Wavenumber Power Spectrum Research Articles

Propagation of capillary waves in two-layer oil–water turbulent flow

We study the dynamics of capillary waves at the interface of a two-layer stratified turbulent channel flow. We use a combined pseudo-spectral/phase field method to solve for the turbulent flow in the two liquid layers and to track the dynamics of the liquid–liquid interface. The two liquid layers have same thickness and same density, but different viscosity. We vary the viscosity of the upper layer (two different values) to mimic a stratified oil–water flow. This allows us to study the interplay between inertial, viscous and surface tension forces in the absence of gravity. In the present set-up, waves are naturally forced by turbulence over a broad range of scales, from the larger scales, whose size is of order of the system scale, down to the smaller dissipative scales. After an initial transient, we observe the emergence of a stationary capillary wave regime, which we study by means of temporal and spatial spectra. The computed frequency and wavenumber power spectra of wave elevation are in line with previous experimental findings and can be explained in the frame of the weak wave turbulence theory. Finally, we show that the dispersion relation, which gives the frequency ( $\omega$ ) as a function of the wavenumber ( $k$ ), is in good agreement with the well-established theoretical prediction, $\omega (k) \sim k^{3/2}$ .

Projected future changes in equatorial wave spectrum in CMIP6

The simulation of the Madden–Julian Oscillation (MJO) and convectively coupled equatorial waves (CCEWs) is considered in 13 state-of-the-art models from phase 6 of the Coupled Model Intercomparison Project (CMIP6). We use frequency–wavenumber power spectra of the models and observations for Outgoing Longwave Radiation (OLR) and zonal winds at 250 hPa (U250), and consider the historical simulations and end of twenty-first century projections for the SSP245 and SSP585 scenarios. The models simulate a spectrum quantitatively resembling that observed, though systematic biases exist. MJO and Kelvin waves (KW) are mostly underestimated, while equatorial Rossby waves (ER) are overestimated. Most models project a future increase in power spectra for the MJO, while nearly all project a robust increase for KW and weaker power values for most other wavenumber–frequency combinations, including higher wavenumber ER. In addition to strengthening, KW also shift toward higher phase speeds (or equivalent depths). Models with a more realistic MJO in their control climate tend to simulate a stronger future intensification.

SARAL/AltiKa data analysis for oceanographic research: Impact of drifting and post star sensor anomaly phases

AltiKa, first ever high frequency Ka-band altimeter on board SARAL (Satellite with ARgos and ALtiKa) has gone through different phases of operations, viz. Exact Repeat Mission, (ERM, March 2013 - July 2016), Drifting phase, (DP, July 2016 - January 2018) and then to Mispointing phase, (MP, February 2018 - till date). A detailed assessment of Sea level anomaly (SLA), Significant Wave Height (SWH) and Ocean Surface Wind Speed (WS) has been carried out during these different phases with a focus on the North Indian Ocean. Crossover analysis using the Jason series of satellites available during various phases of SARAL suggest high quality of SARAL/AltiKa data during the ERM and DP with root mean square differences of the order of 0.080 m, 0.25 m and 1 m/s for SLA, SWH and WS respectively. These differences are more during MP, being 0.095 m, 0.45 m and 1.72 m/s for SLA, SWH and WS respectively. Wavenumber Power spectrum computed from the along-track AltiKa SLA reveals that slopes in the mesoscale band (70–250 km) in different phases of operations are not very different. Errors in gridded SARAL/AltiKa SLA with respect to standard AVISO product remains unchanged during DP, but degrade by nearly 9.3% in the MP as compared to ERM. To assess the effect of assimilating along track SWH and SLA from different phases, two set of wave and circulation model simulations, with and without SARAL AltiKa data assimilation, were performed. Assimilation of SWH improved the wave height simulation by ∼ 12.8% during the DP and ∼ 8% during ERM and MP. As regards to circulation modeling, no significant difference of assimilating SLA from different phases was observed in the mesoscale range. These results indicate the usefulness of SLA from SARAL AltiKa during DP and MP for studying the mesoscale dynamics.

