Rossby Number Research Articles

Immiscible displacement flows in axially rotating pipes

We experimentally study buoyant immiscible displacement flows in an axially rotating pipe, with varying flow parameters, such as the mean imposed flow velocity, density difference, pipe rotation speed, and pipe inclination angle. Via employing image processing and ultrasound Doppler velocimetry techniques, we analyze key flow features, including displacement regimes, interfacial instabilities, interfacial front velocities, and velocity and concentration fields. We find that immiscible displacement flows are distinguished by the emergence of one or two heavy fluid fronts, particularly depending on the rotation speed. Furthermore, our dimensional analysis reveals that the displacement flow is governed by four dimensionless parameters, including the Reynolds, densimetric Froude (or Archimedes), and Rossby numbers, as well as the pipe inclination angle. Using these dimensionless groups, we succeed in categorizing the main flow regimes as efficient and inefficient displacements. Moreover, we classify the interfacial regimes as stable, intermittently unstable, kinks, and separating interfacial patterns. Our analysis shows that the interfacial instabilities observed are indeed characterized by the Kelvin–Helmholtz instability. Our analysis of the velocity fields suggests remarkable differences between displacements in stationary and rotating pipes, especially in terms of the absence and presence of a countercurrent flow, respectively. Finally, our assessment of concentration fields using a Fourier transform approach provides a preliminary fundamental understanding of the characteristics of concentration waves and their corresponding amplitudes.

Intense anticyclones at the global Argentine Basin array of the Ocean Observatory Initiative

Abstract. We analyzed physical oceanic parameters gathered by a mooring array at mesoscale spatial sampling deployed in the Argentine Basin within the Ocean Observatory Initiative, a National Science Foundation major research facility. The array was maintained at 42∘ S and 42∘ W, a historically sparsely sampled region with small ocean variability, over 34 months from March 2015 to January 2018. The data documented four anticyclonic extreme-structure events in 2016. The four anticyclonic structures had different characteristics (size, vertical extension, origin, lifetime and Rossby number). They all featured near-inertial waves (NIWs) trapped at depth and low Richardson values well below the mixed layer. Low Richardson values suggest favorable conditions for mixing. The anticyclonic features likely act as mixing structures at the pycnocline, bringing heat and salt from the South Atlantic Central Water to the Antarctic Intermediate Waters. The intense structures were unique in the 29-year-long satellite altimetry record at the mooring site. The Argentine Basin is populated with many anticyclones, and mixing associated with trapped NIWs probably plays an important role in setting up the upper-water-mass characteristics in the basin.

From shear to veer: theory, statistics, and practical application

Abstract. In the past several years, wind veer – sometimes called “directional shear” – has begun to attract attention due to its effects on wind turbines and their production, particularly as the length of manufactured turbine blades has increased. Meanwhile, applicable meteorological theory has not progressed significantly beyond idealized cases for decades, though veer's effect on the wind speed profile has been recently revisited. On the other hand the shear exponent (α) is commonly used in wind energy for vertical extrapolation of mean wind speeds, as well as being a key parameter for wind turbine load calculations and design standards. In this work we connect the oft-used shear exponent with veer, both theoretically and for practical use. We derive relations for wind veer from the equations of motion, finding the veer to be composed of separate contributions from shear and vertical gradients of crosswind stress. Following from the theoretical derivations, which are neither limited to the surface layer nor constrained by assumptions about mixing length or turbulent diffusivities, we establish simplified relations between the wind veer and shear exponent for practical use in wind energy. We also elucidate the source of commonly observed stress–shear misalignment and its contribution to veer, noting that our new forms allow for such misalignment. The connection between shear and veer is further explored through analysis of one-dimensional (single-column) Reynolds-averaged Navier–Stokes solutions, where we confirm our theoretical derivations as well as the dependence of mean shear and veer on surface roughness and atmospheric boundary layer depth in terms of respective Rossby numbers. Finally we investigate the observed behavior of shear and veer across different sites and flow regimes (including forested, offshore, and hilly terrain cases) over heights corresponding to multi-megawatt wind turbine rotors, also considering the effects of atmospheric stability. From this we find empirical forms for the probability distribution of veer during high-veer (stable) conditions and for the variability in veer conditioned on wind speed. Analyzing observed joint probability distributions of α and veer, we compare the two simplified forms we derived earlier and adapt them to ultimately arrive at more universally applicable equations to predict the mean veer in terms of observed (i.e., conditioned on) shear exponent; lastly, the limitations, applicability, and behavior of these forms are discussed along with their use and further developments for both meteorology and wind energy.

