arXiv subjects

Yuki Yasuda

Publications and source records attributed to Yuki Yasuda.

At least 19 recordsLinked to original sources

arXiv2026

Quantifying internal variability in large-ensemble climate data with the Wasserstein distance

Quantifying forced and internal variability in climate data is fundamental to detecting the climate change signal and assessing uncertainty in climate projections. We propose a metric that quantifies the relative magnitude of internal variability in single-model initial-condition large ensembles (SMILEs). We measure forced and internal variability by two 1-Wasserstein distances, a form of optimal transport cost. The proposed metric is the distance for internal variability divided by the sum of the two. Computing this ratio requires only sorting the samples and differencing the resulting quantiles, with no additional parameters. The metric reflects the entire distribution shape and applies to non-Gaussian variables, because the 1-Wasserstein distance quantifies the difference between any two probability distributions. We validate the metric with synthetic climate data from Gaussian, uniform, and lognormal distributions. Unlike the two existing metrics, the proposed metric gives stable estimates irrespective of the distribution shape and the presence of outliers, provided that the ensemble has about 40 members or more. We then apply the proposed and existing metrics to the 2 m air temperature and total precipitation of the Community Earth System Model Large Ensemble (CESM-LE) under two different forcing scenarios. All the metrics indicate that the relative contribution of internal variability decreases as the forcing increases, but the proposed metric shows this response most clearly. The 1-Wasserstein distance thus provides a simple and useful tool for analyzing large-ensemble datasets.

physics.ao-ph↗

arXiv2026

A Stochastic-Thermodynamic Constraint on the Seasonal Phase Locking of the El Ni\~no-Southern Oscillation

We investigate the seasonal phase locking of the El Ni\~no-Southern Oscillation (ENSO) in a linear stochastic recharge oscillator (SRO), a damped oscillator with additive noise and a time-dependent growth rate. Phase locking is reflected in the seasonality of the variance of the sea surface temperature anomaly (SSTA). In general, energy drives such a change, whereas entropy governs whether it occurs; phase locking is thus subject to both an energy- and an entropy-based constraint. We quantify this entropy-based constraint using a thermodynamic uncertainty relation (TUR), a fundamental inequality in stochastic thermodynamics. The TUR constrains the tendency of the SSTA variance by the partial entropy production rate, which is dominated by the ratio of forward and backward transition probabilities and quantifies the irreversibility of SSTA transitions. The growth rate governs this irreversibility: its extrema occur in boreal autumn and late winter, and the entropy production rate peaks at both times. These peaks relax the TUR constraint on the tendency of the SSTA variance, so that the variance itself can peak in boreal winter, consistent with observed ENSO phase locking. Conversely, when irreversibility is insufficient, ENSO cannot grow or decay. If this irreversibility were interpreted as dissipated energy, the constraint on ENSO growth and decay would require this dissipation to be exported from the equatorial Pacific. A more realistic model is needed to test this hypothesis and to further explore the physical connection between entropy and dissipated energy.

physics.ao-ph↗

arXiv2026

Uncertainty-Aware Spatiotemporal Super-Resolution Data Assimilation with Diffusion Models

Data assimilation (DA) improves prediction of chaotic systems by combining model forecasts with sparse, noisy observations. Many DA methods are inherently probabilistic, but accurate probabilistic DA is often computationally expensive because it requires repeated high-resolution (HR) forecasts and large ensembles. In this study, we develop DiffSRDA, a probabilistic spatiotemporal super-resolution data assimilation framework based on denoising diffusion models, and evaluate it on an idealized barotropic ocean jet instability testbed. DiffSRDA is trained offline to generate short HR analysis windows conditioned on (i) a time series of low-resolution (LR) forecast frames and (ii) sparse HR observations. Repeated reverse diffusion sampling then produces an ensemble of HR analyses, providing both point estimates and uncertainty information. Despite relying only on low-cost LR forecasts, DiffSRDA achieves reconstruction quality close to that of an Ensemble Kalman Filter (EnKF) driven by HR forecasts, while improving over deterministic CNN-based SRDA baselines. The sampled ensemble also yields physically meaningful uncertainty patterns, with spread concentrated in dynamically active regions similarly to EnKF. A key practical result is that accurate base DiffSRDA cycling does not require long reverse chains: most of the full-chain accuracy is retained with only a few reverse steps, making diffusion-based SRDA practical for repeated cycling. Finally, by exploiting the score-based structure of diffusion sampling, we demonstrate training-free observation-consistency guidance for deployment-time sensor-layout shifts, enabling improved use of changed observation configurations without retraining. Overall, diffusion models provide a practical, uncertainty-aware, and computationally efficient approach for spatiotemporal SRDA in chaotic fluid flows.

