Results for “ORTHOGONAL FUNCTION” — Page 8

NASA NTRS2004

Satellite Observations of Spatial and Interannual Variability of Lightning and Radar Reflectivity

The authors use satellite data to examine the relationship between lightning and upper-level radar reflectivity. They find correlations between average flash rates and upper-level reflectivities over both land and ocean, although both flash rates and reflectivities are much lower over ocean than land. Analysis of the data using Empirical Orthogonal Functions (EOFs) shows similar EOFs for averaged lightning and reflectivity. In contrast, the EOFs of the anomalies of lightning and reflectivity have different spatial patterns; however, both have principal component time series that are correlated with the Southern Oscillation Index and, hence, El Nino. Differences in behavior of the lightning and reflectivity anomaly EOFs and principal components suggest that El Nino plays a smaller role in lightning anomaly than precipitation anomaly.

lightning↗

NASA NTRS2006

Annual Cycles of Surface Shortwave Radiative Fluxes

The annual cycles of surface shortwave flux are investigated using the 8-yr dataset of the surface radiation budget (SRB) components for the period July 1983-June 1991. These components include the downward, upward, and net shortwave radiant fluxes at the earth's surface. The seasonal cycles are quantified in terms of principal components that describe the temporal variations and empirical orthogonal functions (EOFs) that describe the spatial patterns. The major part of the variation is simply due to the variation of the insolation at the top of the atmosphere, especially for the first term, which describes 92.4% of the variance for the downward shortwave flux. However, for the second term, which describes 4.1% of the variance, the effect of clouds is quite important and the effect of clouds dominates the third term, which describes 2.4% of the variance. To a large degree the second and third terms are due to the response of clouds to the annual cycle of solar forcing. For net shortwave flux at the surface, similar variances are described by each term. The regional values of the EOFs are related to climate classes, thereby defining the range of annual cycles of shortwave radiation for each climate class.

Wilber, Anne C.↗

NASA NTRS2008

Mechanisms of Interannual Variations of the Meridional Overturning Circulation of the North Atlantic Ocean

The authors investigate the nature of the interannual variability of the meridional overturning circulation (MOC) of the North Atlantic Ocean using an Estimating the Circulation and Climate of the Ocean (ECCO) assimilation product for the period of 1993-2003. The time series of the first empirical orthogonal function of the MOC is found to be correlated with the North Atlantic Oscillation (NAO) index, while the associated circulation anomalies correspond to cells extending over the full ocean depth. Model sensitivity experiments suggest that the wind is responsible for most of this interannual variability, at least south of 40(deg)N. A dynamical decomposition of the meridional streamfunction allows a further look into the mechanisms. In particular, the contributions associated with 1) the Ekman flow and its depth-independent compensation, 2) the vertical shear flow, and 3) the barotropic gyre flowing over zonally varying topography are examined. Ekman processes are found to dominate the shorter time scales (1.5-3 yr), while for longer time scales (3-10 yr) the MOC variations associated with vertical shear flow are of greater importance. The latter is primarily caused by heaving of the pycnocline in the western subtropics associated with the stronger wind forcing. Finally, how these changes in the MOC affect the meridional heat transport (MHT) is examined. It is found that overall, Ekman processes explain a larger part of interannual variability (3-10 yr) for MHT (57%) than for the MOC (33%).

meridional heat transport↗

NASA NTRS2007

TRMM-TMI Satellite Observed Soil Moisture and Vegetation Density (1998-2005) Show Strong Connection with El Nino in Eastern Australia

Spatiotemporal patterns in soil moisture and vegetation water content across mainland Australia were investigated from 1998 through 2005, using TRMMITMI passive microwave observations. The Empirical Orthogonal Function technique was used to extract dominant spatial and temporal patterns in retrieved estimates of moisture content for the top 1-cm of soil (theta) and vegetation moisture content (via optical depth tau). The dominant temporal theta and tau patterns were strongly correlated to El Nino/Southern Oscillation (ENSO) in spring (3 = 0.90), and to a progressively lesser extent autumn, summer and winter. The Indian Ocean Dipole (IOD) index also explained part of the variation in spring 8 and z. Cluster analysis suggested that the regions most affected by ENS0 are mainly located in eastern Australia. The results suggest that the drought conditions experienced in eastern Australia since 2000 an clearly expressed in these satellite observations have a strong connection with ENSO patterns.