Revisiting the cycle-rotation connection for late-type stars

Aims. We analyse the relation between the activity cycle length and the Rossby number, which serves as a ‘normalised’ rotation period and appears to be the natural parameter in any cycle relation. Methods. We collected a sample of 44 main sequence stars with well-known activity cycle periods and rotation periods. To compute the Rossby numbers from the observed rotation periods, we used the respective B − V-dependent empirical turnover-times and derived the empirical relation between the cycle length and Rossby number. Results. We found a linear behaviour in the double-logarithmic relation between the Rossby number and cycle period. The bifurcation into a long and a short period branch is clearly real but it depends, empirically, on the colour index B − V, indicating a physical dependence on effective temperature and position on the main sequence. Furthermore, there is also a correlation between cycle length and convective turnover time with the relative depth of the convection zone. Based on this, we derived empirical relations between cycle period and Rossby number individually for narrow B − V ranges, for both cycle branches, as well as a global relation for the short-period branch. For the short period cycle branch relations, we estimated a scatter of the relative deviation between 14% and 28% on the long-period cycle branch. With these relations derived purely from stellar data, we obtained a good match with the 10.3−1.0+1.1 yr period for the well known 11-year solar Schwabe cycle and a long-period branch value of 104−34+50 yr for the Gleissberg cycle of the Sun. Finally, we suggest that the cycles on the short-period branch appear to be generated in the deeper layers of the convective zone, while long-period branch cycles seem to be related to fewer deep layers in that zone. Conclusions. We show that for a broader B − V range, the Rossby number is a more suitable parameter for universal relation with cycle-rotation than just the rotation period alone. As proof, we demonstrate that our empirical stellar relations are consistent with the 11-year solar Schwabe cycle, in contrast to earlier studies using just the rotation period in their relations. Previous studies have tried to explain the cycle position of the Sun in the cycle-rotation presentation via other kinds of dynamo, however, in our study, no evidence is found that would suggest another type of dynamo for the Sun and other stars.

Stellar Cycles in Fully Convective Stars and a New Interpretation of Dynamo Evolution

An αΩ dynamo, combining shear and cyclonic convection in the tachocline, is believed to generate the solar cycle. However, this model cannot explain cycles in fast rotators (with minimal shear) or in fully convective stars (no tachocline); an analysis of these stars could therefore provide key insights into how these cycles work. We reexamine ASAS data for 15 M dwarfs, 11 of which are presumed fully convective; the addition of newer ASAS-SN data confirms cycles in roughly 12 of them, while presenting new or revised rotation periods for 5 stars. The amplitudes and periods of these cycles follow , with P cyc/P rot ∝ Ro −1.02±0.06 (where Ro is the Rossby number), very similar to P cyc/P rot ∝ Ro −0.81±0.17 that we find for 40 previously studied FGK stars, although P cyc/P rot and α are a factor of ∼20 smaller in the M stars. The very different P cyc/P rot–Ro relation seen here compared to previous work suggests that two types of dynamo, with opposite Ro dependences, operate in cool stars. Initially, a (likely α 2 or α 2Ω) dynamo operates throughout the convective zone in mid- to late-M and fast-rotating FGK stars, but once magnetic breaking decouples the core and convective envelope, a tachocline αΩ dynamo begins and eventually dominates in older FGK stars. A change in α in the tachocline dynamo generates the fundamentally different P cyc/P rot–Ro relation.

Magnetic Braking with MESA Evolutionary Models in the Single Star and Low-mass X-Ray Binary Regimes

Magnetic braking has a prominent role in driving the evolution of close low-mass binary systems and heavily influences the rotation rates of low-mass F- and later-type stars with convective envelopes. Several possible prescriptions that describe magnetic braking in the context of 1D stellar evolution models currently exist. We test four magnetic braking prescriptions against both low-mass X-ray binary orbital periods from the Milky Way and single-star rotation periods observed in open clusters. We find that the data favor a magnetic braking prescription that follows a rapid transition from fast to slow rotation rates, exhibits saturated (inefficient) magnetic braking below a critical Rossby number, and that is sufficiently strong to reproduce ultra-compact X-ray binary systems. Of the four prescriptions tested, these conditions are satisfied by a braking prescription that incorporates the effect of high-order magnetic field topology on angular momentum loss. None of the braking prescriptions tested are able to replicate the stalled spin down observed in open cluster stars aged 700–1000 Myr or so, with masses ≲0.8 M ⊙.

