Eulerian Mean Flow Research Articles

Modeled Stokes drift in the marginal ice zones of the Arctic Ocean

This study modeled Stokes drift in the marginal ice zone (MIZ) of the Arctic Ocean using WAVEWATCH III. Applying two viscoelastic and one empirical frequency-dependent wave–ice models, modeled wave parameters and spectra were compared with field observations from the Beaufort–Chukchi Sea. The three wave–ice parameterizations showed similar ability in reproducing the surface Stokes drift estimated using buoy measurements. Comparison with buoy drift showed the estimation of total surface drift was improved by considering Stokes drift during significant wave events. Based on 5-year (2015–2019) hindcasted directional spectra of the summer-autumn Arctic, we investigated monthly mean surface Stokes drift (1–10 cm/s), e-folding depth (1–14 m), and vertically integrated transport (< 0.23 m2/s) in the MIZ. The results showed that Stokes drift in the MIZ increased in strength from July to October. When bulk wave parameters were adopted to estimate Stokes drift fields, the surface Stokes drift was underestimated by about 34–50%, while the Stokes e-folding depth was overestimated by approximately 1.4–5.0 times, increasing from the interior to the edge of the ice cover. We compared the modeled surface Stokes drift with the Eulerian mean flow from reanalyzed data. It showed the mean surface Stokes drift is typically approximately 30% of the Eulerian current over large parts of the Arctic Ocean MIZ with mean ice concentration < 60%, and is the same order of magnitude as (or even larger than) some areas of the Chukchi, East Siberian, and Laptev Seas. It implies Stokes drift should be considered to improve modeling of the dynamic processes of sea ice, especially ice floe drift.

Why Does the Deep Western Boundary Current “Leak” around Flemish Cap?

AbstractThe southward-flowing deep limb of the Atlantic meridional overturning circulation is composed of both the deep western boundary current (DWBC) and interior pathways. The latter are fed by “leakiness” from the DWBC in the Newfoundland Basin. However, the cause of this leakiness has not yet been explored mechanistically. Here the statistics and dynamics of the DWBC leakiness in the Newfoundland Basin are explored using two float datasets and a high-resolution numerical model. The float leakiness around Flemish Cap is found to be concentrated in several areas (hot spots) that are collocated with bathymetric curvature and steepening. Numerical particle advection experiments reveal that the Lagrangian mean velocity is offshore at these hot spots, while Lagrangian variability is minimal locally. Furthermore, model Eulerian mean streamlines separate from the DWBC to the interior at the leakiness hot spots. This suggests that the leakiness of Lagrangian particles is primarily accomplished by an Eulerian mean flow across isobaths, though eddies serve to transfer around 50% of the Lagrangian particles to the leakiness hot spots via chaotic advection, and rectified eddy transport accounts for around 50% of the offshore flow along the southern face of Flemish Cap. Analysis of the model’s energy and potential vorticity budgets suggests that the flow is baroclinically unstable after separation, but that the resulting eddies induce modest modifications of the mean potential vorticity along streamlines. These results suggest that mean uncompensated leakiness occurs mostly through inertial separation, for which a scaling analysis is presented. Implications for leakiness of other major boundary current systems are discussed.

Wave–vortex interactions and effective mean forces: three basic problems

Three examples of wave–vortex interaction are studied, in analytically tractable weak refraction regimes with attention to the mean recoil forces, local and remote, that are associated with refractive changes in wave pseudomomentum fluxes. Wave-induced mean forces of this kind can be persistent, with cumulative effects, even in the absence of wave dissipation. In each example, a single wavetrain propagates past a single vortex. In the first two examples, in a two-dimensional, non-rotating acoustic or shallow-water setting, the focus is on whether or not the wavetrain overlaps the vortex core. In the overlapping case, the recoil has a local contribution given by the Craik–Leibovich force on the vortex core, the vector product of Stokes drift and mean vorticity. (For a quantum vortex this contribution is called the Iordanskii force arising from the Aharonov–Bohm effect on a phonon current.) However, in all except one special limiting case there are additional “remote” contributions, mediated by Stokes-drift-induced return flows that can intersect the vortex core well away from locations where the waves are refracted. The third example is a non-overlapping, remote-recoil-only example in a rapidly rotating frame, in which the waves are deep-water gravity waves and the mean flow obeys shallow-water quasigeostrophic dynamics. Contrary to what might at first be thought, the Ursell “anti-Stokes flow” induced by the rotation – an Eulerian-mean flow tending to cancel the Stokes drift – fails to suppress remote recoil. There are nontrivial open questions about extending these results to regimes of stronger refraction, especially regarding the scope of the “pseudomomentum rule” for the wave-induced recoil forces.

