Results for “ensemble dynamics” — Page 3

NASA NTRSDate not supplied

Uncertainty Quantification using Deep Ensembles for Decision Making in Cyber-Physical-Human Systems

In this paper and its companion, Differential Equation Approximation Using Gradient-Boosted Quantile Regression, Robison et al., we examine an approach to quantifying model uncertainty with the aim of increasing the trustworthiness of computational models in human-machine interactions. In Differential Equation Approximation Using Gradient-Boosted Quantile Regression, we focus on gradient-boosted decision trees, while in this one, we give more details about deep ensembles. Uncertainty quantification is crucial for building trustworthy autonomous decision-making agents in human-machine teams. There are two types of uncertainties: aleatoric and epistemic. The former is related to the inherent stochasticity (noise) of the process, whereas the latter is associated with the lack of knowledge or representation capability of models, such as neural networks. By lack of knowledge, we mean the model’s inability to accurately predict outputs for all possible inputs. The aleatory uncertainty can be estimated fairly easily with, for example, filters, whereas epistemic uncertainty is challenging to compute. This paper uses deep ensembles to quantify both aleatory and epistemic uncertainty. It can act as an uncertainty-aware surrogate transition model for decision-making frameworks. "Uncertainty-aware" means that the surrogate transition model should make predictions along with confidence in those predictions. In the context of decision-making, the transition models are ordinary differential equations (ODEs). Since ODEs can be simulated to make one-step or multi-step predictions, a good surrogate model for them should perform reasonably well in both modes. In a multi-step approach, the trajectory sampling method TS∞ was used to propagate uncertainty over multiple steps. The cartpole dynamical system was selected to demonstrate the ability of deep ensembles as good surrogate transition models for decision-making frameworks. The deep ensembles modeled the dynamics of cartpole ODEs and made uncertainty-aware predictions in single-step and multi-step transition modes.

CPH systems↗

NASA NTRS1992

Persistence and predictability in a perfect model

A realistic two-level GCM is used to examine the relationship between predictability and persistence. Predictability is measured by the average divergence of ensembles of solutions starting from perturbed initial conditions, and persistence is defined in terms of the autocorrelation function based on a single long-term model integration. The average skill of the dynamical forecasts is compared with the skill of simple persistence-based statistical forecasts. For initial errors comparable in magnitude to present-day analysis errors, the statistical forecast loses all skill after about one week, reflecting the lifetime of the lowest frequency fluctuations in the model. Large ensemble mean dynamical forecasts would be expected to remain skillful for about 3 wk. The disparity between the skill of the statistical and dynamical forecasts is greater for the higher frequency modes, which have little memory beyond 1 d, yet remain predictable for about 2 wk. The results are analyzed in terms of two characteristic time scales.

Schubert, Siegfried D.↗

NASA NTRS2011

Lessons Learned from Assimilating Altimeter Data into a Coupled General Circulation Model with the GMAO Augmented Ensemble Kalman Filter

Satellite altimetry measurements have provided global, evenly distributed observations of the ocean surface since 1993. However, the difficulties introduced by the presence of model biases and the requirement that data assimilation systems extrapolate the sea surface height (SSH) information to the subsurface in order to estimate the temperature, salinity and currents make it difficult to optimally exploit these measurements. This talk investigates the potential of the altimetry data assimilation once the biases are accounted for with an ad hoc bias estimation scheme. Either steady-state or state-dependent multivariate background-error covariances from an ensemble of model integrations are used to address the problem of extrapolating the information to the sub-surface. The GMAO ocean data assimilation system applied to an ensemble of coupled model instances using the GEOS-5 AGCM coupled to MOM4 is used in the investigation. To model the background error covariances, the system relies on a hybrid ensemble approach in which a small number of dynamically evolved model trajectories is augmented on the one hand with past instances of the state vector along each trajectory and, on the other, with a steady state ensemble of error estimates from a time series of short-term model forecasts. A state-dependent adaptive error-covariance localization and inflation algorithm controls how the SSH information is extrapolated to the sub-surface. A two-step predictor corrector approach is used to assimilate future information. Independent (not-assimilated) temperature and salinity observations from Argo floats are used to validate the assimilation. A two-step projection method in which the system first calculates a SSH increment and then projects this increment vertically onto the temperature, salt and current fields is found to be most effective in reconstructing the sub-surface information. The performance of the system in reconstructing the sub-surface fields is particularly impressive for temperature, but not as satisfactory for salt.