Effect of marginal variation of aspect ratio and frequency with kinematic asymmetry on flow field around flapping rigid wing in hover

Phase-locked two dimensional particle image velocimetry (PIV) measurements are used to capture flow field around rigid flat plate wings undergoing main flapping motion (hover) with water as fluid medium in quiescent condition. Experiments are conducted using asymmetric upper-lower stroke single degree of freedom main flapping motion. Two different aspect ratio (AR) 1.5 and 1.0 rectangular wings at 1.5 Hz and 2.0 Hz flapping frequency and chord based Reynolds number of the order of 104 are studied. To achieve the desired aspect ratio, wing span is varied while wing chord remains fixed. Flow features for downstroke and upstroke are comparable for all four cases investigated. This includes vortex shedding from wingtip during wing stroke, vorticity intensification caused due to vortex stretching which occurs more dominantly for AR = 1.0, occurrence of KH instability in shear layer formed due to wing sweep and added mass effect due to the flow dragged by flapping wing. Vortex filamentation and fragmentation phenomena are observed during downstroke and upstroke respectively due to faster downstroke and slower upstroke. Rossby number, which is the ratio of inertial to Coriolis force is significant for rotational flapping wing. With increase in Rossby number there is a reduction in the spanwise flow over wing which influences strength of wingtip vortex at beginning of a stroke. It is found that spanwise flow for AR = 1.5 is stronger than AR = 1.0 and consequently the strength of wingtip vortex is higher for higher aspect ratio. However, peak vorticity is higher for AR = 1.0 due to vortex stretching effect which is expected to be more dominant at lower AR due to three dimensional effects. Effect of marginal variation in wing aspect ratio, flapping frequency with kinematic asymmetry are studied using velocity field, vorticity (ωz) contours, λ2 criterion, peak vorticity variation and its spatial distribution as well as kinetic energy variation of the flow field. Kinetic energy distribution and its connection with wake capture is explored.

Mean flow and Convective instability analysis of the BEK and related rotating boundary layer flows subject to a uniform static magnetic field

This study examines the convective instability of the hydromagnetic Bödewadt, Ekman, and von Kármán (BEK) rotating system under the low magnetic Reynolds number approximation. The computations for the mean flow are carried out for different magnetic parameters using MATLAB for several flows of the BEK family. The equations for the linear instability are derived and then solved through the Chebyshev-spectral method for both Type-I (Inviscid) and Type-II (viscous) modes. It is observed that the Rossby number, the Reynolds number, and the magnetic parameter characterize the stability of the hydromagnetic BEK flows. The Reynolds number is based on the dimensionless radial location at the disk while the Rossby number is the ratio of the difference between the angular speeds of fluid and disk and the angular speed of the disk. The magnetic number is the ratio of magnetic field strength and the system rotation rate. The analysis reveals a stabilizing effect of the magnetic parameter on the Type-I modes for Rossby numbers in the range [−1,1]. Further, the Type-II modes disappear upon elevating the magnetic parameter in the aforementioned range of the Rossby number. The growth rates, |αi|, also diminish upon increasing the magnetic parameter. The computations for mean flow reveal that the radial wall shear stress diminishes upon elevating the strength of the applied magnetic field. However, a reverse trend is noted for the azimuthal wall shear stress which indicates an increase in the torque on the disk with enhancing the intensity of the magnetic field.

Dual-stage radial–tangential vortex tilting reverses radial vorticity and contributes to leading-edge vortex stability on revolving wings

