Sharp Gradients Research Articles

Non-intrusive balancing transformation of highly stiff systems with lightly damped impulse response

Balanced truncation (BT) is a model reduction method that uses a coordinate transformation to retain eigen-directions that are highly observable and reachable. To address realizability and scalability of BT applied to highly stiff and lightly damped systems, a non-intrusive data-driven method is developed for balancing discrete-time linear systems via the eigensystem realization algorithm (ERA). The advantage of ERA for balancing transformation makes full-state outputs tractable. Further, ERA enables balancing despite stiffness, by eliminating computation of balancing modes and adjoint simulations. As a demonstrative example, we create balanced reduced-order models (ROMs) for a one-dimensional reactive flow with pressure forcing, where the stiffness introduced by the chemical source term is extreme (condition number [Formula: see text]), preventing analytical implementation of BT. We investigate the performance of ROMs in prediction of dynamics with unseen forcing inputs and demonstrate stability and accuracy of balanced ROMs in truly predictive scenarios, whereas without ERA, proper orthogonal decomposition-Galerkin and least-squares Petrov-Galerkin projections fail to represent the true dynamics. We show that after the initial transients under unit impulse forcing, the system undergoes lightly damped oscillations, which magnifies the influence of sampling properties on predictive performance of the balanced ROMs. We propose an output domain decomposition approach and couple it with tangential interpolation to resolve sharp gradients at reduced computational costs. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Physics-based sizing functions for ocean-to-estuary shallow-water models

Modeling ocean processes requires a discrete representation of the system geometry, which itself contains multiple degrees of freedom, especially for systems of large geographical scale. This paper assesses the unstructured meshing requirements for a large-scale, astronomic-tide model domain of the western North Atlantic Ocean, Gulf of Mexico and Caribbean Sea with telescopic focus of the South Atlantic Bight (United States southeastern seaboard) and its estuaries. The methodologies of localized truncation error analysis (LTEA), which account only for the model interior and LTEA plus complex derivatives (+CD), which account for the model interior and boundary, are exploited due to their foundations stemming directly from tidal physics. LTEA and LTEA+CD are applied for the M2 tidal harmonics resulting from linear and nonlinear solutions of the ADCIRC (advanced circulation) model based on a highly resolved uniform mesh. The resultant target element size distributions are interrogated distinctively for the inshore (estuarine) region of the South Atlantic Bight, the inlet-punctuated coastline, the shelf-slope-rise triad and deep ocean, and the open-ocean boundary. The relatively large size of the LTEA computational molecule makes it the suitable choice for the deep ocean, offshore and shelf regions where the tidal dynamics are predominantly linear and smoothly changing in space. LTEA+CD is the logical choice for the shallows and estuaries because of the spatial coverage afforded by the combined utility of interior and boundary estimators (computational molecules), as well as its accounting for the nonlinearities in the evaluation of local truncation error. The combination of the two methods enables an almost complete coverage of the domain for generating spacing requirements, with LTEA+CD extending beyond the interior-only definition of LTEA to capture the inshore regions and model boundaries. Implementing the approach with LTEA and LTEA+CD to target finer resolution in areas with sharper gradients in the hydrodynamics, while relaxing resolution in areas with smoother hydrodynamics, resulted in a mesh with a far-reduced number of elements (∼1.3M) relative to an overly resolved uniform mesh (>10M elements) that is comparable in accuracy with ±5% tidal amplitude error for 99% of the domain.

