2025

1. Reframing Generative Models for Physical Systems using Stochastic Interpolants

   Anthony Zhou, Alexander Wikner, Amaury Lancelin, Pedram Hassanzadeh, and Amir Barati Farimani

   arXiv:2509.26282 [cs]

   Abs DOI

   Generative models have recently emerged as powerful surrogates for physical systems, demonstrating increased accuracy, stability, and/or statistical fidelity. Most approaches rely on iteratively denoising a Gaussian, a choice that may not be the most effective for autoregressive prediction tasks in PDEs and dynamical systems such as climate. In this work, we benchmark generative models across diverse physical domains and tasks, and highlight the role of stochastic interpolants. By directly learning a stochastic process between current and future states, stochastic interpolants can leverage the proximity of successive physical distributions. This allows for generative models that can use fewer sampling steps and produce more accurate predictions than models relying on transporting Gaussian noise. Our experiments suggest that generative models need to balance deterministic accuracy, spectral consistency, and probabilistic calibration, and that stochastic interpolants can potentially fulfill these requirements by adjusting their sampling. This study establishes stochastic interpolants as a competitive baseline for physical emulation and gives insight into the abilities of different generative modeling frameworks.

2. An Analytical and AI-discovered Stable, Accurate, and Generalizable Subgrid-scale Closure for Geophysical Turbulence

   Karan Jakhar, Yifei Guan, and Pedram Hassanzadeh

   arXiv:2509.20365 [physics]

   Abs DOI

   By combining AI and fluid physics, we discover a closed-form closure for 2D turbulence from small direct numerical simulation (DNS) data. Large-eddy simulation (LES) with this closure is accurate and stable, reproducing DNS statistics including those of extremes. We also show that the new closure could be derived from a 4th-order truncated Taylor expansion. Prior analytical and AI-based work only found the 2nd-order expansion, which led to unstable LES. The additional terms emerge only when inter-scale energy transfer is considered alongside standard reconstruction criterion in the sparse-equation discovery.

3. Hierarchical Implicit Neural Emulators

   Ruoxi Jiang, Xiao Zhang, Karan Jakhar, Peter Y. Lu, Pedram Hassanzadeh, Michael Maire, and Rebecca Willett

   arXiv:2506.04528 [cs]

   Abs DOI

   Neural PDE solvers offer a powerful tool for modeling complex dynamical systems, but often struggle with error accumulation over long time horizons and maintaining stability and physical consistency. We introduce a multiscale implicit neural emulator that enhances long-term prediction accuracy by conditioning on a hierarchy of lower-dimensional future state representations. Drawing inspiration from the stability properties of numerical implicit time-stepping methods, our approach leverages predictions several steps ahead in time at increasing compression rates for next-timestep refinements. By actively adjusting the temporal downsampling ratios, our design enables the model to capture dynamics across multiple granularities and enforce long-range temporal coherence. Experiments on turbulent fluid dynamics show that our method achieves high short-term accuracy and produces long-term stable forecasts, significantly outperforming autoregressive baselines while adding minimal computational overhead.

4. On the Importance of Learning Non‐Local Dynamics for Stable Data‐Driven Climate Modeling: A 1D Gravity Wave‐QBO Testbed

   Hamid A. Pahlavan, Pedram Hassanzadeh, and M. Joan Alexander

   Geophysical Research Letters,

   Abs DOI

   Abstract Model instability remains a core challenge for data‐driven parameterizations, especially those developed with supervised algorithms, and rigorous methods to address it are lacking. Here, by integrating machine learning (ML) theory with climate physics, we demonstrate the importance of learning spatially non‐local dynamics using a 1D quasi‐biennial oscillation model with parameterized gravity waves (GW) as a testbed. While common offline metrics fail to identify shortcomings in learning non‐local dynamics, we show that the receptive field (RF) can identify instability a‐priori. We find that neural network‐based parameterizations, though predicting GW forcings from wind profiles with 99% accuracy, lead to unstable simulations when RFs are too small to capture non‐local dynamics. Additionally, we demonstrate that learning non‐local dynamics is crucial for the stability of a data‐driven spatiotemporal emulator of the zonal wind field. This work underscores the need to integrate ML theory with physics in designing data‐driven algorithms for climate modeling.

5. Can AI weather models predict out-of-distribution gray swan tropical cyclones?

   Y. Qiang Sun, Pedram Hassanzadeh, Mohsen Zand, Ashesh Chattopadhyay, Jonathan Weare, and Dorian S. Abbot

   Proceedings of the National Academy of Sciences,

   Abs DOI

   Predicting gray swan weather extremes, which are possible but so rare that they are absent from the training dataset, is a major concern for AI weather models and long-term climate emulators. An important open question is whether AI models can extrapolate from weaker weather events present in the training set to stronger, unseen weather extremes. To test this, we train independent versions of the AI weather model FourCastNet on the 1979–2015 ERA5 dataset with all data, or with Category 3–5 tropical cyclones (TCs) removed, either globally or only over the North Atlantic or Western Pacific basin. We then test these versions of FourCastNet on 2018–2023 Category 5 TCs (gray swans). All versions yield similar accuracy for global weather, but the one trained without Category 3–5 TCs cannot accurately forecast Category 5 TCs, indicating that these models cannot extrapolate from weaker storms. The versions trained without Category 3–5 TCs in one basin show some skill forecasting Category 5 TCs in that basin, suggesting that FourCastNet can generalize across tropical basins. This is encouraging and surprising because regional information is implicitly encoded in inputs. Given that current state-of-the-art AI weather and climate models have similar learning strategies, we expect our findings to apply to other models. Other types of weather extremes need to be similarly investigated. Our work demonstrates that novel learning strategies are needed for AI models to reliably provide early warning or estimated statistics for the rarest, most impactful TCs, and, possibly, other weather extremes.

6. Predicting Beyond Training Data via Extrapolation versus Translocation: AI Weather Models and Dubai’s Unprecedented 2024 Rainfall

   Y. Qiang Sun, Pedram Hassanzadeh, Tiffany Shaw, and Hamid A. Pahlavan

   arXiv:2505.10241 [physics]

   Abs DOI

   Artificial intelligence (AI) models have transformed weather forecasting, but their skill for gray swan extreme events is unclear. Here, we analyze GraphCast and FuXi forecasts of the unprecedented 2024 Dubai storm, which had twice the training set’s highest rainfall in that region. Remarkably, GraphCast accurately forecasts this event 8 days ahead. FuXi forecasts the event, but underestimates the rainfall, especially at long lead times. GraphCast’s success stems from "translocation": learning from comparable/stronger dynamically similar events in other regions during training via global effective receptive fields. Evidence of "extrapolation" (learning from training set’s weaker events) is not found. Even events within the global distribution’s tail are poorly forecasted, which is not just due to data imbalance (generalization error) but also spectral bias (optimization error). These findings demonstrate the potential of AI models to forecast regional gray swans and opportunity to improve them through understanding the mechanisms behind their successes and limitations.

7. Fourier analysis of the physics of transfer learning for data-driven subgrid-scale models of ocean turbulence

   Moein Darman, Pedram Hassanzadeh, Laure Zanna, and Ashesh Chattopadhyay

   arXiv:2504.15487 [cs]

   Abs DOI

   Transfer learning (TL) is a powerful tool for enhancing the performance of neural networks (NNs) in applications such as weather and climate prediction and turbulence modeling. TL enables models to generalize to out-of-distribution data with minimal training data from the new system. In this study, we employ a 9-layer convolutional NN to predict the subgrid forcing in a two-layer ocean quasi-geostrophic system and examine which metrics best describe its performance and generalizability to unseen dynamical regimes. Fourier analysis of the NN kernels reveals that they learn low-pass, Gabor, and high-pass filters, regardless of whether the training data are isotropic or anisotropic. By analyzing the activation spectra, we identify why NNs fail to generalize without TL and how TL can overcome these limitations: the learned weights and biases from one dataset underestimate the out-of-distribution sample spectra as they pass through the network, leading to an underestimation of output spectra. By re-training only one layer with data from the target system, this underestimation is corrected, enabling the NN to produce predictions that match the target spectra. These findings are broadly applicable to data-driven parameterization of dynamical systems.

8. Semi-analytical eddy-viscosity and backscattering closures for 2D geophysical turbulence

   Yifei Guan and Pedram Hassanzadeh

   arXiv:2504.09670 [physics]

   Abs DOI

   Physics-based closures such as eddy-viscosity and backscattering models are widely used for large-eddy simulation (LES) of geophysical turbulence for applications including weather and climate prediction. However, these closures have parameters that are often chosen empirically. Here, for the first time, we semi-analytically derive the parameters of the Leith and Smagorinsky eddy-viscosity closures and the Jansen-Held backscattering closure for 2D geophysical turbulence. The semi-analytical derivation provides these parameters up to a constant that can be estimated from the turbulent kinetic energy spectrum of a few snapshots of direct numerical simulation (DNS) or other high-fidelity (eddy resolving) simulations, or even obtained from earlier analytical work based on renormalization group. The semi-analytically estimated closure parameters agree with those obtained from online (a-posteriori) learning in several setups of 2D geophysical turbulence in our earlier work. LES with closures that use these parameters can correctly reproduce the key statistics of DNS, including those of the extreme events and interscale energy and enstrophy transfers, and outperform the baselines (dynamic Leith and Smagorinsky and the latter with standard parameter).

9. Propagation and Periodicity of Mars’s Northern Annular Mode Modulates the Dust Cycle

   J. Michael Battalio, Juan M. Lora, Sandro W. Lubis, and Pedram Hassanzadeh

   Geophysical Research Letters,

   Abs DOI

   Abstract We document the propagation of annular modes—zonally symmetric patterns of variability—in Mars’s atmosphere using a reanalysis dataset. Mars’s Northern Annular Mode (MNAM) sees anomalies of zonal‐mean zonal wind emerge near the subtropics and migrate poleward with a period of 150 days, similarly to Earth’s Southern Annular Mode. The mechanism of propagation involves the interaction of the two leading empirical orthogonal functions that define the MNAM. Moreover, the propagation encourages alternating bands of surface wind stress to migrate polewards with a 150‐day period. In addition, a 150‐day periodicity in anomalous column dust optical depth most likely emerges in response to extrema of the MNAM. The combination of the impact of the MNAM’s internally forced periodicity on the surface wind stress and the seasonal cycle may contribute to the inter‐annual variability of global dust events, as suggested by a Monte Carlo estimate that correctly approximates the observed incidence of global dust events. , Plain Language Summary Some of the largest sources of climate variability for Earth and Mars are annular modes. These phenomena represent shifts of the jet stream north and south in time, and are related to atmospheric eddies, clouds, and/or dust (depending on the planet). Earth’s Southern Hemisphere annular mode migrates with time from lower latitudes toward the pole every 150 days, and this timing has previously been shown to be related to precipitation patterns. Remarkably, we find that Mars’s Northern Hemisphere annular mode similarly migrates poleward with a 150 Mars‐day period. The mechanism controlling this timing stems from different types of patterns of variability interacting with themselves and one another, which can be predicted using a pair of simple prognostic equations. Mars’s annular mode relates to surface wind stress that contributes to dust lifting and the amount of dust in the atmosphere, which both exhibit the same 150 Mars‐day periodicity. This period, which is not related to the seasonal cycle, may contribute to the large differences in the timing of global dust events in different Mars years. , Key Points Mars’s Northern Annular Mode (MNAM), diagnosed from a reanalysis dataset, propagates with a 150‐day period, independent of the season Like Earth’s, MNAM arises from feedbacks between the two leading empirical orthogonal functions of anomalous zonal‐mean zonal wind Surface wind stress and dust opacity vary with a matching 150‐day period, suggesting that the MNAM imparts variability on large dust storms