physics.flu-dyn↗

arXiv2026

Predictor-Driven Diffusion for Spatiotemporal Generation

Multiscale spatial structure complicates temporal prediction because small-scale spatial fluctuations influence large-scale evolution, yet resolving all scales is often intractable. Standard diffusion models do not address this problem effectively since they apply uniform decay across all Fourier modes. We propose Predictor-Driven Diffusion, a framework that combines renormalization-group-based spatial coarse-graining with a path-integral formulation of temporal dynamics. The forward process applies scale-dependent Laplacian damping together with additive noise, producing a hierarchy of coarse-grained fields indexed by diffusion scale $λ$. Training minimizes the Kullback-Leibler divergence between data-induced and predictor-induced path densities, leading to a simple regression loss on temporal derivatives. The resulting predictor captures how eliminated small-scale components statistically influence large-scale evolution. A key insight is that the same predictor provides a path score for reverse-$λ$ sampling, unifying simulation, unconditional generation, and super-resolution in a single framework. Our unified approach is validated through experiments on two multiscale turbulent systems.

physics.flu-dyn↗

arXiv2025

Probabilistic Super-Resolution for Urban Micrometeorology via a Schrödinger Bridge

This study employs a neural network that represents the solution to a Schrödinger bridge problem to perform super-resolution of 2-m temperature in an urban area. Schrödinger bridges generally describe transformations between two data distributions based on diffusion processes. We use a specific Schrödinger-bridge model (SM) that directly transforms low-resolution data into high-resolution data, unlike denoising diffusion probabilistic models (simply, diffusion models; DMs) that generate high-resolution data from Gaussian noise. Low-resolution and high-resolution data were obtained from separate numerical simulations with a physics-based model under common initial and boundary conditions. Compared with a DM, the SM attains comparable accuracy at one-fifth the computational cost, requiring 50 neural-network evaluations per datum for the DM and only 10 for the SM. Furthermore, high-resolution samples generated by the SM exhibit larger variance, implying superior uncertainty quantification relative to the DM. Owing to the reduced computational cost of the SM, our results suggest the feasibility of real-time ensemble micrometeorological prediction using SM-based super-resolution.

physics.ao-ph↗

arXiv2025

Zero-Shot Super-Resolution from Unstructured Data Using a Transformer-Based Neural Operator for Urban Micrometeorology

This study demonstrates that a transformer-based neural operator (TNO) can perform zero-shot super-resolution of two-dimensional temperature fields near the ground in urban areas. During training, super-resolution is performed from a horizontal resolution of 100 m to 20 m, while during testing, it is performed from 100 m to a finer resolution of 5 m. This setting is referred to as zero-shot, since no data with the target 5 m resolution are included in the training dataset. The 20 m and 5 m resolution data were independently obtained by dynamically downscaling the 100 m data using a physics-based micrometeorology model that resolves buildings. Compared to a convolutional neural network, the TNO more accurately reproduces temperature distributions at 5 m resolution and reduces test errors by approximately 33%. Furthermore, the TNO successfully performs zero-shot super-resolution even when trained with unstructured data, in which grid points are randomly arranged. These results suggest that the TNO recognizes building shapes independently of grid point locations and adaptively infers the temperature fields induced by buildings.