Validation of gyrokinetic simulations in NSTX and projections for high-k turbulence measurements in NSTX-U

An extensive validation effort performed for a modest-beta NSTX NBI-heated H-mode discharge predicts that electron thermal transport can be entirely explained by electron-scale turbulence fluctuations driven by the electron temperature gradient mode (ETG), both in conditions of strong and weak ETG turbulence drive. Thermal power-balance estimates computed by TRANSP as well as the shape of the high-k density fluctuation wavenumber spectrum and the fluctuation level ratio between strongly driven and weakly driven ETG-turbulence conditions can be matched by nonlinear gyrokinetic simulations and a synthetic diagnostic for high-k scattering. Linear gyrokinetic simulations suggest that the ion-scale instability in the weak ETG condition is close to the critical threshold for the kinetic ballooning mode instability, and nonlinear ion-scale gyrokinetic simulations show that turbulence might be in a state reminiscent of a Dimits' shift regime, opening speculation on the role that ion-scale turbulence might play for the weak ETG condition. A simulation that matched all experimental constraints is chosen to project high-k turbulence spectra in NSTX-U, revealing that the new high-k system [R. Barchfeld et al., Rev. Sci. Instrum. 89, 10C114 (2018)] should be sensitive to density fluctuations from radially elongated streamer structures. Two schemes are designed to characterize the radial and poloidal wavenumber dependence of the density fluctuation wavenumber power spectrum around the streamer peak, suggesting future high-k fluctuation measurements could be sensitive to an asymmetry in the kr spectrum introduced due to the presence of strong background flow shear.

Four-dimensional Frequency–Wavenumber Power Spectrum of a Strong Turbulence Obtained from Hybrid Kinetic Simulations

Abstract We carry out three-dimensional hybrid kinetic simulations of a strong decaying turbulence. The turbulence is initiated with a seed spectrum that includes Alfvén wave modes at low wavenumbers. From the temporal and spatial distribution of the simulation output in the quasi-steady phase, we calculate a four-dimensional frequency–wavenumber spectrum of the turbulence. Our analysis shows that kinetic Alfvén waves can be identified in the wavenumber–frequency space in the vicinity of the seed turbulence spectrum. They produce a power peak consistent with a linear dispersion relation. However, further away from the seed spectrum, where most of the wave–particle interaction takes place, the signature of the Alfvén modes disappears among other fluctuations that are not described by any dispersion relations. Furthermore, at higher wavenumbers at which the signature is still identifiable, its frequency broadening becomes comparable to the frequency itself. Therefore, the use of linear waves based on the conventional dispersion relation is not necessarily justified to describe the turbulence if their signature is still present and even more so when it disappears. We find that at larger rms amplitudes of the turbulence, the signature of the dispersion relation is confined to lower parallel wavenumbers. In the range where it is visible, the frequency broadening becomes greater at larger amplitudes. This suggests that stronger nonlinearity makes the fluctuations behave less like wave modes until the conventional wave-mode approach is no longer valid.

Succession of Resonances to Achieve Internal Wave Turbulence.

We study experimentally the interaction of nonlinear internal waves in a stratified fluid confined in a trapezoidal tank. The setup has been designed to produce internal wave turbulence from monochromatic and polychromatic forcing through three processes. The first is a linear transfer in wavelength obtained by wave reflection on inclined slopes, leading to an internal wave attractor which has a broad wave number spectrum. Second is the broadbanded time-frequency spectrum of the trapezoidal geometry, as shown by the impulse response of the system. The third one is a nonlinear transfer in frequencies and wave vectors via triadic interactions, which results at large forcing amplitudes in a power law decay of the wave number power spectrum. This first experimental spectrum of internal wave turbulence displays a k^{-3} behavior.