10. Online learning of eddy-viscosity and backscattering closures for geophysical turbulence using ensemble Kalman inversion

    Yifei Guan, Pedram Hassanzadeh, Tapio Schneider, Oliver Dunbar, Daniel Zhengyu Huang, Jinlong Wu, and Ignacio Lopez-Gomez

    arXiv:2409.04985 [physics]

    Abs DOI

    Different approaches to using data-driven methods for subgrid-scale closure modeling have emerged recently. Most of these approaches are data-hungry, and lack interpretability and out-of-distribution generalizability. Here, we use {online} learning to address parametric uncertainty of well-known physics-based large-eddy simulation (LES) closures: the Smagorinsky (Smag) and Leith eddy-viscosity models (1 free parameter) and the Jansen-Held (JH) backscattering model (2 free parameters). For 8 cases of 2D geophysical turbulence, optimal parameters are estimated, using ensemble Kalman inversion (EKI), such that for each case, the LES’ energy spectrum matches that of direct numerical simulation (DNS). Only a small training dataset is needed to calculate the DNS spectra (i.e., the approach is {data-efficient}). We find the optimized parameter(s) of each closure to be constant across broad flow regimes that differ in dominant length scales, eddy/jet structures, and dynamics, suggesting that these closures are {generalizable}. In a-posteriori tests based on the enstrophy spectra and probability density functions (PDFs) of vorticity, LES with optimized closures outperform the baselines (LES with standard Smag, dynamic Smag or Leith), particularly at the tails of the PDFs (extreme events). In a-priori tests, the optimized JH significantly outperforms the baselines and optimized Smag and Leith in terms of interscale enstrophy and energy transfers (still, optimized Smag noticeably outperforms standard Smag). The results show the promise of combining advances in physics-based modeling (e.g., JH) and data-driven modeling (e.g., {online} learning with EKI) to develop data-efficient frameworks for accurate, interpretable, and generalizable closures.

11. Machine Learning for Climate Physics and Simulations

    Ching-Yao Lai, Pedram Hassanzadeh, Aditi Sheshadri, Maike Sonnewald, Raffaele Ferrari, and Venkatramani Balaji

    Annual Review of Condensed Matter Physics,

    Abs DOI

    We discuss the emerging advances and opportunities at the intersection of machine learning (ML) and climate physics, highlighting the use of ML techniques, including supervised, unsupervised, and equation discovery, to accelerate climate knowledge discoveries and simulations. We delineate two distinct yet complementary aspects: ( a ) ML for climate physics and ( b ) ML for climate simulations. Although physics-free ML-based models, such as ML-based weather forecasting, have demonstrated success when data are abundant and stationary, the physics knowledge and interpretability of ML models become crucial in the small-data/nonstationary regime to ensure generalizability. Given the absence of observations, the long-term future climate falls into the small-data regime. Therefore, ML for climate physics holds a critical role in addressing the challenges of ML for climate simulations. We emphasize the need for collaboration among climate physics, ML theory, and numerical analysis to achieve reliable ML-based models for climate applications.

2024

1. Machine learning for the physics of climate

   Annalisa Bracco, Julien Brajard, Henk A. Dijkstra, Pedram Hassanzadeh, Christian Lessig, and Claire Monteleoni

   Nature Reviews Physics,

   Abs DOI

   Climate science has been revolutionized by the combined effects of an exponential growth in computing power, which has enabled more sophisticated and higher-resolution simulations to be made of the climate system, and an exponential increase in observations since the first weather satellite was put in orbit. Big data and associated algorithms, coalesced under the field of machine learning (ML), offer the opportunity to study the physics of the climate system in ways, and with an amount of detail, that were previously infeasible. Additionally, ML can ask causal questions to determine whether one or more variables cause or affect one or more outcomes and improve prediction skills beyond classical limits. Furthermore, when paired with modelling experiments or robust research on model parameterizations, ML can accelerate computations, increasing accuracy and generating very large ensembles with a fraction of the computational cost of traditional systems. In this Review, we outline the accomplishments of ML in climate physics. We discuss how ML has been used to tackle long-standing problems in the reconstruction of observational data, representation of sub-grid-scale phenomena and climate (and weather) prediction. Finally, we consider the benefits and major challenges of exploiting ML in studying complex systems.

2. Volatile atmospheres of lava worlds

   M. Maurice, R. Dasgupta, and P. Hassanzadeh

   Astronomy & Astrophysics,

   Abs DOI

   Context. A magma ocean (MO) is thought to be a ubiquitous stage in the early evolution of rocky planets and exoplanets. During the lifetime of the MO, exchanges between the interior and exterior envelopes of the planet are very efficient. In particular, volatile elements that initially are contained in the solid part of the planet can be released and form a secondary outgassed atmosphere. Aims. We determine trends in the H–C–N–O–S composition and thickness of these secondary atmospheres for varying planetary sizes and MO extents, and the oxygen fugacity of MOs, which provides the main control for the atmospheric chemistry. Methods. We used a model with coupled chemical gas-gas and silicate melt-gas equilibria and mass conservation to predict the composition of an atmosphere at equilibrium with the MO depending on the planet size and the extent and redox state of the MO. We used a self-consistent mass–radius model for the rocky core to inform the structure of the planet, which we combined with an atmosphere model to predict the transit radius of lava worlds. Results. The resulting MOs have potential temperatures ranging from 1415 to 4229 K, and their outgassed atmospheres have total pressures from 3.3 to 768 bar. We find that MOs (especially the shallow ones) on small planets are generally more reduced, and are thus dominated by H 2 -rich atmospheres (whose outgassing is strengthened at low planetary mass), while larger planets and deeper MOs vary from CO to CO 2 –N 2 –SO 2 atmospheres, with increasing {}[f_{}mathrm{O}_2}}]\. In the former case, the low molecular mass of the atmosphere combined with the low gravity of the planets yields a large vertical extension of the atmosphere, while in the latter cases, secondary outgassed atmospheres on super-Earths are likely significantly shrunk. Both N and C are largely outgassed regardless of the conditions, while the S and H outgassing is strongly dependent on the {}[f_{}mathrm{O}_2}}] as well as on the planetary mass and MO extent for the latter. We further use these results to assess how much a secondary outgassed atmosphere may alter the mass–radius relations of rocky exoplanets.

3. Data Imbalance, Uncertainty Quantification, and Transfer Learning in Data‐Driven Parameterizations: Lessons From the Emulation of Gravity Wave Momentum Transport in WACCM

   Y. Qiang Sun, Hamid A. Pahlavan, Ashesh Chattopadhyay, Pedram Hassanzadeh, Sandro W. Lubis, M. Joan Alexander, Edwin P. Gerber, Aditi Sheshadri, and Yifei Guan

   Journal of Advances in Modeling Earth Systems,

   Abs DOI

   Abstract Neural networks (NNs) are increasingly used for data‐driven subgrid‐scale parameterizations in weather and climate models. While NNs are powerful tools for learning complex non‐linear relationships from data, there are several challenges in using them for parameterizations. Three of these challenges are (a) data imbalance related to learning rare, often large‐amplitude, samples; (b) uncertainty quantification (UQ) of the predictions to provide an accuracy indicator; and (c) generalization to other climates, for example, those with different radiative forcings. Here, we examine the performance of methods for addressing these challenges using NN‐based emulators of the Whole Atmosphere Community Climate Model (WACCM) physics‐based gravity wave (GW) parameterizations as a test case. WACCM has complex, state‐of‐the‐art parameterizations for orography‐, convection‐, and front‐driven GWs. Convection‐ and orography‐driven GWs have significant data imbalance due to the absence of convection or orography in most grid points. We address data imbalance using resampling and/or weighted loss functions, enabling the successful emulation of parameterizations for all three sources. We demonstrate that three UQ methods (Bayesian NNs, variational auto‐encoders, and dropouts) provide ensemble spreads that correspond to accuracy during testing, offering criteria for identifying when an NN gives inaccurate predictions. Finally, we show that the accuracy of these NNs decreases for a warmer climate (4 × CO 2 ). However, their performance is significantly improved by applying transfer learning, for example, re‐training only one layer using ∼1% new data from the warmer climate. The findings of this study offer insights for developing reliable and generalizable data‐driven parameterizations for various processes, including (but not limited to) GWs. , Plain Language Summary Scientists increasingly use machine learning methods, especially neural networks (NNs), to improve weather and climate models. However, it can be challenging for an NN to learn rare, large‐amplitude events because they are infrequent in training data. In addition, NNs need to express their confidence (certainty) about a prediction and work effectively across different climates, for example, warmer climates due to increased CO 2 . Traditional NNs often struggle with these challenges. Here, we share insights from emulating known physics (gravity waves) with NNs in a state‐of‐the‐art climate model. We propose specific strategies for effectively learning rare events, quantifying the uncertainty of NN predictions, and making reliable predictions across various climates. For instance, one strategy to address the learning of rare events involves inflating the impact of infrequent events in the training data. We also demonstrate that several methods could be useful in determining the uncertainty of the predictions. Furthermore, we show that NNs trained on simulations of the historical period do not perform as well in warmer climates. We then improve NN performance by employing transfer learning using limited new data from warmer climates. This study provides lessons for developing robust and generalizable approaches for using NNs to improve models in the future. , Key Points Whole Atmosphere Community Climate Model’s orographic, convective, and frontal gravity wave parameterizations are emulated using neural nets to inform future modeling efforts Data imbalance is addressed via resampling and weighted loss; uncertainty quantification via Bayesian, dropout, and variational methods Performance of the neural nets in a warmer climate is improved via transfer learning with ∼1% new data

4. Learning Closed‐Form Equations for Subgrid‐Scale Closures From High‐Fidelity Data: Promises and Challenges

   Karan Jakhar, Yifei Guan, Rambod Mojgani, Ashesh Chattopadhyay, and Pedram Hassanzadeh

   Journal of Advances in Modeling Earth Systems,

   Abs DOI

   Abstract There is growing interest in discovering interpretable, closed‐form equations for subgrid‐scale (SGS) closures/parameterizations of complex processes in Earth systems. Here, we apply a common equation‐discovery technique with expansive libraries to learn closures from filtered direct numerical simulations of 2D turbulence and Rayleigh‐Bénard convection (RBC). Across common filters (e.g., Gaussian, box), we robustly discover closures of the same form for momentum and heat fluxes. These closures depend on nonlinear combinations of gradients of filtered variables, with constants that are independent of the fluid/flow properties and only depend on filter type/size. We show that these closures are the nonlinear gradient model (NGM), which is derivable analytically using Taylor‐series. Indeed, we suggest that with common (physics‐free) equation‐discovery algorithms, for many common systems/physics, discovered closures are consistent with the leading term of the Taylor‐series (except when cutoff filters are used). Like previous studies, we find that large‐eddy simulations with NGM closures are unstable, despite significant similarities between the true and NGM‐predicted fluxes (correlations \textgreater0.95). We identify two shortcomings as reasons for these instabilities: in 2D, NGM produces zero kinetic energy transfer between resolved and subgrid scales, lacking both diffusion and backscattering. In RBC, potential energy backscattering is poorly predicted. Moreover, we show that SGS fluxes diagnosed from data, presumed the “truth” for discovery, depend on filtering procedures and are not unique. Accordingly, to learn accurate, stable closures in future work, we propose several ideas around using physics‐informed libraries, loss functions, and metrics. These findings are relevant to closure modeling of any multi‐scale system. , Plain Language Summary Even in state‐of‐the‐art climate models, the effects of many important small‐scale processes cannot be directly simulated due to limited computing power. Thus, these effects are represented using functions called parameterizations. However, many of the current physics‐based parameterizations have major shortcomings, leading to biases and uncertainties in the models’ predictions. Recently, there has been substantial interest in learning such parameterizations directly from short but very high‐resolution simulations. Most studies have focused on using deep neural networks, which while leading to successful parameterizations in some cases, are hard to interpret and explain. A few more recent studies have focused on another class of machine‐learning methods that discover equations. This approach has resulted in fully interpretable but unsuccessful parameterizations that produce unphysical results. Here, using widely used test cases, we (a) explain the reasons for these unphysical results, (b) connect the discovered equations to well‐known mathematically derived parameterizations, and (c) present ideas for learning successful parameterizations using equation‐discovery methods. Our main finding is that the common loss functions that match patterns representing effects of small‐scale processes are not enough, as important physical phenomena are not properly learned. Based on this, we have proposed a number of physics‐aware metrics and loss functions for future work. , Key Points Subgrid‐scale momentum/heat flux closures discovered using common algorithms are the analytically derivable nonlinear gradient model (NGM) In 2D turbulence/convection, NGM leads to unstable online simulations due to its inability to fully capture key inter‐scale energy transfers We suggest that physics‐informed loss functions, libraries, metrics, and sparsity selections are needed to discover accurate/stable closures

