatial variability on the effectiveness of geogrid in reducing critical strains, considering various placement positions and geogrid moduli. The following main conclusions are drawn: (a) subgrade spatial variability has a substantial amplifying effect on critical pavement strains due to low modulus dominating effect, (b) there exists a worst value of SOF that results in the most unfavorable statistics of critical subgrade strain, (c) the effect of subgrade spatial variability on critical subgrade strain is more pronounced compared to its effect on critical asphalt strain, (d) the mean value of critical subgrade strain in RFFDA can be significantly underestimated when assuming fixed location for the strain, and (e) the effectiveness of geogrid in reducing critical strains is impacted by subgrade spatial variability, with the impact varying with the type of critical strain and geogrid location. Specifically, when placed at the base course–subgrade interface, the ability of geogrid to reduce critical subgrade strain is significantly compromised due to the subgrade spatial variability.

On turbulent burning velocities of V-shaped turbulent premixed flames with circular fractal turbulence generators

The characteristics of turbulent burning velocities of V-shaped turbulent flames are experimentally investigated by varying the shape parameters of cross- and square-type circular fractal turbulence generators (FTG). For both type FTGs, the thickness reduction ratio, Rt, of 0.4, 0.6, 0.8, and 1.0, and the blockage ratios, σ, of 30 and 50% are used. The characteristics of non-reacting turbulent flows such as the mean flow velocity, U¯, velocity fluctuation, u′, and integral length scale, L, are measured using a hot-wire anemometer while the progress variable of reacting flows is evaluated using OH-PLIF images. The turbulent premixed flames are distributed in the Borghi-Peters diagram from the wrinkled flamelet regime to the thin reaction zone regime, depending on the shape parameters of the FTG. The local displacement speed, ST,LD, and the local consumption speed, ST,LC, are evaluated using the measured data of turbulent reacting and non-reacting flows. ST,LD and ST,LC are found to increase with increasing σ and decreasing Rt, and hence, the averaged ST,LD, and ST,LC can be expressed as a function of u′/SL and/or L/δL. Correlations between ST or u′ and the FTG shape parameters (Rt and σ) are also proposed, which can be used to control and predict ST and u′ for fundamental studies and industrial applications.

Direct numerical simulations of supersonic flat-plate turbulent boundary layers with uniform blowing

The effect of uniform blowing on a spatially developing flat-plate turbulent boundary layer at Mach 2.25 is investigated using direct numerical simulations. Two values of the wall blowing ratio are considered, corresponding to low and high blowing rates. Uniform blowing is found to significantly reduce the near-wall turbulence anisotropy, although the turbulent kinetic energy still exhibits near-wall asymptotic behavior and the Reynolds analogy is relatively insensitive to changes in the blowing ratio. The pre-multiplied spectra of turbulent kinetic energy production demonstrate that increasing the blowing ratio significantly energizes the large-scale structures in the outer region, while suppressing the inner small-scale structures. An increase in the blowing ratio also has a strong influence on the behavior of the fluctuating wall pressure, amplifying the fluctuation intensity and reducing the dominant frequencies in the power spectrum. Two-point space–time correlations indicate that the characteristic length scale of the pressure fluctuations increases with increasing blowing ratio, whereas the convection velocity exhibits the opposite trend. Analysis of the reduced mean wall heat flux reveals that it is dominated by the relative balance between the work of the Reynolds stress and the turbulent transport of heat, but is insensitive to uniform blowing. Importantly, bidimensional empirical mode decomposition of the turbulent structures highlights the increasingly dominant contributions related to the significantly energized outer large-scale structures in the blowing region.

Probabilistic design optimization of rectangular skirted mudmats under combined loading in spatially variable soil

Natural soil exhibits inherent spatial variability due to complex physical and chemical formation processes, making it necessary to analyze, evaluation and design foundations using probabilistic method. In this study, the bearing capacity of rectangular skirted mudmats in spatially variable soil is investigated by three-dimensional random finite element analysis (3D-RFEA). Given the high computational cost of 3D-RFEA, an active learning probabilistic method based on Kriging surrogate model is proposed to predict the mean and coefficient of variation of bearing capacity factor of mudmats. Then, considering the effects of the skirt depth ratio of mudmats and the variation coefficient and scales of fluctuation of the undrained shear strength of soil, a prediction model of probabilistic failure envelope is proposed by combining the normalized deterministic failure envelope and the cumulative distribution function (CDF) of the uniaxial bearing capacity factor. Based on the prediction model of probabilistic failure envelope, an example of probabilistic design optimization of mudmats is finally given. Compared with the normative method, the design method based on the prediction model can design mudmats directly with the permissible failure probability.

