Fast Rotation Limit Research Articles

Fast Rotating Non-homogeneous Fluids in Thin Domains and the Ekman Pumping Effect

In this paper, we perform the fast rotation limit varepsilon rightarrow 0^+ of the density-dependent incompressible Navier–Stokes–Coriolis system in a thin strip Omega _varepsilon :=,{mathbb {R}}^2times , left. right] -ell _varepsilon ,ell _varepsilon left[ right. ,, where varepsilon in ,left. right] 0,1left. right] is the size of the Rossby number and ell _varepsilon >0 for any varepsilon >0. By letting ell _varepsilon longrightarrow 0^+ for varepsilon rightarrow 0^+ and considering Navier-slip boundary conditions at the boundary of Omega _varepsilon , we give a rigorous justification of the phenomenon of the Ekman pumping in the context of non-homogeneous fluids. With respect to previous studies (performed for flows of contant density and for compressible fluids), our approach has the advantage of circumventing the complicated analysis of boundary layers. To the best of our knowledge, this is the first study dealing with the asymptotic analysis of fast rotating incompressible fluids with variable density in a 3-D setting. In this respect, we remark that the case ell _varepsilon geqslant ell >0 for all varepsilon >0 remains largely open at present.

Fast rotation limit for the 2-D non-homogeneous incompressible Euler equations

In the present paper, we study the fast rotation limit for the density-dependent incompressible Euler equations in two space dimensions with the presence of the Coriolis force. In the case when the initial densities are small perturbation of a constant profile, we show the convergence of solutions towards the solutions of a quasi-homogeneous incompressible Euler system. The proof relies on a combination of uniform estimates in high regularity norms with a compensated compactness argument for passing to the limit. This technique allows us to treat the case of ill-prepared initial data.

Incompressible and fast rotation limit for barotropic Navier–Stokes equations at large Mach numbers

We study the incompressible and fast rotation limit for the barotropic Navier–Stokes equations with Coriolis force, in the case when the Mach number Ma is large with respect to the Rossby number Ro: namely, we focus on the regime Ro≪Ma. For this, we follow a recent approach in Danchin and Mucha (2017) and take also a large bulk viscosity coefficient. We prove that the limit dynamics is described by an incompressible Navier–Stokes type equation, recasted in the vorticity formulation, where however an additional unknown, linked to density oscillations around a fixed constant reference state, comes into play. The proof of the convergence is based on a compensated compactness argument and on the derivation of sharp decay estimates for solutions to a heat equation with fast diffusion in time.

Mean flow anisotropy without waves in rotating turbulence

We tackle the question of how anisotropy in flows subject to background rotation favours structures elongated along the rotation axis, especially in turbulent flows. A new, wave-free mechanism is identified that challenges the current understanding of the process. Inertial waves propagating near the rotation axis are generally accepted as the most efficient mechanism to transport energy anisotropically. They have been shown to transfer energy to large anisotropic, columnar structures. Nevertheless, they cannot account for the formation of simpler steady anisotropic phenomena such as Taylor columns. Here, we experimentally show that more than one mechanism involving the Coriolis force may promote anisotropy. In particular, in the limit of fast rotation, that is at low Rossby number, anisotropy favouring the direction of rotation of the average of a turbulent flow arises neither because of inertial waves nor following the same mechanism as in steady Taylor columns, but from an interplay between the Coriolis force and average advection.

Asymptotics of fast rotating density-dependent incompressible fluids in two space dimensions

In the present paper we study the fast rotation limit for viscous incompressible fluids with variable density, whose motion is influenced by the Coriolis force. We restrict our analysis to two dimensional flows. In the case when the initial density is a small perturbation of a constant state, we recover in the limit the convergence to the homogeneous incompressible Navier–Stokes equations (up to an additional term, due to density fluctuations). For general non-homogeneous fluids, the limit equations are instead linear, and the limit dynamics is described in terms of the vorticity and the density oscillation function: we lack enough regularity on the latter to prove convergence on the momentum equation itself. The proof of both results relies on a compensated compactness argument, which enables one to treat also the possible presence of vacuum.

