Local Time Research Articles

Disparity in temporal and spatial relationships between resting-state electrophysiological and fMRI signals

Resting-state brain networks (RSNs) have been widely applied in health and disease, but the interpretation of RSNs in terms of the underlying neural activity is unclear. To address this fundamental question, we conducted simultaneous recordings of whole-brain resting-state functional magnetic resonance imaging (rsfMRI) and electrophysiology signals in two separate brain regions of rats. Our data reveal that for both recording sites, spatial maps derived from band-specific local field potential (LFP) power can account for up to 90% of the spatial variability in RSNs derived from rsfMRI signals. Surprisingly, the time series of LFP band power can only explain to a maximum of 35% of the temporal variance of the local rsfMRI time course from the same site. In addition, regressing out time series of LFP power from rsfMRI signals has minimal impact on the spatial patterns of rsfMRI-based RSNs. This disparity in the spatial and temporal relationships between resting-state electrophysiology and rsfMRI signals suggests that electrophysiological activity alone does not fully explain the effects observed in the rsfMRI signal, implying the existence of an rsfMRI component contributed by ‘electrophysiology-invisible’ signals. These findings offer a novel perspective on our understanding of RSN interpretation.

Disparity in temporal and spatial relationships between resting-state electrophysiological and fMRI signals.

Resting-state brain networks (RSNs) have been widely applied in health and disease, but the interpretation of RSNs in terms of the underlying neural activity is unclear. To address this fundamental question, we conducted simultaneous recordings of whole-brain resting-state functional magnetic resonance imaging (rsfMRI) and electrophysiology signals in two separate brain regions of rats. Our data reveal that for both recording sites, spatial maps derived from band-specific local field potential (LFP) power can account for up to 90% of the spatial variability in RSNs derived from rsfMRI signals. Surprisingly, the time series of LFP band power can only explain to a maximum of 35% of the temporal variance of the local rsfMRI time course from the same site. In addition, regressing out time series of LFP power from rsfMRI signals has minimal impact on the spatial patterns of rsfMRI-based RSNs. This disparity in the spatial and temporal relationships between resting-state electrophysiology and rsfMRI signals suggests that electrophysiological activity alone does not fully explain the effects observed in the rsfMRI signal, implying the existence of an rsfMRI component contributed by 'electrophysiology-invisible' signals. These findings offer a novel perspective on our understanding of RSN interpretation.

Large deviations in statistics of the local time and occupation time for a run and tumble particle

Large deviations in statistics of the local time and occupation time for a run and tumble particle

Analysis of Precipitable Water Vapor, Liquid Water Path and Their Variations before Rainfall Event over Northeastern Tibetan Plateau

A ground-based microwave radiometer (MWR) provides continuous atmospheric profiles under various weather conditions. The change in total precipitable water vapor (PWV) and liquid water path (LWP) before rainfall events is particularly important for the improvement in the rainfall forecast. However, the analysis of the PWV and LWP before rainfall event on the plateau is especially worth exploring. In this study, the MWR installed at Xining, a city located over the northeastern Tibetan Plateau, was employed during September 2021 to August 2022. The results reveal that the MWR-retrieved temperature and vapor density demonstrate reliable accuracy, when compared with radiosonde observations; PWV and LWP values during the summer account for over 70% of the annual totals in the Xining area; both PWV and LWP at the initiating time of rainfall events are higher during summer, especially after sunset (during 20-00 local solar time); and notably, PWV and LWP anomalies are enhanced abruptly 8 and 28 min prior to the initiating time, respectively. Furthermore, the mean of LWP anomaly rises after the turning time (the moment rises abruptly) to the initiating time; as the intensity of rainfall events increases, the occurrence of the turning time is delayed, especially for PWV anomalies; while the occurrence of the turning time is similar for both convective cloud and stratiform cloud rainfall events, the PWV and LWP anomalies jump more the initiating time; as the intensity of rainfall events increases, the occurrence of the turning time is delayed, especially for PWV anomalies; while the occurrence of the turning time is similar for both convective cloud and stratiform cloud rainfall events, the PWV and LWP anomalies jump more dramatically after the turning time in convective cloud events. This study aims are to analyze the temporal characteristics of PWV and LWP, and assess their potential in predicting rainfall event.