Volume imaging of aspect sensitivity in VHF radar backscatters: first results inferred from the Advanced Indian MST radar (AIR)

ABSTRACT Very-high frequency (VHF) radar echoes are known to be aspect sensitive in general, which could be either the manifestation of a thin stable layer providing sharp refractive index gradient or shear-driven steep layer structures. The present study for the first time shows the volume imaging of aspect sensitivity in the recently upgraded Advanced Indian Mesosphere Stratosphere Troposphere (MST) radar (AIR). The present configuration of radar can provide a better understanding of three-dimensional structures of turbulence and instabilities, which was not possible when the radar was operated in two orthogonal planes. Both isotropic and anisotropic layers/turbulence are observed at various height levels over the radar volume. Enhanced aspect sensitivity is observed between 6 and 9 km, which is attributed to the presence of dynamic instability arising due to strong wind shear and atmospheric stability. All the scatterers tend to become isotropic when both the square of wind shear and stability parameters are above 0.25 × 10−3 s−2. Due to the high aspect sensitivity, there may be an underestimate of 5–10% in wind velocity measurements, and it is higher for the low off-zenith beam angle.

An artificial damping method for total Lagrangian SPH method with application in biomechanics

Introducing artificial damping into the momentum equation to enhance the numerical stability of the smoothed particle hydrodynamics (SPH) method for large strain dynamics has been well established, however, implementing appropriate damping term in the constitutive model as an alternative stabilization strategy in the context of the SPH method is still not investigated. In this paper, we present a simple stabilization procedure by introducing an artificial damping term into the second Piola–Kirchhoff stress to enhance the numerical stability of the SPH method in total Lagrangian formulation. The key idea is to reformulate the constitutive equation by adding a Kelvin–Voigt (KV) type damper with a scaling factor imitating a von Neumann–Richtmyer type artificial viscosity to alleviate the spurious oscillation in the vicinity of sharp spatial gradients. The proposed method is shown to effectively eliminate the appearance of spurious non-physical instabilities and easy to be implemented into the original total Lagrangian SPH formulation. After validating the numerical stability and accuracy of the present method through a set of benchmark tests with very challenging cases, we demonstrate its applications and potentials in the field of biomechanics by simulating the deformation of complex stent structures and the electromechanical excitation–contraction of the realistic left ventricle.

Increasing footprint of climate warming on flash droughts occurrence in Europe

Flash droughts are caused by a rapid depletion of soil moisture, and they severely affect vegetation growth and agricultural production. Notwithstanding the growing importance of flash droughts under the warming climate, drivers of flash droughts across the Europe are not well understood. Here we estimate the changes in flash droughts characteristics across Europe using the ERA5 reanalysis dataset for 1950–2019 period. We find a substantial increase in the frequency and spatial extent of flash droughts across Europe (with 79% of the total area) during the growing season with at-least one fourth of domain showing two-fold increase in the recent decades. Increased occurrence of flash drought is largely attributed to frequent occurrence of warmer and drier compound extremes, with a sharp gradient of changes being noticed in Mediterranean and Central European regions. Compound meteorological extremes causing the flash drought events across Europe are pre-dominantly driven by the recent climate warming. With unabated greenhouse gas emissions and current pace of climate warming, Europe is likely to face an increased occurrence of flash droughts, requiring prompt response for effective drought adaptation and management strategies.

Junction conditions and sharp gradients in generalized coupling theories

In this article, we develop the formalism for singular hypersurfaces and junction conditions in generalized coupling theories using a variational approach. We then employ this formalism to examine the behavior of sharp matter density gradients in generalized coupling theories. We find that such gradients do not necessarily lead to the pathologies present in other theories of gravity with auxiliary fields. A detailed example, based on a simple instance of a generalized coupling theory called the MEMe model, is also provided. In the static case, we show that sharp boundaries do not generate singularities in the dynamical frame despite the presence of an auxiliary field. Instead, in the case of a collapsing spherical density distribution with a general profile an additional force compresses over-densities and expands underdensities. These results can also be used to deduce additional constraints on the parameter of this model.

