Low Rossby Number Research Articles

The sandpaper theory of flow–topography interaction for homogeneous shallow-water systems

Recent studies reveal the dramatic impact of seafloor roughness on the dynamics and stability of broad oceanic flows. These findings motivate the development of parameterizations that concisely represent the effects of small-scale bathymetric patterns in theoretical and coarse-resolution numerical circulation models. The previously reported quasi-geostrophic ‘sandpaper’ theory of flow–topography interaction a priori assumes gentle topographic slopes and weak flows with low Rossby numbers. Since such conditions are often violated in the ocean, we now proceed to formulate a more general model based on shallow-water equations. The new version of the sandpaper model is validated by comparing roughness-resolving and parametric simulations of the flow over a corrugated seamount.

Kinetic Energy Exchanges between a Two-Dimensional Front and Internal Waves

Abstract Fronts and near-inertial waves (NIWs) are energetic motions in the upper ocean that have been shown to interact and provide a route for kinetic energy (KE) dissipation of balanced oceanic flows. In this paper, we study these KE exchanges using an idealized model consisting of a two-dimensional geostrophically balanced front undergoing strain-induced semigeostrophic frontogenesis and internal wave (IW) vertical modes. The front–IW KE exchanges are quantified separately during two frontogenetic stages: an exponential sharpening stage that is characterized by a low Rossby number and is driven by the imposed strain (i.e., mesoscale frontogenesis), followed by a superexponential sharpening stage that is characterized by an Rossby number and is driven by the convergence of the secondary circulation (i.e., submesoscale frontogenesis). It is demonstrated that high-frequency IWs quickly escape the frontal zone and are very efficient at extracting KE from the imposed geostrophic strain field through the deformation shear production (DSP). Part of the extracted KE is then converted to wave potential energy. On the contrary, NIWs remain locked to the frontal zone and readily exchange energy with the ageostrophic frontal circulation. During the exponential stage, NIWs extract KE from the geostrophic strain through DSP and transfer it to the frontal secondary circulation via the ageostrophic shear production (AGSP) mechanism. During the superexponential stage, a newly identified mechanism, convergence production (CP), plays an important role in the NIW KE budget. The CP transfers KE from the convergent ageostrophic secondary circulation to the NIWs and largely cancels out the KE loss due to the AGSP. This CP may explain previous findings of KE transfer enhancement from balanced motions to IWs in frontal regions of realistic ocean models. We provide analytical estimates for the aforementioned energy exchange mechanisms that match well the numerical results. This highlights that the strength of the exchanges strongly depends on the frontal Rossby and Richardson numbers. Significance Statement Fronts with large horizontal density and velocity gradients are ubiquitous in the upper ocean. They are generated by a process known as frontogenesis, which is often initialized by straining motions of mesoscale balanced circulations. Here we examine the energy exchanges between fronts and internal waves in an idealized configuration, aiming to elucidate the mechanisms that can drain energy from oceanic balanced circulations. We identify a new mechanism for energy transfers from the frontal circulation to near-inertial internal waves called convergence production. This mechanism is especially effective during the later stages of frontogenesis when the convergent ageostrophic secondary circulation that develops is strong.

Spreading or Contraction of an Axisymmetric Viscous Drop between Two Plates in a Rapidly Rotating Frame

We analyse the motions of a axisymmetric drop expanding between two rotating discs. We restrict to the case of a highly viscous fluid and a rapid rate of rotation. Therefore, we make modelling assumptions following from both a low Reynolds number and a low Rossby number. We investigate both the squeezing problem, where the top disc is pushed down on the drop; and the contraction problem, where the top plate is pulled away from the drop. Both problems have similar solutions to the non-rotating case but we find that the rotation term in the contraction problem allows a critical rotation rate that prevents the plates from moving apart. This exists because pressure in the fluid layer is lowered by the rotation and thus there is a suction effect between the two plates which promotes adhesion. We also complete the linear instability analysis of the squeezing problem and determine the critical values where the system shifts from symmetrical to asymmetrical.