Keppenne, Christian↗

NASA NTRS2014

Seasonal Drought Prediction in East Africa: Can National Multi-Model Ensemble Forecasts Help?

The increasing food and water demands of East Africa's growing population are stressing the region's inconsistent water resources and rain-fed agriculture. As recently as in 2011 part of this region underwent one of the worst famine events in its history. Timely and skillful drought forecasts at seasonal scale for this region can inform better water and agro-pastoral management decisions, support optimal allocation of the region's water resources, and mitigate socio-economic losses incurred by droughts. However seasonal drought prediction in this region faces several challenges. Lack of skillful seasonal rainfall forecasts; the focus of this presentation, is one of those major challenges. In the past few decades, major strides have been taken towards improvement of seasonal scale dynamical climate forecasts. The National Centers for Environmental Prediction's (NCEP) National Multi-model Ensemble (NMME) is one such state-of-the-art dynamical climate forecast system. The NMME incorporates climate forecasts from 6+ fully coupled dynamical models resulting in 100+ ensemble member forecasts. Recent studies have indicated that in general NMME offers improvement over forecasts from any single model. However thus far the skill of NMME for forecasting rainfall in a vulnerable region like the East Africa has been unexplored. In this presentation we report findings of a comprehensive analysis that examines the strength and weakness of NMME in forecasting rainfall at seasonal scale in East Africa for all three of the prominent seasons for the region. (i.e. March-April-May, July-August-September and October-November- December). Simultaneously we also describe hybrid approaches; that combine statistical approaches with NMME forecasts; to improve rainfall forecast skill in the region when raw NMME forecasts lack in skill.

Shukla, Shraddhanand↗

NASA NTRS2015

Seasonal Drought Prediction in East Africa: Can National Multi-Model Ensemble Forecasts Help?

The increasing food and water demands of East Africa's growing population are stressing the region's inconsistent water resources and rain-fed agriculture. As recently as in 2011 part of this region underwent one of the worst famine events in its history. Timely and skillful drought forecasts at seasonal scale for this region can inform better water and agro-pastoral management decisions, support optimal allocation of the region's water resources, and mitigate socio-economic losses incurred by droughts. However seasonal drought prediction in this region faces several challenges. Lack of skillful seasonal rainfall forecasts; the focus of this presentation, is one of those major challenges. In the past few decades, major strides have been taken towards improvement of seasonal scale dynamical climate forecasts. The National Centers for Environmental Prediction's (NCEP) National Multi-model Ensemble (NMME) is one such state-of-the-art dynamical climate forecast system. The NMME incorporates climate forecasts from 6+ fully coupled dynamical models resulting in 100+ ensemble member forecasts. Recent studies have indicated that in general NMME offers improvement over forecasts from any single model. However thus far the skill of NMME for forecasting rainfall in a vulnerable region like the East Africa has been unexplored. In this presentation we report findings of a comprehensive analysis that examines the strength and weakness of NMME in forecasting rainfall at seasonal scale in East Africa for all three of the prominent seasons for the region. (i.e. March-April-May, July-August-September and October-November- December). Simultaneously we also describe hybrid approaches; that combine statistical approaches with NMME forecasts; to improve rainfall forecast skill in the region when raw NMME forecasts lack in skill.