5. Recreating Observed Convection‐Generated Gravity Waves From Weather Radar Observations via a Neural Network and a Dynamical Atmospheric Model

   C. G. Kruse, M. J. Alexander, M. Bramberger, A. Chattopadhyay, P. Hassanzadeh, B. Green, A. Grimsdell, and L. Hoffmann

   Journal of Advances in Modeling Earth Systems,

   Abs DOI

   Abstract Convection‐generated gravity waves (CGWs) transport momentum and energy, and this momentum is a dominant driver of global features of Earth’s atmosphere’s general circulation (e.g., the quasi‐biennial oscillation, the pole‐to‐pole mesospheric circulation). As CGWs are not generally resolved by global weather and climate models, their effects on the circulation need to be parameterized. However, quality observations of GWs are spatiotemporally sparse, limiting understanding and preventing constraints on parameterizations. Convection‐permitting or ‐resolving simulations do generate CGWs, but validation is not possible as these simulations cannot reproduce the CGW‐forcing convection at correct times, locations, and intensities. Here, realistic convective diabatic heating, learned from full‐physics convection‐permitting Weather Research and Forecasting simulations, is predicted from weather radar observations using neural networks and a previously developed look‐up table. These heating rates are then used to force an idealized GW‐resolving dynamical model. Simulated CGWs forced in this way closely resembled those observed by the Atmospheric InfraRed Sounder in the upper stratosphere. CGW drag in these validated simulations extends 100s of kilometers away from the convective sources, highlighting errors in current gravity wave drag parameterizations due to the use of the ubiquitous single‐column approximation. Such validatable simulations have significant potential to be used to further basic understanding of CGWs, improve their parameterizations physically, and provide more restrictive constraints on tuning with confidence . , Plain Language Summary Thunderstorms generate waves in the atmosphere that can generate turbulence at commercial aircraft cruising altitudes and further aloft. At these higher altitudes, they eventually break, not only generating turbulence, but also exerting forces that affect the large‐scale flows in the middle atmosphere. While these waves have been known to be important since at least the 1980s, they are difficult to observe. They can be simulated, but weather models do not simulate thunderstorms in the correct locations at the right times, meaning the simulated waves cannot be directly compared against observations. Here, weather radar observations are used as input to a look‐up table and a neural network to force realistic thunderstorm motions and waves within a simplified weather model. This method was able to reproduce a satellite‐observed case with notable skill. In one of the first simulations of thunderstorm‐generated waves comparable to satellite observations, these waves travel 100s of kilometers away from the thunderstorms, conflicting with assumptions made in weather and climate models. , Key Points If realistic convective diabatic heating is supplied at the correct places and times, GW‐resolving models can reasonably reproduce Convection‐generated gravity waves (CGWs) While location and type of convective storm do influence latent heating, these difference are of second order importance for CGW forcing Drag due to convection‐generated GWs from a compact source region is spread over O(1,000) km due to lateral propagation

6. Interpretable Structural Model Error Discovery From Sparse Assimilation Increments Using Spectral Bias‐Reduced Neural Networks: A Quasi‐Geostrophic Turbulence Test Case

   Rambod Mojgani, Ashesh Chattopadhyay, and Pedram Hassanzadeh

   Journal of Advances in Modeling Earth Systems,

   Abs DOI

   Abstract Earth system models suffer from various structural and parametric errors in their representation of nonlinear, multi‐scale processes, leading to uncertainties in their long‐term projections. The effects of many of these errors (particularly those due to fast physics) can be quantified in short‐term simulations, for example, as differences between the predicted and observed states (analysis increments). With the increase in the availability of high‐quality observations and simulations, learning nudging from these increments to correct model errors has become an active research area. However, most studies focus on using neural networks, which while powerful, are hard to interpret, are data‐hungry, and poorly generalize out‐of‐distribution. Here, we show the capabilities of Model Error Discovery with Interpretability and Data Assimilation (MEDIDA), a general, data‐efficient framework that uses sparsity‐promoting equation‐discovery techniques to learn model errors from analysis increments. Using two‐layer quasi‐geostrophic turbulence as the test case, MEDIDA is shown to successfully discover various linear and nonlinear structural/parametric errors when full observations are available. Discovery from spatially sparse observations is found to require highly accurate interpolation schemes. While NNs have shown success as interpolators in recent studies, here, they are found inadequate due to their inability to accurately represent small scales, a phenomenon known as spectral bias. We show that a general remedy, adding a random Fourier feature layer to the NN, resolves this issue enabling MEDIDA to successfully discover model errors from sparse observations. These promising results suggest that with further development, MEDIDA could be scaled up to models of the Earth system and real observations. , Plain Language Summary Numerical models are used to predict the Earth system, for example, from daily weather to the next‐century climate. These models have been developed and validated against observations over decades, however, they still have shortcomings (errors) in their representations of many complex processes, particularly those that are nonlinear and span many scales in time and space. The rapid improvements in the quality and quantity of observational data from the Earth system and advances in machine learning (ML) algorithms provide an opportunity to try to reduce these errors. However, the challenge is that many ML methods require a lot of training data, and it is also often difficult to explain how they are reducing the error. Here, we show the capabilities of a framework called MEDIDA (Model Error Discovery with Interpretability and Data Assimilation), which uses a class of ML methods that provide closed‐form (thus interpretable) equations for what they are learning from the differences between observations and model predictions. We show the success of MEDIDA when applied to a model of atmospheric turbulent circulation. Even when observations are only sparsely (not at every location) available, we show that MEDIDA works accurately once we leverage more recent advances in ML. , Key Points Model error discovery with interpretability and data assimilation is scaled up to geostrophic turbulence and sparse observations Naive use of neural nets (NNs) as interpolator does not capture small scales due to spectral bias, failing discoveries of closed‐form errors Reducing this bias using random Fourier features enables NNs to represent the full range of scales, leading to successful error discoveries

7. Explainable Offline‐Online Training of Neural Networks for Parameterizations: A 1D Gravity Wave‐QBO Testbed in the Small‐Data Regime

   Hamid A. Pahlavan, Pedram Hassanzadeh, and M. Joan Alexander

   Geophysical Research Letters,

   Abs DOI

   Abstract There are different strategies for training neural networks (NNs) as subgrid‐scale parameterizations. Here, we use a 1D model of the quasi‐biennial oscillation (QBO) and gravity wave (GW) parameterizations as testbeds. A 12‐layer convolutional NN that predicts GW forcings for given wind profiles, when trained offline in a big ‐ data regime (100‐year), produces realistic QBOs once coupled to the 1D model. In contrast, offline training of this NN in a small ‐ data regime (18‐month) yields unrealistic QBOs. However, online re‐training of just two layers of this NN using ensemble Kalman inversion and only time‐averaged QBO statistics leads to parameterizations that yield realistic QBOs. Fourier analysis of these three NNs’ kernels suggests why/how re‐training works and reveals that these NNs primarily learn low‐pass, high‐pass, and a combination of band‐pass filters, potentially related to the local and non‐local dynamics in GW propagation and dissipation. These findings/strategies generally apply to data‐driven parameterizations of other climate processes. , Plain Language Summary Due to computational limits, climate models estimate (i.e., parameterize) small‐scale physical processes, such as atmospheric gravity waves (GWs), since they occur on scales smaller than the models’ grid size. Recently, machine learning techniques, especially neural networks (NNs), have emerged as promising tools for learning these parameterizations from data. Offline and online learning are among the main strategies for training these NN‐based parameterizations. Offline learning, while straightforward, requires extensive, high‐quality data from small‐scale processes, which are scarce. Alternatively, online learning only needs time or space‐averaged data based on large‐scale processes, which are more accessible. However, online learning can be computationally expensive. Here, we explore various learning strategies using an NN‐based GW parameterization, within a simple model of the quasi‐biennial oscillation (QBO), an important quasi‐periodic wind pattern in the tropics. When supplied with a large 100‐year data set, the offline‐trained NN accurately replicates wind behaviors once coupled to the QBO model. Yet, when limited to an 18‐month training data set (which is more realistic), its performance degrades. Interestingly, by online re‐training specific parts of this NN using only time‐averaged QBO statistics, its accuracy is restored. We term this approach an “offline‐online” learning strategy. Our findings also benefit parameterization efforts for other climate processes. , Key Points 1D model of quasi‐biennial oscillation (QBO) and gravity waves is used as a testbed for training neural network (NN)‐based parameterizations Offline training NNs in small‐data regimes yields unstable QBOs that are rectified by online re‐training using only time‐averaged statistics Fourier analysis of NNs reveals that they learn specific filters that are consistent with the dynamics of wave propagation and dissipation

8. Challenges of learning multi-scale dynamics with AI weather models: Implications for stability and one solution

   Ashesh Chattopadhyay, Y. Qiang Sun, and Pedram Hassanzadeh

   arXiv:2304.07029 [physics]

   Abs DOI

   Long-term stability and physical consistency are critical properties for AI-based weather models if they are going to be used for subseasonal-to-seasonal forecasts or beyond, e.g., climate change projection. However, current AI-based weather models can only provide short-term forecasts accurately since they become unstable or physically inconsistent when time-integrated beyond a few weeks or a few months. Either they exhibit numerical blow-up or hallucinate unrealistic dynamics of the atmospheric variables, akin to the current class of autoregressive large language models. The cause of the instabilities is unknown, and the methods that are used to improve their stability horizons are ad-hoc and lack rigorous theory. In this paper, we reveal that the universal causal mechanism for these instabilities in any turbulent flow is due to }textit{spectral bias} wherein, }textit{any} deep learning architecture is biased to learn only the large-scale dynamics and ignores the small scales completely. We further elucidate how turbulence physics and the absence of convergence in deep learning-based time-integrators amplify this bias, leading to unstable error propagation. Finally, using the quasi-geostrophic flow and European Center for Medium-Range Weather Forecasting (ECMWF) Reanalysis data as test cases, we bridge the gap between deep learning theory and numerical analysis to propose one mitigative solution to such unphysical behavior. We develop long-term physically-consistent data-driven models for the climate system and demonstrate accurate short-term forecasts, and hundreds of years of time-integration with accurate mean and variability.