Liu, Yi↗

NASA NTRS2010

Nonlinear Dynamics of Cantilever-Sample Interactions in Atomic Force Microscopy

The interaction of the cantilever tip of an atomic force microscope (AFM) with the sample surface is obtained by treating the cantilever and sample as independent systems coupled by a nonlinear force acting between the cantilever tip and a volume element of the sample surface. The volume element is subjected to a restoring force from the remainder of the sample that provides dynamical equilibrium for the combined systems. The model accounts for the positions on the cantilever of the cantilever tip, laser probe, and excitation force (if any) via a basis set of set of orthogonal functions that may be generalized to account for arbitrary cantilever shapes. The basis set is extended to include nonlinear cantilever modes. The model leads to a pair of coupled nonlinear differential equations that are solved analytically using a matrix iteration procedure. The effects of oscillatory excitation forces applied either to the cantilever or to the sample surface (or to both) are obtained from the solution set and applied to the to the assessment of phase and amplitude signals generated by various acoustic-atomic force microscope (A-AFM) modalities. The influence of bistable cantilever modes of on AFM signal generation is discussed. The effects on the cantilever-sample surface dynamics of subsurface features embedded in the sample that are perturbed by surface-generated oscillatory excitation forces and carried to the cantilever via wave propagation are accounted by the Bolef-Miller propagating wave model. Expressions pertaining to signal generation and image contrast in A-AFM are obtained and applied to amplitude modulation (intermittent contact) atomic force microscopy and resonant difference-frequency atomic force ultrasonic microscopy (RDF-AFUM). The influence of phase accumulation in A-AFM on image contrast is discussed, as is the effect of hard contact and maximum nonlinearity regimes of A-AFM operation.

Cantrell, John H.↗

NASA NTRS2010

Warm Season Subseasonal Variability and Climate Extremes in the Northern Hemisphere: The Role of Stationary Rossby Waves

This study examines the nature of boreal summer subseasonal atmospheric variability based on the new NASA Modern-Era Retrospective analysis for Research and Applications (MERRA) for the period 1979-2010. An analysis of the June, July and August subseasonal 250hPa v-wind anomalies shows distinct Rossby wave-like structures that appear to be guided by the mean jets. On monthly subseasonal time scales, the leading waves (the first 10 rotated empirical orthogonal functions or REOFs of the 250hPa v-wind) explain about 50% of the Northern Hemisphere vwind variability, and account for more than 30% (60%) of the precipitation (surface temperature) variability over a number of regions of the northern middle and high latitudes, including the U.S. northern Great Plains, parts of Canada, Europe, and Russia. The first REOF in particular, consists of a Rossby wave that extends across northern Eurasia where it is a dominant contributor to monthly surface temperature and precipitation variability, and played an important role in the 2003 European and 2010 Russian heat waves. While primarily subseasonal in nature, the Rossby waves can at times have a substantial seasonal mean component. This is exemplified by REOF 4 which played a major role in the development of the most intense anomalies of the U.S. 1988 drought (during June) and the 1993 flooding (during July), though differed in the latter event by also making an important contribution to the seasonal mean anomalies. A stationary wave model (SWM) is used to reproduce some of the basic features of the observed waves and provide insight into the nature of the forcing. In particular, the responses to a set of idealized forcing functions are used to map the optimal forcing patterns of the leading waves. Also, experiments to reproduce the observed waves with the SWM using MERRA-based estimates of the forcing indicate that the wave forcing is dominated by sub-monthly vorticity transients.

Schubert, Siegfried↗

NASA NTRS2011

Annual Cycle of Surface Longwave Radiation

The annual cycles of upward and downward longwave fluxes at the Earth s surface are investigated by use of the NASA/GEWEX Surface Radiation Budget Data Set. Because of the immense difference between the heat capacity of land and ocean, the surface of Earth is partitioned into these two categories. Principal component analysis is used to quantify the annual cycles. Over land, the first principal component describes over 95% of the variance of the annual cycle of the upward and downward longwave fluxes. Over ocean the first term describes more than 87% of these annual cycles. Empirical orthogonal functions show the corresponding geographical distributions of these cycles. Phase plane diagrams of the annual cycles of upward longwave fluxes as a function of net shortwave flux show the thermal inertia of land and ocean.

