A waveform skewness index for measuring time series nonlinearity and its applications to the ENSO–Indian monsoon relationship

Many geophysical time series possess nonlinear characteristics that reflect the underlying physics of the phenomena the time series describe. The nonlinear character of times series can change with time, so it is important to quantify time series nonlinearity without assuming stationarity. A common way of quantifying the time evolution of time series nonlinearity is to compute sliding skewness time series, but it is shown here that such an approach can be misleading when time series contain periodicities. To remedy this deficiency of skewness, a new waveform skewness index is proposed for quantifying local nonlinearities embedded in time series. A waveform skewness spectrum is proposed for determining the frequency components that are contributing to time series waveform skewness. The new methods are applied to the El Niño–Southern Oscillation (ENSO) and the Indian monsoon to test a recently proposed hypothesis that states that changes in the ENSO–Indian monsoon relationship are related to ENSO nonlinearity. We show that the ENSO–Indian rainfall relationship weakens during time periods of high ENSO waveform skewness. The results from two different analyses suggest that the breakdown of the ENSO–Indian monsoon relationship during time periods of high ENSO waveform skewness is related to the more frequent occurrence of strong central Pacific El Niño events, supporting arguments that changes in the ENSO–Indian rainfall relationship are not solely related to noise.

Inferring the instability of a dynamical system from the skill of data assimilation exercises

Data assimilation (DA) aims at optimally merging observational data and model outputs to create a coherent statistical and dynamical picture of the system under investigation. Indeed, DA aims at minimizing the effect of observational and model error and at distilling the correct ingredients of its dynamics. DA is of critical importance for the analysis of systems featuring sensitive dependence on the initial conditions, as chaos wins over any finitely accurate knowledge of the state of the system, even in absence of model error. Clearly, the skill of DA is guided by the properties of dynamical system under investigation, as merging optimally observational data and model outputs is harder when strong instabilities are present. In this paper we reverse the usual angle on the problem and show that it is indeed possible to use the skill of DA to infer some basic properties of the tangent space of the system, which may be hard to compute in very high-dimensional systems. Here, we focus our attention on the first Lyapunov exponent and the Kolmogorov–Sinai entropy and perform numerical experiments on the Vissio–Lucarini 2020 model, a recently proposed generalization of the Lorenz 1996 model that is able to describe in a simple yet meaningful way the interplay between dynamical and thermodynamical variables.

Brief communication: Lower-bound estimates for residence time of energy in the atmospheres of Venus, Mars and Titan

The residence time of energy in a planetary atmosphere, τ, which was recently introduced and computed for the Earth's atmosphere (Osácar et al., 2020), is here extended to the atmospheres of Venus, Mars and Titan. τ is the timescale for the energy transport across the atmosphere. In the cases of Venus, Mars and Titan, these computations are lower bounds due to a lack of some energy data. If the analogy between τ and the solar Kelvin–Helmholtz scale is assumed, then τ would also be the time the atmosphere needs to return to equilibrium after a global thermal perturbation.

Reduced non-Gaussianity by 30 s rapid update in convective-scale numerical weather prediction

Non-Gaussian forecast error is a challenge for ensemble-based data assimilation (DA), particularly for more nonlinear convective dynamics. In this study, we investigate the degree of the non-Gaussianity of forecast error distributions at 1 km resolution using a 1000-member ensemble Kalman filter, and how it is affected by the DA update frequency and observation number. Regional numerical weather prediction experiments are performed with the SCALE (Scalable Computing for Advanced Library and Environment) model and the LETKF (local ensemble transform Kalman filter) assimilating phased array radar observations every 30 s. The results show that non-Gaussianity develops rapidly within convective clouds and is sensitive to the DA frequency and the number of assimilated observations. The non-Gaussianity is reduced by up to 40 % when the assimilation window is shortened from 5 min to 30 s, particularly for vertical velocity and radar reflectivity.