2023

1. Extreme Event Prediction with Multi-agent Reinforcement Learning-based Parametrization of Atmospheric and Oceanic Turbulence

   Rambod Mojgani, Daniel Waelchli, Yifei Guan, Petros Koumoutsakos, and Pedram Hassanzadeh

   arXiv:2312.00907 [cs]

   Abs DOI

   Global climate models (GCMs) are the main tools for understanding and predicting climate change. However, due to limited numerical resolutions, these models suffer from major structural uncertainties; e.g., they cannot resolve critical processes such as small-scale eddies in atmospheric and oceanic turbulence. Thus, such small-scale processes have to be represented as a function of the resolved scales via closures (parametrization). The accuracy of these closures is particularly important for capturing climate extremes. Traditionally, such closures are based on heuristics and simplifying assumptions about the unresolved physics. Recently, supervised-learned closures, trained offline on high-fidelity data, have been shown to outperform the classical physics-based closures. However, this approach requires a significant amount of high-fidelity training data and can also lead to instabilities. Reinforcement learning is emerging as a potent alternative for developing such closures as it requires only low-order statistics and leads to stable closures. In Scientific Multi-Agent Reinforcement Learning (SMARL) computational elements serve a dual role of discretization points and learning agents. We leverage SMARL and fundamentals of turbulence physics to learn closures for prototypes of atmospheric and oceanic turbulence. The policy is trained using only the enstrophy spectrum, which is nearly invariant and can be estimated from a few high-fidelity samples (these few samples are far from enough for supervised/offline learning). We show that these closures lead to stable low-resolution simulations that, at a fraction of the cost, can reproduce the high-fidelity simulations’ statistics, including the tails of the probability density functions. The results demonstrate the high potential of SMARL for closure modeling for GCMs, especially in the regime of scarce data and indirect observations.

2. The Intrinsic 150‐Day Periodicity of the Southern Hemisphere Extratropical Large‐Scale Atmospheric Circulation

   Sandro W. Lubis and Pedram Hassanzadeh

   AGU Advances,

   Abs DOI

   Abstract The variability of the Southern Hemisphere (SH) extratropical large‐scale circulation is dominated by the Southern Annular Mode (SAM), whose timescale is extensively used as a key metric in evaluating state‐of‐the‐art climate models. Past observational and theoretical studies suggest that the SAM lacks any internally generated (intrinsic) periodicity. Here, we show, using observations and a climate model hierarchy, that the SAM has an intrinsic 150‐day periodicity. This periodicity is robustly detectable in the power spectra and principal oscillation patterns (aka dynamical mode decomposition) of the zonal‐mean circulation, and in hemispheric‐scale precipitation and ocean surface wind stress. The 150‐day period is consistent with the predictions of a new reduced‐order model for the SAM, which suggests that this periodicity is associated with a complex interaction of turbulent eddies and zonal wind anomalies, as the latter propagate from low to high latitudes. These findings present a rare example of periodic oscillations arising from the internal dynamics of the extratropical turbulent circulations. Based on these findings, we further propose a new metric for evaluating climate models, and show that some of the previously reported shortcomings and improvements in simulating SAM’s variability connect to the models’ ability in reproducing this periodicity. We argue that this periodicity should be considered in evaluating climate models and understanding the past, current, and projected Southern Hemisphere climate variability. , Plain Language Summary The Southern Annular Mode (SAM), which involves hemispheric‐scale north‐south movement of the midlatitude jet stream, dominates the variability of the Southern Hemisphere (SH) large‐scale atmospheric circulation. The SAM has extensive impacts on the Southern Ocean and Antarctica, and the past, current, and future climate of the SH is often viewed through the lens of the SAM. Studies since early 1990s suggested that SAM’s variability lacks any internally generated periodic oscillation, as expected from the turbulent and thus chaotic nature of the midlatitude circulation. However, here we show using observational data, model data, and theory that SAM has an intrinsic 150‐day periodicity arising from the internal dynamics of the extratropical atmosphere. This 150‐day oscillation clearly influences the variability of the hemispheric‐scale precipitation and ocean surface wind stress, suggesting broader impacts of this periodicity on the SH weather and climate. We also found that many state‐of‐the‐art climate models cannot faithfully reproduce this periodicity, providing an explanation for some of the previously reported shortcomings of these models in simulating SAM’s variability. Based on these findings, we propose new metrics and ideas for evaluating these models and understanding their shortcomings, and potentially, improving them. , Key Points We show using theory, reanalysis data, and a model hierarchy that the Southern Annular Mode has an internally generated 150‐day periodicity Periodicity is tied to the annular mode’s propagating regime, and affects hemispheric‐scale precipitation and ocean surface wind stress CMIP models vary in how well they reproduce the periodicity; new metrics are introduced to evaluate these models and understand their biases

3. Quantifying 3D Gravity Wave Drag in a Library of Tropical Convection‐Permitting Simulations for Data‐Driven Parameterizations

   Y. Qiang Sun, Pedram Hassanzadeh, M. Joan Alexander, and Christopher G. Kruse

   Journal of Advances in Modeling Earth Systems,

   Abs DOI

   Abstract Atmospheric gravity waves (GWs) span a broad range of length scales. As a result, the un‐resolved and under‐resolved GWs have to be represented using a sub‐grid scale (SGS) parameterization in general circulation models (GCMs). In recent years, machine learning (ML) techniques have emerged as novel methods for SGS modeling of climate processes. In the widely used approach of supervised (offline) learning, the true representation of the SGS terms have to be properly extracted from high‐fidelity data (e.g., GW‐resolving simulations). However, this is a non‐trivial task, and the quality of the ML‐based parameterization significantly hinges on the quality of these SGS terms. Here, we compare three methods to extract 3D GW fluxes and the resulting drag (Gravity Wave Drag [GWD]) from high‐resolution simulations: Helmholtz decomposition, and spatial filtering to compute the Reynolds stress and the full SGS stress. In addition to previous studies that focused only on vertical fluxes by GWs, we also quantify the SGS GWD due to lateral momentum fluxes. We build and utilize a library of tropical high‐resolution (Δ x  = 3 km) simulations using weather research and forecasting model. Results show that the SGS lateral momentum fluxes could have a significant contribution to the total GWD. Moreover, when estimating GWD due to lateral effects, interactions between the SGS and the resolved large‐scale flow need to be considered. The sensitivity of the results to different filter type and length scale (dependent on GCM resolution) is also explored to inform the scale‐awareness in the development of data‐driven parameterizations. , Plain Language Summary Gravity waves (GWs) present a challenge to climate prediction: waves on scales of O(1)–O(100) km can neither be systematically measured with conventional observational systems, nor properly represented (resolved) in operational climate models, which have a typical grid spacing on the order of 100 km. Therefore, in these climate models, small‐scale GWs must be parameterized, or estimated, based on the resolved (large‐scale) flow. The primary effects of these small‐scale waves on the resolved flow is the so‐called sub‐grid scale drag (Gravity Wave Drag [GWD]), resulting from the propagation and breaking of these waves. Existing GW parameterizations in general circulation models are all highly simplified; for example, they only account for vertical propagation of GWs. With growing computing power, a promising alternative approach is to use machine learning to develop data‐driven parameterizations. However, this requires to first generate reliable high‐resolution computer simulations and then extract GWD from these simulations. This study follows these steps, compares different extraction methods, and describes some challenges and pathways to make advances. Furthermore, our results suggest that the horizontal propagation of GWs should be included in parameterizations too, however, extra care is needed in order to extract the resulting GWD from high‐resolution data. , Key Points In a library of weather research and forecasting model simulations, we compare methods for estimating 3D gravity wave drag force that are un‐ and under‐resolved by general circulation models For drag associated with vertical fluxes, different methods agree on time‐ and zonal‐mean but not on instantaneous spatiotemporal patterns Drag associated with horizontal fluxes is significant but is very sensitive to the estimation methodology

4. Using Neural Networks to Learn the Jet Stream Forced Response from Natural Variability

   Charlotte Connolly, Elizabeth A. Barnes, Pedram Hassanzadeh, and Mike Pritchard

   Artificial Intelligence for the Earth Systems,

   Abs DOI

   Abstract Two distinct features of anthropogenic climate change, warming in the tropical upper troposphere and warming at the Arctic surface, have competing effects on the midlatitude jet stream’s latitudinal position, often referred to as a “tug-of-war.” Studies that investigate the jet’s response to these thermal forcings show that it is sensitive to model type, season, initial atmospheric conditions, and the shape and magnitude of the forcing. Much of this past work focuses on studying a simulation’s response to external manipulation. In contrast, we explore the potential to train a convolutional neural network (CNN) on internal variability alone and then use it to examine possible nonlinear responses of the jet to tropospheric thermal forcing that more closely resemble anthropogenic climate change. Our approach leverages the idea behind the fluctuation–dissipation theorem, which relates the internal variability of a system to its forced response but so far has been only used to quantify linear responses. We train a CNN on data from a long control run of the CESM dry dynamical core and show that it is able to skillfully predict the nonlinear response of the jet to sustained external forcing. The trained CNN provides a quick method for exploring the jet stream sensitivity to a wide range of tropospheric temperature tendencies and, considering that this method can likely be applied to any model with a long control run, could be useful for early-stage experiment design.

5. Explaining the physics of transfer learning in data-driven turbulence modeling

   Adam Subel, Yifei Guan, Ashesh Chattopadhyay, and Pedram Hassanzadeh

   PNAS Nexus,

   Abs DOI

   Abstract Transfer learning (TL), which enables neural networks (NNs) to generalize out-of-distribution via targeted re-training, is becoming a powerful tool in scientific machine learning (ML) applications such as weather/climate prediction and turbulence modeling. Effective TL requires knowing (1) how to re-train NNs? and (2) what physics are learned during TL? Here, we present novel analyses and a framework addressing (1)–(2) for a broad range of multi-scale, nonlinear, dynamical systems. Our approach combines spectral (e.g. Fourier) analyses of such systems with spectral analyses of convolutional NNs, revealing physical connections between the systems and what the NN learns (a combination of low-, high-, band-pass filters and Gabor filters). Integrating these analyses, we introduce a general framework that identifies the best re-training procedure for a given problem based on physics and NN theory. As test case, we explain the physics of TL in subgrid-scale modeling of several setups of 2D turbulence. Furthermore, these analyses show that in these cases, the shallowest convolution layers are the best to re-train, which is consistent with our physics-guided framework but is against the common wisdom guiding TL in the ML literature. Our work provides a new avenue for optimal and explainable TL, and a step toward fully explainable NNs, for wide-ranging applications in science and engineering, such as climate change modeling.

6. Kolmogorov n–width and Lagrangian physics-informed neural networks: A causality-conforming manifold for convection-dominated PDEs

   Rambod Mojgani, Maciej Balajewicz, and Pedram Hassanzadeh

   Computer Methods in Applied Mechanics and Engineering,

   DOI
7. Deep learning-enhanced ensemble-based data assimilation for high-dimensional nonlinear dynamical systems

   Ashesh Chattopadhyay, Ebrahim Nabizadeh, Eviatar Bach, and Pedram Hassanzadeh

   Journal of Computational Physics,

   DOI
8. Long-term stability and generalization of observationally-constrained stochastic data-driven models for geophysical turbulence

   Ashesh Chattopadhyay, Jaideep Pathak, Ebrahim Nabizadeh, Wahid Bhimji, and Pedram Hassanzadeh

   Environmental Data Science,

   Abs DOI

   Abstract Recent years have seen a surge in interest in building deep learning-based fully data-driven models for weather prediction. Such deep learning models, if trained on observations can mitigate certain biases in current state-of-the-art weather models, some of which stem from inaccurate representation of subgrid-scale processes. However, these data-driven models, being over-parameterized, require a lot of training data which may not be available from reanalysis (observational data) products. Moreover, an accurate, noise-free, initial condition to start forecasting with a data-driven weather model is not available in realistic scenarios. Finally, deterministic data-driven forecasting models suffer from issues with long-term stability and unphysical climate drift, which makes these data-driven models unsuitable for computing climate statistics. Given these challenges, previous studies have tried to pre-train deep learning-based weather forecasting models on a large amount of imperfect long-term climate model simulations and then re-train them on available observational data. In this article, we propose a convolutional variational autoencoder (VAE)-based stochastic data-driven model that is pre-trained on an imperfect climate model simulation from a two-layer quasi-geostrophic flow and re-trained, using transfer learning, on a small number of noisy observations from a perfect simulation. This re-trained model then performs stochastic forecasting with a noisy initial condition sampled from the perfect simulation. We show that our ensemble-based stochastic data-driven model outperforms a baseline deterministic encoder–decoder-based convolutional model in terms of short-term skills, while remaining stable for long-term climate simulations yielding accurate climatology.

