48 research outputs found
Local and Nonlocal Dispersive Turbulence
We consider the evolution of a family of 2D dispersive turbulence models. The
members of this family involve the nonlinear advection of a dynamically active
scalar field, the locality of the streamfunction-scalar relation is denoted by
, with smaller implying increased locality. The dispersive
nature arises via a linear term whose strength is characterized by a parameter
. Setting , we investigate the interplay of
advection and dispersion for differing degrees of locality. Specifically, we
study the forward (inverse) transfer of enstrophy (energy) under large-scale
(small-scale) random forcing. Straightforward arguments suggest that for small
the scalar field should consist of progressively larger eddies, while
for large the scalar field is expected to have a filamentary structure
resulting from a stretch and fold mechanism. Confirming this, we proceed to
forced/dissipative dispersive numerical experiments under weakly non-local to
local conditions. For , there is quantitative agreement
between non-dispersive estimates and observed slopes in the inverse energy
transfer regime. On the other hand, forward enstrophy transfer regime always
yields slopes that are significantly steeper than the corresponding
non-dispersive estimate. Additional simulations show the scaling in the inverse
regime to be sensitive to the strength of the dispersive term : specifically,
as decreases, the inertial-range shortens and we also observe that
the slope of the power-law decreases. On the other hand, for the same range of
values, the forward regime scaling is fairly universal.Comment: 19 pages, 8 figures. Significantly revised with additional result
Surface Quasigeostrophic Turbulence : The Study of an Active Scalar
We study the statistical and geometrical properties of the potential
temperature (PT) field in the Surface Quasigeostrophic (SQG) system of
equations. In addition to extracting information in a global sense via tools
such as the power spectrum, the g-beta spectrum and the structure functions we
explore the local nature of the PT field by means of the wavelet transform
method. The primary indication is that an initially smooth PT field becomes
rough (within specified scales), though in a qualitatively sparse fashion.
Similarly, initially 1D iso-PT contours (i.e., PT level sets) are seen to
acquire a fractal nature. Moreover, the dimensions of the iso-PT contours
satisfy existing analytical bounds. The expectation that the roughness will
manifest itself in the singular nature of the gradient fields is confirmed via
the multifractal nature of the dissipation field. Following earlier work on the
subject, the singular and oscillatory nature of the gradient field is
investigated by examining the scaling of a probability measure and a sign
singular measure respectively. A physically motivated derivation of the
relations between the variety of scaling exponents is presented, the aim being
to bring out some of the underlying assumptions which seem to have gone
unnoticed in previous presentations. Apart from concentrating on specific
properties of the SQG system, a broader theme of the paper is a comparison of
the diagnostic inertial range properties of the SQG system with both the 2D and
3D Euler equations.Comment: 26 pages, 11 figures. To appear in Chao
The Decay of Passive Scalars Under the Action of Single Scale Smooth Velocity Fields in Bounded 2D Domains : From non self similar pdf's to self similar eigenmodes
We examine the decay of passive scalars with small, but non zero, diffusivity
in bounded 2D domains. The velocity fields responsible for advection are smooth
(i.e., they have bounded gradients) and of a single large scale. Moreover, the
scale of the velocity field is taken to be similar to the size of the entire
domain. The importance of the initial scale of variation of the scalar field
with respect to that of the velocity field is strongly emphasized. If these
scales are comparable and the velocity field is time periodic, we see the
formation of a periodic scalar eigenmode. The eigenmode is numerically realized
by means of a deterministic 2D map on a lattice. Analytical justification for
the eigenmode is available from theorems in the dynamo literature. Weakening
the notion of an eigenmode to mean statistical stationarity, we provide
numerical evidence that the eigenmode solution also holds for aperiodic flows
(represented by random maps). Turning to the evolution of an initially small
scale scalar field, we demonstrate the transition from an evolving (i.e., {\it
non} self similar) pdf to a stationary (self similar) pdf as the scale of
variation of the scalar field progresses from being small to being comparable
to that of the velocity field (and of the domain). Furthermore, the {\it non}
self similar regime itself consists of two stages. Both the stages are examined
and the coupling between diffusion and the distribution of the Finite Time
Lyapunov Exponents is shown to be responsible for the pdf evolution.Comment: 21 pages (2 col. format), 11 figures. Accepted, to appear in PR
Low frequency modulation of jets in quasigeostrophic turbulence
Quasigeostrophic turbulence on a beta-plane with a finite deformation radius
is studied nu- merically, with particular emphasis on frequency and combined
wavenumber-frequency do- main analyses. Under suitable conditions, simulations
with small-scale random forcing and large-scale drag exhibit a spontaneous
formation of multiple zonal jets. The first hint of wave-like features is seen
in the distribution of kinetic energy as a function of frequency; specifically,
for progressively larger deformation scales there are systematic departures in
the form of isolated peaks (at progressively higher frequencies) from a
power-law scaling. Con- comitantly, there is an inverse flux of kinetic energy
in frequency space which extends to lower frequencies for smaller deformation
scales. The identification of these peaks as Rossby waves is made possible by
examining the energy spectrum in frequency-zonal wavenumber and
frequency-meridional wavenumber diagrams. In fact, the modified Rhines scale
turns out to be a useful measure of the dominant meridional wavenumber of the
modulating Rossby waves; once this is fixed, apart from a spectral peak at the
origin (the steady jet), almost all the energy is contained in westward
propagating disturbances that follow the theoretical Rossby dispersion
relation. Quite consistently, noting that the zonal scale of the modulating
waves is restricted to the first few wavenumbers, the energy spectrum is almost
entirely contained within the corresponding Rossby dispersion curves on a
frequency-meridional wavenumber diagram. Cases when jets do not form are also
considered; once again, there is a hint of Rossby wave activity, though the
spectral peaks are quite muted. Further, the kinetic energy scaling in
frequency domain follows a -5/3 power-law and is distributed much more broadly
in frequency-wavenumber diagramsComment: 18 pages, 9 fig
Quasi-geostrophic dynamics in the presence of moisture gradients
The derivation of a quasi-geostrophic (QG) system from the rotating shallow
water equations on a midlatitude beta-plane coupled with moisture is presented.
Condensation is prescribed to occur whenever the moisture at a point exceeds a
prescribed saturation value. It is seen that a slow condensation time scale is
required to obtain a consistent set of equations at leading order. Further,
since the advecting wind fields are geostrophic, changes in moisture (and
hence, precipitation) occur only via non-divergent mechanisms. Following
observations, a saturation profile with gradients in the zonal and meridional
directions is prescribed. A purely meridional gradient has the effect of
slowing down the dry Rossby waves, through a reduction in the "equivalent
gradient" of the background potential vorticity. A large scale unstable moist
mode results on the inclusion of a zonal gradient by itself, or in conjunction
with a meridional moisture gradient. For gradients that are are representative
of the atmosphere, the most unstable moist mode propagates zonally in the
direction of increasing moisture, matures over an intraseasonal timescale and
has small phase speed.Comment: 9 pages, 8 figures, Quarterly Journal of the Royal Meteorological
Society, DOI:10.1002/qj.2644, 201