A Framework for Constraining Ocean Mixing Rates and Overturning Circulation from Age Tracers

Abstract The age of seawater refers to the amount of time that has elapsed since that water encountered the surface. This age measures the ventilation rate of the ocean, and the spatial distribution of age can be influenced by multiple processes, such as overturning circulation, ocean mixing, and air–sea exchange. In this work, we aim to gain new quantitative insights about how the ocean’s age tracer distribution reflects the strength of the meridional overturning circulation and diapycnal diffusivity. We propose an integral constraint that relates the age tracer flow across an isopycnal surface to the geometry of the surface. With the integral constraint, a relationship between the globally averaged effective diapycnal diffusivity and the meridional overturning strength at an arbitrary density level can be inferred from the age tracer concentration near that level. The theory is tested in a set of idealized single-basin simulations. A key insight from this study is that the age difference between regions of upwelling and downwelling, rather than any single absolute age value, is the best indicator of overturning strength. The framework has also been adapted to estimate the strength of abyssal overturning circulation in the modern North Pacific, and we demonstrate that the age field provides an estimate of the circulation strength consistent with previous studies. This framework could potentially constrain ocean circulation and mixing rates from age-like realistic tracers (e.g., radiocarbon) in both past and present climates. Significance Statement The age of seawater—the local mean time since local water from different pathways was last at the surface—is a valuable indicator of ocean circulation and the transport time scale of heat and carbon. We introduce a novel constraint that relates total age flow across a density surface to its geometry, which provides new insights into constraining ocean circulation and mixing rates from age-like realistic tracers (e.g., radiocarbon).

Clinical outcomes of Proton Beam Therapy for unresectable locally advanced pancreatic cancer: A single-center retrospective study.

PurposeWe retrospectively researched the treatment outcome of proton beam therapy (PBT) and assessed its efficacy for inoperable locally advanced pancreatic cancer (LAPC) at our institution. Materials and MethodsFifty-four patients (28 males and 26 females, median age 67 years of ranges from 40 to 88 years were diagnosed with unresectable stage III LAPC and administered PBT from April 2009 to March 2020. Either patients who could not complete PBT, or had new distant metastases during the treatment, or did not have enough follow-up time were excluded from this study. All patients were clinically staged based on the International Union of Cancer TNM staging system (UICC 8th edition) using CT, MRI, and PET, and were diagnosed as stage III (histologic type: eighteen adenocarcinoma patients, thirty-six clinically diagnosed patients). PBT was performed using the passive method, with a median total dose of 67.5 GyE (range: 50-77 GyE/25-35 fractions). Chemotherapy was used in combination during PBT in forty-six patients (85.2%). Overall survival (OS), local progression-free survival (LPFS), progression-free survival (PFS), and median overall survival time (MST) were analyzed by Kaplan-Meier and log-rank tests. Univariate and multivariate analyses were performed for the following factors: maximum standardized uptake value (SUVmax), ECOG-performance status (PS), tumor site, total irradiation dose, concurrent chemotherapy, and primary tumor site. Cutoff values for SUVmax and tumor diameter were estimated using receiver operating characteristic (ROC) curves and area under the curve (AUC) based on OS. Multivariate analysis was evaluated using the Cox proportional hazards models. Adverse events were evaluated using CTCAE ver. 5.0. ResultsThe median observation period was 17.4 months of ranges from 4.0 to 89.7 months. The median tumor diameter was 36.5 mm of ranges from 15 to 90 mm, the median SUVmax was 5.85 (range: 2.1-27.6), and their cutoff values were estimated to 37 mm and 4.8 mm, respectively. 1-, 2-year OS were 77.8%, 35.2% with MST 18.2 months, respectively and only one patient survived more than 5 years. 12 patients (22.2%) developed local recurrence, and 1-, 2-year LPFS rate were 89.7% and 74.5%, respectively; PFS at 1-year was 58.8%. PS score, Tumor site and irradiation dose were the prognostic factor related to OS that showed a significant difference. On the other hand, there was a significant difference in factors involved in LPFS, 96.7%/77.9% in the first year and 86.6%/54.4% in the second year in the groups with tumor dose ≥67.5 GyE and <67.5 GyE, respectively (p = 0.015). Treatment-related acute toxicities were neutropenia (Grade1/2/3:3.7%/11.1%/31.5%), leukopenia (Grade 1/2/3:1.8%/7.4%/20.4%), and thrombocytopenia (Grade 1/2:1.8%/7.4%), while the late effects including peptic ulcer were captured only grade2+. The late adverse events of Grade 3 or higher were not observed. ConclusionsPBT achieving 67.5 Gy combined with standard chemotherapy showed the excellent local control for unresectable LAPC. Total irradiation dose, Tumor site and PS score at an initial diagnosis could be important prognostic factors. In this study, the dose-effect relationship was found so an increase in dose should be considered to improve prognosis.