Centrifugal instability in the regime of fast rotation

Centrifugal instability, which stems from a difference between the azimuthal angular drift velocity of ions and electrons, is studied in the limit of fast rotation for which ions can rotate up to twice as fast as electrons. As the angular velocity approaches the so-called Brillouin limit, the growth rate for the centrifugal instability in a collisionless solid-body rotating plasma increases markedly and is proportional to the azimuthal mode number. For large wavenumbers, electron inertia effects set in and lead to a cut-off. Interestingly, conditions for the onset of this instability appear to overlap with the operating conditions envisioned for plasma mass separation devices.

Effect of particle rotation on the drift velocity for nonspherical aerosol particles

The theoretical drift velocity of a randomly oriented nonspherical aerosol particles in an external field has been previously computed both in the limit of slow rotation and in the limit of fast rotation but not in the intermediate interval. The low rotation limit has been widely used to calculate the drift velocity for a range of nonspherical particles. The fast rotation limit, which is equivalent to the projected area method, has been used for molecular ions and agglomerates. A 1-D model equation containing the particle acceleration and an orientation dependent friction coefficient is proposed to predict the drift velocity between the two limits. This model has the essential physical phenomena without the complications of the 3-D treatment of the combined translation and rotation behavior. As an example, the drift velocity is computed for model parameters based on the friction tensor and rotational diffusion coefficient for circular cross section nanorods in the free molecular limit. For a momentum accommodation coefficient of 0.9 and a particle density of 1000kg/m3, the largest percent deviation from the low rotation velocity limit is 14% and the deviation is at most 1% for nanorods of any length for diameters of 20nm diameter or larger. Much larger changes in the velocity ratio are shown to occur if the momentum accommodation coefficient is reduced. Also, examples are given where the dimensionless rotation rate increases by about a factor of 7 from either a change in density or a change in the mean free path of the background gas. The results of recent experiments and model calculation of the collision cross section and mobilities of large molecular ions are discussed in regard to our model predictions on the effect of rotation on the drift velocity.

Hydrodynamic theory of rotating ultracold Bose–Einstein condensates in supersolid phase

Within the mean field Gross–Pitaevskii framework, ultracold atomic condensates with long-range interaction are predicted to have a supersolid-like ground state beyond a critical interaction strength. Such a mean field supersolid-like ground state has periodically modulated superfluid density which implies the coexistence of superfluid and crystalline order. An ultracold atomic system in such a mean field ground state can be subjected to an artificial gauge field created either through rotation or by introducing space dependent coupling among hyperfine states of the atoms using Raman lasers. Starting from this Gross–Pitaevskii energy functional that describes such systems at zero temperature, we construct a hydrodynamic theory to describe the low-energy long-wavelength excitations of a rotating supersolid of weakly interacting ultracold atoms in two spatial dimensions for a generic type of long-range interaction. We treat the supersolidity in such a system within the framework of the well known two-fluid approximation. Considering such a system in the fast rotation limit where a vortex lattice in superfluid coexists with the supersolid lattice, we analytically obtain the dispersion relations of collective excitations around this equilibrium state. The dispersion relation gives the modes of the rotating supersolid which can be experimentally measured within the current technology. We point out that this can clearly identify an ultracold atomic supersolid phase in an unambiguous way.

Finite clusters of fast-rotating spinless bosons in a harmonic trap

Rapidly rotating two-dimensional ultracold Bose–Einstein condensates of spinless bosons in a harmonic trap have attracted considerable interest during the recent years. It is expected that, in the fast-rotation limit, the system of bosons will exhibit collective behavior similar to that of two-dimensional electrons in the fractional quantum Hall effect regime. It is predicted that the most robust correlated bosonic state in this regime will be the Bose Laughlin state at a half filling factor. An exact treatment of such a state is generally a formidable task due to the inherent many-particle nature of the wave function. We report in this work that a transformation to Jacobi coordinates allows one to obtain much desirable exact analytic closed-form expressions for various quantities of interest corresponding to a Bose Laughlin wave function for various finite systems of particles.