9. Learning physics-constrained subgrid-scale closures in the small-data regime for stable and accurate LES

   Yifei Guan, Adam Subel, Ashesh Chattopadhyay, and Pedram Hassanzadeh

   Physica D: Nonlinear Phenomena,

   DOI

2022

1. Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto–Sivashinsky test case

   Rambod Mojgani, Ashesh Chattopadhyay, and Pedram Hassanzadeh

   Chaos: An Interdisciplinary Journal of Nonlinear Science,

   Abs DOI

   Models of many engineering and natural systems are imperfect. The discrepancy between the mathematical representations of a true physical system and its imperfect model is called the model error. These model errors can lead to substantial differences between the numerical solutions of the model and the state of the system, particularly in those involving nonlinear, multi-scale phenomena. Thus, there is increasing interest in reducing model errors, particularly by leveraging the rapidly growing observational data to understand their physics and sources. Here, we introduce a framework named MEDIDA: Model Error Discovery with Interpretability and Data Assimilation. MEDIDA only requires a working numerical solver of the model and a small number of noise-free or noisy sporadic observations of the system. In MEDIDA, first, the model error is estimated from differences between the observed states and model-predicted states (the latter are obtained from a number of one-time-step numerical integrations from the previous observed states). If observations are noisy, a data assimilation technique, such as the ensemble Kalman filter, is employed to provide the analysis state of the system, which is then used to estimate the model error. Finally, an equation-discovery technique, here the relevance vector machine, a sparsity-promoting Bayesian method, is used to identify an interpretable, parsimonious, and closed-form representation of the model error. Using the chaotic Kuramoto–Sivashinsky system as the test case, we demonstrate the excellent performance of MEDIDA in discovering different types of structural/parametric model errors, representing different types of missing physics, using noise-free and noisy observations.

2. FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators

   Jaideep Pathak, Shashank Subramanian, Peter Harrington, Sanjeev Raja, Ashesh Chattopadhyay, Morteza Mardani, Thorsten Kurth, David Hall, Zongyi Li, Kamyar Azizzadenesheli, Pedram Hassanzadeh, Karthik Kashinath, and Animashree Anandkumar

   arXiv:2202.11214 [physics]

   Abs DOI

   FourCastNet, short for Fourier Forecasting Neural Network, is a global data-driven weather forecasting model that provides accurate short to medium-range global predictions at \0.25^{}circ} resolution. FourCastNet accurately forecasts high-resolution, fast-timescale variables such as the surface wind speed, precipitation, and atmospheric water vapor. It has important implications for planning wind energy resources, predicting extreme weather events such as tropical cyclones, extra-tropical cyclones, and atmospheric rivers. FourCastNet matches the forecasting accuracy of the ECMWF Integrated Forecasting System (IFS), a state-of-the-art Numerical Weather Prediction (NWP) model, at short lead times for large-scale variables, while outperforming IFS for variables with complex fine-scale structure, including precipitation. FourCastNet generates a week-long forecast in less than 2 seconds, orders of magnitude faster than IFS. The speed of FourCastNet enables the creation of rapid and inexpensive large-ensemble forecasts with thousands of ensemble-members for improving probabilistic forecasting. We discuss how data-driven deep learning models such as FourCastNet are a valuable addition to the meteorology toolkit to aid and augment NWP models.

3. The Summertime Pacific‐North American Weather Regimes and Their Predictability

   Ebrahim Nabizadeh, Sandro W. Lubis, and Pedram Hassanzadeh

   Geophysical Research Letters,

   Abs DOI

   Abstract The forecast skill of numerical weather prediction (NWP) models and the intrinsic predictability can be different among weather regimes. Here, we examine the predictability of distinct Pacific‐North American weather regimes during extended boreal summer. The four identified weather regimes include Pacific trough, Arctic low, Arctic high, and Alaskan ridge. The medium range forecast skill of these regimes is quantified in the ECMWF and the National Centers for Environmental Prediction models from the TIGGE project. Based on anomaly correlation coefficient, persistence, and transition frequency, the highest forecast skill is consistently found for the Arctic high regime. Based on the instantaneous local dimension and persistence from a dynamical systems analysis, the Arctic high regime has the highest intrinsic predictability. The analysis also suggests that overall, the Pacific trough regime has the lowest intrinsic predictability. These findings are consistent with the forecast skills of the NWP models, and highlight the link between prediction skill and intrinsic predictability. , Plain Language Summary Midlatitude circulation is characterized by chaotic dynamics, which makes weather prediction difficult. Due to the impact of the recurring large‐scale patterns (weather regimes) on extreme events, we need to measure (and improve) the numerical weather predictions’ skill for these regimes. In this study, we have identified the summertime weather regimes over the Pacific‐North American region during extended summer (June–September). Then, we measure prediction skill associated with each of the identified regimes in two leading operational weather forecast models. We have found that the forecast models have the highest prediction skill for the regime with the high‐pressure system over the Arctic. Our results indicate a connection between the prediction skill in the forecast models and the complexity and persistence of each weather regime, quantified using a dynamical systems analysis. , Key Points We identified four summertime weather regimes over Pacific‐North America and their corresponding surface impacts Numerical weather prediction models have the highest prediction skills for a regime associated with the Arctic high Prediction skills for these weather regimes are linked to their intrinsic predictability via dynamical systems analysis

4. Stable a posteriori LES of 2D turbulence using convolutional neural networks: Backscattering analysis and generalization to higher Re via transfer learning

   Yifei Guan, Ashesh Chattopadhyay, Adam Subel, and Pedram Hassanzadeh

   Journal of Computational Physics,

   DOI

2021

1. A data-driven, physics-informed framework for forecasting the spatiotemporal evolution of chaotic dynamics with nonlinearities modeled as exogenous forcings

   M.A. Khodkar and Pedram Hassanzadeh

   Journal of Computational Physics,

   DOI
2. Physics-informed machine learning: case studies for weather and climate modelling

   K. Kashinath, M. Mustafa, A. Albert, J-L. Wu, C. Jiang, S. Esmaeilzadeh, K. Azizzadenesheli, R. Wang, A. Chattopadhyay, A. Singh, A. Manepalli, D. Chirila, R. Yu, R. Walters, B. White, H. Xiao, H. A. Tchelepi, P. Marcus, A. Anandkumar, P. Hassanzadeh, and Prabhat

   Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences,

   Abs DOI

   Machine learning (ML) provides novel and powerful ways of accurately and efficiently recognizing complex patterns, emulating nonlinear dynamics, and predicting the spatio-temporal evolution of weather and climate processes. Off-the-shelf ML models, however, do not necessarily obey the fundamental governing laws of physical systems, nor do they generalize well to scenarios on which they have not been trained. We survey systematic approaches to incorporating physics and domain knowledge into ML models and distill these approaches into broad categories. Through 10 case studies, we show how these approaches have been used successfully for emulating, downscaling, and forecasting weather and climate processes. The accomplishments of these studies include greater physical consistency, reduced training time, improved data efficiency, and better generalization. Finally, we synthesize the lessons learned and identify scientific, diagnostic, computational, and resource challenges for developing truly robust and reliable physics-informed ML models for weather and climate processes. This article is part of the theme issue ‘Machine learning for weather and climate modelling’.

3. Data-driven subgrid-scale modeling of forced Burgers turbulence using deep learning with generalization to higher Reynolds numbers via transfer learning

   Adam Subel, Ashesh Chattopadhyay, Yifei Guan, and Pedram Hassanzadeh

   Physics of Fluids,

   Abs DOI

   Developing data-driven subgrid-scale (SGS) models for large eddy simulations (LESs) has received substantial attention recently. Despite some success, particularly in a priori (offline) tests, challenges have been identified that include numerical instabilities in a posteriori (online) tests and generalization (i.e., extrapolation) of trained data-driven SGS models, for example, to higher Reynolds numbers. Here, using the stochastically forced Burgers turbulence as the test-bed, we show that deep neural networks trained using properly pre-conditioned (augmented) data yield stable and accurate a posteriori LES models. Furthermore, we show that transfer learning enables accurate/stable generalization to a flow with 10× higher Reynolds number.

4. The 3D Structure of Northern Hemisphere Blocking Events: Climatology, Role of Moisture, and Response to Climate Change

   Ebrahim Nabizadeh, Sandro W. Lubis, and Pedram Hassanzadeh

   Journal of Climate,

   Abs DOI

   Abstract To better understand the dynamics and impacts of blocking events, their 3D structure needs to be further investigated. We present a comprehensive composite analysis of the 3D structure of blocks and its response to future climate change over North Pacific, North Atlantic, and Russia in summers and winters using reanalysis and two large-ensemble datasets from CESM1 and GFDLCM3. In reanalysis, over both ocean and land, the anomalous winds are equivalent-barotropic in the troposphere and stratosphere, and temperature anomalies are positive throughout the troposphere and negative in the lower stratosphere. The main seasonal and regional differences are that blocks are larger/stronger in winters; over oceans, the temperature anomaly is shifted westward due to latent heating. Analyzing the temperature tendency equation shows that in all three sectors, adiabatic warming due to subsidence is the main driver of the positive temperature anomaly; however, depending on season and region, meridional thermal advection and latent heating might have leading-order contributions too. Both GCMs are found to reproduce the climatological 3D structure remarkably well, but sometimes disagree on future changes. Overall, the future summertime response is weakening of all fields (except for specific humidity), although the impact on near-surface temperature is not necessarily weakened; e.g., the blocking-driven near-surface warming over Russia intensifies. The wintertime response is strengthening of all fields, except for temperature in some cases. Responses of geopotential height and temperature are shifted westward in winters, most likely due to latent heating. Results highlight the importance of process-level analyses of blocks’ 3D structure for improved understanding of the resulting temperature extremes and their future changes.

5. Eddy Length Scale Response to Static Stability Change in an Idealized Dry Atmosphere: A Linear Response Function Approach*

   Pak Wah Chan, Pedram Hassanzadeh, and Zhiming Kuang

   Journal of the Atmospheric Sciences,

   Abs DOI

   Abstract The response of mid-latitude equilibrated eddy length scale to static stability has long been questioned but not investigated in well-controlled experiments with unchanged mean zonal wind and meridional temperature gradient. With iterative use of the linear response function of an idealized dry atmosphere, we obtain a time-invariant and zonally-uniform forcing to decrease the near-surface temperature by over 2 K while keeping the change in zonal wind negligible (within 0.2m s −1 ). In such experiments of increased static stability, energy-containing zonal scale decreases by 3–4%, which matches with Rhines scale decrease near the jet core. Changes in Rossby radius (+2%), maximum baroclinic growth scale (-1%) and Kuo scale (0%) fail to match this change in zonal scale. These findings and well-controlled experiments help with better understanding of eddy–mean flow interactions and hence the mid-latitude circulation and its response to climate change.