Detection of Daytime Ionospheric Irregularities at Low Latitudes With a Multistatic HF Radar

We report a new observation of daytime ionospheric irregularities on December 27, 2009, at low latitudes in China using multistatic HF radar. The transmitter, with a horizontal polarization log-periodic antenna pointing westward, is located at Wanning (18.97° N, 110.50° E, dip angle 19°). Three separate receivers are in Sanya (18.20° N, 109.49° E), Qiongzhong (19.04° N, 109.77° E), and Tunchang (19.42° N, 110.13° E). Doppler spectra, frequency power spectra, and wavenumber power spectra from the three receivers were calculated. The main features of these observations are: 1) the irregularities observed at 12:10 (LT) had typical lengths on the scale of tens of meters and the altitude was in the range of 105–175 km, which is located in the E region and bottom of the F region; 2) the power distributions from the three receivers are consistent with the irregularities being field aligned; and 3) features consistent with a cascade of irregular spatial scales were observed. We suggest that the observed irregularity could be due to the intermediate layer modulated by electrodynamic instability.

Trends in Tropical Wave Activity from the 1980s to 2016

Abstract A frequency–wavenumber power () spectrum was constructed using satellite-derived outgoing longwave radiation (OLR) and brightness temperature for the tropical latitudes. Since the two datasets overlap for over 34 years with nonintersecting sources in error and compare relatively well with each other, it is possible to diagnose trends in the tropical wave activity from the two datasets with confidence. The results suggest a weakening trend in characterized by high interannual variability for wave activity occurring in the low-frequency part of the spectrum and a steady increase in with relatively low interannual variability for wave activity occurring in the high-frequency part of the spectrum. The results show the parts of the spectrum representing the Madden–Julian oscillation and equatorial Rossby wave losing and other parts of the spectrum representing Kelvin waves, mixed Rossby–gravity waves, and tropical disturbance–like wave activity gaining . Similar results were obtained when trends in variance corresponding to the first principal component were produced using spectrally filtered OLR data representative of atmospheric equatorial waves. Spatial trends in the active phase of wave events and the mean duration of events are also shown for the different wave types. Linear trends in for the entire spectrum and regional means in the spectrum corresponding to the abovementioned five wave types with confidence intervals are also presented in the paper. Finally, we demonstrate that El Niño–Southern Oscillation variability does not appear to control the overall spatial patterns and trends observed in the spectrum.

Seasonality of eddy kinetic energy in an eddy permitting global climate model

We examine the seasonal cycle of upper-ocean mesoscale turbulence in a high resolution CESM climate simulation. The ocean model component (POP) has 0.1° resolution, mesoscale resolving at low and middle latitudes. Seasonally and regionally resolved wavenumber power spectra are calculated for sea-surface eddy kinetic energy (EKE). Although the interpretation of the spectral slopes in terms of turbulence theory is complicated by the strong presence of dissipation and the narrow inertial range, the EKE spectra consistently show higher power at small scales during winter throughout the ocean. Potential hypotheses for this seasonality are investigated. Diagnostics of baroclinc energy conversion rates and evidence from linear quasigeostrophic stability analysis indicate that seasonally varying mixed-layer instability is responsible for the seasonality in EKE. The ability of this climate model, which is not considered submesoscale resolving, to produce mixed layer instability although damped by dissipation, demonstrates the ubiquity and robustness of this process for modulating upper ocean EKE.

Direct numerical simulation of turbulent boundary layer over a compliant wall

The performance of compliant wall with different parameters in a spatially developing turbulent boundary layer is investigated by direct numerical simulation. The Reynolds number based on the external flow velocity and the momentum thickness ranges from 300 to approximately 860. The compliant wall is placed in the region starting from 200 to 600 in the streamwise direction, scaled by the inlet momentum thickness. The monoharmonic analysis is carried out for selection of the parameters of compliant wall in numerical simulations. The instantaneous flow fields and the turbulence statistics are presented to show the impact of the compliant wall. A dynamical decomposition of the skin-friction coefficient Cf is performed for examining the sources bringing the variations in Cf. The spatial evolution of compliant wall motion is disclosed, which is a special feature of the boundary layer flow. The spatial–temporal self-correlation of the wall displacement is calculated for obtaining the convection velocity of the wavy motion of compliant wall. The frequency–wavenumber power spectrum of wall pressure fluctuation is presented for studying the interaction of the compliant wall and the above turbulent flow. The effect of the fixed boundaries of the compliant wall is also examined.

