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A note on the modelling of hyper-inflations
In time series macroeconometric models, the first difference in the logarithm of a variable is routinely used to represent the rate of change of that variable. It is often overlooked that the assumed approximation is accurate only if the rates of change are small. Models of hyper-inflation are a case in point, since in these models, by definition, changes in price are large. In this letter, Caganâs model is applied to Hungarian hyper-inflation data. It is then demonstrated that use of the approximation in the formation of the price inflation variable is causing an upward bias in the modelâs key parameter, and therefore an exaggeration of the effect postulated by Cagan
Analytical theory of forced rotating sheared turbulence: The perpendicular case
Rotation and shear flows are ubiquitous features of many astrophysical and geophysical bodies. To understand their origin and effect on turbulent transport in these systems, we consider a forced turbulence and investigate the combined effect of rotation and shear flow on the turbulence properties. Specifically, we study how rotation and flow shear influence the generation of shear flow (e.g., the direction of energy cascade), turbulence level, transport of particles and momentum, and the anisotropy in these quantities. In all the cases considered, turbulence amplitude is always quenched due to strong shear (Ο=Îœky2/AâȘĄ1, where A is the shearing rate, Îœ is the molecular viscosity, and ky is a characteristic wave number of small-scale turbulence), with stronger reduction in the direction of the shear than those in the perpendicular directions. Specifically, in the large rotation limit (ΩâȘąA), they scale as Aâ1 and Aâ1|lnâΟ|, respectively, while in the weak rotation limit (ΩâȘĄA), they scale as Aâ1 and Aâ2/3, respectively. Thus, flow shear always leads to weak turbulence with an effectively stronger turbulence in the plane perpendicular to shear than in the shear direction, regardless of rotation rate. The anisotropy in turbulence amplitude is, however, weaker by a factor of Ο1/3|lnâΟ| (âAâ1/3|lnâΟ|) in the rapid rotation limit (ΩâȘąA) than that in the weak rotation limit (ΩâȘĄA) since rotation favors almost-isotropic turbulence. Compared to turbulence amplitude, particle transport is found to crucially depend on whether rotation is stronger or weaker than flow shear. When rotation is stronger than flow shear (ΩâȘąA), the transport is inhibited by inertial waves, being quenched inversely proportional to the rotation rate (i.e., âΩâ1) while in the opposite case, it is reduced by shearing as Aâ1. Furthermore, the anisotropy is found to be very weak in the strong rotation limit (by a factor of 2) while significant in the strong shear limit. The turbulent viscosity is found to be negative with inverse cascade of energy as long as rotation is sufficiently strong compared to flow shear (ΩâȘąA) while positive in the opposite limit of weak rotation (ΩâȘĄA). Even if the eddy viscosity is negative for strong rotation (ΩâȘąA), flow shear, which transfers energy to small scales, has an interesting effect by slowing down the rate of inverse cascade with the value of negative eddy viscosity decreasing as |ÎœT|âAâ2 for strong shear. Furthermore, the interaction between the shear and the rotation is shown to give rise to a nondiffusive flux of angular momentum (Î effect), even in the absence of external sources of anisotropy. This effect provides a mechanism for the existence of shearing structures in astrophysical and geophysical systems
Periodic magnetorotational dynamo action as a prototype of nonlinear magnetic field generation in shear flows
The nature of dynamo action in shear flows prone to magnetohydrodynamic
instabilities is investigated using the magnetorotational dynamo in Keplerian
shear flow as a prototype problem. Using direct numerical simulations and
Newton's method, we compute an exact time-periodic magnetorotational dynamo
solution to the three-dimensional dissipative incompressible
magnetohydrodynamic equations with rotation and shear. We discuss the physical
mechanism behind the cycle and show that it results from a combination of
linear and nonlinear interactions between a large-scale axisymmetric toroidal
magnetic field and non-axisymmetric perturbations amplified by the
magnetorotational instability. We demonstrate that this large scale dynamo
mechanism is overall intrinsically nonlinear and not reducible to the standard
mean-field dynamo formalism. Our results therefore provide clear evidence for a
generic nonlinear generation mechanism of time-dependent coherent large-scale
magnetic fields in shear flows and call for new theoretical dynamo models.