Shukla, Shraddhanand↗

NASA NTRS1992

Dynamically stratified Monte Carlo forecasting

A new method for performing Monte Carlo forecasts is introduced. The method, called dynamic stratification, selects initial perturbations based on a stratification of the error distribution. A simple implementation is presented in which the error distribution used for the stratification is estimated from a linear model derived from a large ensemble of 12-h forecasts with the full dynamic model. The stratification thus obtained is used to choose a small subsample of initial states with which to perform the dynamical Monte Carlo forecasts. Several test cases are studied using a simple two-level general circulation model with uncertain initial conditions. It is found that the method provides substantial reductions in the sampling error of the forecast mean and variance when compared to the more traditional approach of choosing the initial perturbations at random. The degree of improvement, however, is sensitive to the nature of the initial error distribution and to the base state. In practice the method may be viable only if the computational burden involved in obtaining an adequate estimate of the error distribution is shared with the data-assimilation procedure.

Schubert, Siegfried↗

NASA NTRS2004

Random Matrix Approach to Quantum Adiabatic Evolution Algorithms

We analyze the power of quantum adiabatic evolution algorithms (Q-QA) for solving random NP-hard optimization problems within a theoretical framework based on the random matrix theory (RMT). We present two types of the driven RMT models. In the first model, the driving Hamiltonian is represented by Brownian motion in the matrix space. We use the Brownian motion model to obtain a description of multiple avoided crossing phenomena. We show that the failure mechanism of the QAA is due to the interaction of the ground state with the "cloud" formed by all the excited states, confirming that in the driven RMT models. the Landau-Zener mechanism of dissipation is not important. We show that the QAEA has a finite probability of success in a certain range of parameters. implying the polynomial complexity of the algorithm. The second model corresponds to the standard QAEA with the problem Hamiltonian taken from the Gaussian Unitary RMT ensemble (GUE). We show that the level dynamics in this model can be mapped onto the dynamics in the Brownian motion model. However, the driven RMT model always leads to the exponential complexity of the algorithm due to the presence of the long-range intertemporal correlations of the eigenvalues. Our results indicate that the weakness of effective transitions is the leading effect that can make the Markovian type QAEA successful.

Boulatov, Alexei↗

NASA NTRS2022

A Process-Model Perspective on Recent Changes in the Carbon Cycle of North America

Continental North America has been found to be a carbon (C) sink over recent decades by multiple studies employing a variety of estimation approaches. However, several key questions and uncertainties remain with these assessments. Here we used results from an ensemble of 19 state-of-the-art dynamic global vegetation models from the TRENDYv9 project to improve these estimates and study the drivers of its interannual variability. Our results show that North America has been a C sink with a magnitude of 0.37 ± 0.38 (mean and one standard deviation) PgC year−1 for the period 2000–2019 (0.31 and 0.44 PgC year−1 in each decade); split into 0.18 ± 0.12 PgC year−1 in Canada (0.15 and 0.20), 0.16 ± 0.17 in the United States (0.14 and 0.17), 0.02 ± 0.05 PgC year−1 in Mexico (0.02 and 0.02) and 0.01 ± 0.02 in Central America and the Caribbean (0.01 and 0.01). About 57% of the new C assimilated by terrestrial ecosystems is allocated into vegetation, 30% into soils, and 13% into litter. Losses of C due to fire account for 41% of the interannual variability of the mean net biome productivity for all North America in the model ensemble. Finally, we show that drought years (e.g., 2002) have the potential to shift the region to a small net C source in the simulations (−0.02 ± 0.46 PgC year−1). Our results highlight the importance of identifying the major drivers of the interannual variability of the continental-scale land C cycle along with the spatial distribution of local sink-source dynamics.