The physics of leading-edge vortex (LEV) stability on flapping wings and autorotating seeds is still underexplored due to its complex dependency on Reynolds number ( $\textit {Re}$ ), aspect ratio (AR) and Rossby number (Ro). Our previous study observed an interesting dual-stage vortex tilting between radial and tangential components in a stable LEV. Here, the establishment of this novel mechanism, i.e. dual-stage radial–tangential vortex tilting (DS-VT $_{r-t}$ ), is investigated and explained in detail using numerical methods. The contributions of other tangential vorticity transport terms are also considered. It is shown that the stable LEV region coincides mostly with a constant ratio of tangential and radial vorticity components. The DS-VT $_{r-t}$ mechanism functions as a negative feedback loop for radial vorticity, thereby contributing to the LEV stability at $\textit {Re} > 500$ . Specifically, this mechanism involves a dual-stage vortex tilting starting from negative radial component to positive tangential component, and then back to positive radial component, thereby leading to a $180^{\circ }$ reversal of radial vorticity. The radial Coriolis acceleration can also assist the DS-VT $_{r-t}$ by enhancing the tangential vorticity component and the reduction of radial vorticity inside the LEV via the second stage of DS-VT $_{r-t}$ . The effects of $\textit {Re}$ , AR and Ro on the constant radial–tangential vorticity ratio and DS-VT $_{r-t}$ are then analysed. The coupled effects of AR and Ro are separated into rotational effects and those of tip and root vortices. Our results establish an evident relationship among the LEV stability, the constant radial–tangential vorticity ratio, and the DS-VT $_{r-t}$ , thereby deepening the understanding in the vorticity transport of LEV formation and stability.

A shallow layer laboratory model of large-scale atmospheric circulation

A new shallow layer laboratory model of global atmospheric circulation is realised and studied by experiments and numerical simulations. A shallow rotating cylindrical fluid layer of 30 mm thickness and 690 mm diameter, with a localised heater at the bottom periphery and localised cooler in the central part of the upper boundary is considered. The rim heater imitates the equator heating and disc cooler – the North pole cooling. The flow transforms from the Hadley-like regime to the baroclinic wave regime through transitional states. The decrease in the thermal Rossby number for the fixed value of Taylor number results in the regularisation of the baroclinic waves. All wave regimes, even with regular wave structures, are characterised by strong non-periodic fluctuations. The observed baroclinic wave structures are a combination of temporarily evolving different baroclinic modes. The important outcome of the shallow layer model is a realisation of the Earth-like meridional three-cell structure. It is shown that the three-cell structure with analogs of polar, Ferrel and Hadley cells exist only in a limited range of parameters. A comparison of the results for the water and silicon oil demonstrated that the physical properties of the fluid can have a strong impact on the baroclinic wave structure.

Rotation and activity in late-type members of the young cluster ASCC 123

ABSTRACT ASCC 123 is a little-studied young and dispersed open cluster. Recently, we conducted the first research devoted to it. In this paper, we complement our previous work with Transiting Exoplanet Survey Satellite (TESS) photometry for the 55 likely members of the cluster. We pay special attention to seven of these high-probability members, all with FGK spectral types, for which we have high-resolution spectra from our preceding work. By studying the TESS light curves of the cluster members, we determine the rotational period and the amplitude of the rotational modulation for 29 objects. The analysis of the distribution of the periods allows us to estimate a gyrochronological age for ASCC 123 similar to that of the Pleiades, confirming the value obtained in our previous investigation. A young cluster age is also suggested by the distribution of variation amplitudes. In addition, for those stars with spectroscopic data, we calculate the inclination of their rotation axis. These values appear to follow a random distribution, as already observed in young clusters, with no indication of spin alignment. However, our sample is too small to confirm this on more solid statistical grounds. Finally, for these seven stars, we study the level of magnetic activity from the Hα and Ca ii H&K lines. Despite the small number of data points, we find a correlation of the Hα and Ca ii flux with Rossby number. The position of these stars in flux–flux diagrams follows the general trends observed in other active late-type stars.

Constraints on Magnetic Braking from the G8 Dwarf Stars 61 UMa and τ Cet

During the first half of their main-sequence lifetimes, stars rapidly lose angular momentum to their magnetized winds, a process known as magnetic braking. Recent observations suggest a substantial decrease in the magnetic braking efficiency when stars reach a critical value of the Rossby number, the stellar rotation period normalized by the convective overturn timescale. Cooler stars have deeper convection zones with longer overturn times, reaching this critical Rossby number at slower rotation rates. The nature and timing of the transition to weakened magnetic braking have previously been constrained by several solar analogs and two slightly hotter stars. In this Letter, we derive the first direct constraints from stars cooler than the Sun. We present new spectropolarimetry of the old G8 dwarf τ Cet from the Large Binocular Telescope, and we reanalyze a published Zeeman Doppler image of the younger G8 star 61 UMa, yielding the large-scale magnetic field strengths and morphologies. We estimate mass-loss rates using archival X-ray observations and inferences from Lyα measurements, and we adopt other stellar properties from asteroseismology and spectral energy distribution fitting. The resulting calculations of the wind braking torque demonstrate that the rate of angular momentum loss drops by a factor of 300 between the ages of these two stars (1.4–9 Gyr), well above theoretical expectations. We summarize the available data to help constrain the value of the critical Rossby number, and we identify a new signature of the long-period detection edge in recent measurements from the Kepler mission.