physics.ao-ph↗

arXiv2025

Red Giant Winds Driven by Alfvén Waves with Magnetic Diffusion

We investigate the driving mechanism of Alfvén wave-driven stellar winds from red giant stars, Arcturus ($α$ Boo; K1.5 III) and Aldebaran ($α$ Tau; K5 III), with nonideal MHD simulations in 1D super-radially open flux tubes. Since the atmosphere is not fully ionized, upward propagating Alfvénic waves excited by surface convection are affected by ambipolar diffusion. Our fiducial run with the nonideal MHD effect for $α$ Boo gives a time-averaged mass-loss rate, $\dot{M}=3.3\times 10^{-11}M_{\odot}$/yr, which is more than one order of magnitude reduced from the result in the ideal MHD run and nicely explains the observational value. Magnetized hot bubbles with $T\gtrsim 10^6$ K are occasionally present simultaneously with cool gas with a few $10^3$ K in the atmosphere because of the thermal instability triggered by radiative cooling; there coexist fully ionized plasma emitting soft X-rays and molecules absorbing/emitting infrared radiations. The inhomogeneity in the atmosphere also causes large temporal variations in $\dot{M}$ within an individual magnetic flux tube. We also study the effect of magnetic field strength and metallicity, and find that the wind density, and accordingly the mass-loss rate, positively and sensitively depends on both of them through the ambipolar diffusion of Alfvénic waves. The nonideal MHD simulation for $α$ Tau, which is slightly more evolved than $α$ Boo and has weaker magnetic field, results in weaker wind with $\dot{M}=1.5\times 10^{-12}M_{\odot}$/yr with $T\lesssim 10^5$ K throughout the simulation time. However, given the observations implying the presence of locally strong magnetic fields on the surface of $α$ Tau, we also conduct a simulation with a field strength twice as strong. This results in $\dot{M}=2.0\times 10^{-11}M_{\odot}$/yr - comparable to the observed value - with transient magnetized hot bubbles.

astro-ph.SR↗

arXiv2025

Unsupervised Super-Resolution Data Assimilation Using Conditional Variational Autoencoders with Estimating Background Covariances via Super-Resolution

This study proposes a theory of unsupervised super-resolution data assimilation (SRDA) using conditional variational autoencoders (CVAEs). We derive an evidence lower bound for unsupervised learning, showing that our theory is an extension of a traditional data assimilation (DA) method, namely the three-dimensional variational (3D-Var) formalism. In contrast to 3D-Var, our theory exploits the non-locality of super-resolution (SR) to learn background covariances without explicitly imposing them for assimilating distant observations. For linear SR, SR operators serve as background error covariance matrices,whereas for nonlinear SR, error backpropagation through SR neural networks induces covariance structures in inference. SRDA can naturally be realized with CVAEs because the loss function for CVAEs is generally an evidence lower bound. By incorporating the SR neural network into the CVAE, the encoder estimates the high-resolution (HR) analysis from HR observations and low-resolution forecasts. The decoder acts as the observation operator by reconstructing the HR observations from the estimated HR analysis. The effectiveness of SRDA was evaluated through numerical experiments using an idealized barotropic ocean jet system. Compared to inference with an ensemble Kalman filter, SRDA demonstrated superior accuracy in HR inference. SRDA was also computationally efficient because it does not require HR numerical integration or ensemble calculations. The findings of this study provide a theoretical basis for integrating SR and DA, which will stimulate further research in this direction.

physics.ao-ph↗

arXiv2025

Two-Stage Super-Resolution Simulation Method of Three-Dimensional Street-Scale Atmospheric Flows for Real-Time Urban Micrometeorology Prediction

A two-stage super-resolution simulation method is proposed for street-scale air temperature and wind velocity, which considerably reduces computation time while maintaining accuracy. The first stage employs a convolutional neural network (CNN) to correct large-scale flows above buildings in the input low-resolution simulation results. The second stage uses another CNN to reconstruct small-scale flows between buildings from the output of the first stage, resulting in high-resolution inferences. The CNNs are trained using high-resolution simulation data for the second stage and their coarse-grained version for the first stage as the ground truth, where the high-resolution simulations are conducted independently of the low-resolution simulations used as input. This learning approach separates the spatial scales of inference in each stage. The effectiveness of the proposed method was evaluated using micrometeorological simulations in an actual urban area around Tokyo Station in Japan. The super-resolution simulation successfully inferred high-resolution atmospheric flows, reducing errors by approximately 50% compared to the low-resolution simulations. Furthermore, the two-stage approach enabled localized high-resolution inferences, reducing GPU memory usage to as low as 12% during training. The total wall-clock time for 60-min predictions was reduced to 6.83 min, which was 3.32% of the high-resolution simulation time.

physics.ao-ph↗

arXiv2025

A Mechanism of Stochastic Synchronization in the Climate System: an Interpretation of the Boundary Current Synchronization as a Maxwell's Demon