From isotropic to rapidly rotating anisotropic alpha-squared dynamos

Mean field -dynamos in a sphere with an insulating exterior are considered for a steady -effect of the form , , derived by Moffatt (Dynamo action associated with random inertial waves in a rotating conducting fluid. J. Fluid Mech. 1970, 44, 705–719) for strong rotation. The unit vector is aligned with the angular velocity and as , . We consider the effect on axisymmetric magnetic fields of increasing rotation rate, i.e. , so that the -effect varies from isotropic to anisotropic, for two models. We find that both equatorially symmetric and antisymmetric critical solutions bifurcate from steady to oscillatory at some value of . The bifurcation is associated with the poleward migration of strong magnetic fields, leaving a region of weak field near the equator. This region increases with the rotation rate. No antidynamo action was found at the fast rotation limit . The -effect is re-scaled to for allowing all possible solutions to be studied by restricting to . In particular, we consider the mean field axisymmetric antidynamo limit at . Our results are in complete agreement with those in Phillips (Mean dynamos. Ph.D. Thesis, University of Sydney, 1993), where is considered.

Resolvent estimates for a two-dimensional non-self-adjoint operator

We consider a two-dimensional non-self-adjoint differential operator, originated from a stability problem in the two-dimensional Navier-Stokes equation, given by ${\mathcal L}_\alpha=-\Delta+|x|^2+\alpha \sigma(|x|)\partial_\theta$, where $\sigma(r)=r^{-2}(1-e^{-r^2})$, $\partial_\theta=x_1\partial_2-x_2\partial_1$ and $\alpha$ is a positive parameter tending to $+\infty$. We give a complete study of the resolvent of ${\mathcal L}_\alpha$ along the imaginary axis in the fast rotation limit $\alpha\to+\infty$ and we prove $\sup_{\lambda\in \mathbb{R}}\|({\mathcal L}_\alpha-i\lambda)^{-1}\|_{{\mathcal L}(\tilde L^2(\mathbb{R}^2))}\leq C\alpha^{-1/3}$, which is an optimal estimate. Our proof is based on a multiplier method, metrics on the phase space and localization techniques.

The giant vortex state for a Bose–Einstein condensate in a rotating anharmonic trap: Extreme rotation regimes

We study the fast rotation limit for a Bose–Einstein condensate in a quadratic plus quartic confining potential within the framework of the two-dimensional Gross–Pitaevskii energy functional. As the rotation speed tends to infinity with a proper scaling of the other parameters in the model, a linear limit problem appears for which we are able to derive precise energy estimates. We prove that the energy and density asymptotics of the problem can be obtained by minimizing a simplified one-dimensional energy functional. In the case of a fixed coupling constant we also prove that a giant vortex state appears. It is an annulus with pure irrotational flow encircling a central low-density hole around which there is a macroscopic phase circulation.

Axial breathing mode in rapidly rotating Bose-Einstein condensates and uncertainty of the rotation velocity

Experiments on the axial breathing mode in a rapidly rotating Bose-Einstein condensate are examined. Assuming a cold cloud without thermal component, we show that errors due to defocus of an imaging camera in addition to an inclination of the rotational axis can lead to a significant underestimate of the rotation rate in the fast rotation limit; within these uncertainties, our theoretical prediction agrees with the experimental data. We also show that, in the fast rotation regime, the Thomas-Fermi theory, which is inapplicable there, underestimates the rotation rate. Underestimation of the rotation rate due to these effects would also partly explain a discrepancy between theory and experiment for the Tkachenko mode frequency in the fast rotation regime.

Quantum Hall fractions for spinless bosons

We study the Quantum Hall phases that appear in the fast rotation limit for Bose-Einstein condensates of spinless bosonic atoms. We use exact diagonalization in a spherical geometry to obtain low-lying states of a small number of bosons as a function of the angular momentum. This allows to understand or guess the physics at a given filling fraction nu, ratio of the number of bosons to the number of vortices. This is also the filling factor of the lowest Landau level. In addition to the well-known Bose Laughlin state at nu =1/2 we give evidence for the Jain principal sequence of incompressible states at nu =p/(p+- 1) for a few values of p. There is a collective mode in these states whose phenomenology is in agreement with standard arguments coming e.g. from the composite fermion picture. At filling factor one, the potential Fermi sea of composite fermions is replaced by a paired state, the Moore-Read state. This is most clearly seen from the half-flux nature of elementary excitations. We find that the hierarchy picture does not extend up to the point of transition towards a vortex lattice. While we cannot conclude, we investigate the clustered Read-Rezayi states and show evidence for incompressible states at the expected ratio of flux vs number of Bose particles.