Characterizing the variability of bauxite mine tailings from CPTU test results

In geotechnical engineering, there is a high degree of uncertainty associated with soil parameters, since soil materials are variable due to mineralogy, stress history, and deposition processes in the same layer of soil profile. Given the high degree of variability, there are limitations when calculating and simulating models based on deterministic parameters extracted from the field. Mine tailings are part of this scenario, to which must be added the difficulty of extracting samples and characterizing them in the laboratory, making field investigation crucial for basing geomechanical models. This study aims to quantify statistical properties such as mean, standard deviation, probability density function, and scale of fluctuation obtained from the analysis of direct measurements of piezocone tests, such as tip resistance, lateral friction, and pore pressure. A set of data from a bauxite tailing deposit will be used in the analysis, which which was done by implementing an algorithm in Python to calculate the statistical parameters. Q-Q plots and histograms were presented for the most variable CPTu data to evaluate the probability density function that best fits. The results found indicate a material with high variability with statistical parameters that are close to those pointed out by the literature for other mine tailings. By elucidating the statistical properties of the soil parameters, this study contributes to a better understanding of the geomechanical behavior of mine tailings deposits, aiding in the development of more accurate predictive models. Furthermore, the methodology employed underscores the importance of robust field investigations and advanced statistical analysis techniques in geotechnical engineering research.

Insights into the design principles of JF-ED-VTFET for biosensing application

In this research article, we have designed a junction-free electrostatically doped vertical tunnel field-effect transistor (JF-ED-VTEFT) for label-free biosensing applications. We incorporated a nano-cavity within the gate-oxide layer near the source end of the FET to enable the detection of biomolecules based on the principle of dielectric modulation and without the requirement of external labeling. The proposed biosensor is thoroughly analyzed, considering various aspects such as electric field, energy band, transfer characteristics, and sensitivity parameters including energy band diagram, ON-current, ION/IOFF ratio, electrical analysis, and surface potential characteristics. The investigation of sensitivity encompasses practical challenges, such as different filling factors and step-profiles resulting from steric hindrance. In addition, the performance of the biosensor is evaluated by analyzing the temperature and scaling fluctuation in the integrated nanocavities. Additionally, values of biomolecules that are close to standard have been taken to validate the performance and provide insight into the sensitivity of the biosensor for detecting and analyzing the molecules.

Regional scale evaluation of nitrate fluctuations in groundwater using cluster analysis and standardised hydrometeorological indices

Temporal fluctuations in nitrate in groundwater can result in concentrations temporarily exceeding drinking water standards. This can bring about the need for costly water treatment or blending. Despite this, the extent and potential controls on these fluctuations are poorly understood, particularly at regional to national scales. Applied to Southeast England (UK), here we develop the first application of cluster analysis and standardised hydrometeorological indices to evaluate nitrate fluctuations in groundwater at the regional scale. Hierarchical and K-means cluster analysis of 96 groundwater nitrate time series for the period 1995–2022 showed that nitrate time series can be divided into 4 clusters: (1) long term increasing trends (n = 23, mean trend = 0.26 mg NO3/l/a), (2) long term decreasing trends (n = 19, mean trend = − 0.65 mg NO3/l/a), (3) long term increasing trends with seasonal fluctuations (n = 24, mean trend = 0.29 mg NO3/l/a) and (4) long term increasing trends superimposed on near-decadal scale fluctuations (n = 30, mean trend = 0.22 mg NO3/l/a). Boreholes in cluster 1 appear to be deeper than boreholes in cluster 2. In comparison to shallower boreholes, deeper boreholes are likely to be intersecting longer groundwater flow systems where nitrate concentrations are affected by historic “legacy nitrate” leaching. There is weak spatial coherence in the clustering, with clusters 3 and 4 present in the South and North Downs respectively. Cross-correlation analysis between groundwater nitrate time series with precipitation and groundwater level indices showed that rapid seasonal fluctuations in nitrate concentrations in cluster 3 in the South Downs are associated with rapidly responding groundwater level fluctuation. This is likely due to the highly fractured and faulted nature of the Chalk aquifer in this area. This is in contrast with the slower near-decadal fluctuations in cluster 4 in the North Downs. The strongest correlations between groundwater levels and nitrate concentrations in cluster 3 occurred when cross-correlating at a lag of zero months, which would suggest that matrix diffusion is unlikely to be a significant control on nitrate seasonality. Seasonal fluctuations in nitrate concentrations are likely to be associated with a combination of piston displacement at the water table and changing groundwater flow paths to the borehole. Future climate change may change the magnitude and timing of seasonal fluctuations caused by these processes. The methodology developed here is generic and can be applied wherever there is a large body of groundwater nitrate time series data.