Lateral Circulation in an Elongated Curved Tidal Channel

Abstract The dynamic theory of curvature-induced lateral circulation has been developed for open channel flows but not for oscillatory tides. A linear three-dimensional analytical model was developed to investigate the lateral circulation in an elongated tidal channel with mildly curved bends of which the radius of curvature is larger than the width. The curvature-induced lateral circulation has two components with the same amplitude, namely, a periodic component having an overtide frequency and a steady component. The combination of the two components allows the strength of the lateral circulation to vary periodically and the rotation direction to be unchanged during a tidal period. Friction modifies the strength and structure of the lateral circulation. The phase between the lateral flow and streamwise tidal flow decreases with increasing friction, indicating that the two flows are not necessarily in phase unless friction is strong. The lateral circulations driven by the Coriolis and curvature centrifugal forces augment each other during one phase and compete in the opposite phase, and the relative importance of the two circulations is determined by the Rossby number and friction. The adaptation time is the same for spinup and spindown of the curvature-induced lateral circulation and is determined by water depth and vertical eddy viscosity. The estimation of the adaptation time depends on the leading modes because the transition solution of the curvature-induced lateral circulation comprises a series of cosine modes. These results provide a theoretical basis for interpreting curvature-induced lateral circulation in tidal channels and coastal headlands. Significance Statement The dynamic theory of curvature-induced lateral circulation in a tidal flow remains unexplored. The purpose of this study is to understand the essentials of curvature-induced lateral circulation in an elongated tidal channel using a three-dimensional analytical model. The results showed that the curvature-induced lateral circulation has two components with the same amplitude: a periodic component having an overtide frequency and a steady component. This is significantly different from the curvature-induced lateral circulation associated with open channel flows, which is steady and in phase with the streamwise flow. Future work may show the role of curvature-induced lateral circulation in streamwise dynamics and mass transport.

Nature of Intense Magnetism and Differential Rotation in Convective Dynamos of M-dwarf Stars with Tachoclines

Many of the M-dwarf stars, though they are tiny and dim, are observed to possess strong surface magnetic fields and exhibit remarkably intense flaring. Such magnetism may severely impact habitability on the exoplanets now discovered nearby. The origin of the magnetism must rest with dynamo action achieved by turbulent convection coupled to rotation within the M-dwarfs. To further explore the nature and diversity of the magnetism that can result, we turn here to an extensive set of 45 global MHD simulations to explore dynamos operating within deep convective envelopes of rapidly rotating M2 (0.4 M ⊙) stars. We observe a wide range of cycle periods present in the convection zones, whose durations we find to scale with the Rossby number as Ro−1.66±0.07 in concurrence with scalings identified in simulations of more massive stars. We find a unifying relationship between the ratio of magnetic to convective kinetic energy (ME/CKE) and the degree to which the differential rotation is quenched by magnetic fields. We show that the presence of a tachocline in these model stars enhances their axisymmetric magnetic field components, leading to a surface dipole fraction on average 78% greater than an equivalent star with only a CZ, potentially shedding light on the nature of the tachocline divide through resultant effects on the spin-down rate.

A Simple Family of Tropical Cyclone Models

This review discusses a simple family of models capable of simulating tropical cyclone life cycles, including intensification, the formation of the axisymmetric version of boundary layer shocks, and the development of an eyewall. Four models are discussed, all of which are axisymmetric, f-plane, three-layer models. All four models have the same parameterizations of convective mass flux and air–sea interaction, but differ in their formulations of the radial and tangential equations of motion, i.e., they have different dry dynamical cores. The most complete model is the primitive equation (PE) model, which uses the unapproximated momentum equations for each of the three layers. The simplest is the gradient balanced (GB) model, which replaces the three radial momentum equations with gradient balance relations and replaces the boundary layer tangential wind equation with a diagnostic equation that is essentially a high Rossby number version of the local Ekman balance. Numerical integrations of the boundary layer equations confirm that the PE model can produce boundary layer shocks, while the GB model cannot. To better understand these differences in GB and PE dynamics, we also consider two hybrid balanced models (HB1 and HB2), which differ from GB only in their treatment of the boundary layer momentum equations. Because their boundary layer dynamics is more accurate than GB, both HB1 and HB2 can produce results more similar to the PE model, if they are solved in an appropriate manner.