6. An Eddy–Zonal Flow Feedback Model for Propagating Annular Modes

   Sandro W. Lubis and Pedram Hassanzadeh

   Journal of the Atmospheric Sciences,

   Abs DOI

   Abstract The variability of the zonal-mean large-scale extratropical circulation is often studied using individual modes obtained from empirical orthogonal function (EOF) analyses. The prevailing reduced-order model of the leading EOF (EOF1) of zonal-mean zonal wind, called the annular mode, consists of an eddy–mean flow interaction mechanism that results in a positive feedback of EOF1 onto itself. However, a few studies have pointed out that under some circumstances in observations and GCMs, strong couplings exist between EOF1 and EOF2 at some lag times, resulting in decaying-oscillatory, or propagating, annular modes. Here, we introduce a reduced-order model for coupled EOF1 and EOF2 that accounts for potential cross-EOF eddy–zonal flow feedbacks. Using the analytical solution of this model, we derive conditions for the existence of the propagating regime based on the feedback strengths. Using this model, and idealized GCMs and stochastic prototypes, we show that cross-EOF feedbacks play an important role in controlling the persistence of the annular modes by setting the frequency of the oscillation. We find that stronger cross-EOF feedbacks lead to less persistent annular modes. Applying the coupled-EOF model to the Southern Hemisphere reanalysis data shows the existence of strong cross-EOF feedbacks. The results highlight the importance of considering the coupling of EOFs and cross-EOF feedbacks to fully understand the natural and forced variability of the zonal-mean large-scale circulation.

2020

1. Hurricane Risk Assessment of Petroleum Infrastructure in a Changing Climate

   Majid Ebad Sichani, Katherine A. Anarde, Kendall M. Capshaw, Jamie E. Padgett, Rachel A. Meidl, Pedram Hassanzadeh, Ted P. Loch-Temzelides, and Philip B. Bedient

   Frontiers in Built Environment,

   DOI
2. Effects of climate change on the movement of future landfalling Texas tropical cyclones

   Pedram Hassanzadeh, Chia-Ying Lee, Ebrahim Nabizadeh, Suzana J. Camargo, Ding Ma, and Laurence Y. Yeung

   Nature Communications,

   Abs DOI

   Abstract The movement of tropical cyclones (TCs), particularly around the time of landfall, can substantially affect the resulting damage. Recently, trends in TC translation speed and the likelihood of stalled TCs such as Harvey have received significant attention, but findings have remained inconclusive. Here, we examine how the June-September steering wind and translation speed of landfalling Texas TCs change in the future under anthropogenic climate change. Using several large-ensemble/multi-model datasets, we find pronounced regional variations in the meridional steering wind response over North America, but―consistently across models―stronger June-September-averaged northward steering winds over Texas. A cluster analysis of daily wind patterns shows more frequent circulation regimes that steer landfalling TCs northward in the future. Downscaling experiments show a 10-percentage-point shift from the slow-moving to the fast-moving end of the translation-speed distribution in the future. Together, these analyses indicate increases in the likelihood of faster-moving landfalling Texas TCs in the late 21 st century.

3. Data‐Driven Super‐Parameterization Using Deep Learning: Experimentation With Multiscale Lorenz 96 Systems and Transfer Learning

   Ashesh Chattopadhyay, Adam Subel, and Pedram Hassanzadeh

   Journal of Advances in Modeling Earth Systems,

   Abs DOI

   Abstract To make weather and climate models computationally affordable, small‐scale processes are usually represented in terms of the large‐scale, explicitly resolved processes using physics‐based/semi‐empirical parameterization schemes. Another approach, computationally more demanding but often more accurate, is super‐parameterization (SP). SP involves integrating the equations of small‐scale processes on high‐resolution grids embedded within the low‐resolution grid of large‐scale processes. Recently, studies have used machine learning (ML) to develop data‐driven parameterization (DD‐P) schemes. Here, we propose a new approach, data‐driven SP (DD‐SP), in which the equations of the small‐scale processes are integrated data‐drivenly (thus inexpensively) using ML methods such as recurrent neural networks. Employing multiscale Lorenz 96 systems as the testbed, we compare the cost and accuracy (in terms of both short‐term prediction and long‐term statistics) of parameterized low‐resolution (PLR) SP, DD‐P, and DD‐SP models. We show that with the same computational cost, DD‐SP substantially outperforms PLR and is more accurate than DD‐P, particularly when scale separation is lacking. DD‐SP is much cheaper than SP, yet its accuracy is the same in reproducing long‐term statistics (climate prediction) and often comparable in short‐term forecasting (weather prediction). We also investigate generalization: when models trained on data from one system are applied to a more chaotic system, we find that models often do not generalize, particularly when short‐term prediction accuracies are examined. However, we show that transfer learning, which involves re‐training the data‐driven model with a small amount of data from the new system, significantly improves generalization. Potential applications of DD‐SP and transfer learning in climate/weather modeling are discussed. , Plain Language Summary The weather/climate system involves intertwined physical processes acting on scales from centimeters (or even smaller) to tens of thousands of kilometers. Most weather/climate models used in practice include parameterization schemes that relate small‐scale processes, which are not explicitly resolved (due to coarse spatiotemporal resolution), to large‐scale processes that are resolved. Recently, studies have explored using machine learning for data‐driven parameterization (DD‐P) of small‐scale (subgrid) processes. Here, we first introduce a novel way to leverage recent advances in deep learning to improve the modeling of subgrid processes. In this approach, called data‐driven super‐parameterization (DD‐SP), deep learning is used for fast, data‐driven integration of equations of small‐scale processes, while other equations are integrated using conventional numerical methods. Employing a relatively simple chaotic system, we show the advantages of DD‐SP over DD‐P and conventional parameterizations. Second, we examine how these data‐driven models generalize (extrapolate) from one system to other (e.g., more chaotic) systems. We demonstrate that these models fail to generalize, but then we show that transfer learning (with a small amount of data from the new system) substantially improves their generalization. While encouraging, tests with more complex systems are needed to fully understand the potential/challenges of using DD‐SP and transfer learning for improving weather/climate models. , Key Points Data‐driven super‐parameterization (DD‐SP), in which the equations of small‐scale processes are integrated using deep learning, is proposed DD‐SP is cheaper than numerical SP and tests show that DD‐SP has accuracy superior to data‐driven parameterization and comparable to SP Transfer learning is shown to effectively improve the generalization (extrapolation) of data‐driven models to systems that are more chaotic

4. Predicting clustered weather patterns: A test case for applications of convolutional neural networks to spatio-temporal climate data

   Ashesh Chattopadhyay, Pedram Hassanzadeh, and Saba Pasha

   Scientific Reports,

   Abs DOI

   Abstract Deep learning techniques such as convolutional neural networks (CNNs) can potentially provide powerful tools for classifying, identifying, and predicting patterns in climate and environmental data. However, because of the inherent complexities of such data, which are often spatio-temporal, chaotic, and non-stationary, the CNN algorithms must be designed/evaluated for each specific dataset and application. Yet CNN, being a supervised technique, requires a large labeled dataset to start. Labeling demands (human) expert time which, combined with the limited number of relevant examples in this area, can discourage using CNNs for new problems. To address these challenges, here we (1) Propose an effective auto-labeling strategy based on using an unsupervised clustering algorithm and evaluating the performance of CNNs in re-identifying and predicting these clusters up to 5 days ahead of time; (2) Use this approach to label thousands of daily large-scale weather patterns over North America in the outputs of a fully-coupled climate model and show the capabilities of CNNs in re-identifying and predicting the 4 clustered regimes up to 5 days ahead of time. The deep CNN trained with 1000 samples or more per cluster has an accuracy of 90% or better for both identification and prediction while prediction accuracy scales weakly with the number of lead days. Accuracy scales monotonically but nonlinearly with the size of the training set, e.g. reaching 94% with 3000 training samples per cluster for identification and 93–76% for prediction at lead day 1–5, outperforming logistic regression, a simpler machine learning algorithm, by  ~ 25%. Effects of architecture and hyperparameters on the performance of CNNs are examined and discussed.

5. Data-driven predictions of a multiscale Lorenz 96 chaotic system using machine-learning methods: reservoir computing, artificial neural network, and long short-term memory network

   Ashesh Chattopadhyay, Pedram Hassanzadeh, and Devika Subramanian

   Nonlinear Processes in Geophysics,

   Abs DOI

   Abstract. In this paper, the performance of three machine-learning methods for predicting short-term evolution and for reproducing the long-term statistics of a multiscale spatiotemporal Lorenz 96 system is examined. The methods are an echo state network (ESN, which is a type of reservoir computing; hereafter RC–ESN), a deep feed-forward artificial neural network (ANN), and a recurrent neural network (RNN) with long short-term memory (LSTM; hereafter RNN–LSTM). This Lorenz 96 system has three tiers of nonlinearly interacting variables representing slow/large-scale (X), intermediate (Y), and fast/small-scale (Z) processes. For training or testing, only X is available; Y and Z are never known or used. We show that RC–ESN substantially outperforms ANN and RNN–LSTM for short-term predictions, e.g., accurately forecasting the chaotic trajectories for hundreds of numerical solver’s time steps equivalent to several Lyapunov timescales. The RNN–LSTM outperforms ANN, and both methods show some prediction skills too. Furthermore, even after losing the trajectory, data predicted by RC–ESN and RNN–LSTM have probability density functions (pdf’s) that closely match the true pdf – even at the tails. The pdf of the data predicted using ANN, however, deviates from the true pdf. Implications, caveats, and applications to data-driven and data-assisted surrogate modeling of complex nonlinear dynamical systems, such as weather and climate, are discussed.

6. Analog Forecasting of Extreme‐Causing Weather Patterns Using Deep Learning

   Ashesh Chattopadhyay, Ebrahim Nabizadeh, and Pedram Hassanzadeh

   Journal of Advances in Modeling Earth Systems,

   Abs DOI

   Abstract Numerical weather prediction models require ever‐growing computing time and resources but, still, have sometimes difficulties with predicting weather extremes. We introduce a data‐driven framework that is based on analog forecasting (prediction using past similar patterns) and employs a novel deep learning pattern‐recognition technique (capsule neural networks, CapsNets) and an impact‐based autolabeling strategy. Using data from a large‐ensemble fully coupled Earth system model, CapsNets are trained on midtropospheric large‐scale circulation patterns (Z500) labeled 0–4 depending on the existence and geographical region of surface temperature extremes over North America several days ahead. The trained networks predict the occurrence/region of cold or heat waves, only using Z500, with accuracies (recalls) of 69–45% (77–48%) or 62–41% (73–47%) 1–5 days ahead. Using both surface temperature and Z500, accuracies (recalls) with CapsNets increase to 80% (88%). In both cases, CapsNets outperform simpler techniques such as convolutional neural networks and logistic regression, and their accuracy is least affected as the size of the training set is reduced. The results show the promises of multivariate data‐driven frameworks for accurate and fast extreme weather predictions, which can potentially augment numerical weather prediction efforts in providing early warnings. , Key Points A data‐driven extreme weather prediction framework based on analog forecasting and deep learning pattern‐recognition methods is proposed Extreme surface temperature events over North America are skillfully predicted using only midtropospheric large‐scale circulation patterns More advanced deep learning methods are found to yield better forecasts, encouraging novel methods tailored for climate/weather data