Effect of Intraoperative Ventricular Opening on Recurrence Patterns Following Bis-Chloroethyl-Nitrosourea Wafer Implantation for Newly Diagnosed Glioblastoma.

To evaluate the effect of ventricular opening (VO) on recurrence patterns in patients with newly diagnosed glioblastoma (GBM) treated with bis-chloroethyl-nitrosourea (BCNU) wafer implantation. This single-center retrospective study included 40 patients with newly diagnosed GBM who received BCNU wafer implantation after tumor resection between March 2013 and February 2022. The patients were categorized into two groups based on whether VO occurred during the GBM resection. While 18 patients had VO, 22 did not have VO. In cases with VO, the ventricular wall defect is closed with gelatin or oxidized regenerated cellulose and fibrin glue before BCNU wafer implantation. Recurrence patterns-classified as local, diffuse, distant, or multifocal-and time to recurrence were compared between patients with and without VO. The median follow-up period for the entire cohort was 32.2 months (interquartile range, 16.7-38 months). Median survival time was comparable between patients with VO and patients without VO (38 vs. 26 months, p=0.53). Recurrence occurred in 31/40 patients (77.5%) in entire cohort. The incidence of recurrence was comparable between patients with VO and patients without VO (14 [77.8%] vs. 17 [77.3%], p=1.0). No significant differences were seen between the two groups in time to recurrence (p=0.59) or recurrence patterns (p=0.35). Ventricular opening during surgery with BCNU wafer implantation does not seem to influence the recurrence patterns. Ventricular opening does not induce distant recurrence if appropriate ventricular closure is performed.

Quantifying the Spatiotemporal Evolution of Radiation Belt Electrons Scattered by Lower Band Chorus Waves: An Integrated Model

AbstractWave particle interactions are very important to understand the intricate evolution of the Earth's radiation belt electrons. Kinetic simulations, in terms of solving the Fokker‐Planck equation based on the quasilinear theory, are usually used to simulate the radiation belt electron dynamic evolution. However, the global wave and plasma density distributions adopted in the kinetic simulations are very difficult to be directly obtained by satellites. Here we present a new model, by integrating the machine learning technique and kinetic simulations, to analyze the spatiotemporal evolution of radiation belt electrons scattered by lower band chorus (LBC). Compared to the observations, our integrated model produces effectively the global distribution of plasmapause location, plasma density, and LBC intensity, and assesses quantitatively the scattering effect driven by LBC waves at different magnetic local times (MLT), L‐shell (the Mcllwain L‐parameter), and time. Incorporating the effect of radiation electron drift, we further use the 2‐D Fokker‐Planck equation to simulate the variations of electron phase space density in different MLT sectors at a fixed L, and find that the integrated model replicates reasonably the multi‐MeV electron acceleration at L = 4.5 during the period from the main phase to the early recovery phase of the storm. Our results demonstrate that such an integrated model, on basis of a combination of the machine learning technique and kinetic simulations, provides valuable means for improved understanding of the global dynamic evolution of the Earth's radiation belt electrons.

Deep learning‐based location prediction in VANET

AbstractIn recent years, Vehicular Ad‐hoc Network (VANET) has become an essential component of intelligent transportation systems that, along with the previous systems such as traffic condition, accident alert, automatic parking, and cruise control, use the communication of vehicle to vehicle and vehicle to the roadside unit to facilitate road transportation. Several challenges hinder efforts to improve traffic conditions and reduce traffic fatalities through VANET. A critical challenge is achieving highly accurate and reliable vehicle localization within the VANET. Additionally, the frequent unavailability of Global Positioning System (GPS), particularly in tunnels and parking lots, presents another significant obstacle. Traditional methods like Dead Reckoning offer low accuracy and reliability due to accumulating errors. Similarly, GPS positioning, map matching with mobile phone location services, and other existing solutions struggle with accuracy and economic feasibility. In this article, two Kalman filter approaches are used based on signal statistical information and the other learning‐based networks, including traditional neural network, deep neural network and LSTM (long short‐term memory) to locate the car. The prediction error of car position with root mean square measures. The squared error and distance prediction error are evaluated. It is shown that in terms of prediction time and processing time of vehicle localization, all the vehicle localization methods are efficient in terms of response time for localization, and Kalman filter methods, traditional neural network and deep neural network are faster than LSTM method. Also, in terms of localization error, Kalman filter works better than learning‐based methods, and in learning‐based methods, both deep neural network and LSTM methods perform better than traditional neural network in terms of localization error.