Centrifugal spinning of polymeric solutions: Experiments and modelling

We assess some of the main parameters affecting the performance of Centrifugal Spinning (CS), a method for making nanofibres, using laboratory experiments and modelling. Rheologically characterized polymer solutions of various concentrations are propelled into curved jet trajectories by an in-house CS device and they are visualized via high speed imaging, while final dried fibre morphology diameter distributions are characterized by scanning electron microscopy. Our experiments provide validations of a comprehensive string model, based on the viscoelastic upper convected Maxwell (UCM) constitutive equation for the polymer, allowing us to examine dependence of the CS process on the key dimensionless numbers. We experimentally analyze the effects of the polymer concentration, rotation speed, nozzle diameter, spinneret radius and nozzle angle, and correspondingly use the model to analyze the effects of the Weissenberg number (Wi), Rossby number (Rb), nozzle diameter ratio (ϵa0), spinneret radius ratio (ϵs0), and nozzle angle (α0) on the fibre trajectory and radius. Finally, we use a simple scaling analysis to classify the flow regimes into stable and unstable with respect to jet instabilities and breakup. The results of this analysis agree with the experiments in finding fibres stable at large capillary and Weissenberg numbers.

Explaining Dynamics and Rapid Onset of the Somali Jet through Its Kinetic Energy Budget

Abstract The low-level Somali jet is the primary mechanism of moisture transport for the South Asian monsoon. It precedes monsoon onset over India and shares its key characteristic features such as rapid intensification and slower retreat during seasonal evolution. This study analyzes the kinetic energy (KE) budget of Somali jet region using high-spatiotemporal-resolution reanalysis (ERA5) to explain these key features. The KE budget reveals that in the Southern Hemisphere, the easterly flow that ultimately feeds the jet exhibits a conventional Ekman balance, with KE generation balanced by frictional dissipation. A unique “advective balance”—balance between KE generation in the northward flow and its advection emerges as the jet begins to form near the equatorial region. The fully formed Somali jet exhibits a three-way balance between KE generation, its advection, and dissipation. A nondimensional parameter–boundary layer local Rossby number (Ro) characterizes the transitions across these regimes. The large-Ro regime describes an advective balance under which KE-generating meridional winds become proportional to meridional pressure gradients yielding a nonlinear (quadratic) dependence of KE generation on pressure gradients. This nonlinear relation explains rapid onset of the jet as well as the asymmetry between rapid onset and slower retreat, leading us to propose a simple model for approximating the seasonal evolution of kinetic energy in the Somali jet given the evolution of pressure gradients. In summary, this work shows that Somali jet onset is closely tied to the seasonal evolution of Ro in the region where the advective boundary layer appears. Significance Statement The Somali jet is an important feature of the South Asian monsoon, contributing significantly to enhanced rainfall over the region. We study the dynamics of this jet by focusing on its kinetic energy (KE). Maintenance of the jet at different regions corresponds to different kinds of balances in the kinetic energy budget. A dimensionless parameter characterizes these different regimes of KE balance, which are captured at sufficiently high resolution. The rapid intensification of the jet can be explained as a nonlinear response of KE generation to the seasonal evolution of the north–south pressure gradient. These findings contribute to understanding of the jet and its nonlinear evolution, which is important for accurate representation and simulation of the Somali jet in climate models.