An Efficient Localized Nonlinear Solver for Simulating Natural Depletion of Unconventional Reservoirs with Complex Fracture Networks

Summary Simulating unconventional reservoirs efficiently and accurately poses a big challenge. Transient flow can last for a long period and sharp solution gradients appear because of the severe permeability contrast between fracture and matrix. Although high-resolution well and fracture models are required to achieve sufficient accuracy, they are computationally too demanding for field models with many hydraulic-fracture stages. This paper aims to develop a nonlinear solution method, which is applicable to discrete fracture models (DFMs) for unconventional reservoirs. The localization algorithm takes advantage of solution locality on timestep and Newton iteration levels. We study the new method through multiple natural-depletion cases containing discrete fractures and compositional models. It is found that a large extent of solution locality displays over iterations as well as timesteps. The developed nonlinear solver exhibits outstanding simulation efficiency compared with a standard solver. A significant speedup is achieved by focusing computations to the locales that undergo considerable solution changes. Moreover, nonlinear convergence performance is maintained, with no degradation of the solution results.

Coaxial-Injector Surrogate Modeling Based on Reynolds-Averaged Navier–Stokes Simulations Using Deep Learning

Facing the need to increase the accuracy of rocket engine design tools, the present work introduces an innovative methodology for the design and optimization of rocket engine combustion chambers using numerical simulations and deep learning. An experimental test case of a single coaxial injector is taken as a reference point, and a design of experiments is generated by varying nine parameters (geometrical and operative conditions). Reynolds-averaged Navier–Stokes simulations are carried out to generate the data set. The data are used to train surrogate models of different fidelities, from low-dimensional outputs (zero-dimensional and one-dimensional) toward the two-dimensional temperature field. Attention is given on the selection of the proper machine learning technique. For low-dimensional outputs, results show that deep neural networks outperform other standard machine learning tools, namely, radial basis function and kriging. Regarding high-dimensional outputs, convolutional neural networks with gradient-based loss functions are found effective to capture the large and smooth temperature variations, as well as the thin and sharp temperature gradients at the flame front. Eventually, the models are used in the framework of an optimization problem. Results highlight the benefits of new design and optimization tools based on deep learning, capable of real-time predictions of complex flowfields.

Hydrodynamic model for independent cold and thermo-mineral twin springs in a stratified continental karst aquifer, Camou, Arbailles Massif, Pyrénées, France

The Camou springs (Arbailles Massif, French Western Pyrenees) display an unusual close association of a typically cold karstic spring that drains the Urgonian western limb of the Arbailles, and a thermo-mineral spring (33.5°C; salinity 17.7 g/L). The latter gains its mineralization at the contact of Triassic evaporites mainly through a deep loop in the Apanicé syncline. The fast upflow of this deep water occurs at the cross of large active lines (the North-Pyrenean thrust located at depth, and the Saison transverse fault). Cave diving in the nearby Maddalen Cave allowed reaching the phreatic passage at the origin of the cold spring, which however also crosses the thermal body in the third sump (S3). Both water bodies are separated by a sharp thermocline. 6 pressure-temperature dataloggers were placed in both water bodies along the thermocline for 6 months. The dataloggers located downstream on either side of the thermocline show at the beginning of flood first a rise of the thermal body, then an invasion of the whole phreatic passage by the cold floodwater, controlled by head pressure changes in the karst aquifer. From observation of these mechanisms, we deduce a hydrodynamic model with a warm plume rising into the cold aquifer, without significant mixing. Such independence of water bodies is explained by the decrease of turbulent rate at the interface, due to the sharp density gradient. The relative absence of mixing does not actually require independent “watertight” routes, both water bodies can thus coexist even in the same conduit. This model locally implies the existence of unknown secondary passages close to the spring, which allow an independent draining of each water body toward separate outlets during low stage. Such type of stratified aquifer linked to density differences is common in coastal karst (Florida, French Calanques…), in the continental evaporite karst (Schlotten of the Harz in Germany, Kungur karst in Ural…), but remains poorly identified in continental carbonate karst areas, mainly because of the difficulty of access. Together with the Mescla spring in French Alpes-Maritimes, the Camou twin springs discharging in the same porch are an outstanding example, allowing a direct study of the stratification and the dynamic of highly contrasted water bodies.