Multiple Satellite Observations of the High‐Latitude Cusp Aurora During Northward IMF Conditions

AbstractCusp auroras poleward of the typical auroral oval are ascribed to high‐latitude lobe reconnection when the Interplanetary Magnetic Field (IMF) Bz is predominantly northward. In this study, we further investigate the ionospheric characteristics of a unique high‐latitude cusp region employing multiple satellite observations. A cusp aurora event with wide spatial spread was observed in the postnoon polar cap region. It was found to be associated with northward IMF Bz and positive By components. The cusp aurora was located from 68° to 86° in magnetic latitude and within 15–17 hr in magnetic local time. This broad coverage in the polar cap indicates direct precipitating particles from the magnetosheath. Particle energy is different between the equatorward and poleward edges of the cusp aurora. The precipitating ions at the equatorward side maintain magnetosheath particle characteristics as expected, while ions with higher energies occurred in the poleward side. Further, the poleward edge of the cusp aurora was nearly situated in the center of a convection shear and was associated with an upward field‐aligned current. These observations suggest a lobe cell circulation, hence we attribute the formation of the cusp aurora to the high‐latitude lobe reconnection. Simultaneous observations in the southern hemisphere indicate the absence of cusp aurora. The auroral presence only in the northern hemisphere is probably due to the combination of large dipole tilt angle and positive IMF Bz, which facilitates the lobe reconnection.

Comparison of geomagnetic storm and non-storm periods mid-latitude Pc1 pulsations characteristics

Pc1 pulsations are infrequently observed at mid-latitude ground stations. Previous studies have shown that these pulsations near the ground are commonly observed during dawn and geomagnetic storms, particularly in the storm recovery phase. However, few studies have directly compared the characteristics of Pc1 pulsations during storm and non-storm times using long-term ground-based data. In this study, we conducted a comparative analysis of storm-related and non-storm-related Pc1 pulsations detected at a mid-latitude ground station located at Bohyun mountain in Korea (BOH, L ~ 1.34). Our findings reveal notable distinctions between the two groups. Pc1 pulsations during the non-storm period exhibit a sharper peak in the local time distribution of the occurrence, relatively well indicating daytime wave power attenuation during propagation. Additionally, during the non-storm period, Pc1 pulsations show relatively lower frequencies (0.6~1 Hz) with predominantly linear polarization. In contrast, storm-related Pc1 pulsations exhibit higher and broader frequencies (up to 3 Hz) with more frequent left-handed polarizations. These results suggest that storm-related Pc1 pulsations might originate from a more diverse range of spatial sources.

Research on Pupil Center Localization Detection Algorithm with Improved YOLOv8

Addressing issues such as low localization accuracy, poor robustness, and long average localization time in pupil center localization algorithms, an improved YOLOv8 network-based pupil center localization algorithm is proposed. This algorithm incorporates a dual attention mechanism into the YOLOv8n backbone network, which simultaneously attends to global contextual information of input data while reducing dependence on specific regions. This improves the problem of difficult pupil localization detection due to occlusions such as eyelashes and eyelids, enhancing the model’s robustness. Additionally, atrous convolutions are introduced in the encoding section, which reduce the network model while improving the model’s detection speed. The use of the Focaler-IoU loss function, by focusing on different regression samples, can improve the performance of detectors in various detection tasks. The performance of the improved Yolov8n algorithm was 0.99971, 1, 0.99611, and 0.96495 in precision, recall, MAP50, and mAP50-95, respectively. Moreover, the improved YOLOv8n algorithm reduced the model parameters by 7.18% and the computational complexity by 10.06%, while enhancing the environmental anti-interference ability and robustness, and shortening the localization time, improving real-time detection.

