Internal waves in fluid flows. Possible coexistence with turbulence

Waves in fluid flows represents the underlying theme of this research work. Wave interactions in fluid flows are part of multidisciplinary physics. It is known that many ideas and phenomena recur in such apparently diverse fields, as solar physics, meteorology, oceanography, aeronautical and hydraulic engineering, optics, and population dynamics. In extreme synthesis, waves in fluids include, on the one hand, surface and internal waves, their evolution, interaction and associated wave-driven mean flows; on the other hand, phenomena related to nonlinear hydrodynamic stability and, in particular, those leading to the onset of turbulence. Close similarities and key differences exist between these two classes of phenomena. In the hope to get hints on aspects of a potential overall vision, this study considers two different systems located at the opposite limits of the range of existing physical fluid flow situations: first, sheared parallel continuum flows - perfect incompressibility and charge neutrality - second, the solar wind - extreme rarefaction and electrical conductivity. Therefore, the activity carried out during the doctoral period consists of two parts. The first is focused on the propagation properties of small internal waves in parallel flows. This work was partly carried out in the framework of a MISTI-Seeds MITOR project proposed by Prof. D. Tordella (PoliTo) and Prof. G. Staffilani (MIT) on the long term interaction in fluid flows. The second part regards the analysis of solar-wind fluctuations from in situ measurements by the Voyagers spacecrafts at the edge of the heliosphere. This work was supported by a second MISTI-Seeds MITOR project, proposed by D. Tordella (PoliTo), J. D. Richardson (MIT, Kavli Institute), with the collaboration of M. Opher (BU)

Dispersive to non-dispersive transition and phase velocity transient for linear waves in plane wake and channel flows

In this study we analyze the phase and group velocity of three-dimensional linear traveling waves in two sheared flows, the plane channel and the wake flows. This was carried out by varying the wave number over a large interval of values at a given Reynolds number inside the ranges 20-100, 1000-8000, for the wake and channel flow, respectively. Evidence is given about the possible presence of both dispersive and non-dispersive effects which are associated with the long and short ranges of wavelength. We solved the Orr-Sommerfeld and Squire eigenvalue problem and observed the least stable mode. It is evident that, at low wave numbers, the least stable eigenmodes in the left branch of the spectrum beave in a dispersive manner. By contrast, if the wavenumber is above a specific threshold, a sharp dispersive to non-dispersive transition can be observed. Beyond this transition, the dominant mode belongs to the right branch of the spectrum. The transient behavior of the phase velocity of small three-dimensional traveling waves was also considered. Having chosen the initial conditions, we then show that the shape of the transient highly depends on the transition wavelength threshold value. We show that the phase velocty can oscillate with a frequency which is equal to the frequency width of the eigenvalue spectrum. Furthermore, evidence of intermediate self-similarity is given for the perturbation field.Comment: 19 pages, 11 figures. Text and discussion improved with respect to the first version. Accepted for publication on Physical Review

Wave focusing and related multiple dispersion transitions in plane Poiseuille flows

Motivated by the recent discovery of a dispersive-to-nondispersive transition for linear waves in shear flows, we accurately explored the wavenumber-Reynolds number parameter map of the plane Poiseuille flow, in the limit of least-damped waves. We have discovered the existence of regions of the map where the dispersion and propagation features vary significantly from their surroundings. These regions are nested in the dispersive, low-wavenumber part of the map. This complex dispersion scenario demonstrates the existence of linear dispersive focusing in wave envelopes evolving out of an initial, spatially localized, three-dimensional perturbation. An asymptotic wave packet's representation, based on the saddle-point method, allows to enlighten the nature of the packet's morphology, in particular the arrow-shaped structure and spatial spreading rates. A correlation is also highlighted between the regions of largest dispersive focusing and the regions which are most subject to strong nonlinear coupling in observations

Existence of non-dispersion niches of long perturbation waves in the plane Poiseuille flow. Impact on wave packets morphology.

We consider the dispersion of 3D wavy perturbations in the plane Poiseuille flow. We focus on the wavenumbers-Reynolds numbers map. By considering the long-term evolution of these linear traveling waves, we found a sub-region nested in the dispersive part of the map where dispersion is abruptly inhibited. This region is observed at the bottom right dial of the map (Re>29840 and k<0.35) and includes non-dispersive waves moving as the basic flow. Two other regions were observed with a dispersion substantially different with respect to the surroundings. In one case, the dispersion level measured as the difference between the group speed and the phase speed is enhanced. In the other, the dispersion level is damped. Such regions contain waves with higher phase speed than waves in the surrounding area of the parameter space. This study builds on a previous one (PRE 93, 2016) where, by moving in the map from small to high wavenumbers, we show that a dispersive-to-nondispersive transition occurs in sheared flows under fixed flow conditions. The transition takes place at a specific wavenumber threshold, which splits the map in two main regions: the lower one, the dispersive one, being that hosting the nested regions above. An inference on the morphology of wave packets is presented

Perturbation enstrophy decay in Poiseuille and Couette flows according to Synge's method

In this work we derive the conditions for no enstrophy growth for bidimensional perturbations in the plane Couette and Poiseuille flows. We follow the method of vorticity proposed by Synge in 1938 (see the Semi-Centennial Puplication of the Amer. Math. Soc., equation 12.13, and the more detailed version in the Proc. of the Fifth Inter. Congress of Applied Mechanics, pages 326-332), which is actually based on the analysis of the spatially averaged enstrophy. We find that the limit curve in the perturbation wavenumber-Reynolds number map differs from the limit for no energy growth (see e.g. Reddy 1993). In particular, the absolute stability region for the enstrophy is wider than that of the kinetic energy, and the maximum Reynolds number giving the monotonic enstrophy decay, at all wavenumbers, is 155 and 80 for the Poiseuille and Couette flows, respectively. It should be noted that in past literature the energy-based analysis was preferred to Synge's enstrophy analysis. This, possibly, for two reasons: the low diffusivity of the 1938 Vth ICAM proceedings and the objectively very complicated analytical treatment required. Nevertheless, the potentiality of this method seems high and therefore it is interesting nowadays to exploit it by means of the symbolic calculus

Voyager observations of magnetic field turbulence in the far heliosheath and in the local interstellar medium. Power spectra from high-resolution data.