2019

1. Evaluating Indices of Blocking Anticyclones in Terms of Their Linear Relations With Surface Hot Extremes

   Pak‐Wah Chan, Pedram Hassanzadeh, and Zhiming Kuang

   Geophysical Research Letters,

   Abs DOI

   Abstract Changes in frequencies of blocking anticyclones are sometimes used to explain changes in surface hot extremes. However, there is no consensus on the definition of blocking anticyclones, and several indices have been developed to measure them. Here we linearly regress interannual variations of hemispheric continental summer surface hot extreme area on the corresponding variations of blocking anticyclones in the ERA‐Interim reanalysis data and use cross‐validation test error to measure the blocking‐extreme link. Relative mean square validation error is at least 0.91 (relative to no knowledge of blocking) for existing indices defined on 500‐hPa geopotential height, when summed over land and ocean. This can be reduced to 0.55, largely by excluding blocks over ocean and optimizing parameters/thresholds and partly by using anomaly‐based or anomaly‐reversal hybrid indices. This framework helps to quantify the association of hot extremes and blocks with an uncertainty estimate and can be used for other extremes as well. , Plain Language Summary Blocking anticyclones refer to a type of high pressure system that blocks the atmospheric jet stream. They often cause weather extremes such as heat waves, but there is no consensus on how to measure them. Here we present a framework to evaluate different measures of blocking anticyclones by examining their linear relation with heat waves. This framework helps to quantify the association of heat waves and blocking anticyclones along with an uncertainty estimate and can be used for other extremes as well. , Key Points Multiple indices of blocking anticyclones are evaluated in terms of their linear relations with surface hot extremes using cross‐validation Blocking anticyclones over ocean should be excluded when surface hot extremes on continents are concerned Performance of blocking anticyclone indices under different thresholds and parameters is quantified

2. A hierarchical classification of adolescent idiopathic scoliosis: Identifying the distinguishing features in 3D spinal deformities

   Saba Pasha, Pedram Hassanzadeh, Malcolm Ecker, and Victor Ho

   PLOS ONE,

   DOI
3. Quantifying the Annular Mode Dynamics in an Idealized Atmosphere

   Pedram Hassanzadeh and Zhiming Kuang

   Journal of the Atmospheric Sciences,

   Abs DOI

   Abstract The linear response function (LRF) of an idealized GCM, the dry dynamical core with Held–Suarez physics, is used to accurately compute how eddy momentum and heat fluxes change in response to the zonal wind and temperature anomalies of the annular mode at the quasi-steady limit. Using these results and knowing the parameterizations of surface friction and thermal radiation in Held–Suarez physics, the contribution of each physical process (meridional and vertical eddy fluxes, surface friction, thermal radiation, and meridional advection) to the annular mode dynamics is quantified. Examining the quasigeostrophic potential vorticity balance, it is shown that the eddy feedback is positive and increases the persistence of the annular mode by a factor of more than 2. Furthermore, how eddy fluxes change in response to only the barotropic component of the annular mode, that is, vertically averaged zonal wind (and no temperature) anomaly, is also calculated similarly. The response of eddy fluxes to the barotropic-only component of the annular mode is found to be drastically different from the response to the full (i.e., barotropic + baroclinic) annular mode anomaly. In the former, the eddy generation is significantly suppressed, leading to a negative eddy feedback that decreases the persistence of the annular mode by nearly a factor of 3. These results suggest that the baroclinic component of the annular mode anomaly, that is, the increased low-level baroclinicity, is essential for the persistence of the annular mode, consistent with the baroclinic mechanism but not the barotropic mechanism proposed in the previous studies.

4. Size of the Atmospheric Blocking Events: Scaling Law and Response to Climate Change

   Ebrahim Nabizadeh, Pedram Hassanzadeh, Da Yang, and Elizabeth A. Barnes

   Geophysical Research Letters,

   Abs DOI

   Abstract Understanding the response of atmospheric blocking events to climate change has been of great interest in recent years. However, potential changes in the blocking area (size), which can affect the spatiotemporal characteristics of the resulting extreme events, have not received much attention. Using two large‐ensemble, fully coupled general circulation model (GCM) simulations, we show that the size of blocking events increases with climate change, particularly in the Northern Hemisphere (by as much as 17%). Using a two‐layer quasi‐geostrophic model and a dimensional analysis technique, we derive a scaling law for the size of blocking events, which shows that area mostly scales with width of the jet times the Kuo scale (i.e., the length of stationary Rossby waves). The scaling law is validated in a range of idealized GCM simulations. Predictions of this scaling law agree well with changes in blocking events’ size under climate change in fully coupled GCMs in winters but not in summers. , Key Points Size of blocking events robustly increases with climate change in most regions in two sets of large‐ensemble fully coupled GCM simulations A scaling law for blocking area is derived in a QG model using the Buckingham‐ π theorem and is verified using idealized GCM simulations The scaling law is area ~ width of the jet × Kuo scale and partially explains the projected changes in the fully coupled GCM simulations

5. Reduced-order modeling of fully turbulent buoyancy-driven flows using the Green’s function method

   M. A. Khodkar, Pedram Hassanzadeh, Saleh Nabi, and Piyush Grover

   Physical Review Fluids,

   DOI

2018

1. A Barotropic Mechanism for the Response of Jet Stream Variability to Arctic Amplification and Sea Ice Loss

   Bryn Ronalds, Elizabeth Barnes, and Pedram Hassanzadeh

   Journal of Climate,

   Abs DOI

   Previous studies have found that the most consistent response of the eddy-driven jet to sea ice loss and Arctic amplification in fully coupled general circulation models (GCMs) is a broad region of anomalous easterlies on the poleward flank. In this study, a similar response is noted in a dry dynamical core GCM with imposed surface heating at the pole, and it is shown that in both a fully coupled GCM’s North Atlantic basin and the dry dynamical core, the anomalous easterlies cause an asymmetrical narrowing of the jet on the poleward flank of the climatological jet. A suite of barotropic model simulations run with polar forcing shows decreased jet positional variability consistent with a narrowing of the jet profile, and it is proposed that this narrowing decreases the distance Rossby waves can propagate away from the jet core, which drives changes in jet variability. Since Rossby wave propagation and dissipation is intrinsic to the development and maintenance of the eddy-driven jet, and is tightly coupled to a jet’s variability, this acts as a meridional constraint on waves’ ability to propagate outside of the jet core, leading to the decreased variability in zonal-mean jet position. The results from all three models demonstrates that this relationship is present across a model hierarchy.

2. Data-driven reduced modelling of turbulent Rayleigh–Bénard convection using DMD-enhanced fluctuation–dissipation theorem

   M. A. Khodkar and Pedram Hassanzadeh

   Journal of Fluid Mechanics,

   Abs DOI

   A data-driven model-free framework is introduced for the calculation of reduced-order models (ROMs) capable of accurately predicting time-mean responses to external forcings, or forcings needed for specified responses, e.g. for control, in fully turbulent flows. The framework is based on using the fluctuation–dissipation theorem (FDT) in the space of a limited number of modes obtained from dynamic mode decomposition (DMD). Use of the DMD modes as the basis functions, rather than the commonly used proper orthogonal decomposition modes, resolves a previously identified problem in applying FDT to high-dimensional non-normal turbulent flows. Employing this DMD-enhanced FDT method ( {}text{FDT}_{DMD}\ ), a linear ROM with horizontally averaged temperature as state vector is calculated for a 3D Rayleigh–Bénard convection system at a Rayleigh number of \10^{6}\ using data obtained from direct numerical simulation. The calculated ROM performs well in various tests for this turbulent flow, suggesting {}text{FDT}_{DMD}\ as a promising method for developing ROMs for high-dimensional turbulent systems.

2017

1. Persistent anomalies of the extratropical Northern Hemisphere wintertime circulation as an initiator of El Niño/Southern Oscillation events

   Bruce T. Anderson, Pedram Hassanzadeh, and Rodrigo Caballero

   Scientific Reports,

   Abs DOI

   Abstract Climates across both hemispheres are strongly influenced by tropical Pacific variability associated with the El Niño/Southern Oscillation (ENSO). Conversely, extratropical variability also can affect the tropics. In particular, seasonal-mean alterations of near-surface winds associated with the North Pacific Oscillation (NPO) serve as a significant extratropical forcing agent of ENSO. However, it is still unclear what dynamical processes give rise to year-to-year shifts in these long-lived NPO anomalies. Here we show that intraseasonal variability in boreal winter pressure patterns over the Central North Pacific (CNP) imparts a significant signature upon the seasonal-mean circulations characteristic of the NPO. Further we show that the seasonal-mean signature results in part from year-to-year variations in persistent, quasi-stationary low-pressure intrusions into the subtropics of the CNP, accompanied by the establishment of persistent, quasi-stationary high-pressure anomalies over high latitudes of the CNP. Overall, we find that the frequency of these persistent extratropical anomalies (PEAs) during a given winter serves as a key modulator of intraseasonal variability in extratropical North Pacific circulations and, through their influence on the seasonal-mean circulations in and around the southern lobe of the NPO, the state of the equatorial Pacific 9–12 months later.

2. Quantifying the Eddy–Jet Feedback Strength of the Annular Mode in an Idealized GCM and Reanalysis Data

   Ding Ma, Pedram Hassanzadeh, and Zhiming Kuang

   Journal of the Atmospheric Sciences,

   Abs DOI

   Abstract A linear response function (LRF) that relates the temporal tendency of zonal-mean temperature and zonal wind to their anomalies and external forcing is used to accurately quantify the strength of the eddy–jet feedback associated with the annular mode in an idealized GCM. Following a simple feedback model, the results confirm the presence of a positive eddy–jet feedback in the annular mode dynamics, with a feedback strength of 0.137 day−1 in the idealized GCM. Statistical methods proposed by earlier studies to quantify the feedback strength are evaluated against results from the LRF. It is argued that the mean-state-independent eddy forcing reduces the accuracy of these statistical methods because of the quasi-oscillatory nature of the eddy forcing. Assuming the mean-state-independent eddy forcing is sufficiently weak at the low-frequency limit, a new method is proposed to approximate the feedback strength as the regression coefficient of low-pass-filtered eddy forcing onto the low-pass-filtered annular mode index. When time scales longer than 200 days are used for the low-pass filtering, the new method produces accurate results in the idealized GCM compared to the value calculated from the LRF. The estimated feedback strength in the southern annular mode converges to 0.121 day−1 in reanalysis data using the new method. This work also highlights the significant contribution of medium-scale waves, which have periods less than 2 days, to the annular mode dynamics. Such waves are filtered out if eddy forcing is calculated from daily mean data. The present study provides a framework to quantify the eddy–jet feedback strength in GCMs and reanalysis data.