Meteorological Conditions That Promote Persistent Contrails

Climate-impacting contrails need ice (super-)saturation to persist longer than a few minutes. However, this simple criterion cannot be easily applied for the prediction of persistent contrails. The current weather forecast models, which lack humidity data for assimilation in the upper troposphere, have difficulties coping with the enormous variability and sharp gradients in the relative humidity field. Thus, ice supersaturation, which is an extremal state of relative humidity, is hard to forecast at a precise location and time to allow contrail-avoiding flight routing. In this paper, we investigate the possibility of using dynamical proxy variables for improved contrail prediction. This idea is guided by the fact that the probability of ice supersaturation differs in different dynamical regimes. Therefore, we determine probability distributions of temperature, water vapour concentration, vertical velocity, divergence, relative and potential vorticity, geopotential height, and lapse rate conditioned on three situations: (a) contrail persistence not possible; (b) contrail persistence possible; and (c) strongly warming persistent contrails possible. While the atmospheric variables are taken from reanalysis data, the conditions (a–c) are based on airborne measurement data and radiation quantities from the reanalysis. It turns out that the vorticity variables, and in particular geopotential and lapse rate, show quite distinct conditional probabilities, suggesting a possibility to base an improved forecast of persistent contrails not only on the traditional quantities of temperature and relative humidity, but on these dynamical proxies as well. Furthermore, we show the existence of long flight tracks with the formation of strongly warming contrails, which are probably embedded in larger ice-supersaturated regions with conditions that foster such contrails. For forecasting purposes, this is a beneficial property since the humidity forecast is easier on large, rather than small, spatial scales.

Flow Velocity Gradient of Refined Hydrocarbon and Viscous Fluid in A Single Stage Symmetrical Bifurcated Channel

A single stage bifurcated system in its wide spectrum of applications in the petroleum and gas processing industry, chemical industry and civil engineering industry, in the distribution and recovery of processed and unprocessed fluid and materials. Considered in this experimental investigation of the impact of a single stage symmetrical bifurcated system on the flow of viscous fluid and refined hydrocarbon to truly establish a relationship between specific dimensions of symmetrical bifurcated flow channels and the physical properties of various classes of fluid samples. Peanut oil and diesel were selected to respectively represent viscous fluid and refined hydrocarbon, as they are allowed to flow through some selected angles of a single stage symmetrical bifurcated channel until designated volumes of each of the fluid samples are recovered into a recovery beaker. Profile of results presented from the experimental study reveals a trend with a sharp negative velocity gradient for small flow time corresponding to smaller
 recovery volume and a much lower velocity gradient for relatively higher flow time for higher recovery volume for less viscous and dense diesel oil representing refined hydrocarbon and a more viscous and dense peanut oil represented by the trend with a lower negative velocity gradient. The implication of the results obtained is that bifurcated systems offer flow stability in a continuous flow situation, and also observed from the results is the confirmation of the existence of velocity gradient between opposite wall of the branched channels, which further reveals where the flow state will eventually begin to graduate into turbulence. The study can be extended by investigating the impact of two-stage or multiple level bifurcated network system on flow structure and stability, and how the pressure based on the level of fluid in the reservoir is related to the flow velocity and stability.

Distinct Mid-Latitude Eurasian Rossby Wave Trains Preceding Strong and Weak Cold Surges in Southern China

This study investigates the large-scale circulation anomalies of strong and weak cold surges in southern China. Results show that the strong and weak surges are associated with a distinct Rossby wave train across Eurasia projecting onto the Scandinavia (SCAND) and Eurasian (EU) patterns. On one hand, the strong surge is preceded by a positive SCAND pattern and a transition of the EU pattern from its negative to positive phase. The corresponding wave train propagates southeastward from East Asia to the North Pacific. This is associated with a strong intensification of the Siberian high, a deepened East Asian trough, and a sharp meridional pressure gradient between northern and southern China; these advect intense cold air masses from Siberia to southern China. On the other hand, the weak surge is preceded by a negative SCAND pattern and a positive EU pattern. The corresponding wave train does not propagate from East Asia to the North Pacific. This wave train has a weak impact on the East Asian winter monsoon circulation and corresponds to weak cold air advections towards southern China. Overall, the intra-seasonal variation of the Eurasian teleconnection patterns is crucial for a comprehensive understanding of the cold surge in southern China.