C 13 nuclear Overhauser polarization nuclear magnetic resonance in rotating solids: Replacement of cross polarization in uniformly C13 labeled molecules with methyl groups

A new C13 polarization technique in solids is presented on the basis of a recently proposed C13–C13 recoupling sequence [13C–1H dipolar-assisted rotational resonance (DARR), K. Takegoshi, S. Nakamura, and T. Terao, Chem. Phys. Lett. 344, 631 (2001)] operative under fast magic angle spinning (MAS), in which a rf field is applied to H1H with a rotary resonance condition but none to C13. The H1 irradiation in DARR saturates H1 signals, leading to the C13 signal enhancement due to the nuclear Overhauser effect for fast rotating methyl groups, if any. If we use a uniformly C13 labeled sample, C13–C13 polarization transfer enhanced by DARR successively distributes the enhanced methyl carbon polarization to the other C13 spins, leading to uniform enhancement for all C13 spins even under very fast MAS. In uniformly C13 labeled rotating samples, the enhancement factor in cross polarization (CP) is about 2.4, while in the present nuclear Overhauser polarization (NOP), it is 3.0 in the fast rotation limit of the methyl groups. While the CP enhancement becomes smaller for molecules with short T1ρ of H1 or C13, NOP would work well for such mobile molecules, and also NOP enables us to acquire a signal with a short repetition time even if H1 T1 is long. Further, NOP has the advantage of quantitativeness, and is very easy to carry out, being insensitive to the adjustment of rf field intensity and requiring only very low rf power. These features are demonstrated for uniformly C13, N15-labeled L-threonine and uniformly C13, N15-labeled glycylisoleucine. NOP-MAS is also applied for a naturally abundant C13 sample.

Strongly nonlinear convection cells in a rapidly rotating fluid layer: the tilted f-plane

Investigation of the linear stability problem for rapidly rotating convection on an f-plane has revealed the existence of two distinct scales in the vertical structure of the critical eigenfunctions: a small length scale whose vertical wavenumber kz is comparable with the large horizontal wavenumber k⊥ selected at onset, and a large-scale modulation which forms an envelope on the order of the layer depth d. The small-scale structure in the vertical results from a geostrophic balance imposed by the Taylor–Proudman constraint. This primary balance forces rotational alignment and confines fluid motions to planes perpendicular to the rotation axis. For convective transport in the vertical this constraint must be relaxed. This is achieved by molecular dissipation which allows weak upward (downward) spiralling of hot (cold) fluid elements across the Taylor–Proudman planes and results in a large-scale vertical modulation of the Taylor columns.In the limit of fast rotation (i.e. large Taylor number) a multiple-scales analysis leads to the determination of a critical Rayleigh number as a function of wavenumber, roll orientation and the tilt angle of the f-plane. The corresponding critical eigenfunction represents the core solution; matching to passive Ekman boundary layers is required for a complete solution satisfying boundary conditions.An extension of this analysis, introduced by Bassom & Zhang (1994), is used to describe strongly nonlinear two-dimensional convection, characterized by significant departures of the mean thermal field from its conduction profile. The analysis requires the solution of a nonlinear eigenvalue problem for the Nusselt number (for steady convection) and the Nusselt number and oscillation frequency (for the overstable problem). The solutions of this problem are used to calculate horizontal and vertical heat fluxes, as well as Reynolds stresses, as functions of both the latitude and roll orientation in the horizontal, and these are used to calculate self-consistently north–south and east–west mean flows. These analytical predictions are in good agreement with the results of three-dimensional simulations reported by Hathaway & Somerville (1983).