These findings may offer important clues to understand the transitional and
statistical properties of subcritical magnetorotational turbulence.Comment: 10 pages, 6 figures, accepted for publication in Physical Review
Vorticity alignment results for the three-dimensional Euler and Navier-Stokes equations
We address the problem in Navier-Stokes isotropic turbulence of why the
vorticity accumulates on thin sets such as quasi-one-dimensional tubes and
quasi-two-dimensional sheets. Taking our motivation from the work of Ashurst,
Kerstein, Kerr and Gibson, who observed that the vorticity vector
{\boldmath\omega} aligns with the intermediate eigenvector of the strain
matrix , we study this problem in the context of both the three-dimensional
Euler and Navier-Stokes equations using the variables \alpha =
\hat{{\boldmath\xi}}\cdot S\hat{{\boldmath\xi}} and {\boldmath\chi} =
\hat{{\boldmath\xi}}\times S\hat{{\boldmath\xi}} where
\hat{{\boldmath\xi}} = {\boldmath\omega}/\omega. This introduces the
dynamic angle , which lies between
{\boldmath\omega} and S{\boldmath\omega}. For the Euler equations a
closed set of differential equations for and {\boldmath\chi} is
derived in terms of the Hessian matrix of the pressure . For
the Navier-Stokes equations, the Burgers vortex and shear layer solutions turn
out to be the Lagrangian fixed point solutions of the equivalent
(\alpha,{\boldmath\chi}) equations with a corresponding angle .
Under certain assumptions for more general flows it is shown that there is an
attracting fixed point of the (\alpha,\bchi) equations which corresponds to
positive vortex stretching and for which the cosine of the corresponding angle
is close to unity. This indicates that near alignment is an attracting state of
the system and is consistent with the formation of Burgers-like structures.Comment: To appear in Nonlinearity Nov. 199
``Smoke Rings'' in Ferromagnets
It is shown that bulk ferromagnets support propagating non-linear modes that
are analogous to the vortex rings, or ``smoke rings'', of fluid dynamics. These
are circular loops of {\it magnetic} vorticity which travel at constant
velocity parallel to their axis of symmetry. The topological structure of the
continuum theory has important consequences for the properties of these
magnetic vortex rings. One finds that there exists a sequence of magnetic
vortex rings that are distinguished by a topological invariant (the Hopf
invariant). We present analytical and numerical results for the energies,
velocities and structures of propagating magnetic vortex rings in ferromagnetic
materials.Comment: 4 pages, 3 eps-figures, revtex with epsf.tex and multicol.sty. To
appear in Physical Review Letters. (Postscript problem fixed.
On the three-dimensional temporal spectrum of stretched vortices
The three-dimensional stability problem of a stretched stationary vortex is
addressed in this letter. More specifically, we prove that the discrete part of
the temporal spectrum is only associated with two-dimensional perturbations.Comment: 4 pages, RevTeX, submitted to PR
On magnetic field generation in Kolmogorov turbulence
We analyze the initial, kinematic stage of magnetic field evolution in an
isotropic and homogeneous turbulent conducting fluid with a rough velocity
field, v(l) ~ l^alpha, alpha<1. We propose that in the limit of small magnetic
Prandtl number, i.e. when ohmic resistivity is much larger than viscosity, the
smaller the roughness exponent, alpha, the larger the magnetic Reynolds number
that is needed to excite magnetic fluctuations. This implies that numerical or
experimental investigations of magnetohydrodynamic turbulence with small
Prandtl numbers need to achieve extremely high resolution in order to describe
magnetic phenomena adequately.Comment: 4 pages, revised, new material adde
Large Scale Structures a Gradient Lines: the case of the Trkal Flow
A specific asymptotic expansion at large Reynolds numbers (R)for the long
wavelength perturbation of a non stationary anisotropic helical solution of the
force less Navier-Stokes equations (Trkal solutions) is effectively constructed
of the Beltrami type terms through multi scaling analysis. The asymptotic
procedure is proved to be valid for one specific value of the scaling
parameter,namely for the square root of the Reynolds number (R).As a result
large scale structures arise as gradient lines of the energy determined by the
initial conditions for two anisotropic Beltrami flows of the same helicity.The
same intitial conditions determine the boundaries of the vortex-velocity tubes,
containing both streamlines and vortex linesComment: 27 pages, 2 figure
Comment on "Growth of a weak magnetic field in a turbulent conducting fluid with large magnetic Prandtl number"
The effect of a uniform straining motion u = (α1x1, α2x2, α3x3), where α1 + α2 + α3 = 0, and α1 > α2 > α3, on what may be called a "blob" of weak magnetic field, that is a magnetic field which is exponentially small outside a certain bounded region, has recently been examined by Pao.(1
Resolving singular forces in cavity flow: Multiscale modeling from atoms to millimeters
A multiscale approach for fluid flow is developed that retains an atomistic
description in key regions. The method is applied to a classic problem where
all scales contribute: The force on a moving wall bounding a fluid-filled
cavity. Continuum equations predict an infinite force due to stress
singularities. Following the stress over more than six decades in length in
systems with characteristic scales of millimeters and milliseconds allows us to
resolve the singularities and determine the force for the first time. The
speedup over pure atomistic calculations is more than fourteen orders of
magnitude. We find a universal dependence on the macroscopic Reynolds number,
and large atomistic effects that depend on wall velocity and interactions.Comment: 4 pages,3 figure
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