A Linear Stochastic Field Model of Midlatitude Mesoscale Variability

AbstractA semiempirical model of midlatitude sea surface height (SSH) variability is formulated and tested against two decades of weekly global fields of merged altimeter data. The model is constrained to match approximately the observed SSH wavenumber power spectrum, but it predicts the spatiotemporal SSH field structure as a propagating, damped, linear response to a stochastic forcing field. An objective, coherent-eddy identification and tracking procedure is applied to the model and altimeter SSH fields, with a focus on eddies with lifetimes L ≥ 16 weeks. The model eddy dataset reproduces the basic global-mean characteristics of the altimeter eddy dataset, including the structure of mean amplitude and scale life cycles, the number distributions versus lifetime, and the distributions of all eddy length scale realizations. The model underpredicts the numbers of eddy realizations with large amplitudes and large scales, overpredicts the growth of mean amplitude and scale with lifetime, and modestly overpredicts the curvature of the mean amplitude life cycle and the number of eddies with intermediate lifetimes. The stochastic forcing evidently represents nonlinear dynamical interactions, implying that eddy splitting and merging events are equally likely, and that mesoscale nonlinearity is weaker than longwave linearity but as strong as shortwave dispersion. The time-reversal symmetry of the life cycles is explained by the time reversibility of the underlying stochastic model. The random SSH increment processes are effectively continuous on the derived 25-week damping time scale, with SSH-increment standard deviation σW ≈ 2.5 × 10−3 cm s−1/2.

A Parabolic Equation Approach to Modeling Acousto-Gravity Waves for Local Helioseismology

A wide-angle parabolic-wave-equation algorithm is developed and validated for local-helioseismic wave propagation. The parabolic equation is derived from a factorization of the linearized acousto-gravity wave equation. We apply the parabolic-wave equation to modeling acoustic propagation in a plane-parallel waveguide with physical properties derived from helioseismic data. The wavenumber power spectrum and wave-packet arrival-time structure for receivers in the photosphere with separation up to \(30^{\circ}\) is computed, and good agreement is demonstrated with measured values and a reference spectral model.

Phase Speed Cross Spectra of Eddy Heat Fluxes in the Eastern Pacific

AbstractThis study investigates the observed spectral character of eddy heat fluxes near the ocean surface, focusing on the distribution in wavenumber and phase speed space. Eddy heat fluxes in the eastern Pacific are calculated from concurrent satellite sea surface height and sea surface temperature data. A high-resolution coupled climate model is also analyzed in order to verify the physical mechanisms involved and to validate the model against observations. Wavenumber, frequency, and phase speed power spectra and cross spectra are constructed and presented as a function of latitude. These spectra reveal the dominance of coherent mesoscale eddies in both the length scale and phase speed of eddy heat fluxes. The breadths of the spectra are characterized via spectral moments; these moments show that the eddy fluxes are relatively concentrated around the dominant wavenumber and phase speed. Good agreement is found between the model and the observed spectra. The integrated heat transport and corresponding eddy diffusivity are shown to compare well with previous studies, but the results give a deeper insight into what determines the heat flux. Implications for eddy parameterization are discussed.

