Abstract: Parameters in climate models are usually calibrated manually, exploiting only small subsets of the available data. This precludes an optimal calibration and quantification of uncertainties. Traditional Bayesian calibration methods that allow uncertainty quantification are too expensive for climate models; they are also not robust in the presence of internal climate variability. For example, Markov chain Monte Carlo (MCMC) methods typically require O(10⁵) model runs, rendering them infeasible for climate models. Here we demonstrate an approach to model calibration and uncertainty quantification that requires only O(10²) model runs and can accommodate internal climate variability. The approach consists of three stages: (i) a calibration stage uses variants of ensemble Kalman inversion to calibrate a model by minimizing mismatches between model and data statistics; (ii) an emulation stage emulates the parameter-to-data map with Gaussian processes (GP), using the model runs in the calibration stage for training; (iii) a sampling stage approximates the Bayesian posterior distributions by using the GP emulator and then samples using MCMC. We demonstrate the feasibility and computational efficiency of this calibrate-emulate-sample (CES) approach in a perfect-model setting. Using an idealized general circulation model, we estimate parameters in a simple convection scheme from data surrogates generated with the model. The CES approach generates probability distributions of the parameters that are good approximations of the Bayesian posteriors, at a fraction of the computational cost usually required to obtain them. Sampling from this approximate posterior allows the generation of climate predictions with quantified parametric uncertainties.

ID: CaltechAUTHORS:20210113-143919927

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Abstract: Convection parameterizations such as eddy-diffusivity mass-flux (EDMF) schemes require a consistent closure formulation for the perturbation pressure, which arises in the equations for vertical momentum and turbulence kinetic energy (TKE). Here we derive an expression for the perturbation pressure from approximate analytical solutions for 2D and 3D rising thermal bubbles. The new closure combines a modified pressure drag and virtual mass effects with a new momentum advection term. This momentum advection is an important source in the lower half of the thermal bubble and at cloud base levels in convective systems. It represents the essential physics of the perturbation pressure, that is, to ensure the 3D non-divergent properties of the flow. Moreover, the new formulation modifies the pressure drag to be inversely proportional to updraft depth. This is found to significantly improve simulations of the diurnal cycle of deep convection, without compromising simulations of shallow convection. It is thus a key step toward a unified scheme for a range of convective motions. By assuming that the pressure only redistributes TKE between plumes and the environment, rather than vertically, a closure for the velocity pressure-gradient correlation is obtained from the perturbation pressure closure. This novel pressure closure is implemented in an extended EDMF scheme and is shown to successfully simulate a rising bubble test case as well as shallow and deep convection cases in a single column model.

ID: CaltechAUTHORS:20201204-110354763

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Abstract: Clouds cover on average nearly 70% of Earth’s surface and are important for the global albedo. The magnitude of the shortwave reflection by clouds depends on their location, optical properties, and 3D structure. Earth system models are unable to perform 3D radiative transfer calculations and thus partially neglect the effect of cloud morphology on albedo. We show how the resulting radiative flux bias depends on cloud morphology and solar zenith angle. Using large-eddy simulations to produce 3D cloud fields, a Monte Carlo code for 3D radiative transfer, and observations of cloud climatology, we estimate the effect of this flux bias on global climate. The flux bias is largest at small zenith angles and for deeper clouds, while the albedo bias is largest (and negative) for large zenith angles. Globally, the radiative flux bias is estimated to be 1.6 W m⁻² and locally can be on the order of 5 W m⁻².

ID: CaltechAUTHORS:20201023-133020582

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Abstract: Enforcing sparse structure within learning has led to significant advances in the field of data-driven discovery of dynamical systems. However, such methods require access not only to time-series of the state of the dynamical system, but also to the time derivative. In many applications, the data are available only in the form of time-averages such as moments and autocorrelation functions. We propose a sparse learning methodology to discover the vector fields defining a (possibly stochastic or partial) differential equation, using only time-averaged statistics. Such a formulation of sparse learning naturally leads to a nonlinear inverse problem to which we apply the methodology of ensemble Kalman inversion (EKI). EKI is chosen because it may be formulated in terms of the iterative solution of quadratic optimization problems; sparsity is then easily imposed. We then apply the EKI-based sparse learning methodology to various examples governed by stochastic differential equations (a noisy Lorenz 63 system), ordinary differential equations (Lorenz 96 system and coalescence equations), and a partial differential equation (the Kuramoto-Sivashinsky equation). The results demonstrate that time-averaged statistics can be used for data-driven discovery of differential equations using sparse EKI. The proposed sparse learning methodology extends the scope of data-driven discovery of differential equations to previously challenging applications and data-acquisition scenarios.

Publication: arXiv
ID: CaltechAUTHORS:20201109-141011032

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Abstract: The uncertainty in polar cloud feedbacks calls for process understanding of the cloud response to climate warming. As an initial step, we investigate the seasonal cycle of polar clouds in the current climate by adopting a novel modeling framework using large eddy simulations (LES), which explicitly resolve cloud dynamics. Resolved horizontal and vertical advection of heat and moisture from an idealized GCM are prescribed as forcing in the LES. The LES are also forced with prescribed sea ice thickness, but surface temperature, atmospheric temperature, and moisture evolve freely without nudging. A semigray radiative transfer scheme, without water vapor or cloud feedbacks, allows the GCM and LES to achieve closed energy budgets more easily than would be possible with more complex schemes; this allows the mean states in the two models to be consistently compared, without the added complications from interaction with more comprehensive radiation. We show that the LES closely follow the GCM seasonal cycle, and the seasonal cycle of low clouds in the LES resembles observations: maximum cloud liquid occurs in late summer and early autumn, and winter clouds are dominated by ice in the upper troposphere. Large-scale advection of moisture provides the main source of water vapor for the liquid clouds in summer, while a temperature advection peak in winter makes the atmosphere relatively dry and reduces cloud condensate. The framework we develop and employ can be used broadly for studying cloud processes and the response of polar clouds to climate warming.

ID: CaltechAUTHORS:20201027-125955966

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Abstract: Although the governing equations of many systems, when derived from first principles, may be viewed as known, it is often too expensive to numerically simulate all the interactions within the first principles description. Therefore researchers often seek simpler descriptions that describe complex phenomena without numerically resolving all the interacting components. Stochastic differential equations (SDEs) arise naturally as models in this context. The growth in data acquisition provides an opportunity for the systematic derivation of SDE models in many disciplines. However, inconsistencies between SDEs and real data at small time scales often cause problems, when standard statistical methodology is applied to parameter estimation. The incompatibility between SDEs and real data can be addressed by deriving sufficient statistics from the time-series data and learning parameters of SDEs based on these. Following this approach, we formulate the fitting of SDEs to sufficient statistics from real data as an inverse problem and demonstrate that this inverse problem can be solved by using ensemble Kalman inversion (EKI). Furthermore, we create a framework for non-parametric learning of drift and diffusion terms by introducing hierarchical, refineable parameterizations of unknown functions, using Gaussian process regression. We demonstrate the proposed methodology for the fitting of SDE models, first in a simulation study with a noisy Lorenz 63 model, and then in other applications, including dimension reduction starting from various deterministic chaotic systems arising in the atmospheric sciences, large-scale pattern modeling in climate dynamics, and simplified models for key observables arising in molecular dynamics. The results confirm that the proposed methodology provides a robust and systematic approach to fitting SDE models to real data.

Publication: arXiv
ID: CaltechAUTHORS:20201109-140955956

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