This study has applied information thermodynamics to a bivariate linear stochastic differential equation (SDE) that describes a synchronization phenomenon of sea surface temperatures (SSTs) between the Gulf Stream and the Kuroshio Current, which is referred to as the boundary current synchronization (BCS). Information thermodynamics divides the entire system fluctuating with stochastic noise into subsystems and describes the interactions between these subsystems from the perspective of information transfer. The SDE coefficients have been estimated through regression analysis using observational and numerical simulation data. In the absence of stochastic noise, the solution of the estimated SDE shows that the SSTs relax toward zero without oscillation. The estimated SDE can be interpreted as a Maxwell's demon system, with the Gulf Stream playing the role of the "Particle" and the Kuroshio Current playing the role of the "Demon." The Gulf Stream forces the SST of the Kuroshio Current to be in phase. By contrast, the Kuroshio Current maintains the phase by interfering with the relaxation of the Gulf Stream SST. In the framework of Maxwell's demon, the Gulf Stream is measured by the Kuroshio Current, whereas the Kuroshio Current performs feedback control on the Gulf Stream. When the Gulf Stream and the Kuroshio Current are coupled in an appropriate parameter regime, synchronization is realized with atmospheric and oceanic fluctuations as the driving source. This new mechanism, "stochastic synchronization," suggests that such fluctuations can be converted into directional variations in a subsystem of the climate system through synchronization by utilizing information (i.e., information-to-energy conversion).

physics.ao-ph↗

arXiv2023

Spatio-Temporal Super-Resolution Data Assimilation (SRDA) Utilizing Deep Neural Networks with Domain Generalization

Deep learning has recently gained attention in the atmospheric and oceanic sciences for its potential to improve the accuracy of numerical simulations or to reduce computational costs. Super-resolution is one such technique for high-resolution inference from low-resolution data. This paper proposes a new scheme, called four-dimensional super-resolution data assimilation (4D-SRDA). This framework calculates the time evolution of a system from low-resolution simulations using a physics-based model, while a trained neural network simultaneously performs data assimilation and spatio-temporal super-resolution. The use of low-resolution simulations without ensemble members reduces the computational cost of obtaining inferences at high spatio-temporal resolution. In 4D-SRDA, physics-based simulations and neural-network inferences are performed alternately, possibly causing a domain shift, i.e., a statistical difference between the training and test data, especially in offline training. Domain shifts can reduce the accuracy of inference. To mitigate this risk, we developed super-resolution mixup (SR-mixup)--a data augmentation method for domain generalization. SR-mixup creates a linear combination of randomly sampled inputs, resulting in synthetic data with a different distribution from the original data. The proposed methods were validated using an idealized barotropic ocean jet with supervised learning. The results suggest that the combination of 4D-SRDA and SR-mixup is effective for robust inference cycles. This study highlights the potential of super-resolution and domain-generalization techniques, in the field of data assimilation, especially for the integration of physics-based and data-driven models.

physics.ao-ph↗

arXiv2023

Super-Resolution of Three-Dimensional Temperature and Velocity for Building-Resolving Urban Micrometeorology Using Physics-Guided Convolutional Neural Networks with Image Inpainting Techniques

Atmospheric simulations for urban cities can be computationally intensive because of the need for high spatial resolution, such as a few meters, to accurately represent buildings and streets. Deep learning has recently gained attention across various physical sciences for its potential to reduce computational cost. Super-resolution is one such technique that enhances the resolution of data. This paper proposes a convolutional neural network (CNN) that super-resolves instantaneous snapshots of three-dimensional air temperature and wind velocity fields for urban micrometeorology. This super-resolution process requires not only an increase in spatial resolution but also the restoration of missing data caused by the difference in the building shapes that depend on the resolution. The proposed CNN incorporates gated convolution, which is an image inpainting technique that infers missing pixels. The CNN performance has been verified via supervised learning utilizing building-resolving micrometeorological simulations around Tokyo Station in Japan. The CNN successfully reconstructed the temperature and velocity fields around the high-resolution buildings, despite the missing data at lower altitudes due to the coarseness of the low-resolution buildings. This result implies that near-surface flows can be inferred from flows above buildings. This hypothesis was assessed via numerical experiments where all input values below a certain height were made missing. This research suggests the possibility that building-resolving micrometeorological simulations become more practical for urban cities with the aid of neural networks that enhance computational efficiency.

physics.ao-ph↗

arXiv2022

Rotationally Equivariant Super-Resolution of Velocity Fields in Two-Dimensional Fluids Using Convolutional Neural Networks