Laboratory study of the wave-induced mean flow and set-down in unidirectional surface gravity wave packets on finite water depth

We derive a multiple-scales solution for the Eulerian mean flow under wavepackets, which is driven by the divergence of Stokes drift and the setdown. This solution is valid for all water depths and is validated by flume experiments and recovers all previous solutions in their respective limits.

An inviscid analysis of the Prandtl azimuthal mass transport during swirl-type sloshing

An inviscid analytical theory of a slow steady liquid mass rotation during the swirl-type sloshing in a vertical circular cylindrical tank with a fairly deep depth is proposed by utilising the asymptotic steady-state wave solution by Faltinsen et al. (J. Fluid Mech., vol. 804, 2016, pp. 608–645). The tank performs a periodic horizontal motion with the forcing frequency close to the lowest natural sloshing frequency. The azimuthal mass transport (first observed in experiments by Prandtl (Z. Angew. Math. Mech., vol. 29(1/2), 1949, pp. 8–9)) is associated with the summarised effect of a vortical Eulerian-mean flow, which, as we show, is governed by the inviscid Craik–Leibovich equation, and an azimuthal non-Eulerian mean. Suggesting the mass-transport velocity tends to zero when approaching the vertical wall (supported by existing experiments) leads to a unique non-trivial solution of the Craik–Leibovich boundary problem and, thereby, gives an analytical expression for the summarised mass-transport velocity within the framework of the inviscid hydrodynamic model. The analytical solution is validated by comparing it with suitable experimental data.

On mean flow generation due to oblique reflection of internal waves at a slope

AbstractSmall‐amplitude expansions are utilized to discuss the mean flow induced by the reflection of a weakly nonlinear internal gravity wave beam at a uniform rigid slope, in the case where the beam planes of constant phase meet the slope at an arbitrary direction, not necessarily parallel to the isobaths, and the flow cannot be taken as two dimensional. Along the vertical, the Eulerian mean flow, due to such an oblique reflection, is equal and opposite to the Stokes drift so the Lagrangian mean flow vanishes, similar to a two‐dimensional reflection. The horizontal Eulerian mean flow, however, is controlled by the mean potential vorticity (PV) and the corresponding Lagrangian mean flow is generally nonzero, in contrast to two‐dimensional flow where PV identically vanishes. For an oblique reflection, furthermore, viscous dissipation can trigger generation of horizontal mean flow via irreversible production of mean PV, a phenomenon akin to streaming.

Vertically varying Eulerian mean currents induced by internal coastal Kelvin waves

The lost momentum in spatially damped internal Kelvin waves reappears as Eulerian mean currents through the action of the nonlinear wave-wave interaction terms. A novel expression is derived for the steady balance between the frictional force on the coastally trapped horizontal Eulerian mean flow, and the forcing from the wave field in terms of the mean wave Reynolds stresses and the horizontal divergence of the Stokes drift. The forcing can be expressed in terms of orthogonal eigenfunctions for internal waves, yielding the vertical variation of the Eulerian mean flow. For arbitrary values of the Brunt-Vaisala frequency N, it is shown that the wave forcing on the Eulerian mean is always negative, yielding a Poiseuille type flow. Therefore, unlike the Stokes drift velocity in internal Kelvin waves which exhibits a backward drift for the first mode in the region of maximum N, the wave-induced horizontal Eulerian mean current is always in the direction of the waves. The results are illustrated by an example from Van Mijenfjorden in Svalbard, which is an arctic sill fjord where internal waves are generated by the action of the barotropic semi-diurnal tide. This article is protected by copyright. All rights reserved.

