157 research outputs found
Weakly nonlinear surface waves on vertically sheared currents
No full textExtreme waves, also known as rogue waves, have much larger heights than surrounding waves, which are of great danger to ships and coastal structures. A subsurface shear current is able to change the properties of the surface waves significantly. Hence, rogue waves may be altered by a shear current. This thesis aims to investigate the effects of a vertically sheared background flow on weakly nonlinear deep-water waves with both a statistical study for irregular random waves and a deterministic study for a focused wave group.
First, we extend the theory of Longuet-Higgins [J. Fluid Mech. 12, 321 (1962)] to allow for a background shear flow that varies with depth. The theory is valid up to second order in wave steepness. It is applicable to arbitrarily strong shear currents and broadband waves. The linear dispersion relation and wave field are solved by the Direct Numerical Method (DIM) proposed by Li and Ellingsen [J. Geophys. Res.: Oceans, 124, 2527 (2019)]. The second-order corrections are solved numerically from the Rayleigh equation with the finite volume method.
To examine the effects of different shear currents on the statistics of weakly nonlinear waves, we perform simulations and generate random waves with both numerical cases and real-world data from the mouth of the Columbia River. For numerical cases, a JONSWAP spectrum and several different linear and exponential shear currents with reasonable strength are adopted. The real-world data is measured by Zippel and Thomson [J. Geophys. Res.: Oceans 122, 3311 (2017)]. We investigate the probability density function of wave surface elevation and maxima, exceedance probability distribution of crest height, maximum wave crest among a number of consecutive waves, and the skewness of the surface elevation. The simulating results of both numerical cases and real-world data demonstrate that an opposing shear enhances the probability of rogue waves and skewness, while a following shear has opposite influences compared with the no-current case.
Knowing that the shear current can have a considerable influence on rogue waves, we proceed to examine the focused wave group on different shear currents, which acts as a representation of the rogue wave event. The investigations are carried out for both long-crested linear and weakly nonlinear waves.
Linear waves: For a prescribed initial wave group surface elevation, different techniques are used to derive explicit approximate expressions of wave surface elevation and orbital velocity profiles. The approximate expressions agree well with the numerical results from DIM. Compared to the case without a current, orbital velocities are significantly amplified by a following shear and suppressed by an opposing shear; an amplification factor is defined from here, and the approximate expression of the amplification factor with respect to the parameters of a shear current is derived explicitly. Compared to the no-current case, a reasonably strong shear alters the orbital velocities substantially. The present expressions can be tested by future experiments at the NTNU fluid mechanics laboratory.
Weakly nonlinear waves: The nonlinear waves are solved numerically similar to the statistical study. We use Gaussian energy spectra to generate linear wave groups. The exponential shear currents are assumed to propagate at an arbitrary angle to the wave groups. The measured wave spectra and shear currents are also used in the calculation for comparison. The wave surface elevation and horizontal velocity beneath the focused point are analyzed. Our results suggest that in regions of strong oceanic currents, substantial error occurs in predicting wave impact on ocean structures if the shear currents are ignored
Error Sources in Wave-Based Remote Sensing and Free-Surface Synthetic Schlieren
No full textPhysical processes at and under the water surface of oceans play a key role in the Earth’s climate. For the models that are used to make predictions of temperature, gas contents, humidity, etc., we rely on data from laboratory experiments and field observations. This thesis presents numerical studies of error sources in two measurement techniques commonly used in the context of waves at the air-water interface. Findings include conditions under which results obtained from these methods become unreliable as well as mitigation strategies.
In laboratory studies, it is often desirable to obtain the surface topography in an area of interest. An appropriate technique in this context is the Free Surface Synthetic Schlieren (FS-SS) method. It utilizes the apparent distortions of a pattern seen through the surface due to refraction at the air-water interface, to infer the free-surface topography.
In the context of field observations of currents, remote sensing techniques are of appreciable interest due to their ability to cover large areas at low cost compared to in-situ methods. The second technique investigated in this work uses the wavenumber-frequency spectrum obtained from, e.g., airborne video footage, to measure Doppler shifts in the waves’ frequencies due to a background current, from which the current can be inferred.
Both methods rely on certain assumptions or approximations that may not always be applicable. In order to test their limits, numerical schemes were developed that create ideal input data, which, when analyzed with the respective method, allow the isolation of error sources.
Systematic errors in the measured surface gradients obtained using the FSSS method were investigated for two configurations. In the standard configuration, the camera is placed above the water surface and a pattern visualizing the distortions is placed underwater, typically at the bottom. In the “flipped”configuration, the camera is placed underneath the transparent water container and the pattern above the free surface. The errors found for the standard geometry stay within a few percent for typical setups, whereas the flipped configuration shows errors that can be much larger, exceeding 50%. With proper adjustment of the setup, these can be reduced to below 10%, retaining the usability of this configuration.