3. A perspective on climate model hierarchies

   Nadir Jeevanjee, Pedram Hassanzadeh, Spencer Hill, and Aditi Sheshadri

   Journal of Advances in Modeling Earth Systems,

   Abs DOI

   Abstract To understand Earth’s climate, climate modelers employ a hierarchy of climate models spanning a wide spectrum of complexity and comprehensiveness. This essay, inspired by the World Climate Research Programme’s recent “Model Hierarchies Workshop,” attempts to survey and synthesize some of the current thinking on climate model hierarchies, especially as presented at the workshop. We give a few formal descriptions of the hierarchy and survey the various ways it is used to generate, test, and confirm hypotheses. We also discuss some of the pitfalls of contemporary climate modeling, and how the “elegance” advocated for by Held (2005) has (and has not) been used to address them. We conclude with a survey of current activity in hierarchical modeling, and offer suggestions for its continued fruitful development. , Key Points The model hierarchy can be seen as a cartesian product of individual hierarchical axes The hierarchy facilitates the generation and testing of hypotheses The diversity of models still poses issues, which will likely require further model “elegance”

4. Stability of three-dimensional Gaussian vortices in an unbounded, rotating, vertically stratified, Boussinesq flow: linear analysis

   Mani Mahdinia, Pedram Hassanzadeh, Philip S. Marcus, and Chung-Hsiang Jiang

   Journal of Fluid Mechanics,

   Abs DOI

   The linear stability of three-dimensional vortices in rotating, stratified flows has been studied by analysing the non-hydrostatic inviscid Boussinesq equations. We have focused on a widely used model of geophysical and astrophysical vortices, which assumes an axisymmetric Gaussian structure for pressure anomalies in the horizontal and vertical directions. For a range of Rossby numbers ( −0.5<𝑅𝑜<0.5 ) and Burger numbers ( 0.02<𝐵𝑢<2.3 ) relevant to observed long-lived vortices, the growth rate and spatial structure of the most unstable eigenmodes have been numerically calculated and presented as a function of 𝑅𝑜−𝐵𝑢 . We have found neutrally stable vortices only over a small region of the 𝑅𝑜−𝐵𝑢 parameter space: cyclones with 𝑅𝑜∼0.02−0.05 and 𝐵𝑢∼0.85−0.95 . However, we have also found that anticyclones in general have slower growth rates compared to cyclones. In particular, the growth rate of the most unstable eigenmode for anticyclones in a large region of the parameter space (e.g. 𝑅𝑜<0 and 0.5≲𝐵𝑢≲1.3 ) is slower than 50 turnaround times of the vortex (which often corresponds to several years for ocean eddies). For cyclones, the region with such slow growth rates is confined to 0<𝑅𝑜<0.1 and 0.5≲𝐵𝑢≲1.3 . While most calculations have been done for 𝑓/𝑁¯=0.1 (where 𝑓 and 𝑁¯ are the Coriolis and background Brunt–Väisälä frequencies), we have numerically verified and explained analytically, using non-dimensionalized equations, the insensitivity of the results to reducing 𝑓/𝑁¯ to the more ocean-relevant value of 0.01. The results of our stability analysis of Gaussian vortices both support and contradict the findings of earlier studies with QG or multilayer models or with other families of vortices. The results of this paper provide a stepping stone to study the more complicated problems of the stability of geophysical (e.g. those in the atmospheres of giant planets) and astrophysical vortices (in accretion disks).

2016

1. The Linear Response Function of an Idealized Atmosphere. Part I: Construction Using Green’s Functions and Applications

   Pedram Hassanzadeh and Zhiming Kuang

   Journal of the Atmospheric Sciences,

   Abs DOI

   Abstract A linear response function (LRF) determines the mean response of a nonlinear climate system to weak imposed forcings, and an eddy flux matrix (EFM) determines the eddy momentum and heat flux responses to mean-flow changes. Neither LRF nor EFM can be calculated from first principles owing to the lack of a complete theory for turbulent eddies. Here the LRF and EFM for an idealized dry atmosphere are computed by applying numerous localized weak forcings, one at a time, to a GCM with Held–Suarez physics and calculating the mean responses. The LRF and EFM for zonally averaged responses are then constructed using these forcings and responses through matrix inversion. Tests demonstrate that LRF and EFM are fairly accurate. Spectral analysis of the LRF shows that the most excitable dynamical mode, the neutral vector, strongly resembles the model’s annular mode. The framework described here can be employed to compute the LRF and EFM for zonally asymmetric responses and more complex GCMs. The potential applications of the LRF and EFM constructed here are (i) forcing a specified mean flow for hypothesis testing, (ii) isolating/quantifying the eddy feedbacks in complex eddy–mean flow interaction problems, and (iii) evaluating/improving more generally applicable methods currently used to construct LRFs or diagnose eddy feedbacks in comprehensive GCMs or observations. As an example for (iii), in Part II, the LRF is also computed using the fluctuation–dissipation theorem (FDT), and the previously calculated LRF is exploited to investigate why FDT performs poorly in some cases. It is shown that dimension reduction using leading EOFs, which is commonly used to construct LRFs from the FDT, can significantly degrade the accuracy owing to the nonnormality of the operator.

2. The Linear Response Function of an Idealized Atmosphere. Part II: Implications for the Practical Use of the Fluctuation–Dissipation Theorem and the Role of Operator’s Nonnormality

   Pedram Hassanzadeh and Zhiming Kuang

   Journal of the Atmospheric Sciences,

   Abs DOI

   Abstract A linear response function (LRF) relates the mean response of a nonlinear system to weak external forcings and vice versa. Even for simple models of the general circulation, such as the dry dynamical core, the LRF cannot be calculated from first principles owing to the lack of a complete theory for eddy–mean flow feedbacks. According to the fluctuation–dissipation theorem (FDT), the LRF can be calculated using only the covariance and lag-covariance matrices of the unforced system. However, efforts in calculating the LRFs for GCMs using FDT have produced mixed results, and the reason(s) behind the poor performance of the FDT remain(s) unclear. In Part I of this study, the LRF of an idealized GCM, the dry dynamical core with Held–Suarez physics, is accurately calculated using Green’s functions. In this paper (Part II), the LRF of the same model is computed using FDT, which is found to perform poorly for some of the test cases. The accurate LRF of Part I is used with a linear stochastic equation to show that dimension reduction by projecting the data onto the leading EOFs, which is commonly used for FDT, can alone be a significant source of error. Simplified equations and examples of 2 × 2 matrices are then used to demonstrate that this error arises because of the nonnormality of the operator. These results suggest that errors caused by dimension reduction are a major, if not the main, contributor to the poor performance of the LRF calculated using FDT and that further investigations of dimension-reduction strategies with a focus on nonnormality are needed.

2015

1. Blocking variability: Arctic Amplification versus Arctic Oscillation: Blocking: −ao Versus Arctic Amplification

   Pedram Hassanzadeh and Zhiming Kuang

   Geophysical Research Letters,

   DOI
2. ZOMBIE VORTEX INSTABILITY. I. A PURELY HYDRODYNAMIC INSTABILITY TO RESURRECT THE DEAD ZONES OF PROTOPLANETARY DISKS

   Philip S. Marcus, Suyang Pei, Chung-Hsiang Jiang, Joseph A. Barranco, Pedram Hassanzadeh, and Daniel Lecoanet

   The Astrophysical Journal,

   DOI

2014

1. Responses of midlatitude blocks and wave amplitude to changes in the meridional temperature gradient in an idealized dry GCM

   Pedram Hassanzadeh, Zhiming Kuang, and Brian F. Farrell

   Geophysical Research Letters,

   Abs DOI

   Abstract The response of atmospheric blocks and the wave amplitude of midlatitude jets to changes in the midlatitude to pole, near‐surface temperature difference ( Δ T ), is studied using an idealized dry general circulation model (GCM) with Held‐Suarez forcing. Decreasing Δ T results in slower zonal winds, a mean state with reduced meridional gradient of the 500 hPa geopotential height ( Z 500), a smaller variance of Z 500 anomalies, and a robust decrease in blocks and meridional amplitude of waves. Neglecting the decrease of variance associated with reduced Δ T would lead to the incorrect expectation that mean states with smaller Z 500 gradients produce more blocks and higher wave amplitudes. Our results suggest further investigation of the hypothesis that reduced Δ T due to Arctic Amplification would increase blocking events and wave amplitude, hence leading to more midlatitude extreme weather events. , Key Points Robust decline in blocks and waviness as meridional temperature gradient reduces A dynamical link between Arctic warming and weather extremes needs further study Blocks occur in the absence of topography or zonally asymmetric forcings

2. Wall to wall optimal transport

   Pedram Hassanzadeh, Gregory P. Chini, and Charles R. Doering

   Journal of Fluid Mechanics,

   Abs DOI

   The calculus of variations is employed to find steady divergence-free velocity fields that maximize transport of a tracer between two parallel walls held at fixed concentration for one of two constraints on flow strength: a fixed value of the kinetic energy (mean square velocity) or a fixed value of the enstrophy (mean square vorticity). The optimizing flows consist of an array of (convection) cells of a particular aspect ratio 𝛤 . We solve the nonlinear Euler–Lagrange equations analytically for weak flows and numerically – as well as via matched asymptotic analysis in the fixed energy case – for strong flows. We report the results in terms of the Nusselt number Nu , a dimensionless measure of the tracer transport, as a function of the Péclet number Pe , a dimensionless measure of the strength of the flow. For both constraints, the maximum transport NuMAX(Pe) is realized in cells of decreasing aspect ratio 𝛤opt(Pe) as Pe increases. For the fixed energy problem, NuMAX∼Pe and 𝛤opt∼Pe−1/2 , while for the fixed enstrophy scenario, NuMAX∼Pe10/17 and 𝛤opt∼Pe−0.36 . We interpret our results in the context of buoyancy-driven Rayleigh–Bénard convection problems that satisfy the flow intensity constraints, enabling us to investigate how the transport scalings compare with upper bounds on Nu expressed as a function of the Rayleigh number Ra . For steady convection in porous media, corresponding to the fixed energy problem, we find NuMAX∼Ra and 𝛤opt∼Ra−1/2 , while for steady convection in a pure fluid layer between stress-free isothermal walls, corresponding to fixed enstrophy transport, NuMAX∼Ra5/12 and 𝛤opt∼Ra−1/4.

2013

1. Three-Dimensional Vortices Generated by Self-Replication in Stably Stratified Rotating Shear Flows

   Philip S. Marcus, Suyang Pei, Chung-Hsiang Jiang, and Pedram Hassanzadeh

   Physical Review Letters,

   DOI

2012

1. The universal aspect ratio of vortices in rotating stratified flows: theory and simulation

   Pedram Hassanzadeh, Philip S. Marcus, and Patrice Le Gal

   Journal of Fluid Mechanics,

   Abs DOI

   We derive a relationship for the vortex aspect ratio (vertical half-thickness over horizontal length scale) for steady and slowly evolving vortices in rotating stratified fluids, as a function of the Brunt–Väisälä frequencies within the vortex and in the background fluid outside the vortex , the Coriolis parameter and the Rossby number of the vortex: . This relation is valid for cyclones and anticyclones in either the cyclostrophic or geostrophic regimes; it works with vortices in Boussinesq fluids or ideal gases, and the background density gradient need not be uniform. Our relation for has many consequences for equilibrium vortices in rotating stratified flows. For example, cyclones must have ; weak anticyclones (with ) must have ; and strong anticyclones must have . We verify our relation for with numerical simulations of the three-dimensional Boussinesq equations for a wide variety of vortices, including: vortices that are initially in (dissipationless) equilibrium and then evolve due to an imposed weak viscous dissipation or density radiation; anticyclones created by the geostrophic adjustment of a patch of locally mixed density; cyclones created by fluid suction from a small localized region; vortices created from the remnants of the violent breakups of columnar vortices; and weakly non-axisymmetric vortices. The values of the aspect ratios of our numerically computed vortices validate our relationship for , and generally they differ significantly from the values obtained from the much-cited conjecture that in quasi-geostrophic vortices.

2009

1. The Efficient Iterative Solution of the P1 Equation

   P. Hassanzadeh and G. D. Raithby

   Journal of Heat Transfer,

   Abs DOI

   The P1 model is often used to obtain approximate solutions of the radiative transfer equation for heat transfer in a participating medium. For large problems, the algebraic equations used to obtain the P1 solution are solved by iteration, and the convergence rate can be very slow. This paper compares the performance of the corrective acceleration scheme of and Li and Modest (2002, “A Method to Accelerate Convergence and to Preserve Radiative Energy Balance in Solving the P1 Equation by Iterative Methods,” ASME J. Heat Transfer, 124, pp. 580–582), and the additive correction multigrid method, to that of the Gauss–Seidel solver alone. Additive correction multigrid is found to outperform the other solvers. Hence, multigrid is a superior solver for the P1 equation.

2008

1. Efficient Calculation of Radiation Heat Transfer in Participating Media

   P. Hassanzadeh, G. D. Raithby, and E. H. Chui

   Journal of Thermophysics and Heat Transfer,

   DOI
2. Finite-Volume Solution of the Second-Order Radiative Transfer Equation: Accuracy and Solution Cost

   P. Hassanzadeh and G. D. Raithby

   Numerical Heat Transfer, Part B: Fundamentals,

   DOI