A Cost-efficient Shadow Particle Tracking Velocimetry Setup Suitable for Tracking Small Objects in a Large Volume

We use a Shadow Particle Tracking Velocimetry technique (S-PTV) with collimated light to investigate the dynamics of a turbulent von Kármán (VK) flow in water. Such a PTV technique permits easy calibration since the apparent particle positions on the images do not depend on the camera-particle distance and tracking of small objects (100 μm particles, fibers, ...) with low power LEDs light sources is made possible in a large volume (approximately (6 cm)3). This technique provides an ensemble of Lagrangian trajectories which are conditioned in space on a grid in order to reconstruct the 3d Eulerian mean flow together with its fluctuations. Typical Lagrangian statistics can also be obtained with this technique, such as velocity and acceleration auto-correlation functions.

Lagrangian flows within reflecting internal waves at a horizontal free-slip surface

In this paper sequel to Zhou and Diamessis [“Reflection of an internal gravity wave beam off a horizontal free-slip surface,” Phys. Fluids 25, 036601 (2013)], we consider Lagrangian flows within nonlinear internal waves (IWs) reflecting off a horizontal free-slip rigid lid, the latter being a model of the ocean surface. The problem is approached both analytically using small-amplitude approximations and numerically by tracking Lagrangian fluid particles in direct numerical simulation (DNS) datasets of the Eulerian flow. Inviscid small-amplitude analyses for both plane IWs and IW beams (IWBs) show that Eulerian mean flow due to wave-wave interaction and wave-induced Stokes drift cancels each other out completely at the second order in wave steepness A, i.e., O(A2), implying zero Lagrangian mean flow up to that order. However, high-accuracy particle tracking in finite-Reynolds-number fully nonlinear DNS datasets from the work of Zhou and Diamessis suggests that the Euler-Stokes cancelation on O(A2) is not c...

Prognostic Residual Mean Flow in an Ocean General Circulation Model and its Relation to Prognostic Eulerian Mean Flow

AbstractRecent work has shown that taking the thickness-weighted average (TWA) of the Boussinesq equations in buoyancy coordinates results in exact equations governing the prognostic residual mean flow where eddy–mean flow interactions appear in the horizontal momentum equations as the divergence of the Eliassen–Palm flux tensor (EPFT). It has been proposed that, given the mathematical tractability of the TWA equations, the physical interpretation of the EPFT, and its relation to potential vorticity fluxes, the TWA is an appropriate framework for modeling ocean circulation with parameterized eddies. The authors test the feasibility of this proposition and investigate the connections between the TWA framework and the conventional framework used in models, where Eulerian mean flow prognostic variables are solved for. Using the TWA framework as a starting point, this study explores the well-known connections between vertical transfer of horizontal momentum by eddy form drag and eddy overturning by the bolus velocity, used by Greatbatch and Lamb and Gent and McWilliams to parameterize eddies. After implementing the TWA framework in an ocean general circulation model, the analysis is verified by comparing the flows in an idealized Southern Ocean configuration simulated using the TWA and conventional frameworks with the same mesoscale eddy parameterization.

Spectral wave-driven sediment transport across a fringing reef

A laboratory experiment was conducted to investigate the dynamics of cross-shore sediment transport across a fringing coral reef. The aim was to quantify how a highly bimodal spectrum of high-frequency (sea-swell) and low-frequency (infragravity and seiching) waves that is typically present on coral reef flats, influences the various sediment transport mechanisms. The experiments were conducted in a 55m wave flume, using a 1:15 scale fringing reef model that had a 1:5 forereef slope, a 14m long reef flat, and a 1:12 sloping beach. The initial 7m of reef flat had a fixed bed, whereas the back 7m of the reef and the beach had a moveable sandy bed. Four seven-hour irregular wave cases were conducted both with and without bottom roughness elements (schematically representing bottom friction by coral roughness), as well as for both low and high still water levels. We observed that the wave energy on the reef flat was partitioned between two primary frequency bands (high and low), and the proportion of energy within each band varied substantially across the reef flat, with the low-frequency waves becoming increasingly important near the shore. The offshore transport of suspended sediment by the Eulerian mean flow was the dominant transport mechanism near the reef crest, but a wide region of onshore transport prevailed on the reef flat where low-frequency waves were very important to the overall transport. Ripples developed over the movable bed and their properties were consistent with the local high-frequency wave orbital excursion lengths despite substantial low-frequency wave motions also present on the reef flat. This study demonstrated that while a proportion of the sediment was transported by bedload and mean flow, the greatest contributions to cross-shore transport were due to the skewness and asymmetry of the high and low-frequency waves.

