69 research outputs found
Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows
We consider the flow of a Newtonian fluid in a three-dimensional domain,
rotating about a vertical axis and driven by a vertically invariant horizontal
body-force. This system admits vertically invariant solutions that satisfy the
2D Navier-Stokes equation. At high Reynolds number and without global rotation,
such solutions are usually unstable to three-dimensional perturbations. By
contrast, for strong enough global rotation, we prove rigorously that the 2D
(and possibly turbulent) solutions are stable to vertically dependent
perturbations: the flow becomes 2D in the long-time limit.
These results shed some light on several fundamental questions of rotating
turbulence: for arbitrary Reynolds number and small enough Rossby number, the
system is attracted towards purely 2D flow solutions, which display no energy
dissipation anomaly and no cyclone-anticyclone asymmetry. Finally, these
results challenge the applicability of wave turbulence theory to describe
stationary rotating turbulence in bounded domains.Comment: To be published in Journal of Fluid Mechanic
Exact two-dimensionalization of low-magnetic-Reynolds-number flows subject to a strong magnetic field
We investigate the behavior of flows, including turbulent flows, driven by a
horizontal body-force and subject to a vertical magnetic field, with the
following question in mind: for very strong applied magnetic field, is the flow
mostly two-dimensional, with remaining weak three-dimensional fluctuations, or
does it become exactly 2D, with no dependence along the vertical?
We first focus on the quasi-static approximation, i.e. the asymptotic limit
of vanishing magnetic Reynolds number Rm << 1: we prove that the flow becomes
exactly 2D asymptotically in time, regardless of the initial condition and
provided the interaction parameter N is larger than a threshold value. We call
this property "absolute two-dimensionalization": the attractor of the system is
necessarily a (possibly turbulent) 2D flow.
We then consider the full-magnetohydrodynamic equations and we prove that,
for low enough Rm and large enough N, the flow becomes exactly two-dimensional
in the long-time limit provided the initial vertically-dependent perturbations
are infinitesimal. We call this phenomenon "linear two-dimensionalization": the
(possibly turbulent) 2D flow is an attractor of the dynamics, but it is not
necessarily the only attractor of the system. Some 3D attractors may also exist
and be attained for strong enough initial 3D perturbations.
These results shed some light on the existence of a dissipation anomaly for
magnetohydrodynamic flows subject to a strong external magnetic field.Comment: Journal of Fluid Mechanics, in pres
Refraction of swell by surface currents
Using recordings of swell from pitch-and-roll buoys, we have reproduced the
classic observations of long-range surface wave propagation originally made by
Munk et al. (1963) using a triangular array of bottom pressure measurements. In
the modern data, the direction of the incoming swell fluctuates by about on a time scale of one hour. But if the incoming direction is
averaged over the duration of an event then, in contrast with the observations
by Munk et al. (1963), the sources inferred by great-circle backtracking are
most often in good agreement with the location of large storms on weather maps
of the Southern Ocean. However there are a few puzzling failures of
great-circle backtracking e.g., in one case, the direct great-circle route is
blocked by the Tuamoto Islands and the inferred source falls on New Zealand.
Mirages like this occur more frequently in the bottom-pressure observations of
Munk et al. (1963), where several inferred sources fell on the Antarctic
continent.
Using spherical ray tracing we investigate the hypothesis that the refraction
of waves by surface currents produces the mirages. With reconstructions of
surface currents inferred from satellite altimetry, we show that mesoscale
vorticity significantly deflects swell away from great-circle propagation so
that the source and receiver are connected by a bundle of many rays, none of
which precisely follow a great circle. The directional
fluctuations at the receiver result from the arrival of wave packets that have
travelled along the different rays within this multipath. The occasional
failure of great-circle backtracking, and the associated mirages, probably
results from partial topographic obstruction of the multipath, which biases the
directional average at the receiver.Comment: Journal of Marine Research, in pres
Wave turbulence description of interacting particles: Klein-Gordon model with a Mexican-hat potential
In field theory, particles are waves or excitations that propagate on the
fundamental state. In experiments or cosmological models one typically wants to
compute the out-of-equilibrium evolution of a given initial distribution of
such waves. Wave Turbulence deals with out-of-equilibrium ensembles of weakly
nonlinear waves, and is therefore well-suited to address this problem. As an
example, we consider the complex Klein-Gordon equation with a Mexican-hat
potential. This simple equation displays two kinds of excitations around the
fundamental state: massive particles and massless Goldstone bosons. The former
are waves with a nonzero frequency for vanishing wavenumber, whereas the latter
obey an acoustic dispersion relation. Using wave turbulence theory, we derive
wave kinetic equations that govern the coupled evolution of the spectra of
massive and massless waves. We first consider the thermodynamic solutions to
these equations and study the wave condensation transition, which is the
classical equivalent of Bose-Einstein condensation. We then focus on nonlocal
interactions in wavenumber space: we study the decay of an ensemble massive
particles into massless ones. Under rather general conditions, these massless
particles accumulate at low wavenumber. We study the dynamics of waves
coexisting with such a strong condensate, and we compute rigorously a nonlocal
Kolmogorov-Zakharov solution, where particles are transferred non-locally to
the condensate, while energy cascades towards large wave numbers through local
interactions. This nonlocal cascading state constitute the intermediate
asymptotics between the initial distribution of waves and the thermodynamic
state reached in the long-time limit
Surface gravity waves propagating in a rotating frame: the Ekman-Stokes instability
We report on an instability arising when surface gravity waves propagate in a
rotating frame. The Stokes drift associated to the uniform wave field, together
with global rotation, drives a mean flow in the form of a horizontally
invariant Ekman-Stokes spiral. We show that the latter can be subject to an
instability that triggers the appearance of an additional
horizontally-structured cellular flow. We determine the instability threshold
numerically, in terms of the Rossby number Ro associated to the Stokes drift of
the waves and the Ekman number E. We confirm the numerical results through
asymptotic expansions at both large and low Ekman number. At large E the
instability reduces to that of a standard Ekman spiral driven by the
wave-induced surface stress instead of a wind stress, while at low E the
Stokes-drift profile crucially determines the shape of the unstable mode. In
both limits the instability threshold asymptotes to an Ekman-number-independent
critical Rossby number, which in both cases also corresponds to a critical
Reynolds number associated to the Lagrangian base-flow velocity profile.