Turbulent Transition of a Flow from Small to O(1) Rossby Numbers

Abstract Oceanic flows are energetically dominated by low vertical modes. However, disturbances in the form of atmospheric storms, eddy interactions with various forms of boundaries, or spontaneous emission by coherent structures can generate weak high-baroclinic modes. The feedback of the low-energy high-baroclinic modes on large-scale energetically dominant low modes may be weak or strong depending on the flow Rossby number. In this paper we study this interaction using an idealized setup by constraining the flow dynamics to a high-energy barotropic mode and a single low-energy high-baroclinic mode. Our investigation points out that at low Rossby numbers the barotropic flow organizes into large-scale coherent vortices via an inverse energy flux while the baroclinic flow accumulates predominantly in anticyclonic barotropic vortices. In contrast, with increasing Rossby number, the baroclinic flow catalyzes a forward flux of barotropic energy. The barotropic coherent vortices decrease in size and number, with a strong preference for cyclonic coherent vortices at higher Rossby numbers. On partitioning the flow domain into strain-dominant and vorticity-dominant regions based on the barotropic flow, we find that at higher Rossby numbers baroclinic flow accumulates in strain-dominant regions, away from vortex cores. Additionally, a major fraction of the forward energy flux of the flow takes place in strain-dominant regions. Overall, one of the key outcomes of this study is the finding that even a low-energy high-baroclinic flow can deplete and dissipate large-scale coherent structures at O(1) Rossby numbers.

Large Eddy Simulation Investigation of Low Rossby Number Buoyant Flow in Rotating Cavities

Abstract Flow and heat transfer in axial compressor disk cavities involve strong interaction of axial throughflow at the disk bores with centrifugal buoyant flow in the cavities. This paper presents large eddy simulation (LES) of flow and heat transfer in rotating cavities with a heated shroud and a relatively weak axial cooling throughflow. The conditions considered for a single cavity configuration correspond to Rossby numbers Ro=0.2 and 0.3, rotational Reynolds numbers ReΩ=3.2× 105 and 7.7×105, and buoyancy parameters βΔT=0.24 and 0.26. Reasonable agreement of the results with shroud heat transfer measurements was confirmed for the Ro=0.2 condition for which test data were available. A dual cavity configuration for Ro=0.3 and ReΩ=3.2× 105 is also modeled. The simulations show that, at low Ro conditions, flow reversals occur along the length of the bore flow path, upstream and downstream of the rotating cavities. With the dual cavity strong, unsteady interactions between the flows in the two cavities occur. These flow interactions result in less stable flow structures, higher air temperatures within the cavities and lower shroud and disk heat transfer compared to the single cavity case. FFT analysis reveals a complex phase-locking mechanism between flows in the two cavities.

Generation of Solar-like Differential Rotation

We analyze the simulation result shown in Hotta & Kusano (2021) in which the solar-like differential rotation is reproduced. The Sun is rotating differentially with the fast equator and the slow pole. It is widely thought that the thermal convection maintains the differential rotation, but recent high-resolution simulations tend to fail to reproduce the fast equator. This fact is an aspect of one of the biggest problems in solar physics called the convective conundrum. Hotta & Kusano succeed in reproducing the solar-like differential rotation without using any manipulation with an unprecedentedly high-resolution simulation. In this study, we analyze the simulation data to understand the maintenance mechanism of the fast equator. Our analyses lead to conclusions that are summarized as follows. (1) The superequipatition magnetic field is generated by the compression, which can indirectly convert the massive internal energy to magnetic energy. (2) The efficient small-scale energy transport suppresses large-scale convection energy. (3) Non-Taylor–Proudman differential rotation is maintained by the entropy gradient caused by the anisotropic latitudinal energy transport enhanced by the magnetic field. (4) The fast equator is maintained by the meridional flow mainly caused by the Maxwell stress. The Maxwell stress itself also has a role in the angular momentum transport for the fast near-surface equator (we call it the P unching ball effect). The fast equator in the simulation is reproduced not due to the low Rossby number regime but due to the strong magnetic field. This study newly finds the role of the magnetic field in the maintenance of differential rotation.