A Formulation of Three-Dimensional Residual Mean Flow and Wave Activity Flux Applicable to Equatorial Waves

Abstract The large-scale waves that are known to be trapped around the equator are called equatorial waves. The equatorial waves cause mean zonal wind acceleration related to quasi-biennial and semiannual oscillations. The interaction between equatorial waves and the mean wind has been studied by using the transformed Eulerian mean (TEM) equations in the meridional cross section. However, to examine the three-dimensional (3D) structure of the interaction, the 3D residual mean flow and wave activity flux for the equatorial waves are needed. The 3D residual mean flow is expressed as the sum of the Eulerian mean flow and Stokes drift. The present study derives a formula that is approximately equal to the 3D Stokes drift for equatorial waves on the equatorial beta plane (EQSD). The 3D wave activity flux for equatorial waves whose divergence corresponds to the wave forcing is also derived using the EQSD. It is shown that the meridionally integrated 3D wave activity flux for equatorial waves is proportional to the group velocity of equatorial waves.

A Formulation of Three-Dimensional Residual Mean Flow Applicable Both to Inertia–Gravity Waves and to Rossby Waves

Abstract The three-dimensional (3D) residual mean flow is expressed as the sum of the Eulerian-mean flow and the Stokes drift. The present study derives formulas that are approximately equal to the 3D Stokes drift for the primitive equation (PRSD) and for the quasigeostrophic equation (QGSD) using small-amplitude theory for a slowly varying time-mean flow. The PRSD has a broad utility that is applicable to both Rossby waves and inertia–gravity waves. The 3D wave activity flux whose divergence corresponds to the wave forcing is also derived using PRSD. The PRSD agrees with QGSD under the small-Rossby-number assumption, and it agrees with the 3D Stokes drift derived by S. Miyahara and by T. Kinoshita et al. for inertia–gravity waves under the constant-Coriolis-parameter assumption. Moreover, a phase-independent 3D Stokes drift is derived under the QG approximation. The 3D residual mean flow in the upper troposphere in April is investigated by applying the new formulas to the European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Re-Analysis (ERA-Interim) data. It is observed that the PRSD is strongly poleward (weakly equatorward) upstream (downstream) of the storm track. A case study was also made for dominant gravity waves around the southern Andes in the simulation by a gravity wave–resolving general circulation model. The 3D residual mean flow associated with the gravity waves is poleward (equatorward) in the western (eastern) region of the southern Andes. This flow is due to the horizontal structure of the variance in the zonal component of the mountain waves, which do not change much while they propagate upward.

The residual circulation of the Southern Ocean: Which spatio-temporal scales are needed?

The Southern Ocean circulation consists of a complicated mixture of processes and phenomena that arise at different time and spatial scales which need to be parametrized in the state-of-the-art climate models. The temporal and spatial scales that give rise to the present-day residual mean circulation are here investigated by calculating the Meridional Overturning Circulation (MOC) in density coordinates from an eddy-permitting global model. The region sensitive to the temporal decomposition is located between 38°S and 63°S, associated with the eddy-induced transport. The “Bolus” component of the residual circulation corresponds to the eddy-induced transport. It is dominated by timescales between 1month and 1year. The temporal behavior of the transient eddies is examined in splitting the “Bolus” component into a “Seasonal”, an “Eddy” and an “Inter-monthly” component, respectively representing the correlation between density and velocity fluctuations due to the average seasonal cycle, due to mesoscale eddies and due to large-scale motion on timescales longer than one month that is not due to the seasonal cycle. The “Seasonal” bolus cell is important at all latitudes near the surface. The “Eddy” bolus cell is dominant in the thermocline between 50°S and 35°S and over the whole ocean depth at the latitude of the Drake Passage. The “Inter-monthly” bolus cell is important in all density classes and is maximal in the Brazil–Malvinas Confluence and the Agulhas Return Current. The spatial decomposition indicates that a large part of the Eulerian mean circulation is recovered for spatial scales larger than 11.25°, implying that small-scale meanders in the Antarctic Circumpolar Current (ACC), near the Subantarctic and Polar Fronts, and near the Subtropical Front are important in the compensation of the Eulerian mean flow.