Downscaling Satellite-Based Estimates of Ocean Bottom Pressure for Tracking Deep Ocean Mass Transport

Gravimetry measurements from the GRACE and GRACE-Follow-On satellites provide observations of ocean bottom pressure (OBP), which can be differenced between basin boundaries to infer mass transport variability at a given level in the deep ocean. However, GRACE data products are limited in spatial resolution, and conflate signals from many depth levels along steep continental slopes. To improve estimates of OBP variability near steep bathymetry, ocean bottom pressure observations from a JPL GRACE mascon product are downscaled using an objective analysis procedure, with OBP covariance information from an ocean model with horizontal grid spacing of ∼18 km. In addition, a depth-based adjustment was applied to enhance correlations at similar depths. Downscaled GRACE OBP shows realistic representations of sharp OBP gradients across bathymetry contours and strong currents, albeit with biases in the shallow ocean. In validations at intraannual (3–12 month) timescales, correlations of downscaled GRACE data (with depth adjustment) and in situ bottom pressure recorder time series were improved in ∼79% of sites, compared to correlations that did not involve downscaled GRACE. Correlations tend to be higher at sites where the amplitude of the OBP signal is larger, while locations where surface eddy kinetic energy is high (e.g., Gulf Stream extension) are more likely to have no improvement from the downscaling procedure. The downscaling procedure also increases the amplitude (standard deviation) of OBP variability compared to the non-downscaled GRACE at most sites, resulting in standard deviations that are closer to in situ values. A comparison of hydrography-based transport from RAPID with estimates based on downscaled GRACE data suggests substantial improvement from the downscaling at intraannual timescales, though this improvement does not extend to longer interannual timescales. Possible efforts to improve the downscaling technique through process studies and analysis of alongtrack GRACE/GRACE-FO observations are discussed.

Atmosphere-ionosphere coupling induced by volcanoes eruption and dust storms and role of GEC as the agent of geospheres interaction

In this publication we demonstrate how interdependent are our geophysical shells. The large-scale air pollution provided by the intensive volcano eruptions and dust storms from desert areas modify the electric properties of the troposphere creating the low conducting layers and increasing the column resistance for the vertical current flowing between the ionosphere and ground within the Global Electric Circuit (GEC). This modification leads to the local modulation of the ionospheric potential (IP) magnitude and hence to formation of the large-scale irregularities of electron concentration in the ionosphere. To reveal these phenomena, we use the differential GIM TEC mapping procedure showing specific features of the ionosphere reaction. In low latitude and equatorial regions we observe the formation of positive TEC anomalies not only over the area of air pollution but in the magnetically conjugated area as well. We can conclude that these phenomena pose the double error for Precise Point Positioning (PPP). To the danger of aircraft engines destroying is added the increase of navigational PPP errors due to sharp TEC gradients on the borders of the formed large-scale positive irregularities of electron concentration.

Robust Nonequilibrium Edge Currents with and without Band Topology.

We study two-dimensional bosonic and fermionic lattice systems under nonequilibrium conditions corresponding to a sharp gradient of temperature imposed by two thermal baths. In particular, we consider a lattice model with broken time-reversal symmetry that exhibits both topologically trivial and nontrivial phases. Using a nonperturbative Green function approach, we characterize the nonequilibrium current distribution in different parameter regimes. For both bosonic and fermionic systems, we find chiral edge currents that are robust against coupling to reservoirs and to the presence of defects on the boundary or in the bulk. This robustness not only originates from topological effects at zero temperature but, remarkably, also persists as a result of dissipative symmetries in regimes where band topology plays no role. Chirality of the edge currents implies that energy locally flows against the temperature gradient without any external work input. In the fermionic case, there is also a regime with topologically protected boundary currents, which nonetheless do not circulate around all system edges.

