Small Rossby Number Research Articles

The drag on a rising sphere along the axis in a short rotating cylinder of fluid: revisiting the data and theory

We revisit the problem of a solid sphere rising slowly in a rotating short container filled with a slightly viscous fluid, with emphasis on the drag force. The data of the classical experiments of Maxworthy (J. Fluid Mech., vol. 31, 1968, pp. 643–655) and recent experiments of Kozlov et al. (Fluids, vol. 8 (2), 2023, paper 49), and the available geostrophic and quasi-geostrophic theories, are subjected to a novel scrutiny by combined reprocessing and comparisons. The measured drag is, consistently, about 20 % lower than the geostrophic prediction (assuming that flow is dominated by the Ekman layers, while in the inviscid cores the Coriolis acceleration is supported by the pressure gradient). The major objective is the interpretation and improvement of the gap between data and predictions. We show that the data cover a small range of relevant parameters (in particular the Taylor number $T$ and the height ratio $H$ of cylinder to particle diameter) that precludes a thorough and reliable assessment of the theories. However, some useful insights and improvements can be derived. The hypothesis that the discrepancy between data and the geostrophic prediction is due to inertial effects (not sufficiently small Rossby number $Ro$ in the experiments) is dismissed. We show that the major reason for the discrepancy is the presence of relatively thick Stewartson layers about the cylinder (Taylor column) attached to the sphere. The $1/3$ layer displaces the boundary condition of the angular velocity ( $\omega = 0$ ) outside the radius of the particle. This observation suggests a semi-empirical correction to the theoretical quasi-geostrophic predictions (which takes into account the Ekman layers and the $1/4$ Stewartson layers); the corrected drag is in fair agreement with the data. We demonstrate that the inertial terms are negligible for $Ro\,T^{1/2} <0.4$ . We consider curve-fit approximations, and point out some persistent gaps of knowledge that require further experiments and simulations.

Focusing of weakly nonlinear inertial waves by a vertically oscillating annular forcing

We investigate the focusing phenomenon of inertial waves generated by a vertically oscillating slender torus in a uniformly rotating fluid. Building on the previous research on linear aspects of inertial wave focusing by Liu et al. [“Focusing of inertial waves by a vertically annular forcing,” Phys. Fluids 34, 086601 (2022)], we analyze the focusing phenomenon in a fluid with a small Rossby number using the perturbation method. Our analytical solution to the weakly nonlinear problem reveals that, in addition to the primary inertial waves predicted by the linear theory, secondary inertial waves (SIW) are excited due to wave interaction if the forcing frequency is lower than the fluid rotation rate. Particularly, in our case, the SIWs induced directly by the oscillating torus form a relatively weak secondary focal area on the rotation axis below the primary focal point. The rays of the SIWs form a new double cone with a different angle than the primary cone. In addition, the SIWs are also excited at the primary focal point with a much stronger effect than that generated by the oscillating torus in an inviscid limit, but significantly suppressed even in low-viscosity fluids. A fully nonlinear analysis, especially in the primary focal zone, is necessary for further study.

Von Kármán vortex streets around invisible bluff bodies

Satellite images captured during the past decade identify clearly von Kármán vortex streets in the atmosphere above certain islands in the Atlantic and Pacific Oceans. The vortex streets are visible due to the cloud formations since the latter behave as unintended tracers of the fluid motion driven by winds. Experimental as well as numerical simulation results of von Kármán vortex shedding for flow around a cylinder or other bluff body identify the latter in a certain range of Reynolds numbers. The satellite images, however, show these von Kármán vortex structures at elevations well above the ground level inviting the appealing question of where is the cylinder or the bluff body? This short paper identifies the answer to this question as the creation of Taylor–Proudman columns above the islands that are invisible until a tracer makes them detectable at a certain height above the island. The latter also provides the clarification that the flow is geostrophic at the leading order and consequently two-dimensional (i.e., variations occur in the horizontal directions with no leading order changes in the vertical direction). In addition, this paper shows that a small Rossby number is not a necessary requirement for the creation of Taylor–Proudman columns. The latter can emerge also if a certain modified Beltrami condition is satisfied even when Rossby number is not small.