Parameter values typical of ocean swell fall into the low-E unstable regime:
the corresponding "anti-Stokes" flows are unstable, with possible consequences
for particle dispersion and mixing
Disentangling inertial waves from eddy turbulence in a forced rotating turbulence experiment
We present a spatio-temporal analysis of a statistically stationary rotating
turbulence experiment, aiming to extract a signature of inertial waves, and to
determine the scales and frequencies at which they can be detected. The
analysis uses two-point spatial correlations of the temporal Fourier transform
of velocity fields obtained from time-resolved stereoscopic particle image
velocimetry measurements in the rotating frame. We quantify the degree of
anisotropy of turbulence as a function of frequency and spatial scale. We show
that this space-time-dependent anisotropy is well described by the dispersion
relation of linear inertial waves at large scale, while smaller scales are
dominated by the sweeping of the waves by fluid motion at larger scales. This
sweeping effect is mostly due to the low-frequency quasi-two-dimensional
component of the turbulent flow, a prominent feature of our experiment which is
not accounted for by wave turbulence theory. These results question the
relevance of this theory for rotating turbulence at the moderate Rossby numbers
accessible in laboratory experiments, which are relevant to most geophysical
and astrophysical flows
Onset of three-dimensionality in rapidly rotating turbulent flows
Turbulent flows driven by a vertically invariant body force were proven to
become exactly two-dimensional above a critical rotation rate, using upper
bound theory. This transition in dimensionality of a turbulent flow has key
consequences for the energy dissipation rate. However, its location in
parameter space is not provided by the bounding procedure. To determine this
precise threshold between exactly 2D and partially 3D flows, we perform a
linear stability analysis over a fully turbulent 2D base state. This requires
integrating numerically a quasi-2D set of equations over thousands of turnover
times, to accurately average the growth rate of the 3D perturbations over the
statistics of the turbulent 2Dbase flow. We leverage the capabilities of modern
GPUs to achieve this task, which allows us to investigate the parameter space
up to Re = 10^5. At Reynolds numbers typical of 3D DNS and laboratory
experiments, Re in [10^2, 5x10^3], the turbulent 2D flow becomes unstable to 3D
motion through a centrifugal-type instability. However, at even higher Reynolds
number another instability takes over. A candidate mechanism for the latter
instability is the parametric excitation of inertial waves by the modulated 2D
flow, a phenomenon that we illustrate with an oscillatory 2D Kolmogorov flow
Dynamo saturation down to vanishing viscosity: strong-field and inertial scaling regimes
We present analytical examples of fluid dynamos that saturate through the
action of the Coriolis and inertial terms of the Navier-Stokes equation. The
flow is driven by a body force and is subject to global rotation and uniform
sweeping velocity. The model can be studied down to arbitrarily low viscosity
and naturally leads to the strong-field scaling regime for the magnetic energy
produced above threshold: the magnetic energy is proportional to the global
rotation rate and independent of the viscosity. Depending on the relative
orientations of global rotation and large-scale sweeping, the dynamo
bifurcation is either supercritical or subcritical. In the supercritical case,
the magnetic energy follows the scaling-law for supercritical strong-field
dynamos predicted on dimensional grounds by Petrelis & Fauve (2001). In the
subcritical case, the system jumps to a finite-amplitude dynamo branch. The
magnetic energy obeys a magneto-geostrophic scaling-law (Roberts & Soward
1972), with a turbulent Elsasser number of the order of unity, where the
magnetic diffusivity of the standard Elsasser number appears to be replaced by
an eddy diffusivity. In the absence of global rotation, the dynamo bifurcation
is subcritical and the saturated magnetic energy obeys the equipartition
scaling regime. We consider both the vicinity of the dynamo threshold and the
limit of large distance from threshold to put these various scaling behaviors
on firm analytical ground
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