Guillermo Murray-Tortarolo↗

NASA NTRS2017

Cross - Scale Intercomparison of Climate Change Impacts Simulated by Regional and Global Hydrological Models in Eleven Large River Basins

Ideally, the results from models operating at different scales should agree in trend direction and magnitude of impacts under climate change. However, this implies that the sensitivity to climate variability and climate change is comparable for impact models designed for either scale. In this study, we compare hydrological changes simulated by 9 global and 9 regional hydrological models (HM) for 11 large river basins in all continents under reference and scenario conditions. The foci are on model validation runs, sensitivity of annual discharge to climate variability in the reference period, and sensitivity of the long-term average monthly seasonal dynamics to climate change. One major result is that the global models, mostly not calibrated against observations, often show a considerable bias in mean monthly discharge, whereas regional models show a better reproduction of reference conditions. However, the sensitivity of the two HM ensembles to climate variability is in general similar. The simulated climate change impacts in terms of long-term average monthly dynamics evaluated for HM ensemble medians and spreads show that the medians are to a certain extent comparable in some cases, but have distinct differences in other cases, and the spreads related to global models are mostly notably larger. Summarizing, this implies that global HMs are useful tools when looking at large-scale impacts of climate change and variability. Whenever impacts for a specific river basin or region are of interest, e.g. for complex water management applications, the regional-scale models calibrated and validated against observed discharge should be used.

Hydrological models↗

NASA NTRS2018

Impact of Satellite Sea Surface Salinity Observations on ENSO Predictions from the GMAO S2S Forecast System

El Nino/Southern Oscillation (ENSO) has far reaching global climatic impacts and so extending useful ENSO forecasts would be of great benefit for society. However, one key variable that has yet to be fully exploited within coupled forecast systems is accurate estimation of near-surface ocean density. Satellite Sea Surface Salinity (SSS), combined with temperature, help to identify ocean density changes and associated mixing near the ocean surface. We assess the impact of satellite SSS observations for improving near-surface dynamics within ocean analyses and how these impact dynamical ENSO forecasts using the NASA GMAO (Global Modeling and Assimilation Office) Sub-seasonal to Seasonal (S2S_v2.1) coupled forecast system (Molod et al. 2018 - i.e. NASA's contribution to the NMME (National Multi-Model Ensemble) project). For all initialization experiments, all available along-track absolute dynamic topography and in situ observations are assimilated using the LETKF ( Local Ensemble Transform Kalman Filter) scheme similar to Penny et al., 2013. A separate reanalysis additionally assimilates Aquarius V5 (September 2011 to June 2015) and SMAP (Soil Moisture Active Passive) V4 (March 2015 to present) along-track data.We highlight the impact of satellite SSS on ocean reanalyses by comparing validation statistics of experiments that assimilate SSS versus our current prediction system that withholds SSS. We find that near-surface validation versus observed statistics for salinity are slightly degraded when assimilating SSS. This is an expected result due to known biases between SSS (measured by the satellite at approximately 1 centimeter) and in situ measurements (typically measured by Argo floats at 3 meters). On the other hand, a very encouraging result is that both temperature, absolute dynamic topography, and mixed layer statistics are improved with SSS assimilation. Previous work has shown that correcting near-surface density structure via gridded SSS assimilation can improve coupled forecasts. Here we present results of coupled forecasts that are initialized from the GMAO S2S reanalyses that assimilates/withholds along-track (L2) SSS. In particular, we contrast forecasts of the overestimated 2014 El Nino, the big 2015 El Nino, and the minor 2016 La Nina. For each of these ENSO scenarios, assimilation of satellite SSS improves the forecast validation. Improved SSS and density upgrades the mixed layer depth leading to more accurate coupled air/sea interaction.