This paper investigates the super-resolution (SR) of velocity fields in two-dimensional fluids from the viewpoint of rotational equivariance. SR refers to techniques that estimate high-resolution images from those in low resolution and has lately been applied in fluid mechanics. The rotational equivariance of SR models is defined as the property in which the super-resolved velocity field is rotated according to a rotation of the input, which leads to the inference covariant to the orientation of fluid systems. Generally, the covariance in physics is related to symmetries. To clarify a relationship to symmetries, the rotational consistency of datasets for SR is newly introduced as the invariance of pairs of low- and high-resolution velocity fields with respect to rotation. This consistency is sufficient and necessary for SR models to acquire rotational equivariance from large datasets with supervised learning. Such a large dataset is not required when rotational equivariance is imposed on SR models through weight sharing of convolution kernels as prior knowledge. Even if a fluid system has rotational symmetry, this symmetry may not carry over to a velocity dataset, which is not rotationally consistent. This inconsistency can occur when the rotation does not commute with the generation of low-resolution velocity fields. These theoretical suggestions are supported by the results from numerical experiments, where two existing convolutional neural networks (CNNs) are converted into rotationally equivariant CNNs and the inferences of the four CNNs are compared after the supervised training.

physics.flu-dyn↗

arXiv2021

Super-Resolution of Near-Surface Temperature Utilizing Physical Quantities for Real-Time Prediction of Urban Micrometeorology

The present paper proposes a super-resolution (SR) model based on a convolutional neural network and applies it to the near-surface temperature in urban areas. The SR model incorporates a skip connection, a channel attention mechanism, and separated feature extractors for the inputs of temperature, building height, downward shortwave radiation, and horizontal velocity. We train the SR model with sets of low-resolution (LR) and high-resolution (HR) images from building-resolving large-eddy simulations (LESs) in a city, where the horizontal resolutions of LR and HR are 20 and 5 m, respectively. The generalization capability of the SR model is confirmed with LESs in another city. The estimated HR temperature fields are more accurate than those of the bicubic interpolation and image SR model that takes only the temperature as its input. Except for the temperature input, the building height is the most important to reconstruct the HR temperature and enables the SR model to reduce errors in temperature near building boundaries. The SR model considers the appropriate boundary for each building from its height information. The analysis of attention weights indicates that the importance of the building height increases as the downward shortwave radiation becomes larger. The contrast between sun and shade is strengthened with the increase in solar radiation, which may affect the temperature distribution. The short inference time suggests the potential of the proposed SR model to facilitate a real-time HR prediction in metropolitan areas by combining it with an LR building-resolving LES model.

physics.ao-ph↗

arXiv2019

X-ray pumping of the Th-229 nuclear clock isomer

Thorium-229 is a unique case in nuclear physics: it presents a metastable first excited state Th-229m, just a few electronvolts above the nuclear ground state. This so-called isomer is accessible by VUV lasers, which allows transferring the amazing precision of atomic laser spectroscopy to nuclear physics. Being able to manipulate the Th-229 nuclear states at will opens up a multitude of prospects, from studies of the fundamental interactions in physics to applications as a compact and robust nuclear clock. However, direct optical excitation of the isomer or its radiative decay back to the ground state has not yet been observed, and a series of key nuclear structure parameters such as the exact energies and half-lives of the low-lying nuclear levels of Th-229 are yet unknown. Here we present the first active optical pumping into Th-229m. Our scheme employs narrow-band 29 keV synchrotron radiation to resonantly excite the second excited state, which then predominantly decays into the isomer. We determine the resonance energy with 0.07 eV accuracy, measure a half-life of 82.2 ps, an excitation linewidth of 1.70 neV, and extract the branching ratio of the second excited state into the ground and isomeric state respectively. These measurements allow us to re-evaluate gamma spectroscopy data that have been collected over 40~years.

nucl-ex↗

arXiv2018

Towards an energy measurement of the internal conversion electron in the de-excitation of the Th-229 isomer