Radiation stress and mean drift in continental shelf waves

The time- and depth-averaged mean drift induced by barotropic continental shelf waves (CSW’s) is studied theoretically for idealized shelf topography by calculating the mean volume fluxes to second order in wave amplitude. The waves suffer weak spatial damping due to bottom friction, which leads to radiation stress forcing of the mean fluxes. In terms of the total wave energy density E̅̅ over the shelf region, the radiation stress tensor component S̅11 for CSW’s is found to be different from that of shallow water surface waves in a non-rotating ocean. For CSW’s, the ratio S̅11/E̅ depends strongly on the wave number. The mean Lagrangian flow forced by the radiation stress can be subdivided into a Stokes drift and a mean Eulerian drift current. The magnitude of latter depends on ratio between the radiation stress and the bottom stress acting on the mean flow. When the effect of bottom friction acts equally strong on the waves and the mean current, calculations for short CSW’s show that the Stokes drift and the friction-dependent wave-induced mean Eulerian current varies approximately in anti-phase over the shelf, and that the latter is numerically the largest. For long CSW’s they are approximately in phase. In both cases the mean Lagrangian current, which is responsible for the net particle drift, has its largest numerical value at the coast on the shallow part of the shelf. Enhancing the effect of bottom friction on the Eulerian mean flow, results in a general current speed reduction, as well as a change in spatial structure for long waves. Applying realistic physical parameters for the continental shelf west of Norway, calculations yield along-shelf mean drift velocities for short CSW’s that may be important for the transport of biological material, neutral tracers, and underwater plumes of dissolved oil from deepwater drilling accidents.

Stokes drift dynamos

Fluid particles can have a mean motion in time, even when the Eulerian mean flow disappears everywhere in space. In the present article, we demonstrate that this phenomenon, known as the Stokes drift, plays an essential role in the problem of magnetic field generation by fluctuation flows (kinematic dynamo) at high Rm. At leading order, the dynamo is generated by the Stokes drift that acts as if it were a mean flow. This result is derived from a mean-field dynamo theory in terms of time averages, which reveals how classical expressions for alpha and beta tensors actually recombine into a single Stokes drift contribution. In a test case, we find fluctuation flows that have a G. O. Roberts flow as Stokes drift and this allows to confront our model to direct integration of the induction equation. We find an excellent quantitative agreement between the prediction of our model and the results of our simulations. We finally apply our Stokes drift model to prove that a broad class of inertial waves in rapidly rotating flows cannot drive a dynamo.

Streaming by leaky surface acoustic waves

Acoustic streaming, the generation of mean flow by dissipating acoustic waves, provides a promising method for flow pumping in microfluidic devices. In recent years, several groups have been experimenting with acoustic streaming induced by leaky surface waves: (Rayleigh) surface waves excited in a piezoelectric solid interact with a small volume of fluid where they generate acoustic waves and, as result of the viscous dissipation of these waves, a mean flow. We discuss the computation of the corresponding Lagrangian mean flow, which controls the trajectories of fluid particles and hence the mixing properties of the flows generated by this method. The problem is formulated using the averaged vorticity equation which extracts the dominant balance between wave dissipation and mean-flow dissipation. Particular attention is paid to the thin boundary layer that forms at the solid/liquid interface, where the flow is best computed using matched asymptotics. This leads to an explicit expression for a slip velocity, which includes the effect of the oscillations of the boundary. The Lagrangian mean flow is naturally separated into three contributions: an interior-driven Eulerian mean flow, a boundary-driven Eulerian mean flow and the Stokes drift. A scale analysis indicates that the latter two contributions can be neglected in devices much larger than the acoustic wavelength but need to be taken into account in smaller devices. A simple two-dimensional model of mean flow generation by surface acoustic waves is discussed as an illustration.