Multivariate localization functions for strongly coupled data assimilation in the bivariate Lorenz 96 system

Localization is widely used in data assimilation schemes to mitigate the impact of sampling errors on ensemble-derived background error covariance matrices. Strongly coupled data assimilation allows observations in one component of a coupled model to directly impact another component through the inclusion of cross-domain terms in the background error covariance matrix. When different components have disparate dominant spatial scales, localization between model domains must properly account for the multiple length scales at play. In this work, we develop two new multivariate localization functions, one of which is a multivariate extension of the fifth-order piecewise rational Gaspari–Cohn localization function; the within-component localization functions are standard Gaspari–Cohn with different localization radii, while the cross-localization function is newly constructed. The functions produce positive semidefinite localization matrices which are suitable for use in both Kalman filters and variational data assimilation schemes. We compare the performance of our two new multivariate localization functions to two other multivariate localization functions and to the univariate and weakly coupled analogs of all four functions in a simple experiment with the bivariate Lorenz 96 system. In our experiments, the multivariate Gaspari–Cohn function leads to better performance than any of the other multivariate localization functions.

Identification of linear response functions from arbitrary perturbation experiments in the presence of noise – Part 1: Method development and toy model demonstration

Existent methods to identify linear response functions from data require tailored perturbation experiments, e.g., impulse or step experiments, and if the system is noisy, these experiments need to be repeated several times to obtain good statistics. In contrast, for the method developed here, data from only a single perturbation experiment at arbitrary perturbation are sufficient if in addition data from an unperturbed (control) experiment are available. To identify the linear response function for this ill-posed problem, we invoke regularization theory. The main novelty of our method lies in the determination of the level of background noise needed for a proper estimation of the regularization parameter: this is achieved by comparing the frequency spectrum of the perturbation experiment with that of the additional control experiment. The resulting noise-level estimate can be further improved for linear response functions known to be monotonic. The robustness of our method and its advantages are investigated by means of a toy model. We discuss in detail the dependence of the identified response function on the quality of the data (signal-to-noise ratio) and on possible nonlinear contributions to the response. The method development presented here prepares in particular for the identification of carbon cycle response functions in Part 2 of this study (Torres Mendonça et al., 2021a). However, the core of our method, namely our new approach to obtaining the noise level for a proper estimation of the regularization parameter, may find applications in also solving other types of linear ill-posed problems.

Identification of linear response functions from arbitrary perturbation experiments in the presence of noise – Part 2: Application to the land carbon cycle in the MPI Earth System Model

The response function identification method introduced in the first part of this study is applied here to investigate the land carbon cycle in the Max Planck Institute for Meteorology Earth System Model. We identify from standard C4MIP 1 % experiments the linear response functions that generalize the land carbon sensitivities β and γ. The identification of these generalized sensitivities is shown to be robust by demonstrating their predictive power when applied to experiments not used for their identification. The linear regime for which the generalized framework is valid is estimated, and approaches to improve the quality of the results are proposed. For the generalized γ sensitivity, the response is found to be linear for temperature perturbations until at least 6 K. When this sensitivity is identified from a 2×CO2 experiment instead of the 1 % experiment, its predictive power improves, indicating an enhancement in the quality of the identification. For the generalized β sensitivity, the linear regime is found to extend up to CO2 perturbations of 100 ppm. We find that nonlinearities in the β response arise mainly from the nonlinear relationship between net primary production and CO2. By taking as forcing the resulting net primary production instead of CO2, the response is approximately linear until CO2 perturbations of about 850 ppm. Taking net primary production as forcing also substantially improves the spectral resolution of the generalized β sensitivity. For the best recovery of this sensitivity, we find a spectrum of internal timescales with two peaks, at 4 and 100 years. Robustness of this result is demonstrated by two independent tests. We find that the two-peak spectrum can be explained by the different characteristic timescales of functionally different elements of the land carbon cycle. The peak at 4 years results from the collective response of carbon pools whose dynamics is governed by fast processes, namely pools representing living vegetation tissues (leaves, fine roots, sugars, and starches) and associated litter. The peak at 100 years results from the collective response of pools whose dynamics is determined by slow processes, namely the pools that represent the wood in stem and coarse roots, the associated litter, and the soil carbon (humus). Analysis of the response functions that characterize these two groups of pools shows that the pools with fast dynamics dominate the land carbon response only for times below 2 years. For times above 25 years the response is completely determined by the pools with slow dynamics. From 100 years onwards only the humus pool contributes to the land carbon response.