Zeeman-Doppler imaging of five young solar-type stars

Context. The magnetic activity of the Sun changes with the solar cycle. Similar cycles are found in other stars as well, but their details are not known to a similar degree. Characterising stellar magnetic cycles is important for the understanding of the stellar and solar dynamos that are driving the magnetic activity. Aims. We present spectropolarimetric observations of five young, solar-type stars and compare them to previous observations, with the aim to identify and characterise stellar equivalents of the solar cycle. Methods. We use Zeeman-Doppler imaging (ZDI) to map the surface magnetic field and brightness of our targets. The magnetic field is decomposed into spherical harmonic expansions, from which we report the strengths of the axisymmetric versus non-axisymmetric and poloidal versus toroidal components, and we compare them to the Rossby numbers of the stars. Results. We present five new ZDI maps of young, solar-type stars from December 2017. Of special interest is the case of V1358 Ori, which had gone through a polarity reversal between our observations and earlier ones. A less evident polarity reversal might also have occurred in HD 35296. There is a preference for a more axisymmetric field, and possibly a more toroidal field, for the more active stars with lower Rossby number, but a larger sample should be studied to draw any strong conclusions from this. For most of the individual stars, the amounts of toroidal and poloidal field have stayed on levels similar to those in earlier observations. Conclusions. We find evidence for a magnetic polarity reversal having occurred in V1358 Ori. An interesting target for future observations is χ1 Ori, which may have a short magnetic cycle of a few years. The correlation between the brightness maps and the magnetic field is mostly poor, which could indicate the presence of small-scale magnetic features of different polarities that cancel one another out and are thus not resolved in our maps.

Unsteady Pressure Measurements in a Heated Rotating Cavity

Abstract The flow in the heated rotating cavity of an aero-engine compressor is driven by buoyancy forces, which result in pairs of cyclonic and anticyclonic vortices. The resultant cavity flow field is three-dimensional, unsteady, and unstable, which makes it challenging to model the flow and heat transfer. In this paper, properties of the vortex structures are determined from novel unsteady pressure measurements collected on the rotating disk surface over a range of engine-representative parameters. These measurements are the first of their kind with practical significance to the engine designer and for validation of computational fluid dynamics. One cyclonic/anticyclonic vortex pair was detected over the experimental range, despite the measurement of harmonic modes in the frequency spectra at low Rossby numbers. It is shown that these modes were caused by unequal size vortices, with the cyclonic vortex the larger of the pair. The structures slipped relative to the disks at a speed typically around 10%–15% of that of the rotor, but the speed of precession was often unsteady. The coherency, strength, and slip of the vortex pair increased with the buoyancy parameter, due to the stronger buoyancy forces, but they were largely independent of the rotational Reynolds number.

On the Origin of Pulsar and Magnetar Magnetic Fields

Abstract In order to address the generation of neutron star magnetic fields, with particular focus on the dichotomy between magnetars and radio pulsars, we consider the properties of dynamos as inferred from other astrophysical systems. With sufficiently low (modified) Rossby number, convective dynamos are known to produce dipole-dominated fields whose strength scales with convective flux, and we argue that these expectations should apply to the convective protoneutron stars (PNSs) at the centers of core-collapse supernovae. We analyze a suite of three-dimensional simulations of core collapse, featuring a realistic equation of state and full neutrino transport, in this context. All our progenitor models, ranging from 9 M ⊙ to 25 M ⊙, including one with initial rotation, have sufficiently vigorous PNS convection to generate dipole fields of order ∼1015 Gauss, if the modified Rossby number resides in the critical range. Thus, the magnetar/radio pulsar dichotomy may arise naturally in part from the distribution of core rotation rates in massive stars.

Generation mechanism of solar-like differential rotation in high-resolution simulations

We analyze the simulation result shown in Hotta & Kusano, 2021 in which the solar-like differential rotation is reproduced. The Sun is rotating differentially with the fast equator and the slow pole. It is widely thought that the thermal convection maintains the differential rotation, but recent high-resolution simulations tend to fail to reproduce the fast equator. This fact is an aspect of one of the biggest problems in solar physics called the convective conundrum. Hotta & Kusano, 2021 succeed in reproducing the solar-like differential rotation without using any manipulation with unprecedentedly high-resolution simulation. In this study, we analyze the simulation data to understand the maintenance mechanism of the fast equator. Our analyses lead to conclusions that are summarized as follows. 1. Superequipatition magnetic field is generated by the compression, which can indirectly convert the massive internal energy to magnetic energy. 2. The efficient small-scale energy transport suppresses large-scale convection energy. 3. Non-Taylor--Proudman differential rotation is maintained by the entropy gradient caused by the anisotropic latitudinal energy transport enhanced by the magnetic field. 4. The fast equator is maintained by the meridional flow mainly caused by the Maxwell stress. The Maxwell stress itself also has a role in the angular momentum transport for fast near-surface equator (we call it the Punching ball effect). The fast equator in the simulation is reproduced not due to the low Rossby number regime but due to the strong magnetic field. This study newly finds the role of the magnetic field in the maintenance of differential rotation.