On the Three-Dimensional Residual Mean Circulation and Wave Activity Flux of the Primitive Equations

The transformed Eulerian-mean (TEM) equations are useful in examining how the generation and/or dissipation of atmospheric waves drives the mean meridional circulation. However, the TEM equations do not provide a three-dimensional view of the transport. Several previous studies extended the TEM equation system to three dimensions but usually under the quasi-geostrophic assumption, which excludes small-scale phenomena such as gravity waves. Miyahara recently derived three-dimensional wave activity flux and the corresponding residual circulation applicable to gravity waves. However, his formulation has two flaws. First, the three-dimensional residual mean circulation does not satisfy the continuity equation. Second, the Eulerian-mean flow appears in the advection terms and the residual circulation appears in the Coriolis force term of the zonal momentum equation, unlike in the TEM one. The present study developed theoretical formulae of a three-dimensional residual mean circulation and wave activity flux on the basis of primitive equations that overcome these flaws. It is confirmed that the three-dimensional residual mean circulation accords with the sum of the Eulerian time-mean flow and the Stokes drift and that the three-dimensional wave activity flux accords with the mean tangential forces across material surfaces corrugated by the waves under an assumption similar to the TEM equations. A simple physical meaning is given for the terms including the shear of time-mean flow in the three-dimensional wave activity flux. Moreover, the time mean tracer transport equation is derived using the three-dimensional residual mean circulation. A simple case study using the new formulae was made on the three-dimensional transport of stratospheric ozone in the Southern Hemisphere. It is shown that the product of the Coriolis parameter and the strong poleward/equatorward Stokes drifts also balances the divergence/convergence of the three-dimensional wave-activity flux.

Eddy Modulation of Air–Sea Interaction and Convection

Abstract Eddy modulation of the air–sea interaction and convection that occurs in the process of mode water formation is analyzed in simulations of a baroclinically unstable wind- and buoyancy-driven jet. The watermass transformation analysis of Walin is used to estimate the formation rate of mode water and to characterize the role of eddies in that process. It is found that diabatic eddy heat flux divergences in the mixed layer are comparable in magnitude, but of opposite sign, to the surface air–sea heat flux and largely cancel the direct effect of buoyancy loss to the atmosphere. The calculations suggest that mode water formation estimates based on climatological air–sea heat flux data and outcrops, which do not fully resolve ocean eddies, may neglect a large opposing term in the heat budget and are thus likely to significantly overestimate true formation rates. In Walin’s watermass transformation framework, this manifests itself as a sensitivity of formation rate estimates to the averaging period over which the outcrops and air–sea fluxes are subjected. The key processes are described in terms of a transformed Eulerian-mean formalism in which eddy-induced mean flow tends to cancel the Eulerian-mean flow, resulting in weaker residual mean flow, subduction, and mode water formation rates.

Effects of mesoscale eddies on global ocean distributions of CFC-11, CO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, and Δ&amp;lt;sup&amp;gt;14&amp;lt;/sup&amp;gt;C

Abstract. Global-scale tracer simulations are typically made at coarse resolution without explicitly modelling eddies. Here we ask what role do eddies play in ocean uptake, storage, and meridional transport of transient tracers. We made global anthropogenic transient-tracer simulations in coarse-resolution (2°cosφ×2°, ORCA2) and eddy-permitting (½°cosφ×½°, ORCA05) versions of the ocean general circulation model OPA9. Our focus is on surface-to-intermediate waters of the southern extratropics where air-sea tracer fluxes, tracer storage, and meridional tracer transport are largest. Eddies have little effect on global and regional bomb Δ14C uptake and storage. Yet for anthropogenic CO2 and CFC-11, refining the horizontal resolution reduced southern extratropical uptake by 25% and 28%, respectively. There is a similar decrease in corresponding inventories, which yields better agreement with observations. With higher resolution, eddies strengthen upper ocean vertical stratification and reduce excessive ventilation of intermediate waters by 20% between 60° S and 40° S. By weakening the residual circulation, i.e., the sum of Eulerian mean flow and the opposed eddy-induced flow, eddies reduce the supply of tracer-impoverished deep waters to the surface near the Antarctic divergence, thus reducing the air-sea tracer flux. Thus in the eddy permitting model, surface waters in that region have more time to equilibrate with the atmosphere before they are transported northward and subducted. As a result, the eddy permitting model's inventories of CFC-11 and anthropogenic CO2 are lower in that region because mixed-layer concentrations of both tracers equilibrate with the atmosphere on relatively short time scales (15 days and 6 months, respectively); conversely, bomb Δ14C's air-sea equilibration time of 6 years is so slow that, even in the eddy permitting model, there is little time for surface concentrations to equilibrate with the atmosphere, i.e., before surface waters are subducted.