Hackert, E.↗

NASA NTRS2019

Impact of Satellite Sea Surface Salinity Observations on ENSO Predictions from the GMAO Seasonal Forecast System

El Nino/Southern Oscillation (ENSO) has far reaching global climatic impacts and so extending useful ENSO forecasts would be of great benefit for society. However, one key variable that has yet to be fully exploited within coupled forecast systems is accurate estimation of near-surface ocean density. Satellite sea surface salinity (SSS), combined with temperature, help to identify ocean density changes and associated mixing near the ocean surface. We assess the impact of satellite SSS observations for improving near-surface dynamics within ocean analyses and how these impact dynamical ENSO forecasts using the NASA GMAO (Global Modeling and Assimilation Office) Sub-seasonal to Seasonal (S2S_v2.1) coupled forecast system (Molod et al. 2018 - i.e. NASA's contribution to the NMME (North American Multi-Model Ensemble) project). For all initialization experiments, all available along-track absolute dynamic topography and in situ observations are assimilated using the LETKF (Local Ensemble Transform Kalman Filter) scheme similar to Penny et al., 2013. A separate reanalysis additionally assimilates Aquarius V5 (September 2011 to June 2015) and SMAP (Soil Moisture Active Passive satellite) V4.1 (March 2015 to present) along-track data.We highlight the impact of satellite SSS on ocean reanalyses by comparing validation statistics of experiments that assimilate SSS versus our current prediction system that withholds SSS. We find that near-surface validation versus observed statistics for salinity are slightly degraded when assimilating SSS. This is an expected result due to known biases between SSS (measured by satellite at approximately 1-centimeter depth) and in situ measurements (typically measured by Argo floats at 3-meters depth). On the other hand, a very encouraging result is that both temperature, absolute dynamic topography, and mixed layer statistics are improved with SSS assimilation. Previous work has shown that correcting near-surface density structure via gridded SSS assimilation can improve coupled forecasts. Here we present results of coupled forecasts that are initialized from GMAO S2S spring reanalyses that assimilate/withhold along-track (L2) SSS. In particular, we contrast forecasts of the big 2015 El Nino, the 2017 La Nina and the 2018 weak El Nino. For each of these ENSO scenarios, assimilation of satellite SSS improves the forecast validation. Improved SSS and density upgrade the mixed layer depth leading to more accurate coupled air/sea interaction. From March to June 2015, the availability of two overlapping satellite SSS instruments, Aquarius and SMAP, allows a unique opportunity to compare and contrast forecasts initialized with the benefit of these two satellite SSS observation types. We assess the impact of gridded satellite sea surface salinity observations on dynamical ENSO forecasts for the big 2015 El Nino.

Hackert, E.↗

NASA NTRS2007

Symmetry, Statistics and Structure in MHD Turbulence

Here, we examine homogeneous MHD turbulence in terms of truncated Fourier series. The ideal MHD equations and the associated statistical theory of absolute equilibrium ensembles are symmetric under P, C and T. However, the presence of invariant helicities, which are pseudoscalars under P and C, dynamically breaks this symmetry. This occurs because the surface of constant energy in phase space has disjoint parts, called components: while ensemble averages are taken over all components, a dynamical phase trajectory is confined to only one component. As the Birkhoff-Khinchin theorem tells us, ideal MHD turbulence is thus non-ergodic. This non-ergodicity manifests itself in low-wave number Fourier modes that have large mean values (while absolute ensemble theory predicts mean values of zero). Therefore, we have coherent structure in ideal MHD turbulence. The level of non-ergodicity and amount of energy contained in the associated coherent structure depends on the values of the helicities, as well as on the presence, or not, of a mean magnetic field and/or overall rotation. In addition to the well known cross and magnetic helicities, we also present a new invariant, which we call the parallel helicity, since it occurs when mean field and rotation axis are aligned. The question of applicability of these results to real (i.e., dissipative) MHD turbulence is also examined. Several long-time numerical simulations on a 64(exp 3) grid are given as examples. It is seen that coherent structure begins to form before decay dominates over nonlinearity. The connection of these results with inverse spectral cascades, selective decay, and magnetic dynamos is also discussed.