Garnet zoning patterns record multiple processes of chemical transfer during subduction

Subduction is the principal mechanism by which volatiles are transferred from the Earth's surface to its interior. In garnets from eclogites and blueschists formed within the subduction setting, fine-scale, oscillatory elemental zoning is a common feature, sometimes considered to record open-system fluid exchange during prograde metamorphism. We present oxygen isotope data for garnets with such zoning from five exhumed subduction zone complexes. Short length scale fluctuations in elemental and oxygen isotope zoning (which are themselves spatially decoupled) cannot be linked to open-system fluid exchange during garnet crystallization in all samples; these data do not provide evidence for a genetic relationship between elemental oscillations and fluid fluxing. However, garnets from one setting do provide clear evidence for syn-growth ingress of elementally and isotopically buffering fluids, a process that operated simultaneously with the formation of elemental oscillations. Our findings indicate multiple mechanisms of chemical transfer operate at the grain–rock scale during subduction, and that some subduction zone rocks may experience only limited interaction with external prograde fluids. These results are consistent with a picture of highly heterogenous volatile transfer during subduction, and suggest that some proportion of the fluid inventory inherited at shallow depths may be transferred to sub-arc depths.

Extracting electromagnetic signatures of spacetime fluctuations

We present a formalism to discern the effects of fluctuations of the spacetime metric on electromagnetic radiation. The formalism works via the measurement of electromagnetic field correlations, while allowing a clear assessment of the assumptions involved. As an application of the formalism, we present a model of spacetime fluctuations that appear as random fluctuations of the refractive index of the vacuum in single, and two co-located Michelson interferometers. We compare an interferometric signal predicted using this model to experimental data from the Holometer and aLIGO. We show that if the signal manifests at a frequency at which the interferometers are sensitive, the strength and scale of possible spacetime fluctuations can be constrained. The bounds, thus obtained, on the strength and scale of the spacetime fluctuations, are also shown to be more stringent than the bounds obtained previously using astronomical observation at optical frequencies. The formalism enables us to evaluate proposed experiments such as QUEST for constraining quantum spacetime fluctuations and to design new ones.

Stability of Regularized Hastings–Levitov Aggregation in the Subcritical Regime

We prove bulk scaling limits and fluctuation scaling limits for a two-parameter class ALE(α,η)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\alpha },\\eta )$$\\end{document} of continuum planar aggregation models. The class includes regularized versions of the Hastings–Levitov family HL(α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\alpha })$$\\end{document} and continuum versions of the family of dielectric-breakdown models, where the local attachment intensity for new particles is specified as a negative power -η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\eta $$\\end{document} of the density of arc length with respect to harmonic measure. The limit dynamics follow solutions of a certain Loewner–Kufarev equation, where the driving measure is made to depend on the solution and on the parameter ζ=α+η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\zeta }={\\alpha }+\\eta $$\\end{document}. Our results are subject to a subcriticality condition ζ⩽1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\zeta }\\leqslant 1$$\\end{document}: this includes HL(α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\alpha })$$\\end{document} for α⩽1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\alpha }\\leqslant 1$$\\end{document} and also the case α=2,η=-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\alpha }=2,\\eta =-1$$\\end{document} corresponding to a continuum Eden model. Hastings and Levitov predicted a change in behaviour for HL(α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\alpha })$$\\end{document} at α=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\alpha }=1$$\\end{document}, consistent with our results. In the regularized regime considered, the fluctuations around the scaling limit are shown to be Gaussian, with independent Ornstein–Uhlenbeck processes driving each Fourier mode, which are seen to be stable if and only if ζ⩽1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\zeta }\\leqslant 1$$\\end{document}.