Corrections of Mesoscale Eddies and Kuroshio Extension Surface Velocities Derived from Satellite Altimeters

Oceanic datasets derived from satellite altimeters are of great significance to physical oceanography and ocean dynamics research and the protection of marine environmental resources. Ageostrophic velocity induced by centrifugal force is not considered in altimeter products. This study introduces an iterative method to perform cyclogeostrophic corrections of mesoscale eddies’ surface velocities derived from satellite altimeters. The corrected eddy velocity field and geostrophic velocity field were compared by combining eddy detection and mathematical statistics methods. The results show that eddies with small curvature radii, high roundness, or Rossby number larger than 0.1 illustrate that cyclogeostrophic correction is required. The cyclogeostrophic velocity is greater (less) than the geostrophic velocity in anticyclonic (cyclonic) eddies. Additionally, the iterative method is applied to cyclogeostrophic-corrected multi-year (1998–2012) Kuroshio surface velocities. The effect of cyclogeostrophic correction is significant for the Kuroshio Extension region, where the maximum relative difference of velocities with and without correction is about 10% and the eddy kinetic energy is 20%.

Is Core Angular Momentum Key to the Giant Dynamo?

The pros and cons of core angular momentum dissipation into the convective envelopes of giants as a driver of giant activity is discussed in face of the observational evidence, which points to two ”magnetic strips“, in the HRD, where in the first, at the base of the RGB, activity of moderate mass stars is freshly started and rejuvenated in the second strip, ascending along the mid-AGB. It remains unclear, though, which depths the giant dynamo is operating. Both concentrations of active giants in the HRD are related to stellar evolution phases with core contraction and spin-up, and presumably the dissipation of angular momentum into the convective envelope above. At the same time, the latter has a small Rossby number by virtue of its increasing convective turn-over time—i.e., favourable conditions to run an alpha-omega dynamo. Since coronal X-ray emission appears to give an incomplete picture of stellar activity across the HR diagramme, we here focus on the observed chromospheric emissions across the giant branches and find good agreement with the magnetic field Zeeman-detections there. Stable evolution phases—solar-type main sequence stars with central hydrogen burning and moderate mass, central Helium burning K giants—by contrast demonstrate a decline in activity, apparently imposed by magnetic braking, as such stars are also slower rotators. In that sense, the observed picture of two magnetic activity strips across the HR diagramme could empirically be explained as an interplay of magnetic braking during the stable phase of core helium burning and supply by internal angular momentum during episodes of fast core contraction with core spin-up and angular momentum dredge-up, while meeting dynamo-friendly envelope conditions. At the same time, the sporadic external supply of angular momentum by the engulfment events of a planet, in the course of the evolutionary envelope expansion, may explain some cases of exceptional activity outside the here-described general picture.

Asymmetry in the spin-up of a stratified fluid in a sliced, square cylinder

Previously, we have reported theory and experiments on the spin-up of a linearly stratified fluid contained in a sealed square cylinder with a bottom sloped at angle alpha (Munro & Foster, Phys. Fluids, 26, 026603, 2014). The motion is characterized by the small Rossby number, epsilon, the fractional change in the rotation rate, the Ekman number, E, and the Burger number, S, the square of the ratio of the buoyancy frequency (or Brunt-Väisälä frequency) to the base rotation rate. This paper, supplementing what has gone before, is focused on the horizontal and vertical structure of the propagating vorticity waves (Rossby waves), with alternating anti-cyclonic and cyclonic cells, which move across the tank from east to west, thereby accomplishing at least the early stages of the spin-up. We find strong dependence of the modal structure of these Rossby waves, and their speed, on the Burger number. However, what is particularly noteworthy is the observed strong north-south asymmetry of the wave field. We find that the asymmetry is not consistent with typical Rossby-wave theories (Pedlosky and Greenspan, J. Fluid Mech., 27, p. 291, 1967), but is attributable to second-order effects in the asymptotic expansion of the streamfuntion in powers of alpha.

