skip to main content


The NSF Public Access Repository (NSF-PAR) system and access will be unavailable from 11:00 PM ET on Thursday, May 23 until 2:00 AM ET on Friday, May 24 due to maintenance. We apologize for the inconvenience.

Title: Paraxial Wave Propagation in Random Media with Long-Range Correlations
We study the paraxial wave equation with a randomly perturbed index of refraction, which can model the propagation of a wave beam in a turbulent medium. The random perturbation is a stationary and isotropic process with a general form of the covariance that may or may not be integrable. We focus attention mostly on the nonintegrable case, which corresponds to a random perturbation with long-range correlations, that is, relevant for propagation through a cloudy turbulent atmosphere. The analysis is carried out in a high-frequency regime where the forward scattering approximation holds. It reveals that the randomization of the wave field is multiscale: The travel time of the wave front is randomized at short distances of propagation, and it can be described by a fractional Brownian motion. The wave field observed in the random travel time frame is affected by the random perturbations at long distances, and it is described by a Schr\"odinger-type equation driven by a standard Brownian field. We use these results to quantify how scattering leads to decorrelation of the spatial and spectral components of the wave field and to a deformation of the pulse emitted by the source. These are important questions for applications, such as imaging and free space communications with pulsed laser beams through a turbulent atmosphere. We also compare the results with those used in the optics literature, which are based on the Kolmogorov model of turbulence. We show explicitly that the commonly used approximations for the decorrelation of spatial and spectral components are appropriate for the Kolmogorov model but fail for models with long-range correlations.  more » « less
Award ID(s):
Author(s) / Creator(s):
; ;
Publisher / Repository:
Date Published:
Journal Name:
SIAM Journal on Applied Mathematics
Page Range / eLocation ID:
25 to 51
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    Atmospheric gravity waves (AGWs) are low-frequency, buoyancy-driven waves that are generated by turbulent convection and propagate obliquely throughout the solar atmosphere. Their proposed energy contribution to the lower solar atmosphere and sensitivity to atmospheric parameters (e.g., magnetic fields and radiative damping) highlight their diagnostic potential. We investigate AGWs near a quiet-Sun disk center region using multiwavelength data from the Interferometric Bidimensional Spectrometer and the Solar Dynamics Observatory. These observations showcase the complex wave behavior present in the entire acoustic-gravity wave spectrum. Using Fourier spectral analysis and local helioseismology techniques on simultaneously observed line core Doppler velocity and intensity fluctuations, we study both the vertical and horizontal properties of AGWs. Propagating AGWs with perpendicular group and phase velocities are detected at the expected temporal and spatial scales throughout the lower solar atmosphere. We also find previously unobserved, varied phase difference distributions among our velocity and intensity diagnostic combinations. Time–distance analysis indicates that AGWs travel with an average group speed of 4.5 km s−1, which is only partially described by a simple simulation, suggesting that high-frequency AGWs dominate the signal. Analysis of the median magnetic field (4.2 G) suggests that propagating AGWs are not significantly affected by quiet-Sun photospheric magnetic fields. Our results illustrate the importance of multiheight observations and the necessity of future work to properly characterize this observed behavior.