MJO structure associated with the higher-order CEOF modes

The real-time multivariate Madden–Julian oscillation (RMM; MJO) index has been widely employed to monitor the amplitude, phase, and time evolution of MJO events, as the index is formulated from the leading two combined-empirical orthogonal function (CEOF) modes of daily anomalous OLR and 850- and 200-hPa zonal winds, and the modes describe the MJO dynamics well. These two CEOF modes, however, are known to dominate in power spectra at zonal wavenumber one and may underestimate the power and structure at wavenumbers 2–5 where many MJO events are also prominent. This study approximated a baseline for MJO by applying band-pass filters to daily anomalies on 30–100 day periods and at 1–5 eastward propagating waves, as slightly different bands led to the same conclusions. Following the procedures to develop the RMM index, the daily anomalous data were derived and subjected to the CEOF analysis with all modes archived for diagnosis. Different numbers of the leading modes were compared in explained variance, standard deviation (STD), and wavenumber power spectra to describe the overall MJO magnitude and structure, and on the Hovmoller diagrams to represent the evolution of three distinct MJO events. Results show that the two leading CEOF modes explain only a small portion of the power spectra at wavenumbers 2–5. This spectral leakage notably reduces the MJO amplitude, particularly of the OLR in the western Pacific. The CEOF modes 3–10 can withhold power sufficiently such that the anomalies reconstructed by the first 10 modes contribute most of the baseline variance; their structures agree well with the baseline by constituting nearly the same proportion in the region from the central Indian Ocean to the dateline and by providing more complete evolutions of the three MJO events on the Hovmoller diagrams. Meanwhile, these modes introduce a notable amount of power for the equatorial Rossby and Kelvin waves that are partially embedded in the evolution of MJO. The first 50 of the total 432 CEOF modes retain all variance of the baseline MJO, while those higher than 10 contain less information and more noise and can be discarded. Furthermore, this study indicated that the longitudinal STD of the reconstructed anomalies detects the MJO phases and magnitudes in the western Pacific with more physical meaning and in better agreement with the Hovmoller diagrams than the RMM-like amplitude. The results provide an integral figure of the MJO structure from the CEOF analysis and a more robust RMM framework for monitoring the MJO’s evolution in real time and for validating its numerical forecast and simulations.

Influence of rough seabed surface on statistics of modal energy flux

When sound propagates through a random media, the wave properties of the acoustic field may be viewed as stochastic variables. It is thus natural to investigate the relationship between the statistical properties of the acoustic field and the random fluctuations of the waveguide. The interest here is a 2-D ocean waveguide with a rough seabed. Numerical solutions to the 2-way integral coupled mode equations (ICME), for random realizations of the roughness from a wavenumber power spectrum, provide the statistics of the modal intensity and cross-mode coherence with range. The roughness induces mode coupling within the trapped spectrum, between the trapped and the continuum spectrum, and to the back propagating modal spectrum. Instead of a master equation for the modal intensities to study the connection between the statistics of the acoustics and the fluctuations in the waveguide as has been advocated in previous studies, the conservation law for acoustic energy flux is used to develop an expression for the individual modal Poynting vectors. In addition to exact numerical computations of the range derivatives of the modal energy flux vectors for both forward and backward propagation, a Poynting vector master equation is derived for the case where the Born approximation is valid.

Role of gravity waves in the spatial and temporal variability of stratospheric temperature measured by COSMIC/FORMOSAT‐3 and Rayleigh lidar observations

This study utilizes COSMIC satellite and lidar observations to examine the spatial and temporal variability of stratospheric temperature at a number of scales. The geographic variation of the RMS temperature difference between pairs of COSMIC profiles shows a strong correspondence to previous climatologies of gravity wave activity. In addition, the second‐order structure functions we form can be directly related to the horizontal wave number power spectrum. These structure functions for different seasons and altitudes display a close correspondence to previous studies which examined the form of the horizontal wave number power spectra. Our analysis suggests that the wavefield may be particularly affected by changes in the zonal wind between 15 and 25 km and that the wind reversal between tropospheric westerlies and stratospheric easterlies in summer strongly contributes to critical‐level filtering. Inspection also shows that longer horizontal wavelength waves are preferentially removed in this region. At low altitudes, the variability related to gravity waves shows a remarkably similar pattern as a function of horizontal separation in both hemispheres but is quite different at higher altitudes. Such contrast implies that seasonal variability at higher altitudes may be dominated by changes in propagation conditions in the lower stratosphere. Examination of temperature variability as a function of spatial and temporal separation indicates that gravity wave activity dominates stratospheric temperature variability, and this has impacts on validation study site selection. For example, validation exercises in the summer hemisphere stratosphere are likely to be less affected by geophysical variability than those in the winter hemisphere.