Dynamics of Ocean Worlds Likely Controlled by Their Rotation

New simulations suggest that subsurface oceans on icy moons with small natural Rossby numbers may be dominated by rotational effects.

Measurements of chromospheric Mg i emission lines of zero-age main-sequence stars

Abstract The chromosphere is the active atmosphere in which energetic eruption events, such as flares, occur. Chromospheric activity is driven by the magnetic field generated by stellar rotation and convection. The relationship between chromospheric activity and the Rossby number, the ratio of the rotational period to the convective turnover time, has been extensively examined for many types of stars, by using narrow chromospheric emission lines, such as the Ca ii lines and the Mg ii H and K lines. However, the stars with small Rossby numbers, i.e., stars with rapid rotations and/or long convective turnover times, show constant strengths of such lines against the Rossby number. In this study, we investigate the infrared Mg i emission lines at 8807 Å of 47 zero-age main-sequence (ZAMS) stars in IC 2391 and IC 2602 using the archive data of the Anglo-Australian Telescope at the University College London Echelle Spectrograph. After subtracting the photospheric absorption component, the Mg i line is detected as an emission line for 45 ZAMS stars, the equivalent widths of which are between 0.02 Å and 0.52 Å. A total of 42 ZAMS stars show the narrower Mg i emission lines instead of the Ca ii infrared triplet emission lines, suggesting that they are formed at different depths. The ZAMS stars with smaller Rossby numbers show stronger Mg i emission lines. The Mg i emission line is not saturated even in the saturated regime of the Ca ii emission lines, i.e., Rossby number <10−1.1. The Mg i emission line is considered to be a good indicator of chromospheric activity, particularly for active objects.

Coherent magnetic modon solutions in quasi-geostrophic shallow water magnetohydrodynamics

A class of exact solutions of the magnetohydrodynamic quasi-geostrophic equations (MQG), which result from rotating shallow water magnetohydrodynamics in the limit of small Rossby and magnetic Rossby numbers is constructed analytically. These solutions are magnetic modons, steady-moving dipolar vortices, which are generalizations of the well-known quasi-geostrophic modons. It is shown that various configurations of magnetic modons are possible: with or without external magnetic field, and with or without internal magnetic field trapped inside the dipole. By using the modon solutions as initial conditions for direct numerical simulations of the MQG equations, it is shown that they remain coherent for a long time, running over about a hundred deformation radii without change of form, provided the external and internal magnetic fields are not too strong, and even if a small-amplitude noise is added to initial conditions.

Point-vortex dynamics in three-dimensional ageostrophic balanced flows

Geophysical turbulent flows, characterized by rapid rotation, quantified by small Rossby number, and stable stratification, often self-organize into a collection of coherent vortices, referred to as a vortex gas. The lowest-order asymptotic expansion in Rossby number is quasigeostrophy, which has purely horizontal velocities and cyclone–anticyclone antisymmetry. Ageostrophic effects are important components of many geophysical flows and, as such, these phenomena are not well modelled by quasigeostrophy. The next-order correction in Rossby number, which includes ageostrophic effects, is the so-called balanced dynamics. Balanced dynamics includes ageostrophic vertical velocity and breaks the geostrophic cyclone–anticyclone antisymmetry. Point-vortex solutions are well known in two-dimensional and quasigeostrophic dynamics and are useful for studying the vortex-gas regime of geophysical turbulence. Here, we find point-vortex solutions in fully three-dimensional continuously stratified QG $^{+1}$ dynamics, a particular formulation of balanced dynamics. Simulations of QG $^{+1}$ point vortices show several interesting features not captured by quasigeostrophic point vortices including significant vertical transport on long time scales. The ageostrophic component of QG $^{+1}$ point vortex point-vortex dynamics renders them useful in modelling flows where quasigeostrophy filters out important physical processes.