    more » « less
  2. Dynamic and steady-state aspects of wave propagation are deeply connected in lossless open systems ‎in which the scattering matrix is unitary. There is then an equivalence among the energy excited within ‎the medium through all channels, the Wigner time delay, which is the sum of dwell times in all ‎channels coupled to the medium, and the density of states. But these equivalences fall away in the ‎presence of material loss or gain. In this paper, we use microwave measurements, numerical ‎simulations, and theoretical analysis to discover the changing relationships among fundamental wave ‎properties with loss and gain, and their dependence upon dimensionality and spectral overlap. We ‎begin with the demonstrations that the transmission time in random 1D media is equal to the density ‎of states even in the presence of ultrastrong absorption and that its ensemble average is independent ‎of the strengths of scattering and absorption. In contrast, the Wigner time becomes imaginary in the ‎presence of loss, with real and imaginary parts that fall with absorption. In multichannel media, the ‎transmission time remains equal to the density of states and is independent of the scattering strength ‎in unitary systems but falls with absorption to a degree that increases with the strengths of absorption ‎and scattering, and the number of channels coupled to the medium. We show that the relationships ‎between key propagation variables in non-Hermitian systems can be understood in terms of the ‎singularities of the phase of the determinant of the transmission matrix. The poles of the transmission ‎matrix are the same as those of the scattering matrix, but the transmission zeros are fundamentally ‎different. Whereas the zeros of the scattering matrix are the complex conjugates of the poles, the ‎transmission zeros are topological: in unitary systems they occur only singly on the real axis or as ‎conjugate pairs. We follow the evolution and statistics of zeros in the complex plane as random ‎samples are deformed. The sensitivity of the spacing of zeros in the complex plane with deformation ‎of the sample has a square-root singularity at a zero point at which two single zeros and a complex ‎pair interconvert. The transmission time is a sum of Lorentzian functions associated with poles and ‎zeros. The sum over poles is the density of states with an average that is independent of scattering ‎and dissipation. But the sum over zeros changes with loss, gain, scattering strength and the number of ‎channels in ways that make it possible to control ultranarrow spectral features in transmission and ‎transmission time. We show that the field, including the contribution of the still coherent incident ‎wave, is a sum over modal partial fractions with amplitudes that are independent of loss and gain. The ‎energy excited may be expressed in terms of the resonances of the medium and is equal to the dwell ‎time even in the presence of loss or gain.‎ 
    more » « less
  3. There are few observational techniques for measuring the distribution of kinetic energy within the mesosphere with a wide range of spatial and temporal scales. This study describes a method for estimating the three‐dimensional mesospheric wind field correlation function from specular meteor trail echoes. Each radar echo provides a measurement of a one‐dimensional projection of the wind velocity vector at a randomly sampled point in space and time. The method relies on using pairs of such measurements to estimate the correlation function of the wind with different spatial and temporal lags. The method is demonstrated using a multistatic meteor radar data set that includes ≈105meteor echoes observed during a 24‐hr time period. The new method is found to be in good agreement with the well‐established technique for estimating horizontal mean winds. High‐resolution correlation functions with temporal, horizontal, and vertical lags are also estimated from the data. The temporal correlation function is used to retrieve the kinetic energy spectrum, which includes the semidiurnal mode and a 3‐hr period wave. The horizontal and vertical correlation functions of the wind are then used to derive second‐order structure functions, which are found to be compatible with the Kolmogorov prediction for spectral distribution of kinetic energy in the turbulent inertial range. The presented method can be used to extend the capabilities of specular meteor radars. It is relatively flexible and has a multitude of applications beyond what has been shown in this study.

    more » « less
  4. null (Ed.)
    ABSTRACT We analysed line-of-sight magnetic fields and magnetic power spectra of an undisturbed photosphere using magnetograms acquired by the Helioseismic and Magnetic Imager (HMI) on-board the Solar Dynamic Observatory and the Near InfraRed Imaging Spectrapolarimeter (NIRIS) operating at the Goode Solar Telescope of the Big Bear Solar Observatory. In the NIRIS data, we revealed thin flux tubes of 200–400 km in diameter and of 1000–2000 G field strength. The HMI power spectra determined for a coronal hole, a quiet sun, and a plage areas exhibit the same spectral index of −1 on a broad range of spatial scales from 10–20 Mm down to 2.4 Mm. This implies that the same mechanism(s) of magnetic field generation operate everywhere in the undisturbed photosphere. The most plausible one is the local turbulent dynamo. When compared to the HMI spectra, the −1.2 slope of the NIRIS spectrum appears to be more extended into the short spatial range until the cut-off at 0.8–0.9 Mm, after which it continues with a steeper slope of −2.2. Comparison of the observed and Kolmogorov-type spectra allowed us to infer that the Kolmogorov turbulent cascade cannot account for more than 35 per cent of the total magnetic energy observed in the scale range of 3.5–0.3 Mm. The energy excess can be attributed to other mechanisms of field generation such as the local turbulent dynamo and magnetic superdiffusivity observed in an undisturbed photosphere that can slow down the rate of the Kolmogorov cascade leading to a shallower resulting spectrum. 
    more » « less
  5. A model for the structure function tensor is proposed, incorporating the e↵ect of anisotropy as a linear perturbation to the standard isotropic form. The analysis extends the spectral approach of Ishihara et al. (2002) to physical space based on Kolmogorov’s theory and is valid in the inertial range of turbulence. Previous results for velocity co-spectra are used to obtain estimates of the model coe"cients. Structure functions measured from direct numerical simulations of channel flow and from experimental measurements in turbulent boundary layers are compared with predicted behaviour and reasonable agreement is found. We note that power-law scaling is more evident in the co-spectra than for the mixed structure functions. New observations are made about countergradient correlation between Fourier modes of wall normal and streamwise velocity components for wavenumbers approaching the Kolmogorov scale. 
    more » « less