Statistical Distribution of the Peak Frequency of ECH Waves in the Outer Magnetosphere From Magnetospheric Multiscale Satellite Observations

AbstractElectron cyclotron harmonic (ECH) waves are electrostatic emissions with frequencies between the harmonics of the electron gyrofrequencies. Their frequency properties provide clues for understanding their generation and are keys to evaluating their scattering efficiency. Based on Magnetospheric Multiscale satellite observations, we explored the statistical frequency properties of first‐harmonic band ECH waves in the outer magnetosphere. The frequencies at the peak power of ECH waves are found to be day‐night and dawn‐dusk asymmetries, with higher values in the regions from dawn to post‐noon, and these asymmetries are more evident during weaker geomagnetic activity. Furthermore, the frequencies at the peak power of ECH waves decrease gradually with increasing |MLAT| and are positively correlated with their amplitudes at each magnetic local time or |MLAT|. Information on the frequency properties of ECH waves presented in this study can be crucial for future modeling of their contributions to magnetospheric electron dynamics.

Inferences of water–ice cloud physics determined from MAVEN/IUVS aerosol retrievals and Mars GCM comparisons

We performed retrievals of dust and water–ice optical depths at select times spanning Mars years (MYs) 33 through 36. We used data taken by the Imaging Ultraviolet Spectrograph (IUVS) instrument aboard the Mars Atmosphere and Volatile Evolution (MAVEN) spacecraft, whose precessing orbit allowed IUVS to image the diurnal evolution of water–ice clouds at local times throughout this MY range where they have not been studied. We created a radiative-transfer retrieval algorithm and used it to simultaneously fit the dust and water–ice optical depth of each usable spectrum in IUVS apoapse data for the majority of the MAVEN mission. We compared our results to other datasets to ensure our retrievals produced plausible values and discuss potential reasons for discrepancies. We obtained global climate model (GCM) simulations from two state-of-the-art models and compared our results to both sets of simulations. We show that there are significant differences between our water–ice retrievals and both sets of simulations, both in the spatial cloud patterns and column-integrated optical depths. We discuss the different approaches both models used for simulating these clouds and the implications for accurately modeling these clouds in future GCM work.

Ionospheric upwelling and the level of associated noise at solar minimum

Abstract. We have studied the ionospheric upwelling with a magnitude of above 1013 m−2 s−1 using the data during the European Incoherent Scatter Scientific Association (EISCAT) Svalbard Radar International Polar Year (IPY-ESR) 2007 campaign, which coincides with the solar minimum. The noise level in low-, medium- and high-flux upflows is investigated. We found that the noise level in high-flux upflow is about 93 %, while in the low and medium categories it is 62 % and 80 %, respectively. This shows that robust and stringent filtering techniques must be ensured when analysing incoherent data in order not to introduce bias to the result. Analysis reveals that the frequency of the low-flux upflow events is about 8 and 73 times the medium- and high-flux upflow events, respectively. Seasonal observation shows that the noise level in the upflow classes is predominantly high during winter. The noise is minimal in summer, with a notable result indicating occurrence of actual data above noise in the low-flux class. Moreover, the percentage occurrence of the noise level in the data increases with increasing flux strength, irrespective of the season. Further analysis reveals that the noise level in the local time variation peaked around 17:00–18:00 LT (local time) and minimum around 12:00 LT.

Resolving the Location and Magnitude of the 1918 Queensland (Bundaberg), Australia, Earthquake

ABSTRACT Eastern Queensland (Australia) was struck by a major earthquake at ≈04:14 a.m. local time on 7 June 1918. Most previous studies have suggested that the epicenter of this earthquake lies off the coast of Bundaberg, between the port cities of Gladstone and Rockhampton. This epicentral location was based upon instrumental observations from the Riverview College observatory in Sydney. However, this epicenter lies ≈250 km to the northeast of an inland region that experienced both the strongest shaking effects and numerous felt aftershocks. We revisited available macroseismic data from 224 geographic locations and surviving instrumental observations for the 1918 Queensland earthquake to show that the most likely epicentral location was inland at ≈24.93° S and ≈150.88° E in the Banana Shire and North Burnett region. The re-estimated instrumental magnitude of Mw 6.0 ± 0.3 (1σ) makes it one of the largest onshore earthquakes in eastern Australia in the past century. Our observations also offer support for a viewpoint proposed in 1935 by an eminent Queensland geologist, Walter Heywood Bryan, that the 1918 earthquake was inland. Our study highlights the benefit of the critical evaluation of primary source materials, both archival and seismological, to study historical earthquakes in Australia that are relevant for modern seismic hazard analysis.