Coastal imbalance: generation of oceanic Kelvin waves by atmospheric perturbations

The response of a semi-infinite ocean to a slowly travelling atmospheric perturbation crossing the coast provides a simple example of the breakdown of nearly geostrophic balance induced by a boundary. We examine this response in the linear shallow-water model at small Rossby number$\varepsilon \ll 1$. Using matched asymptotics, we show that a long Kelvin wave, with$O(\varepsilon ^{-1})$length scale and$O(\varepsilon )$amplitude relative to quasi-geostrophic response, is generated as the perturbation crosses the coast. Accounting for this Kelvin wave restores the conservation of mass that is violated in the quasi-geostrophic approximation.

High-Resolution Ocean Currents from Sea Surface Temperature Observations: The Catalan Sea (Western Mediterranean)

Current observations of ocean currents are mainly based on altimetric measurements of Sea Surface Heights (SSH), however the characteristics of the present-day constellation of altimeters are only capable to retrieve surface currents at scales larger than 50–70 km. By contrast, infrared and visible radiometers reach spatial resolutions thirty times higher than altimeters under cloud-free conditions. During the last years, it has been shown how the Surface Quasi-Geostrophic (SQG) approximation is able to reconstruct surface currents from measured Sea Surface Temperature (SST), but it has not been yet used to retrieve velocities at scales shorter than those provided by altimeters. In this study, the velocity field of ocean structures with characteristic lengths between 10 and 20 km has been derived from infrared SST using the SQG approach and compared to the velocities derived from the trajectories of Lagrangian drifters. Results show that the SQG approach is able to reconstruct the direction of the velocity field with observed RMS errors between 8 and 15 degrees and linear correlations between 0.85 and 0.99. The reconstruction of the modulus of the velocity is more problematic due to two limitations of the SQG approach: the need to calibrate the level of energy and the ageostrophic contributions. If drifter trajectories are used to calibrate velocities and the analysis is restricted to small Rossby numbers, the RMS error in the range of 10 to 16 cm/s and linear correlations can be as high as 0.97.

On the Structure and Kinematics of an Algerian Eddy in the Southwestern Mediterranean Sea

An Algerian Eddy, anticyclonic vortex generated by the instability of the Algerian Current in the southwestern Mediterranean Sea, is studied using data provided by drifters (surface currents), Argo floats (temperature and salinity profiles), environmental satellites (absolute dynamic topography maps and ocean color images) and operational oceanography products. The eddy was generated in May 2018 and lasted as an isolated vortex until November 2018. Its morphology and kinematics are described in June–July 2018 when drifters were trapped in its core. During that period, the eddy was slowly moving to the NE (~2 km/day), with an overall diameter of about 200 km (slowly growing with time) and maximal surface swirl velocity of ~50 cm/s at a radius of ~50 km. Geostrophic currents derived from satellite altimetry data compare well with low-pass filtered drifter velocities, with only a slight overestimation, which is expected as its maximum vorticity corresponds to a small Rossby number of ~0.6. Satellite ocean color images and some drifters show that the eddy has an elliptical spiral structure. The looping tracks of the drifters trapped in the eddy were analyzed using two statistical methods: least-squares ellipse fitting and wavelet ridge analysis, revealing a typical eccentricity of about 0.5, a wide range of inclination and a rotation period between 3 and 10 days. Clusters of drifters on the northeastern limb of the eddy were also considered to estimate divergence and vorticity. The results indicate convergence (divergence) and downwelling (upwelling) at scales of 20–50 km near the northeastern (northwestern) edge of the eddy, in agreement with the quasi-geostrophic theory. Vertically, the eddy extends mostly down to 250 m depth, with a warm, low-salinity and low-density signature and with geostrophic currents near 50 cm/s in the top layer (down to ~80 m) reducing to less than 10 cm/s near 250 m. Near the surface, colder water is advected into it.