Global GNSS-RO Electron Density in the Lower Ionosphere

Lack of instrument sensitivity to low electron density (Ne) concentration makes it difficult to measure sharp Ne vertical gradients (four orders of magnitude over 30 km) in the D/E-region. A robust algorithm is developed to retrieve global D/E-region Ne from the high-rate GNSS radio occultation (RO) data, to improve spatiotemporal coverage using recent SmallSat/CubeSat constellations. The new algorithm removes F-region contributions in the RO excess phase profile by fitting a linear function to the data below the D-region. The new GNSS-RO observations reveal many interesting features in the diurnal, seasonal, solar-cycle, and magnetic-field-dependent variations in the Ne morphology. While the D/E-region Ne is a function of solar zenith angle (χ), it exhibits strong latitudinal variations for the same χ with a distribution asymmetric about noon. In addition, large longitudinal variations are observed along the same magnetic field pitch angle. The summer midlatitude Ne and sporadic E (Es) show a distribution similar to each other. The distribution of auroral electron precipitation correlates better with the pitch angle from the magnetosphere than from one at 100 km. Finally, a new TEC retrieval technique is developed for the high-rate RO data with a top reaching at least 120 km. For better characterization of the E- to F-transition in Ne and more accurate TEC retrievals, it is recommended to have all GNSS-RO acquisition routinely up to 220 km.

Time-discontinuous state-based peridynamics for elasto-plastic dynamic fracture problems

A time-discontinuous state-based peridynamic method (SBPD-TD) is developed in this paper to model the transient problems of wave and crack propagation in solids. In this method, the non-local deformation gradient is derived in a consistent way using the least square method, so that the classical elasto-plastic constitutive relationship can be directly adopted in the SBPD based on the correspondence material modeling. Then the displacement field and the velocity field are interpolated independently in the temporal domain respectively, in which the jump terms representing the discontinuities of variables are introduced between the adjacent time steps. After that, an integral weak form in the temporal domain of the spatially discrete governing equations is constructed and the basic formula of SBPD-TD is derived. The above-mentioned characteristics ensure that the SBPD-TD can accurately capture the sharp gradient characteristics in the propagation of elastic and plastic stress waves and effectively control the spurious numerical oscillations. Several representative numerical examples show that the SBPD-TD provides more accurate and satisfactory results when compared with the conventional SBPD, and has broad applications to predict the contact impact fracture behaviors of various materials.

Polyphase growth history of peritectic garnet from a granite: Trace-element zonation, Lu-Hf ages and their significance for the duration of granite-forming processes