Enhanced Indoor Positioning Using RSSI and Time-Distributed Auto Encoder-Gated Recurrent Unit Model

This study presents a novel approach to indoor positioning leveraging radio frequency identification (RFID) technology based on received signal strength indication (RSSI). The proposed methodology integrates Gaussian Kalman filtering for effective signal preprocessing and a time-distributed auto encoder-gated recurrent unit (TAE-GRU) model for precise location prediction. Addressing the prevalent challenges of low accuracy and extended localization times in current systems, the proposed method significantly enhances the preprocessing of RSSI data and effectively captures the temporal relationships inherent in the data. Experimental validation demonstrates that the proposed approach achieves a 75.9% improvement in localization accuracy over simple neural network methods and markedly enhances the speed of localization, thereby proving its practical applicability in real-world indoor localization scenarios.

Prolonging a discrete time crystal by quantum-classical feedback

Nonequilibrium phases of quantum matter featuring time crystalline eigenstate order have been realized recently on noisy intermediate-scale quantum (NISQ) devices. While ideal quantum time crystals exhibit collective subharmonic oscillations and spatiotemporal long-range order persisting for infinite times, the decoherence time of current NISQ devices sets a natural limit to the survival of these phases, restricting their observation to a shallow quantum circuit. Here we propose a time-periodic scheme that leverages quantum-classical feedback protocols in subregions of the system to enhance a time crystal signal significantly exceeding the decoherence time of the device. As a case of study, we demonstrate the survival of the many-body localized discrete time crystal phase in the one-dimensional periodically kicked Ising model, accounting for decoherence of the system with an environment. Based on classical simulation of quantum circuit realizations we find that this approach is suitable for implementation on existing quantum hardware and presents a prospective path to simulate complex quantum many-body dynamics that transcend the low depth limit of current digital quantum computers. Published by the American Physical Society 2024

Effect of current intensity on molten pool and solidification in vacuum arc remelting process of Ti60 alloy

Current intensity is a critical parameter in the vacuum arc remelting (VAR) process, significantly impacting the molten pool, solidification structure, and ultimately, the quality of the ingot. This study establishes a transient electromagnetic-fluid-thermal-solidified coupled model to examine the effects of current intensity on the shape of molten pool and the solidification process of Ti60 alloy ingots produced by VAR process. Findings indicate that with the increase in current intensity, there is a corresponding rise in the magnetic induction intensity, current density, and Lorentz force. This enhancement leads to more vigorous flow in the liquid phase, creating a vortex flow opposing to the ingot's center at the boundary, influenced by the strong current. As a consequence, the molten pool's shape tends to shift from a “V” shape to a “U” shape, and the depth of molten pool increases while the mushy zone thickness diminishes. For instance, at 17000 A, the depth of molten pool surges by 57.43%, and the mushy zone thickness contracts by 62.01% compared to 5000 A. Over time, the depth of molten pool gradually decreases, with its reduction range ( Δh) within 400 to 6000 s inversely related to current intensity. At 17000 A, Δh is 23.71% less than at 5000 A. Furthermore, elevating the current intensity results in a higher temperature gradient within the ingot and a marked reduction in local solidification time, particularly in the ingot's central area.

Necessary and sufficient conditions for Kolmogorov’s flux laws on T2 and T3

Necessary and sufficient conditions for the third order Kolmogorov universal scaling flux laws are derived for the stochastically forced incompressible Navier Stokes equations on the torus in 2D and 3D. This paper rigorously generalises the result of (Bedrossian 2019 Commun. Math. Phys. 367 1045–75) to functions which are heavy-tailed in Fourier space or have local finite time singularities in the inviscid limit. In other words, we have rigorously derived the existence of the well known physical relationships, the direct and inverse cascades. Furthermore we show that the rate of the direct cascade is proportional to the amount of energy ‘escaping to infinity’ in spectral space as well as a measure of the total singularities within the solution. Similarly, an inverse cascade is proportional to the amount of energy that moves towards the k = 0 Fourier mode in the invisicid limit.