Shebalin, John V.↗

NASA NTRS2003

Large-eddy Simulation of Stratocumulus-topped Atmospheric Boundary Layers with Dynamic Subgrid-scale Models

The objective of the present study is to evaluate the dynamic procedure in LES of stratocumulus topped atmospheric boundary layer and assess the relative importance of subgrid-scale modeling, cloud microphysics and radiation modeling on the predictions. The simulations will also be used to gain insight into the processes leading to cloud top entrainment instability and cloud breakup. In this report we document the governing equations, numerical schemes and physical models that are employed in the Goddard Cumulus Ensemble model (GCEM3D). We also present the subgrid-scale dynamic procedures that have been implemented in the GCEM3D code for the purpose of the present study.

Senocak, Inane↗

NASA NTRS2022

Projected Changes in Early Summer Ridging and Drought Over the Central Plains

Early summer (May–June–July; MJJ) droughts over the Central Plains are often caused by atmospheric ridging, but it is uncertain if these events will increase in frequency or if their influence on drought severity will change in a warming world. Here, we use tree-ring based reconstructions (1500–2020 CE) of MJJ ridging and 0–200 cm soil moisture with six CMIP6 model ensembles to investigate the response of Central Plains drought dynamics to a moderate warming scenario (SSP2-4.5). By the end of the 21st century (2071–2100), precipitation increases in most models during the preceding months (February–March–April), especially over the northern part of the Central Plains, while changes during MJJ are non-robust. By contrast, vapor pressure deficit increases strongly in all models, resulting in five of the six models projecting robust median soil moisture drying and all six models projecting more rapid seasonal soil moisture declines during the transition into the summer. Major ridging events increase in frequency in some models, and there is strong agreement across all models that when ridging events do occur, they will cause more severe soil moisture drought and seasonal drying at the end of the 21st century. The median multi-model response also indicates, by the end of the 21st century, that the Central Plains will experience a three-fold increase in the risk of drought events equivalent to the most extreme droughts of the last 500 years. Our results demonstrate that even moderate warming is likely to increase early summer soil moisture drought severity and risk over the Central Plains, even in the absence of robust precipitation declines, and that drought responses to major atmospheric ridging events will be significantly stronger.

Drought↗

NASA NTRS2018

The Impact of Satellite Sea Surface Salinity for Prediction of the Coupled Indo-Pacific System

We assess the impact of satellite sea surface salinity (SSS) observations on seasonal to interannual variability of tropical Indo-Pacific Ocean dynamics as well as on dynamical ENSO forecasts. Our coupled model is composed of a primitive equation ocean model for the tropical Indo-Pacific region that is coupled with the global SPEEDY atmospheric model (Molteni, 2003). The Ensemble Reduced Order Kalman Filter is used to assimilate observations to constrain dynamics and thermodynamics for initialization of the coupled model. The baseline experiment assimilates satellite sea level, SST, and in situ subsurface temperature and salinity observations. This baseline is then compared with experiments that additionally assimilate Aquarius (version 4.0) and SMAP (version 2.0) SSS. Twelve-month forecasts are initialized for each month from Sep. 2011 to Dec. 2016. We find that including satellite SSS significantly improves NINO 3.4 sea surface temperature anomaly validation after 1 out to 12 month forecast lead times. For initialization of the coupled forecast, the positive impact of SSS assimilation is brought about by surface freshening near the eastern edge of the western Pacific warm pool and density changes that lead to shallower mixed layer between 10 degrees South latitude-5 degrees North latitude. SST differences at initialization force wide-spread downwelling favorable curl over most of the tropical Pacific. Over an average forecast, SST remains warmer with SSS assimilation at the eastern edge of the warm pool. This warm SST propagates into the eastern Pacific and drags westerly wind anomalies eastward into the NINO 3.4 region. In addition, salting near the ITCZ (Intertropical Convergence Zone) leads to a deepening of the mixed layer and thermocline near 8 degrees North latitude. These patterns together lead to a funneling effect that provides the background state to amplify equatorial Kelvin waves. We show that the downwelling Kelvin waves are amplified by assimilating satellite SSS and lead to significantly improved forecasts particularly for the 2015 El Nino.