Hydrostatic and Non‐Hydrostatic Baroclinic Instability in the Dynamical Core of the DOE Global Climate Model

AbstractThe dynamical core of the Department of Energy global climate model is used to understand the role of non‐hydrostatic dynamics in the simulation of dry and moist baroclinic waves. To reduce computational cost, the Diabatic Acceleration and REscaling approach is adopted. Scale analysis and numerical simulations suggest that the model solution is not distorted by the change of spherical metric terms due to the change of Earth radius, neither are the inter‐scale interactions strongly altered as indicated by the analysis of spectral flux. Compared with hydrostatic simulations, the onset of baroclinic instability is delayed under non‐hydrostatic dynamics as the associated weaker vertical motions tend to increase the critical wavelength and narrow the range of waves that can be baroclinically unstable. During the development of baroclinic waves, non‐hydrostatic dynamics tends to induce vertical motions in the upper troposphere, accelerating the eastward propagation of upper‐level ridges through their impact on local vorticity tendency. These processes reduce the westward tilt of the vertical ridge axes and suppress the conversion of mean flow available potential energy to eddy kinetic energy, leading to weaker baroclinic eddies than in hydrostatic simulations. The contrasts between hydrostatic and non‐hydrostatic settings hold in supplemental experiments that vary the background flow Rossby number, the amount of water vapor content and vertical resolution. We also find that mesoscale and smaller‐scale activities can be considerably under‐represented when the vertical resolution is limited to that typically used in global climate models.

Weakly nonlinear versus semi-linear models of the nonlinear evolution of the centrifugal instability

We carry out a weakly nonlinear analysis of the centrifugal instability for a columnar vortex in a rotating fluid, and compare the results to those of the semi-linear model derived empirically by Yim et al. (J. Fluid Mech., vol. 897, 2020, A34). The asymptotic analysis assumes that the Reynolds number is close to the instability threshold so that the perturbation is only marginally unstable. This leads to two coupled equations that govern the evolutions of the amplitude of the perturbation and of the mean flow under the effect of the Reynolds stresses due to the perturbation. These equations differ from the Stuart–Landau amplitude equation or coupled amplitude equations involving a mean field that have been derived previously. In particular, the amplitude does not saturate to a constant as in the supercritical Stuart–Landau equation, but decays afterwards reflecting the instability disappearance when the mean flow tends toward a neutrally stable profile in the direct numerical simulations (DNS). These equations resemble those of the semi-linear model except that the perturbation in the weakly nonlinear model keeps at leading order the structure of the eigenmode of the unperturbed base flow. The predictions of the weakly nonlinear equations are compared to those of the semi-linear model and to DNS for the Rossby number $Ro=-4$ and various Reynolds numbers and wavenumbers. They are in good agreement with the DNS when the growth rate is sufficiently small. However, the agreement deteriorates and becomes only qualitative for parameters away from the marginal values, whereas the semi-linear model continues to be in better agreement with the DNS.

Unsteady dissipation scaling in static- and active-grid turbulence

A new time-dependent analysis of the global and local fluctuating velocity signals in grid turbulence is conducted to assess the scaling laws for non-equilibrium turbulence. Experimental datasets of static- and active-grid turbulence with different Rossby numbers $R_o({=}U/\varOmega M$ : $U$ is the mean velocity, $\varOmega$ is the mean rotation rate and $M$ is the grid mesh size) are considered. Although the global (long-time-averaged) non-dimensional dissipation rate $C_\varepsilon$ is independent of the Reynolds number $Re_\lambda$ based on the global Taylor microscale, the local (short-time-averaged) non-dimensional dissipation rate $\left \langle C_\varepsilon (t_i) \right \rangle$ ( $t_i$ is the local time) both in the static- and active-grid turbulence clearly show the non-equilibrium scaling $\left \langle C_\varepsilon (t_i)\right \rangle / \sqrt {Re_0} \propto \left \langle Re_\lambda (t_i) \right \rangle ^{-1}$ ( $\left \langle Re_\lambda (t_i) \right \rangle$ and $Re_0$ are the Reynolds numbers based on the local Taylor microscale $\lambda (t_i)$ and the global integral length scale, respectively), which has only been confirmed for global statistics in the near field of grid turbulence. The local value of $\left \langle L(t_i) / \lambda (t_i) \right \rangle$ ( $L(t_i)$ is the local integral length scale) shifts from the equilibrium to non-equilibrium scaling as $\left \langle Re_\lambda (t_i) \right \rangle$ increases, further confirming that the non-equilibrium scalings are recovered for local statistics both in the static- and active-grid turbulence. The local values of $\left \langle C_\varepsilon (t_i) \right \rangle$ and $\left \langle L(t_i) / \lambda (t_i) \right \rangle$ follow the theoretical predictions for global statistics (Bos & Rubinstein, Phys. Rev. Fluids, vol. 2, 2017, 022601).