Determining the ages and duration of high-grade metamorphic episodes during orogenesis and concomitant melt extraction events is pivotal in reconstructing and quantifying the chemical and petrological evolution of the Earth's continental crust over time. Major and trace elements combined with multichronometric garnet data from the Pan-African Damara belt of Namibia place absolute time constraints on high-grade regional metamorphism and melting in the lower crust that led to the production of widespread granites emplaced in the upper crust. Some of these granites contain inherited peritectic garnet with inferred igneous rims that record their genesis and evolution from formation by partially melting in the deep crust to the emplacement of their host rock at shallower levels. Particular characteristics of these garnets are textural and chemical disequilibrium features such as resorption, local development of partly euhedral crystal faces and occurrence of quartz-feldspar stringers in the rim of the garnet and trace element abundances with sharp concentration gradients. These features document a previous melting event followed by garnet crystallization in an ascending melt. With the exception of the outermost rim, which records late-stage diffusional features, the garnet crystals have constant Mg and Mn concentrations. In contrast, Ca shows pronounced chemical zoning. Trace elements (HREE, LREE, Zr, Sr, Y) are depleted in the core and enriched in the rim region indicating disequilibrium conditions during garnet growth. A pseudosection calculated with the garnet rim composition and the composition of the host granite yields P-T conditions of 0.62 GPa and 760 °C for the formation of the rim. This result is consistent with a P-T estimate of 0.68 GPa and 760 °C derived from the conventional garnet-biotite thermobarometer. Regional metamorphic conditions were above the cordierite-andalusite stability field with estimated conditions of ~550-600 °C and 0.4–0.5 GPa. This P-T regime indicates that the garnet originally crystallized at deep crustal levels. Diffusion modelling suggests that the observed concentration gradient of Sm in garnet is consistent with a maximum duration of ca. 0.5–1.5 Ma for the inferred melting interval during peak metamorphism in the lower crust. High-precision multi-point Lu-Hf garnet-whole rock data yield dates of 522.0 ± 1.0 and 522.3 ± 0.9 Ma for two samples. Because of presumed short intervals between melting in the lower crust and intrusion of the granite, these dates are interpreted as the age of the melting event during high-grade metamorphism and are temporally indistinguishable from the crystallization interval. Thus, dehydration melting in the deeper crust during peak metamorphism produced peritectic garnet that was incorporated into the granitic melt followed by subsequent new growth of igneous garnet around peritectic cores during intrusion. Previously obtained Sm-Nd garnet whole rock ages are 20–25 Ma younger and likely represent cooling ages. This study shows that granites enriched in mafic components such as garnet and biotite may intrude as mixtures of melt and the peritectic assemblage produced by incongruent melting involving dehydration-melting of biotite in the lower crust. Melting episodes represent short-lived processes relative to the absolute duration of orogenic cycles.

Hydrothermal Factors Influence on Spatial-Temporal Variation of Evapotranspiration-Precipitation Coupling over Climate Transition Zone of North China

As a land–atmosphere coupling “hot spot”, the northern China climate transition zone has a sharp spatial gradient of hydrothermal conditions, which plays an essential role in shaping the spatial and temporal pattern of evapotranspiration-precipitation coupling, but whose mechanisms still remain unclear. This study analyzes the spatial and temporal variation in land–atmosphere coupling strength (CS) in the climate transitional zone of northern China and its relationship with soil moisture and air temperature. Results show that CS gradually transitions from strong positive in the northwest to negative in the southeast and northeast corners. The spatial distribution of CS is closely related to climatic hydrothermal conditions, where soil moisture plays a more dominant role: CS increases first, and then decreases with increasing soil moisture, with the threshold of soil moisture at 0.2; CS gradually transitions from positive to negative at soil moisture between 0.25 and 0.35; CS shows an exponential decreasing trend with increasing temperature. In terms of temporal variation, CS is strongest in spring and weakens sequentially in summer, autumn, and winter, and has significant interdecadal fluctuations. The trend in CS shifts gradually from significantly negative in the west to a non-significant positive in the east. Soil moisture variability dominates the intra-annual variability of CS in the study regions, and determines the interannual variation of CS in arid and semi-arid areas. Moreover, the main reason for the positive and negative spatial differences in CS in the study area is the different driving regime of evapotranspiration (ET). ET is energy-limited in the southern part of the study area, leading to a positive correlation between ET and lifting condensation level (LCL), while in most of the northern part, ET is water-limited and is negatively correlated with LCL; LCL has a negative correlation with P across the study area, thus leading to a negative ET-P coupling in the south and a positive coupling in the north.

Artificial neural networks for solving elliptic differential equations with boundary layer

In this paper, we consider the artificial neural networks for solving the elliptic differential equation with boundary layer, in which the gradient of the solution changes sharply near the boundary layer. The solution of the boundary layer problems poses a huge challenge to both traditional numerical methods and artificial neural network methods. By theoretically analyzing the changing rate of the weights of the first hidden layer near the boundary layer, a mapping strategy is added in traditional neural network to improve the convergence of the loss function. Numerical examples are carried out for the 1D and 2D convection–diffusion equation with boundary layer. The results demonstrate that the modified neural networks significantly improve the ability in approximating the solutions with sharp gradient.