El Nino↗

NASA NTRS2018

The Impact of Satellite Sea Surface Salinity for Prediction of the Coupled Indo-Pacific System

We assess the impact of satellite sea surface salinity (SSS) observations on seasonal to interannual variability of tropical Indo-Pacific Ocean dynamics as well as on dynamical ENSO forecasts. Our coupled model is composed of a primitive equation ocean model for the tropical Indo-Pacific region that is coupled with the global SPEEDY atmospheric model (Molteni, 2003). The Ensemble Reduced Order Kalman Filter is used to assimilate observations to constrain dynamics and thermodynamics for initialization of the coupled model. The baseline experiment assimilates satellite sea level, SST, and in situ subsurface temperature and salinity observations. This baseline is then compared with experiments that additionally assimilate Aquarius (version 4.0) and SMAP (version 2.0) SSS. Twelve-month forecasts are initialized for each month from Sep. 2011 to Dec. 2016. We find that including satellite SSS significantly improves NINO3.4 sea surface temperature anomaly validation after 1 out to 12 month forecast lead times. For initialization of the coupled forecast, the positive impact of SSS assimilation is brought about by surface freshening near the eastern edge of the western Pacific warm pool and density changes that lead to shallower mixed layer between 10S-5N. SST differences at initialization force wide-spread downwelling favorable curl over most of the tropical Pacific. Over an average forecast, SST remains warmer with SSS assimilation at the eastern edge of the warm pool. This warm SST propagates into the eastern Pacific and drags westerly wind anomalies eastward into the NINO3.4 region. In addition, salting near the ITCZ leads to a deepening of the mixed layer and thermocline near 8N. These patterns together lead to a funneling effect that provides the background state to amplify equatorial Kelvin waves. We show that the downwelling Kelvin waves are amplified by assimilating satellite SSS and lead to significantly improved forecasts particularly for the 2015 El Nino.

SST↗

NASA NTRS2014

Statistical Mechanics of Turbulent Dynamos

Incompressible magnetohydrodynamic (MHD) turbulence and magnetic dynamos, which occur in magnetofluids with large fluid and magnetic Reynolds numbers, will be discussed. When Reynolds numbers are large and energy decays slowly, the distribution of energy with respect to length scale becomes quasi-stationary and MHD turbulence can be described statistically. In the limit of infinite Reynolds numbers, viscosity and resistivity become zero and if these values are used in the MHD equations ab initio, a model system called ideal MHD turbulence results. This model system is typically confined in simple geometries with some form of homogeneous boundary conditions, allowing for velocity and magnetic field to be represented by orthogonal function expansions. One advantage to this is that the coefficients of the expansions form a set of nonlinearly interacting variables whose behavior can be described by equilibrium statistical mechanics, i.e., by a canonical ensemble theory based on the global invariants (energy, cross helicity and magnetic helicity) of ideal MHD turbulence. Another advantage is that truncated expansions provide a finite dynamical system whose time evolution can be numerically simulated to test the predictions of the associated statistical mechanics. If ensemble predictions are the same as time averages, then the system is said to be ergodic; if not, the system is nonergodic. Although it had been implicitly assumed in the early days of ideal MHD statistical theory development that these finite dynamical systems were ergodic, numerical simulations provided sufficient evidence that they were, in fact, nonergodic. Specifically, while canonical ensemble theory predicted that expansion coefficients would be (i) zero-mean random variables with (ii) energy that decreased with length scale, it was found that although (ii) was correct, (i) was not and the expected ergodicity was broken. The exact cause of this broken ergodicity was explained, after much investigation, by greatly extending the statistical theory of ideal MHD turbulence. The mathematical details of broken ergodicity, in fact, give a quantitative explanation of how coherent structure, dynamic alignment and force-free states appear in turbulent magnetofluids. The relevance of these ideal results to real MHD turbulence occurs because broken ergodicity is most manifest in the ideal case at the largest length scales and it is in these largest scales that a real magnetofluid has the least dissipation, i.e., most closely approaches the behavior of an ideal magnetofluid. Furthermore, the effects grow stronger when cross and magnetic helicities grow large with respect to energy, and this is exactly what occurs with time in a real magnetofluid, where it is called selective decay. The relevance of these results found in ideal MHD turbulence theory to the real world is that they provide at least a qualitative explanation of why confined turbulent magnetofluids, such as the liquid iron that fills the Earth's outer core, produce stationary, large-scale magnetic fields, i.e., the geomagnetic field. These results should also apply to other planets as well as to plasma confinement devices on Earth and in space, and the effects should be manifest if Reynolds numbers are high enough and there is enough time for stationarity to occur, at least approximately. In the presentation, details will be given for both theoretical and numerical results, and references will be provided.

