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Understanding how finescale turbulent motions and 0.1–10 km submesoscale processes contribute to the large-scale budgets of nutrients, oxygen, carbon, and heat and affect sea surface temperature, the air–sea exchange of gases, and the carbon cycle is one of the key challenges in oceanography. The ocean, as the atmosphere, is largely in geostrophic balance at mesoscales (10–100 km) or larger scales. Since the horizontal pressure gradientforce (per unit mass) is balanced by Coriolis acceleration and the ocean is density stratified, vertical velocities are typically 1,000 to 10,000 times smaller than horizontal velocities at these scales. [...]the majority of vertical motion in the surface mixed layer changes direction before water parcels cross the base of the mixed layer and only a small fraction of trajectories cross the base of the mixed layer along outcropping isopycnals. Observing, understanding and predicting the three-dimensional pathways by which water from the surface ocean makes its way into the interior is the goal of an Office of Naval Research Departmental Research Initiative, “CALYPSO” (Coherent Lagrangian Pathways from the Surface Ocean to Interior).

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.

The effects of the evolution of vortices on the aeroacoustics generated by a hovering wing are numerically investigated by using a hybrid method of an immersed boundary–finite difference method for the three-dimensional incompressible flows and a simplified model based on the Ffowcs Williams-Hawkings acoustic analogy. A low-aspect-ratio ($AR=1.5$) rectangular wing at low Reynolds ($Re=1000$) and Mach ($M=0.04$) numbers is investigated. Based on the simplified model, the far-field acoustics is shown to be dominated by the time derivative of the pressure on the wing surface. Results show that vortical structure evolution in the flow fields, which is described by the divergence of the convection term of the incompressible Navier–Stokes equations in a body-fixed reference frame, determines the time derivative of the surface pressure and effectively the far-field acoustics. It dominates over the centrifugal acceleration and Coriolis acceleration terms in determining the time derivative of the surface pressure. The position of the vortex is also found to affect the time derivative of the surface pressure. A scaling analysis reveals that the vortex acoustic source is scaled with the cube of the flapping frequency.

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.

Plenoptic particle image velocimetry and surface pressure measurements were used to analyse the early development of leading-edge vortices (LEVs) created by a flat-plate wing of aspect ratio 2 rolling in a uniform flow parallel to the roll axis. Four cases were constructed by considering two advance coefficients, $J=0.54$ and 1.36, and two wing radii of gyration, $R_g/c=2.5$ and 3.25. In each case, the wing pitch angle was articulated such as to achieve an angle of attack of $33^{\\circ }$ at the radius of gyration of the wing. The sources and sinks of vorticity were quantified for a chordwise rectangular control region, using a vorticity transport framework in a non-inertial coordinate system attached to the wing. Within this framework, terms associated with Coriolis acceleration provide a correction to tilting and spanwise convective fluxes measured in the rotating frame and, for the present case, have insignificant values. For the baseline case ($J=0.54, R_g/c=3.25$), three distinct spanwise regions were observed within the LEV, with distinct patterns of vortex evolution and vorticity transport mechanisms in each region. Reducing the radius of gyration to $R_g/c=2.5$ resulted in a more stable vortex with the inboard region extending over a broader spanwise range. Increasing advance ratio eliminated the conical vortex, resulting in transport processes resembling the mid-span region of the baseline case. Although the circulation of the LEV system was generally stronger at the larger advance coefficient, the shear-layer contribution was diminished.