Mlynczak, Pamela E.↗

NASA NTRS2010

Forced and Unforced Variability of Twentieth Century North American Droughts and Pluvials

Research on the forcing of drought and pluvial events over North America is dominated by general circulation model experiments that often have operational limitations (e.g., computational expense, ability to simulate relevant processes, etc). We use a statistically based modeling approach to investigate sea surface temperature (SST) forcing of the twentieth century pluvial (1905-1917) and drought (1932-1939, 1948-1957, 1998-2002) events. A principal component (PC) analysis of Palmer Drought Severity Index (PDSI) from the North American Drought Atlas separates the drought variability into five leading modes accounting for 62% of the underlying variance. Over the full period spanning these events (1900-2005), the first three PCs significantly correlate with SSTs in the equatorial Pacific (PC 1), North Pacific (PC 2), and North Atlantic (PC 3), with spatial patterns (as defined by the empirical orthogonal functions) consistent with our understanding of North American drought responses to SST forcing. We use a large ensemble statistical modeling approach to determine how successfully we can reproduce these drought/pluvial events using these three modes of variability. Using Pacific forcing only (PCs 1-2), we are able to reproduce the 1948-1957 drought and 1905-1917 pluvial above a 95% random noise threshold in over 90% of the ensemble members; the addition of Atlantic forcing (PCs 1-2-3) provides only marginal improvement. For the 1998-2002 drought, Pacific forcing reproduces the drought above noise in over 65% of the ensemble members, with the addition of Atlantic forcing increasing the number passing to over 80%. The severity of the drought, however, is underestimated in the ensemble median, suggesting this drought intensity can only be achieved through internal variability or other processes. Pacific only forcing does a poor job of reproducing the 1932-1939 drought pattern in the ensemble median, and less than one third of ensemble members exceed the noise threshold (28%). Inclusion of Atlantic forcing improves the ensemble median drought pattern and nearly doubles the number of ensemble members passing the noise threshold (52%). Even with the inclusion of Atlantic forcing, the intensity of the simulated 1932-1939 drought is muted, and the drought itself extends too far into the southwest and southern Great Plains. To an even greater extent than the 1998-2002 drought, these results suggest much of the variance in the 1932-1939 drought is dependent on processes other than SST forcing. This study highlights the importance of internal noise and non SST processes for hydroclimatic variability over North America, complementing existing research using general circulation models.

Cook, Benjamin I.↗

NASA NTRS2011

Experiments in Reconstructing Twentieth-Century Sea Levels

One approach to reconstructing historical sea level from the relatively sparse tide-gauge network is to employ Empirical Orthogonal Functions (EOFs) as interpolatory spatial basis functions. The EOFs are determined from independent global data, generally sea-surface heights from either satellite altimetry or a numerical ocean model. The problem is revisited here for sea level since 1900. A new approach to handling the tide-gauge datum problem by direct solution offers possible advantages over the method of integrating sea-level differences, with the potential of eventually adjusting datums into the global terrestrial reference frame. The resulting time series of global mean sea levels appears fairly insensitive to the adopted set of EOFs. In contrast, charts of regional sea level anomalies and trends are very sensitive to the adopted set of EOFs, especially for the sparser network of gauges in the early 20th century. The reconstructions appear especially suspect before 1950 in the tropical Pacific. While this limits some applications of the sea-level reconstructions, the sensitivity does appear adequately captured by formal uncertainties. All our solutions show regional trends over the past five decades to be fairly uniform throughout the global ocean, in contrast to trends observed over the shorter altimeter era. Consistent with several previous estimates, the global sea-level rise since 1900 is 1.70 +/- 0.26 mm/yr. The global trend since 1995 exceeds 3 mm/yr which is consistent with altimeter measurements, but this large trend was possibly also reached between 1935 and 1950.

Ray, Richard D.↗

NASA NTRS2011

Method of performing computational aeroelastic analyses

Computational aeroelastic analyses typically use a mathematical model for the structural modes of a flexible structure and a nonlinear aerodynamic model that can generate a plurality of unsteady aerodynamic responses based on the structural modes for conditions defining an aerodynamic condition of the flexible structure. In the present invention, a linear state-space model is generated using a single execution of the nonlinear aerodynamic model for all of the structural modes where a family of orthogonal functions is used as the inputs. Then, static and dynamic aeroelastic solutions are generated using computational interaction between the mathematical model and the linear state-space model for a plurality of periodic points in time.