Shebalin, John V.↗

NASA NTRS2014

MHD Turbulence and Magnetic Dynamos

Incompressible magnetohydrodynamic (MHD) turbulence and magnetic dynamos, which occur in magnetofluids with large fluid and magnetic Reynolds numbers, will be discussed. When Reynolds numbers are large and energy decays slowly, the distribution of energy with respect to length scale becomes quasi-stationary and MHD turbulence can be described statistically. In the limit of infinite Reynolds numbers, viscosity and resistivity become zero and if these values are used in the MHD equations ab initio, a model system called ideal MHD turbulence results. This model system is typically confined in simple geometries with some form of homogeneous boundary conditions, allowing for velocity and magnetic field to be represented by orthogonal function expansions. One advantage to this is that the coefficients of the expansions form a set of nonlinearly interacting variables whose behavior can be described by equilibrium statistical mechanics, i.e., by a canonical ensemble theory based on the global invariants (energy, cross helicity and magnetic helicity) of ideal MHD turbulence. Another advantage is that truncated expansions provide a finite dynamical system whose time evolution can be numerically simulated to test the predictions of the associated statistical mechanics. If ensemble predictions are the same as time averages, then the system is said to be ergodic; if not, the system is nonergodic. Although it had been implicitly assumed in the early days of ideal MHD statistical theory development that these finite dynamical systems were ergodic, numerical simulations provided sufficient evidence that they were, in fact, nonergodic. Specifically, while canonical ensemble theory predicted that expansion coefficients would be (i) zero-mean random variables with (ii) energy that decreased with length scale, it was found that although (ii) was correct, (i) was not and the expected ergodicity was broken. The exact cause of this broken ergodicity was explained, after much investigation, by greatly extending the statistical theory of ideal MHD turbulence. The mathematical details of broken ergodicity, in fact, give a quantitative explanation of how coherent structure, dynamic alignment and force-free states appear in turbulent magnetofluids. The relevance of these ideal results to real MHD turbulence occurs because broken ergodicity is most manifest in the ideal case at the largest length scales and it is in these largest scales that a real magnetofluid has the least dissipation, i.e., most closely approaches the behavior of an ideal magnetofluid. Furthermore, the effects grow stronger when cross and magnetic helicities grow large with respect to energy, and this is exactly what occurs with time in a real magnetofluid, where it is called selective decay. The relevance of these results found in ideal MHD turbulence theory to the real world is that they provide at least a qualitative explanation of why confined turbulent magnetofluids, such as the liquid iron that fills the Earth's outer core, produce stationary, large-scale magnetic fields, i.e., the geomagnetic field. These results should also apply to other planets as well as to plasma confinement devices on Earth and in space, and the effects should be manifest if Reynolds numbers are high enough and there is enough time for stationarity to occur, at least approximately. In the presentation, details will be given for both theoretical and numerical results, and references will be provided.

Shebalin, John V↗