Ocean gravity currents flow along the inclined ocean floor for long times compared to the planet's rotation period. Their shape and motion is governed by the gravity buoyancy, Coriolis acceleration and friction-induced Ekman-layer spinup circulation. In order to understand this process, we consider the flow of a dense-fluid Boussinesq gravity current of fixed volume over an inclined bottom in a rotating system, in the framework of Cartesian 2.5-dimensional geometry (no dependency on the lateral direction $y$, but with a non-trivial $y$-component velocity $v$ due to Coriolis coupling with the main $u$ along the bottom $x$). After release from rest in a lock (co-rotating, with two gates creating propagation in ${\\pm }x$-directions), the current forms a quasi-steady geostrophic ‘vein’ of parabolic height profile with a significant lateral velocity $v$. Subsequently, a spinup process, driven by the Ekman layers on the bottom and interface, appears and prevails for many revolutions, during which $v$ decays and the shape of the interface changes dramatically. We investigate the spinup motion, using an approximate model, for the case of large Rossby number, small Ekman number and small slope $\\gamma$ (relevant to oceanic currents). We show that the initial shape of the natural geostrophic vein can be calculated rigorously (not an arbitrary parabola), and the initial lateral velocity $v(x,t=0)$ is counter-rotation about a fixed point (pivot) $x_{\\rm \\pi}$ at which $v(x_{\\rm \\pi},t) =0$ (at the beginning and during spinup). This point is placed excentrically, in the upper part, and this excentre, $\\propto \\gamma$, plays a significant role in the process. The spinup in a rigid container is developed as the prototype process; an essential component is the edge (outer wall) where the flux of the Ekman layer is arrested (and then returned to the centre via the inviscid core). While the upper part of the vein adopts this spinup pattern, the lower part (most of the vein, $x< x_{\\rm \\pi}$) develops a leak (drainage) at the edge that (a) modifies the spinup of the vein, and (b) generates a thin tail extension downslope. The tail consists of two merged non-divergent Ekman layers, which chokes the drainage flow rate. The present model provides clear-cut insights and some quantitative predictions of the major spinup stage by analytical algebraic solutions. A comparison with a previously published simple model (Wirth, Ocean Dyn., vol. 59, 2009, pp. 551–563) is presented. We also discuss briefly stability of the initial vein.

The traditional approximation neglects the cosine components of the Coriolis acceleration, and this approximation has been widely used in the study of geophysical phenomena. However, the justification of the traditional approximation is questionable under a few circumstances. In particular, dynamics with substantial vertical velocities or geophysical phenomena in the tropics have non-negligible cosine Coriolis terms. Such cases warrant investigations with the non-traditional setting, i.e. the full Coriolis acceleration. In this manuscript, we study the effect of the non-traditional setting on an isothermal, hydrostatic and compressible atmosphere assuming a meridionally homogeneous flow. Employing linear stability analysis, we show that, given appropriate boundary conditions, i.e. a bottom boundary condition that allows for a vertical energy flux and non-reflecting boundary at the top, the atmosphere at rest becomes prone to a novel unstable mode. The validity of assuming a meridionally homogeneous flow is investigated via scale analysis. Numerical experiments were conducted, and Rayleigh damping was used as a numerical approximation for the non-reflecting top boundary. Our three main results are as follows: (i) experiments involving the full Coriolis terms exhibit an exponentially growing instability, yet experiments subjected to the traditional approximation remain stable; (ii) the experimental instability growth rate is close to the theoretical value; (iii) a perturbed version of the unstable mode arises even under sub-optimal bottom boundary conditions. Finally, we conclude our study by discussing the limitations, implications and remaining open questions. Specifically, the influence on numerical deep-atmosphere models and possible physical interpretations of the unstable mode are discussed.

The near-bottom mixing that allows abyssal waters to upwell tilts isopycnals and spins up flow over the flanks of midocean ridges. Meso- and large-scale currents along sloping topography are subjected to a delicate balance of Ekman arrest and spindown. These two seemingly disparate oceanographic phenomena share a common theory, which is based on a one-dimensional model of rotating, stratified flow over a sloping, insulated boundary. This commonly used model, however, lacks rapid adjustment of interior flows, limiting its ability to capture the full physics of spinup and spindown of along-slope flow. Motivated by two-dimensional dynamics, the present work extends the one-dimensional model by constraining the vertically integrated cross-slope transport and allowing for a barotropic cross-slope pressure gradient. This produces a closed secondary circulation by forcing Ekman transport in the bottom boundary layer to return in the interior. The extended model can thus capture Ekman spinup and spindown physics: the interior return flow is turned by the Coriolis acceleration, leading to rapid rather than slow diffusive adjustment of the along-slope flow. This transport-constrained one-dimensional model accurately describes two-dimensional mixing-generated spinup over an idealized ridge and provides a unified framework for understanding the relative importance of Ekman arrest and spindown of flow along a slope.