The first excited isomeric state of Th-229 has an exceptionally low energy of only a few eV and could form the gateway to high-precision laser spectroscopy of nuclei. The excitation energy of the isomeric state has been inferred from precision gamma spectroscopy, but its uncertainty is still too large to commence laser spectroscopy. Reducing this uncertainty is one of the most pressing challenges in the field. Here we present an approach to infer the energy of the isomer from spectroscopy of the electron which is emitted when the isomer de-excites through internal conversion (IC). The experiment builds on U-233, which decays to Th-229 and populates the isomeric state with a 2% fraction. A film of U-233 is covered by a stopping layer of few-nm thickness and placed between an alpha detector and an electron detector, such that the alpha particle and the IC electron can be detected in coincidence. Retarding field electrodes allow for an energy measurement. In the present design, the signal of the Th-229m IC electrons is masked by low-energy electrons emitted from the surface of the metallic stopping layer. We perform reference measurements with U-232 and U-234 to study systematic effects, and we study various means to reduce the background of low-energy electrons. Our study gives guidelines to the design of an experiment that is capable of detecting the IC electrons and measuring the isomer energy.

nucl-ex↗

arXiv2017

A New Interpretation of Vortex-Split Sudden Stratospheric Warmings in Terms of Equilibrium Statistical Mechanics

Vortex-split sudden stratospheric warmings (S-SSWs) are investigated by using the Japanese 55-year Reanalysis (JRA-55), a spherical barotropic quasi-geostrophic (QG) model, and equilibrium statistical mechanics. The QG model reproduces well the evolution of the composite potential vorticity (PV) field obtained from JRA-55 by considering a time-dependent effective topography given by the composite height field of the 550 K potential temperature surface. The zonal-wavenumber-2 component of the effective topography is the most essential feature required to observe the vortex splitting. The statistical-mechanics theory predicts a large-scale steady state as the most probable outcome of turbulent stirring, and such a state can be computed without solving the QG dynamics. The theory is applied to a disk domain, which is modeled on the north polar cap in the stratosphere. The equilibrium state is obtained by computing the maximum of an entropy functional. In the range of parameters relevant to the winter stratosphere, this state is anticyclonic. By contrast, cyclonic states are quasi-stationary states corresponding to saddle points of the entropy functional. The theoretical calculations are compared with the results of the quasi-static experiment in which the wavenumber-2 topographic amplitude is increased linearly and slowly with time. The results suggest that S-SSWs can be qualitatively interpreted as the transition from the cyclonic quasi-stationary state toward the anticyclonic equilibrium state. The polar vortex splits during the transition toward the equilibrium state. Without any forcing such as radiative cooling, the anticyclonic equilibrium state would be realized sufficiently after an S-SSW.

physics.ao-ph↗

arXiv2016

Dust formation and mass loss around intermediate-mass AGB stars with initial metallicity $Z_{\rm ini} \leq 10^{-4}$ in the early Universe I: Effect of surface opacity on the stellar evolution and dust-driven wind

Dust formation and resulting mass loss around Asymptotic Giant Branch (AGB) stars with initial metallicity in the range of $0 \leq Z_{\rm ini} \leq 10^{-4}$ and initial mass $2\leq M_{\rm ini}/M_{\odot} \leq 5$ are explored by the hydrodynamical calculations of dust-driven wind (DDW) along the AGB evolutionary tracks. We employ the MESA code to simulate the evolution of stars, assuming an empirical mass-loss rate in the post-main sequence phase, and considering the three types of low-temperature opacities (scaled-solar, CO-enhanced, and CNO-enhanced opacities) to elucidate the effect on the stellar evolution and the DDW. We find that the treatment of low-temperature opacity strongly affects the dust formation and resulting DDW; in the carbon-rich AGB phase, the maximum $\dot{M}$ of $M_{\rm ini} \geq$ 3 $M_{\odot}$ star with the CO-enhanced opacity is at least one order of magnitude smaller than that with the CNO-enhanced opacity. A wide range of stellar parameters being covered, a necessary condition for driving efficient DDW with $\dot{M} \ge 10^{-6}$ $M_{\odot}$ yr$^{-1}$ is expressed as the effective temperature $T_{\rm eff} \lesssim 3850$ K and $\log(δ_{\rm C}L/κ_{\rm R} M) \gtrsim 10.43\log T_{\rm eff}-32.33 $ with the carbon excess $δ_{\rm C}$ defined as $ε_{\rm C} - ε_{\rm O}$ and the Rosseland mean opacity $κ_{\rm R}$ in units of cm$^2$g$^{-1}$ in the surface layer, and the stellar mass (luminosity) $M$ $(L)$ in solar units. The derived fitting formulae of gas and dust mass-loss rates in terms of input stellar parameters could be useful for investigating the dust yield from AGB stars in the early Universe being consistent with the stellar evolution calculations.

astro-ph.SR↗