Silva, Walter A.↗

NASA NTRS2005

Observed Land Impacts on Clouds, Water Vapor, and Rainfall at Continental Scales

How do the continents affect large-scale hydrological cycles? How important can one continent be to the climate system? To address these questions, five years of National Aeronautics and Space Administration (NASA) Terra Moderate Resolution Imaging Spectroradiometer (MODIS) observations, Tropical Rainfall Measuring Mission (TRMM) observations, and the Global Precipitation Climatology Project (GPCP) global precipitation analysis, were used to assess the land impacts on clouds, ~ainfall, and water vapor at continental scales. At these scales, Empirical Orthogonal Function (EOF) and continentally averaged analyses illustrate that continents as integrated regions enhance the seasonality of atmospheric and surface hydrological parameters. Specifically, the continents of Eurasia and North America enhance the seasonality of cloud optical thickness, cirrus fraction, rainfall, and water vapor. Over land, both liquid water and ice cloud effective radii are smaller than over oceans, primarily because land has more aerosol particles. In addition, different continents have similar impacts on hydrological variables in terms of seasonality, but differ in magnitude. For example, in winter, North America and Eurasia increase cloud optical thickness to 17.5 and 16, respectively, while in summer, Eurasia has much smaller cloud optical thicknesses than North An1erica. Such different land impacts are determined by each continent's geographical condition, land cover, and land use. These new understandings help further address the land-ocean contrasts on global climate, help validate global climate model simulated land-atmosphere interactions, and shed light on interpreting climate change over land.

Jin, Menglin↗

NASA NTRS2012

A Technique Using Principal Component Analysis to Compare Seasonal Cycles of Earth Radiation from CERES and Model Computations

A method for quantitatively comparing the seasonal cycles of two global data sets is presented. The seasonal cycles of absorbed solar radiation (ASR) and outgoing longwave radiation (OLR) have been computed from an eight-year data set from the Clouds and Earth's Radiant Energy System (CERES) scanning radiometers and from a model data set produced by the NASA Goddard Space Flight Center Global Modeling and Assimilation Office. To compare the seasonal cycles from these two data sets, principal component (PC) analysis is used, where the PCs express the time variations and the corresponding empirical orthogonal functions (EOFs) describe the geographic variations. Ocean has a long thermal response time compared to land, so land and ocean are separated for the analysis. The root-mean square values for the seasonal cycles of ASR and OLR are extremely close for the two data sets. The first three PCs are quite close, showing that the time responses and magnitudes over the globe are very similar. The agreement between the two sets of PCs is quantified by computing the matrix of inner products of the two sets. For ASR over land, the first PCs of CERES and the model agree to better than 99.9%. The EOF maps are similar for most of the globe, but differ in a few places, and the agreement of the EOF maps is likewise quantified. Maps of differences between the annual cycles show regions of agreement and disagreement.

Smith, G. Louis↗

NASA NTRS2013

Application of Spectral Analysis Techniques in the Intercomparison of Aerosol Data: 1. an EOF Approach to the Spatial-Temporal Variability of Aerosol Optical Depth Using Multiple Remote Sensing Data Sets

Many remote sensing techniques and passive sensors have been developed to measure global aerosol properties. While instantaneous comparisons between pixel-level data often reveal quantitative differences, here we use Empirical Orthogonal Function (EOF) analysis, also known as Principal Component Analysis, to demonstrate that satellite-derived aerosol optical depth (AOD) data sets exhibit essentially the same spatial and temporal variability and are thus suitable for large-scale studies. Analysis results show that the first four EOF modes of AOD account for the bulk of the variance and agree well across the four data sets used in this study (i.e., Aqua MODIS, Terra MODIS, MISR, and SeaWiFS). Only SeaWiFS data over land have slightly different EOF patterns. Globally, the first two EOF modes show annual cycles and are mainly related to Sahara dust in the northern hemisphere and biomass burning in the southern hemisphere, respectively. After removing the mean seasonal cycle from the data, major aerosol sources, including biomass burning in South America and dust in West Africa, are revealed in the dominant modes due to the different interannual variability of aerosol emissions. The enhancement of biomass burning associated with El Niño over Indonesia and central South America is also captured with the EOF technique.