We consider a theoretical model for the settling of rod-shaped particles of a dilute, initially homogeneous, suspension in rapid rotation. The particle Reynolds number and the particle Taylor number of the detailed flow around the particles are assumed small, representing a relevant limit for an industrial centrifugal separation process. By applying a statistical approach using the Fokker–Planck equation, and neglecting particle–particle interactions, we obtain an explicit, analytical solution for the time dependent, spatially uniform particle orientation distribution function. Not only does the volume fraction in the bulk of the suspension decrease with time due to the divergent centrifugal field, as similarly described in the literature for suspensions of spherical particles, the orientation of the rod particles also changes with time from an initially uniform distribution to one where the particles tend to align with a plane perpendicular to the axis of rotation. The corresponding particle trajectories, as also influenced by first-order effects from the Coriolis acceleration and gyroscopic effects, are obtained numerically for different initial particle orientation angles.

The flow of a gravity current of finite volume and density $\\unicode[STIX]{x1D70C}_{1}$ released from rest from a rectangular lock (of height  $h_{0}$ ) into an ambient fluid of density $\\unicode[STIX]{x1D70C}_{0}$ ( ${<}\\unicode[STIX]{x1D70C}_{1}$ ) in a system rotating with $\\unicode[STIX]{x1D6FA}$ about the vertical  $z$ is investigated by means of fully resolved direct numerical simulations (DNS) and a theoretical model (based on shallow-water and Ekman layer spin-up theories, including mixing). The motion of the dense fluid includes several stages: propagation in the $x$ -direction accompanied by Coriolis acceleration/deflection in the $-y$ -direction, which produces a quasi-steady wedge-shaped structure with significant anticyclonic velocity  $v$ , followed by a spin-up reduction of $v$ accompanied by a slow $x$ drift, and oscillation. The theoretical model aims to provide useful insights and approximations concerning the formation time and shape of wedge, and the subsequent spin-up effect. The main parameter is the Coriolis number, ${\\mathcal{C}}=\\unicode[STIX]{x1D6FA}h_{0}/(g^{\\prime }h_{0})^{1/2}$ , where $g^{\\prime }=(\\unicode[STIX]{x1D70C}_{1}/\\unicode[STIX]{x1D70C}_{0}-1)g$ is the reduced gravity. The DNS results are focused on a range of relatively small Coriolis numbers, $0.1\\leqslant {\\mathcal{C}}\\leqslant 0.25$ (i.e. Rossby number $Ro=1/(2{\\mathcal{C}})$ in the range $2\\leqslant Ro\\leqslant 5$ ), and a large range of Schmidt numbers $1\\leqslant Sc<\\infty$ ; the Reynolds number is large in all cases. The current spreads out in the $x$ direction until it is arrested by the Coriolis effect (in ${\\sim}1/4$ revolution of the system). A complex motion develops about this state. First, we record oscillations on the inertial time scale $1/\\unicode[STIX]{x1D6FA}$ (which are a part of the geostrophic adjustment), accompanied by vortices at the interface. Second, we note the spread of the wedge on a significantly longer time scale; this is an indirect spin-up effect – mixing and entrainment reduce the lateral (angular) velocity, which in turn decreases the Coriolis support to the $\\unicode[STIX]{x2202}h/\\unicode[STIX]{x2202}x$ slope of the wedge shape. Contrary to non-rotating gravity currents, the front does not remain sharp as it is subject to (i) local stretching along the streamwise direction and (ii) convective mixing due to Kelvin–Helmholtz vortices generated by shear along the spanwise direction and stemming from Coriolis effects. The theoretical model predicts that the length of the wedge scales as ${\\mathcal{C}}^{-2/3}$ (in contrast to the Rossby radius $\\propto 1/{\\mathcal{C}}$ which is relevant for large  ${\\mathcal{C}}$ ; and in contrast to ${\\mathcal{C}}^{-1/2}$ for the axisymmetric lens).