Voyager 2 (V2) is in the heliosheath (HS) since the termination shock crossing in Aug 2007, while V1 is in the local interstellar medium (LISM) since Aug 2012. The fundamental processes at the basis of the observed solar wind's disordered fluctuations are still unclear. Open points regard the nature of compressive turbulence within the sectored and unipolar HS in proximity of the heliopause and in the LISM. Possibility that MHD waves give origin to turbulence in the LISM has been recently suggested by Zank, Du & Hunana [APJ 842,2017]. However, addressing these issues is a challenging task because of the data sparsity. We provide the first collection of magnetic field power spectra computed in consecutive periods after 2009 from 48s resolution data in the HS (V1, V2) and in the LISM (V1). A description of the fluctuations evolution with the heliocentric distance is given in terms of spectral decay law and anisotropy. In the HS, our observations are consistent with an anisotropic mainly inertial cascade in the frequency range [ 10^-4 , 5 ⋅ 10^-5 ] Hz, with spectral slopes from -1.7 to -1.9. Larger scales may be featured by wavy fluctuations leading to a f ^-1 decay for f < 10^-5 Hz. LISM spectra show a f^-1 power law in the whole observed range [ 10^-7 , 10^-2 ] Hz

A Lower Bound for Transient Enstrophy Growth of Two-Dimensional Internal Traveling Waves

This study provides the temporal monotonic decay region of the wavenumber-Reynolds number stability map, for the enstrophy of any two-dimensional perturbations traveling in the incompressible and viscous plane Poiseuille and plane Couette flows. Mathematical difficulty related to this problem was due to the unknown boundary conditions on the perturbation vorticity, which left the problem open since the first historical studies conducted by J. L. Synge 234 (1930s). By extending Synge’s work to the non-modal approach, we provide the smallest Reynolds number, Re Ω , allowing transient growth of perturbations’ integral-enstrophy. As a noticeable result, the enstrophy monotonic decay region inside the parameters space is wider than the kinetic energy one. The shape, evolution and wall vorticity of optimal-enstrophy streamfunctions will also be discussed. Concurrently, this study considers the dispersive nature of wavy perturbations. Building on our previous study, we show how the coexistence of dispersion andnondispersion at fixed value of the flow control parameter can affect the morphology and evolution of wave packets in the plane Poiseuille flow. Short waves experience mild growth but travel nondispersively and generate compact structures. Dispersive wave components show the largest enstrophy growth and are responsible for the morphology of the spot core. Both components are relevant in the dynamics of transitional structures

Voyager 2 solar plasma and magnetic field spectral analysis for intermediate data sparsity

The Voyager probes are the furthest, still active, spacecraft ever launched from Earth. During their 38-year trip, they have collected data regarding solar wind properties (such as the plasma velocity and magnetic field intensity). Unfortunately, a complete time evolution of the measured physical quantities is not available. The time series contains many gaps which increase in frequency and duration at larger distances. The aim of this work is to perform a spectral and statistical analysis of the solar wind plasma velocity and magnetic field using Voyager 2 data measured in 1979, when the gaps/signal ratio is of order of unity. This analysis is achieved using four different data reconstruction techniques: averages on linearly interpolated subsets, correlation of linearly interpolated data, compressed sensing spectral estimation, and maximum likelihood data reconstruction. With five frequency decades, the spectra we obtained have the largest frequency range ever computed at 5 astronomical units from the Sun; spectral exponents have been determined for all the components of the velocity and magnetic field fluctuations. Void analysis is also useful in recovering other spectral properties such as integral scales (see for instance Table 4) and, if the confidence level of the measurements is sufficiently high, the decay variation in the small scale range due, for instance, to dissipative effects.Comment: 11 pages, 7 figure

The structure of magnetic turbulence in the heliosheath region observed by Voyager 2 at 106 AU

It is currently believed that the turbulent fluctuations pervade the outermost heliosphere. Turbulence, magnetic reconnection, and their link may be responsible for magnetic energy conversion in these regions. The governing mechanisms of such anisotropic and compressible magnetic turbulence in the inner heliosheath (IHS) and in the local interstellar medium (LISM) still lack a thorough description. The present literature mainly concerns large scales which are not representative of the inertial-cascade dynamics of turbulence. Moreover, lack of broadband spectral analysis makes the IHS dynamics critically understudied. Our recent study [1] shows that 48 s magnetic-field data from the Voyager mission are appropriate for a spectral analysis over a frequency range of six decades, from 5 x 10-8 Hz to 10-2 Hz. Here, focusing on the Voyager 2 observation interval from 2013.824 to 2016.0, we describe the structure of turbulence in a sector zone of the IHS. A spectral break around 7 x 10-7 Hz (magnetic structures with size l ~ 1.3 Astronomical Units) separates the energy-injection regime from the inertial-cascade regime of turbulence. A second scale is observed around 6 x 10-5 Hz (l~ 0.017 AU) and corresponds to a peak of compressibility and intermittency of fluctuations