Variational balance models for the three-dimensional Euler–Boussinesq equations with full Coriolis force

We derive a semi-geostrophic variational balance model for the three-dimensional Euler–Boussinesq equations on the nontraditional f-plane under the rigid lid approximation. The model is obtained by a small Rossby number expansion in the Hamilton principle, with no other approximations made. We allow for a fully non-hydrostatic flow and do not neglect the horizontal components of the Coriolis parameter; that is, we do not make the so-called “traditional approximation.” The resulting balance models have the same structure as the “L1 balance model” for the primitive equations: a kinematic balance relation, the prognostic equation for the three-dimensional tracer field, and an additional prognostic equation for a scalar field over the two-dimensional horizontal domain, which is linked to the undetermined constant of integration in the thermal wind relation. The balance relation is elliptic under the assumption of stable stratification and sufficiently small fluctuations in all prognostic fields.

The Seasonal Variability of the Ocean Energy Cycle from a Quasi-Geostrophic Double Gyre Ensemble

With the advent of submesoscale O(1km) permitting basin-scale ocean simulations, the seasonality of mesoscale O(50km) eddies with kinetic energies peaking in summer has been commonly attributed to submesoscale eddies feeding back onto the mesoscale via an inverse energy cascade under the constraint of stratification and Earth’s rotation. In contrast, by running a 101-member, seasonally forced, three-layer quasi-geostrophic (QG) ensemble configured to represent an idealized double-gyre system of the subtropical and subpolar basin, we find that the mesoscale kinetic energy shows a seasonality consistent with the summer peak without resolving the submesoscales; by definition, a QG model only resolves small Rossby and Froude number dynamics (O(Ro)≪1,O(Fr)≪1) while submesoscale dynamics are associated with O(Ro)∼1,O(Fr)≳1. Here, by quantifying the Lorenz cycle of the mean and eddy energy, defined as the ensemble mean and fluctuations about the mean, respectively, we propose a different mechanism from the inverse energy cascade. During summer, when the Western Boundary Current is stabilized and strengthened due to increased stratification, stronger mesoscale eddies are shed from the separated jet. Conversely, the opposite occurs during the winter; the separated jet destablizes and results in overall lower mean and eddy kinetic energies despite the domain being more susceptible to baroclinic instability from weaker stratification.

Iterative Methods for Solving the Nonlinear Balance Equation with Optimal Truncation

Two types of existing iterative methods for solving the nonlinear balance equation (NBE) are revisited. In the first type, the NBE is rearranged into a linearized equation for a presumably small correction to the initial guess or the subsequent updated solution. In the second type, the NBE is rearranged into a quadratic form of the absolute vorticity with the positive root of this quadratic form used in the form of a Poisson equation to solve NBE iteratively. The two methods are rederived by expanding the solution asymptotically upon a small Rossby number, and a criterion for optimally truncating the asymptotic expansion is proposed to obtain the super-asymptotic approximation of the solution. For each rederived method, two iterative procedures are designed using the integral-form Poisson solver versus the over-relaxation scheme to solve the boundary value problem in each iteration. Upon testing with analytically formulated wavering jet flows on the synoptic, sub-synoptic and meso-α scales, the iterative procedure designed for the first method with the Poisson solver, named M1a, is found to be the most accurate and efficient. For the synoptic wavering jet flow in which the NBE is entirely elliptic, M1a is extremely accurate. For the sub-synoptic wavering jet flow in which the NBE is mostly elliptic, M1a is sufficiently accurate. For the meso-α wavering jet flow in which the NBE is partially hyperbolic so its boundary value problem becomes seriously ill-posed, M1a can effectively reduce the solution error for the cyclonically curved part of the wavering jet flow, but not for the anti-cyclonically curved part.

Fast-Wave Averaging with Phase Changes: Asymptotics and Application to Moist Atmospheric Dynamics

Many systems involve the coupled nonlinear evolution of slow and fast components, where, for example, the fast waves might be acoustic (sound) waves with a small Mach number or inertio-gravity waves with small Froude and Rossby numbers. In the past, for some such systems, an interesting property has been shown: the slow component actually evolves independently of the fast waves, in a singular limit of fast wave oscillations. Here, a fast-wave averaging framework is developed for a moist Boussinesq system with additional complexity beyond past cases, now including phase changes between water vapor and liquid water. The main question is: Do phase changes induce coupling between the slow component and fast waves? Or does the slow component evolve independently, according to moist quasi-geostrophic equations? Compared to the dry dynamics, a substantial challenge is that the method needs to be adapted to a piecewise operator with variable coefficients, due to phase changes. A formal asymptotic analysis is presented here. For purely saturated flow without phase changes, it is shown that precipitation does not induce coupling, and the slow modes evolve independently. With phase changes present, the limiting equations show that phase boundaries could possibly induce coupling between the slow modes and fast waves.