A study of capturing Atlantic meridional overturning circulation (AMOC) regime transition through observation-constrained model parameters

The multiple equilibria are an outstanding characteristic of the Atlantic meridional overturning circulation (AMOC) that has important impacts on the Earth climate system appearing as regime transitions. The AMOC can be simulated in different models, but the behavior deviates from the real world due to the existence of model errors. Here, we first combine a general AMOC model with an ensemble Kalman filter to form an ensemble coupled model data assimilation and parameter estimation (CDAPE) system and derive the general methodology to capture the observed AMOC regime transitions through utilization of observational information. Then we apply this methodology designed within a “twin” experiment framework with a simple conceptual model that simulates the transition phenomenon of AMOC multiple equilibria as well as a more physics-based MOC box model to reconstruct the “observed” AMOC multiple equilibria. The results show that the coupled model parameter estimation with observations can significantly mitigate the model deviations, thus capturing regime transitions of the AMOC. This simple model study serves as a guideline when a coupled general circulation model is used to incorporate observations to reconstruct the AMOC historical states and make multi-decadal climate predictions.

Calibrated ensemble forecasts of the height of new snow using quantile regression forests and ensemble model output statistics

Height of new snow (HN) forecasts help to prevent critical failures of infrastructures in mountain areas, e.g. transport networks and ski resorts. The French national meteorological service, Météo-France, operates a probabilistic forecasting system based on ensemble meteorological forecasts and a detailed snowpack model to provide ensembles of HN forecasts. These forecasts are, however, biased and underdispersed. As for many weather variables, post-processing methods can be used to alleviate these drawbacks and obtain meaningful 1 to 4 d HN forecasts. In this paper, we compare the skill of two post-processing methods. The first approach is an ensemble model output statistics (EMOS) method, which can be described as a nonhomogeneous regression with a censored shifted Gamma distribution. The second approach is based on quantile regression forests, using different meteorological and snow predictors. Both approaches are evaluated using a 22 year reforecast. Thanks to a larger number of predictors, the quantile regression forest is shown to be a powerful alternative to EMOS for the post-processing of HN ensemble forecasts. The gain of performance is large in all situations but is particularly marked when raw forecasts completely miss the snow event. This type of situation happens when the rain–snow transition elevation is overestimated by the raw forecasts (rain instead of snow in the raw forecasts) or when there is no precipitation in the forecast. In that case, quantile regression forests improve the predictions using the other weather predictors (wind, temperature, and specific humidity).

Enhanced diapycnal mixing with polarity-reversing internal solitary waves revealed by seismic reflection data

Shoaling internal solitary waves near the Dongsha Atoll in the South China Sea dissipate their energy and enhance diapycnal mixing, which have an important impact on the oceanic environment and primary productivity. The enhanced diapycnal mixing is patchy and instantaneous. Evaluating its spatiotemporal distribution requires comprehensive observation data. Fortunately, seismic oceanography meets the requirements, thanks to its high spatial resolution and large spatial coverage. In this paper, we studied three internal solitary waves in reversing polarity near the Dongsha Atoll and calculated their spatial distribution of diapycnal diffusivity. Our results show that the average diffusivities along three survey lines are 2 orders of magnitude larger than the open-ocean value. The average diffusivity in internal solitary waves with reversing polarity is 3 times that of the non-polarity reversal region. The diapycnal diffusivity is higher at the front of one internal solitary wave and gradually decreases from shallow to deep water in the vertical direction. Our results also indicate that (1) the enhanced diapycnal diffusivity is related to reflection seismic events, (2) convective instability and shear instability may both contribute to the enhanced diapycnal mixing in the polarity-reversing process, and (3) the difference between our results and Richardson-number-dependent turbulence parameterizations is about 2–3 orders of magnitude, but its vertical distribution is almost the same.