Energy cascades in rapidly rotating and stratified turbulence within elongated domains

We study forced, rapidly rotating and stably stratified turbulence in an elongated domain using an asymptotic expansion at simultaneously low Rossby number $\mathit {Ro}\ll 1$ and large domain height compared with the energy injection scale, $h=H/\ell _{in}\gg 1$ . The resulting equations depend on the parameter $\lambda =(h \mathit {Ro} )^{-1}$ and the Froude number $\mathit {Fr}$ . An extensive set of direct numerical simulations (DNS) is performed to explore the parameter space $(\lambda,\mathit {Fr})$ . We show that a forward energy cascade occurs in one region of this space, and a split energy cascade outside it. At weak stratification (large $\mathit {Fr}$ ), an inverse cascade is observed for sufficiently large $\lambda$ . At strong stratification (small $\mathit {Fr}$ ) the flow becomes approximately hydrostatic and an inverse cascade is always observed. For both weak and strong stratification, we present theoretical arguments supporting the observed energy cascade phenomenology. Our results shed light on an asymptotic region in the phase diagram of rotating and stratified turbulence, which is difficult to attain by brute-force DNS.

Estimating Ocean Surface Currents With Machine Learning

Global surface currents are usually inferred from directly observed quantities like sea-surface height, wind stress by applying diagnostic balance relations (like geostrophy and Ekman flow), which provide a good approximation of the dynamics of slow, large-scale currents at large scales and low Rossby numbers. However, newer generation satellite altimeters (like the upcoming SWOT mission) will capture more of the high wavenumber variability associated with the unbalanced components, but the low temporal sampling can potentially lead to aliasing. Applying these balances directly may lead to an incorrect un-physical estimate of the surface flow. In this study we explore Machine Learning (ML) algorithms as an alternate route to infer surface currents from satellite observable quantities. We train our ML models with SSH, SST, and wind stress from available primitive equation ocean GCM simulation outputs as the inputs and make predictions of surface currents (u,v), which are then compared against the true GCM output. As a baseline example, we demonstrate that a linear regression model is ineffective at predicting velocities accurately beyond localized regions. In comparison, a relatively simple neural network (NN) can predict surface currents accurately over most of the global ocean, with lower mean squared errors than geostrophy + Ekman. Using a local stencil of neighboring grid points as additional input features, we can train the deep learning models to effectively “learn” spatial gradients and the physics of surface currents. By passing the stenciled variables through convolutional filters we can help the model learn spatial gradients much faster. Various training strategies are explored using systematic feature hold out and multiple combinations of point and stenciled input data fed through convolutional filters (2D/3D), to understand the effect of each input feature on the NN's ability to accurately represent surface flow. A model sensitivity analysis reveals that besides SSH, geographic information in some form is an essential ingredient required for making accurate predictions of surface currents with deep learning models.

Rigorous derivation and well-posedness of a quasi-homogeneous ideal MHD system

The goal of this paper is twofold. On the one hand, we introduce a quasi-homogeneous version of the classical ideal MHD system and study its well-posedness in critical Besov spaces Bp,rs(Rd), d≥2, with 1<p<+∞ and under the Lipschitz condition s>1+d∕p and r∈[1,+∞], or s=1+d∕p and r=1. A key ingredient is the reformulation of the system via the so-called Elsässer variables. On the other hand, we give a rigorous justification of quasi-homogeneous MHD models, both in the ideal and in the dissipative cases: when d=2, we will derive them from a non-homogeneous incompressible MHD system with Coriolis force, in the regime of low Rossby number and for small density variations around a constant state. Our method of proof relies on a relative entropy inequality for the primitive system, and yields precise rates of convergence, depending on the size of the initial data, on the order of the Rossby number and on the regularity of the viscosity and resistivity coefficients.

Quantitative Experimental Observation of Weak Inertial-Wave Turbulence.

We report the quantitative experimental observation of the weak inertial-wave turbulence regime of rotating turbulence. We produce a statistically steady homogeneous turbulent flow that consists of nonlinearly interacting inertial waves, using rough top and bottom boundaries to prevent the emergence of a geostrophic flow. As the forcing amplitude increases, the temporal spectrum evolves from a discrete set of peaks to a continuous spectrum. Maps of the bicoherence of the velocity field confirm such a gradual transition between discrete wave interactions at weak forcing amplitude and the regime described by weak turbulence theory (WTT) for stronger forcing. In the former regime, the bicoherence maps display a near-zero background level, together with sharp localized peaks associated with discrete resonances. By contrast, in the latter regime, the bicoherence is a smooth function that takes values of the order of the Rossby number in line with the infinite-domain and random-phase assumptions of WTT. The spatial spectra then display a power-law behavior, both the spectral exponent and the spectral level being accurately predicted by WTT at high Reynolds number and low Rossby number.