variability↗

NASA NTRS2013

Northern North Atlantic Sea Surface Height and Ocean Heat Content Variability

The evolution of nearly 20 years of altimetric sea surface height (SSH) is investigated to understand its association with decadal to multidecadal variability of the North Atlantic heat content. Altimetric SSH is dominated by an increase of about 14 cm in the Labrador and Irminger seas from 1993 to 2011, while the opposite has occurred over the Gulf Stream region over the same time period. During the altimeter period the observed 0-700 m ocean heat content (OHC) in the subpolar gyre mirrors the increased SSH by its dominantly positive trend. Over a longer period, 1955-2011, fluctuations in the subpolar OHC reflect Atlantic multidecadal variability (AMV) and can be attributed to advection driven by the wind stress ''gyre mode'' bringing more subtropical waters into the subpolar gyre. The extended subpolar warming evident in SSH and OHC during the altimeter period represents transition of the AMV from cold to warm phase. In addition to the dominant trend, the first empirical orthogonal function SSH time series shows an abrupt change 2009-2010 reaching a new minimum in 2010. The change coincides with the change in the meridional overturning circulation at 26.5N as observed by the RAPID (Rapid Climate Change) project, and with extreme behavior of the wind stress gyre mode and of atmospheric blocking. While the general relationship between northern warming and Atlantic meridional overturning circulation (AMOC) volume transport remains undetermined, the meridional heat and salt transport carried by AMOC's arteries are rich with decade-to-century timescale variability.

Sea-surface↗

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

A Generic Nonlinear Aerodynamic Model for Aircraft

A generic model of the aerodynamic coefficients was developed using wind tunnel databases for eight different aircraft and multivariate orthogonal functions. For each database and each coefficient, models were determined using polynomials expanded about the state and control variables, and an othgonalization procedure. A predicted squared-error criterion was used to automatically select the model terms. Modeling terms picked in at least half of the analyses, which totalled 45 terms, were retained to form the generic nonlinear aerodynamic (GNA) model. Least squares was then used to estimate the model parameters and associated uncertainty that best fit the GNA model to each database. Nonlinear flight simulations were used to demonstrate that the GNA model produces accurate trim solutions, local behavior (modal frequencies and damping ratios), and global dynamic behavior (91% accurate state histories and 80% accurate aerodynamic coefficient histories) under large-amplitude excitation. This compact aerodynamics model can be used to decrease on-board memory storage requirements, quickly change conceptual aircraft models, provide smooth analytical functions for control and optimization applications, and facilitate real-time parametric system identification.

Grauer, Jared A.↗

NASA NTRS2014

The East Atlantic - West Russia Teleconnection in the North Atlantic: Climate Impact and Relation to Rossby Wave Propagation

Large-scale winter teleconnection of the East Atlantic - West Russia (EA-WR) over the Atlantic and surrounding regions is examined in order to quantify its impacts on temperature and precipitation and identify the physical mechanisms responsible for its existence. A rotated empirical orthogonal function (REOF) analysis of the upper-tropospheric monthly height field captures successfully the EA-WR pattern and its interannual variation, with the North Atlantic Oscillation as the first mode. EA-WRs climate impact extends from eastern North America to Eurasia. The positive (negative) EA-WR produces positive (negative) temperature anomalies over the eastern US, western Europe and Russia east of Caspian Sea, with negative (positive) anomalies over eastern Canada, eastern Europe including Ural Mountains and the Middle East. These anomalies are largely explained by lower-tropospheric temperature advections. Positive (negative) precipitation anomalies are found over the mid-latitude Atlantic and central Russia around 60E, where lower-level cyclonic (anticyclonic) circulation anomaly is dominant. The eastern Canada and the western Europe are characterized by negative (positive) precipitation anomalies.The EA-WR is found to be closely associated with Rossby wave propagation. Wave activity fluxes show that it is strongly tied to large-scale stationary waves. Furthermore, a stationary wave model (SWM) forced with vorticity transients in the mid-latitude Atlantic (approximately 40N) or diabatic heat source over the subtropical Atlantic near the Caribbean Sea produces well-organized EA-WR-like wave patterns, respectively. Sensitivity tests with the SWM indicate improvement in the simulation of the EA-WR when the mean state is modified to have a positive NAO component that enhances upper-level westerlies between 40-60N.

EA/WR↗

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↗