On the fast rotation asymptotics of a non-homogeneous incompressible MHD system

This paper is devoted to the analysis of a singular perturbation problem for a 2D incompressible MHD system with density variations and Coriolis force, in the limit of small Rossby numbers. Two regimes are considered. The first one is the quasi-homogeneous regime, where the densities are small perturbations around a constant state. The limit dynamics is identified as an incompressible homogeneous MHD system, coupled with an additional transport equation for the limit of the density variations. The second case is the fully non-homogeneous regime, where the densities vary around a general non-constant profile. In this case, in the limit, the equation for the magnetic field combines with an underdetermined linear equation, which links the limit density variation function with the limit velocity field. The proof is based on a compensated compactness argument, which enables us to consider general ill-prepared initial data. An application of Di Perna–Lions theory for transport equations allows to treat the case of density-dependent viscosity and resistivity coefficients.

A Multiscale Problem for Viscous Heat-Conducting Fluids in Fast Rotation

In the present paper, we study the combined incompressible and fast rotation limits for the full Navier-Stokes-Fourier system with Coriolis, centrifugal and gravitational forces, in the regime of small Mach, Froude and Rossby numbers and for general ill-prepared initial data. We consider both the isotropic scaling (where all the numbers have the same order of magnitude) and the multi-scale case (where some effect is predominant with respect to the others). In the case when the Mach number is of higher order than the Rossby number, we prove that the limit dynamics is described by an incompressible Oberbeck-Boussinesq system, where the velocity field is horizontal (according to the Taylor-Proudman theorem), but vertical effects on the temperature equation are not negligible. Instead, when the Mach and Rossby numbers have the same order of magnitude, and in absence of the centrifugal force, we show convergence to a quasi-geostrophic equation for a stream function of the limit velocity field, coupled with a transport-diffusion equation for a new unknown, which links the target density and temperature profiles. The proof of the convergence is based on a compensated compactness argument. The key point is to identify some compactness properties hidden in the system of acoustic-Poincar\'e waves. Compared to previous results, our method enables first of all to treat the whole range of parameters in the multi-scale problem, and also to consider a low Froude number regime with the somehow critical choice $Fr=\sqrt{Ma}$, where $Ma$ is the Mach number. This allows us to capture some (low) stratification effects in the limit.

TOI 540 b: A Planet Smaller than Earth Orbiting a Nearby Rapidly Rotating Low-mass Star

Abstract We present the discovery of TOI 540 b, a hot planet slightly smaller than Earth orbiting the low-mass star 2MASS J05051443-4756154. The planet has an orbital period of P = 1.239149 days (±170 ms) and a radius of , and is likely terrestrial based on the observed mass–radius distribution of small exoplanets at similar insolations. The star is 14.008 pc away and we estimate its mass and radius to be and , respectively. The star is distinctive in its very short rotational period of hr and correspondingly small Rossby number of 0.007 as well as its high X-ray-to-bolometric luminosity ratio of based on a serendipitous XMM-Newton detection during a slew operation. This is consistent with the X-ray emission being observed at a maximum value of as predicted for the most rapidly rotating M dwarfs. TOI 540 b may be an alluring target to study atmospheric erosion due to the strong stellar X-ray emission. It is also among the most accessible targets for transmission and emission spectroscopy and eclipse photometry with the James Webb Space Telescope, and may permit Doppler tomography with high-resolution spectroscopy during transit. This discovery is based on precise photometric data from the Transiting Exoplanet Survey Satellite and ground-based follow-up observations by the MEarth team.