Evidence of the Zakharov-Kolmogorov spectrum in numerical simulations of inertial wave turbulence

Rotating turbulence is commonly known for being dominated by geostrophic vortices that are invariant along the rotation axis and undergo an inverse cascade. Yet, it has recently been shown to sustain fully three-dimensional states with a downscale energy cascade. In this letter, we investigate the statistical properties of three-dimensional rotating turbulence by the means of direct numerical simulations in a triply periodic box where geostrophic vortices are specifically damped. The resulting turbulent flow is an inertial wave turbulence that verifies the Zakharov-Kolmogorov spectrum derived analytically by Galtier (Galtier S., Phys. Rev. E, 68 (2003) 015301), thus offering numerical proof of the relevance of wave turbulence theory for three-dimensional, anisotropic waves. Lastly, we show that the same forcing leads to either geostrophic or wave turbulence depending on the initial conditions. Our results thus bring further evidence for bi-stability in rotating turbulent flows at low Rossby numbers.

On a Singular Limit for the Compressible Rotating Euler System

The work addresses a singular limit for a rotating compressible Euler system in the low Mach number and low Rossby number regime. Based on the concept of dissipative measure-valued solution, the quasi-geostrophic system is identified as the limit problem in the case of ill-prepared initial data. The ill-prepared initial data will cause rapidly oscillating acoustic waves. Using dispersive estimates of Strichartz type, the effect of the acoustic waves in the asymptotic limit is eliminated.

Critical transition in fast-rotating turbulence within highly elongated domains

Abstract

Energetics of Seamount Wakes. Part II: Wave Fluxes

AbstractSeamounts are thought to facilitate ocean mixing through unsteady wake processes, and through the generation of internal waves, which propagate away from the seamount and later break. The relative importance of these processes is examined for idealized, isolated seamounts (with characteristic width D and height H) in uniform barotropic flow U. A range of Coriolis parameters f and buoyancy frequencies N are used such that a broad parameter space of low Froude numbers (U/NH) and low Rossby numbers (U/fD) is considered. Results indicate that eddy processes energetically dominate the internal wave energy flux in this range of parameter space. The internal wave field is specifically examined and partitioned into steady lee waves and unsteady, wake-generated waves. It is found that the lee wave energy flux cannot be explained by existing analytical theories. A lee wave model by Smith is then extended into the low-Froude-number regime and the effect of rotation is included. While strongly stratified experiments have previously indicated that only the top U/N of an obstacle generates internal waves, the effect of rotation appears to modify this wavemaking height. Once the U/N height is revised to account for rotation, the lee wave energy flux can be reasonably accurately reproduced by the extended Smith model.

Influence of Coriolis Force Upon Bottom Boundary Layers in a Large‐Scale Gravity Current Experiment: Implications for Evolution of Sinuous Deep‐Water Channel Systems

AbstractOceanic density currents in many deep‐water channels are strongly influenced by the Coriolis force. The dynamics of the bottom boundary layer in large geostrophic flows and low Rossby number turbidity currents are very important for determining the erosion and deposition of sediment in channelized contourite currents and many large‐scale turbidity currents. However, these bottom boundary layers are notoriously difficult to resolve with oceanic field measurements or in previous small‐scale rotating laboratory experiments. We present results from a large, 13‐m diameter, rotating laboratory platform that is able to achieve both stratified and highly turbulent flows in regimes where the rotation is sufficiently rapid that the Coriolis force can potentially dominate. By resolving the dynamics of the turbulent bottom boundary in straight and sinuous channel sections, we find that the Coriolis force can overcome centrifugal force to switch the direction of near‐bed flows in channel bends. This occurs for positive Rossby numbers less than +0.8, defined as RoR = /Rf, where is the depth and time‐averaged velocity, R is the radius of channel curvature, and f is the Coriolis parameter. Density and velocity fields decoupled in channel bends, with the densest fluid of the gravity current being deflected to the outer bend of the channel by the centrifugal force, while the location of velocity maximum shifted with the Coriolis force, leading to asymmetries between left‐ and right‐turning bends. These observations of Coriolis effects on gravity currents are synthesized into a model of how sedimentary structures might evolve in sinuous turbidity current channels at various latitudes.

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.