Kelvin-Helmholtz instability of the magnetopause of disc-accreting stars

This work investigates the short wavelength stability of the magnetopause between a rapidly-rotating, supersonic, dense accretion disc and a slowly-rotating low-density magnetosphere of a magnetized star. The magnetopause is a strong shear layer with rapid changes in the azimuthal velocity, the density, and the magnetic field over a short radial distance and thus the Kelvin-Helmholtz (KH) instability may be important. The plasma dynamics is treated using non-relativistic, compressible (isentropic) magnetohydrodynamics. It is necessary to include the displacement current in order that plasma wave velocities remain less than the speed of light. We focus mainly on the case of a star with an aligned dipole magnetic field so that the magnetic field is axial in the disc midplane and perpendicular to the disc flow velocity. However, we also give results for cases where the magnetic field is at an arbitrary angle to the flow velocity. For the aligned dipole case the magnetopause is most unstable for KH waves propagating in the azimuthal direction perpendicular to the magnetic field which tends to stabilize waves propagating parallel to it. The wave phase velocity is that of the disc matter. A quasi-linear theory of the saturation of the instability leads to a wavenumber ($k$) power spectrum $\propto k^{-1}$ of the density and temperature fluctuations of the magnetopause, and it gives the mass accretion and angular momentum inflow rates across the magnetopause. For self-consistent conditions this mass accretion rate will be equal to the disc accretion rate at large distances from the magnetopause.

CHAMP observations of global gravity wave fields in the troposphere and stratosphere

This paper presents the analysis of 56 months of CHAMP (Challenging Minisatellite Payload) RO (radio occultation) temperature profiles to study the global gravity wave fields in the troposphere and lower stratosphere. The relatively new GPS (Global Positioning System) RO technique is suitable for global monitoring and provides high vertical resolution, long‐term stability, and opportunity of all‐weather viewing. The CHAMP data set analyzed in this paper is the largest RO data set used for the investigation of global structure of gravity waves. The analysis also makes use of the global radiosonde data provided by the British Atmospheric Data Centre (BADC). First, the global winter temperature pattern and temperature fluctuations in the troposphere and lower stratosphere are discussed. The global structure of gravity wave variances in the upper troposphere and lower stratosphere for the summer and winter seasons indicate the dominance in the tropical region. The analysis reveals that the maximum potential energy (Ep) is observed around the equator with considerable longitudinal variation. Comparison of the wave energy derived from the CHAMP and radiosonde data sets for the low‐latitude region (20°S–20°N) shows good similarity. The observed vertical wave number power spectra were compared with a model spectrum. The low‐latitude power spectra show considerable similarities with the model spectrum. However, the disparity increases in the midlatitude or high‐latitude sectors; particularly, in the high latitudes the observed spectral power is much underestimated for longer wavelengths. The results demonstrate the usefulness of GPS RO data in determining the global gravity wave fields.

The effects of finite lifetime of passive scalars and vorticity on their power spectra

The power law exponent for the wavenumber power spectrum of a passive scalar field with finite lifetime in Lagrangian chaotic flows and the enstrophy spectrum of two-dimensional Navier–Stokes turbulence with drag are investigated. The power law exponents are related to the distribution of finite-time Lyapunov exponents and the finite lifetime (or the drag coefficient).

Gravity Wave Spectra from GPS/MET Occultation Observations

Abstract The potential utility of radio occultation data in general, and of data from the Global Positioning System/Meteorology (GPS/MET) experiment in particular, for studying atmospheric gravity waves is discussed. Based on a validated set of ∼270 GPS/MET-derived temperature profiles, the authors produced and analyzed mean vertical wavenumber power spectra of normalized temperature fluctuations in three latitude bands (low, middle, high) within the lower stratosphere (∼15–30 km), where data accuracy was proven highest. The Fresnel diffraction limited vertical resolution and the limited height range of the dataset restricted this initial investigation to medium- to large-scale waves with vertical wavelengths of about 2–5 km. The deduced vertical wavenumber power spectra were compared with a saturation spectrum predicted by gravity wave saturation theory and generally found consistent with the theoretical saturation limit. The low-latitude power spectra exhibited almost saturation, with spectral power abo...