For the investigation of biases in the effective Doppler shift extraction from wave elevation data, a framework was developed to generate synthetic surface wave “videos”, where the background current can be both vertically and horizontally sheared. These were subsequently analyzed using the muchused Normalized Scalar Product method. The key findings of this part of the study include that both spectral leakage and horizontal shear can cause significant errors in the current measurements. Their severity was found to depend strongly on a range of parameters, most notably the wave spectrum shape in terms of angular spread and peakedness, and the current variation and direction. Knowledge of how the respective biases emerge enables the detection and mitigation of these errors, which are discussed in this thesis as well
Surface water waves on depth dependent flows
No full textThe present thesis provides essential insights into surface water waves propagating atop a horizontal current whose magnitude and direction may vary arbitrarily with water depth. A comprehensive theory in this regard is developed in the framework of linear wave theory in three dimensions, being readily applied to a wide range of realistic circumstances. General theoretical solutions to different boundary value problems are presented. In particular, explicit expressions with regard to the surface elevation and the vertical velocity are derived. The boundary value problems include the Cauchy-Poisson problem, surface disturbances generated by an initial
impulsive and a time-dependent pressure, and a steady pressure that normally works as the model of moving vessels and oscillating travelling sources. Efforts focus especially on the dispersion relation and the effects of a subsurface shear current on surface waves. A subsurface shear current is most often found to have significant effects on surface waves. In particular, the presence of a current of uniform vorticity is analysed in detail for the problems of ship waves, an oscillating advancing source and wave interferences.
A theory is especially presented to calculate waves from a general, time-dependent applied surface pressure acting on the surface of a horizontally directed shear current which may vary arbitrarily with depth in both direction and magnitude. It is based on deriving the response function in the context of waves generated by an impulsive applied pressure. Effective approaches to calculate wave resistance without undue difficulty are presented. Strikingly, a lateral radiation force – that is defined towards the starboard (right) – is firstly found apart from the well-known wave resistance along the stern-wise direction due to the presence of a shear current when a ship is making an oblique angle with the shear current. The lateral radiation force may amount to 20 percent of the normal wave resistance in some specific situations.
As for waves on a current in arbitrary variation of water depth, an implicit dispersion relation is derived, which poses potential challenges in obtaining analytical solutions. Several semi-analytical approaches to solve/approximate the dispersion relation are hence derived. In particular, a direct integration approach – that solves the linearised Rayleigh equation and implicit dispersion relation in a coupled way – and approximations based on a perturbation method are presented. The proper criteria under which different perturbed approximate dispersion relations are applicable
are determined. Furthermore, the analytical solutions of the dispersion relation under limited circumstances are derived, e.g. for a shear current of uniform vorticity and stationary waves for a specific class of shear profiles of non-zero curvature.
Despite the fact that linear waves in the presence of a linear shear current have been extensively analysed in two dimensions, studies in three dimensions are scarce. This means that realistic three dimensional effects may be in some cases overlooked or yet discovered. The present thesis hence attempts to fill this gap based on theoretical as well as numerical analysis. Effects of a uniform vorticity are specially analysed in the context of ship waves, waves generated by an oscillating travelling source and interferences of waves generated by a two-point wavemaker of
monohull ships. Fascinating and novel features are found due to the uniform vorticity S that is known either as the ’intrinsic shear Froude number’ Frsb = S_L/g or the ’shear Froude number’ Frs = |V|S/g (L: the reference length; |V|: moving speed of a wavemaker; g: the gravitational acceleration). In particular, asymmetrical ship wave patterns, the critical shear velocity above which the transverse ship waves vanish, the transitions between the sub-critical and supercritical situations due to the complex interplays of the shear current and seabed, non-constant Kelvin
angles, and a somewhat similar effect as a finite water depth on wave interferences are shown for the problem of ship wakes. All of those novel features would not have been found if theoretical studies are constrained to 2D. Furthermore, the classical Doppler resonance occurs when the non-dimensional frequency τ = |V|ω/g (ω : the oscillating frequency) is equal to 1/4 in the absence of a shear current, while there may be multiple Doppler resonances – as many as 4 – for Frs > 1/3 in deep water due to the presence of a linear shear current. It is also indicated that the
Doppler resonance may be profoundly modified even for a linear current of weak vorticity
Dispersion forces in Micromechanics: Casimir and Casimir-Polder forces affected by geometry and non-zero temperature
No full textThe present thesis focuses on several topics within three separate but related branches of the overall field of dispersion forces. The three branches are: temperature corrections to the Casimir force between real materials (Part 1), explicit calculation of Casimir energy in wedge geometries (Part 2), and Casimir-Polder forces on particles out of thermal equilibrium (Part 3).
Part 1 deals primarily with analysis of a previously purported thermodynamic inconsistency in the Casimir-Lifshitz free energy of the interaction of two plane mirrors – violation of the third law of thermodynamics – when the latter’s dielectric response is described with dissipative models. It is shown analytically and numerically that the Casimir entropy of the interaction between two metallic mirrors described by the Drude model does tend to zero at zero temperature, provided electronic relaxation does not vanish. The leading order terms at low temperature are found. A similar calculation is carried out for the interaction of semiconductors with small but non-zero DC conductivity. In a generalisation, it is shown that a violation of the third law can only occur for permittivities whose low-frequency behaviour is temperature dependent near zero temperature. A calculation using path integral methods shows that the low temperature behaviour of the interaction of fluctuating Foucault currents in two mirrors of Drude metal is identical to that of the full Casimir-Lifshitz free energy, reasserting a previous finding by Intravaia and Henkel that such fluctuating bulk currents are the physical reason for the anomalous entropy behaviour.
In a related effort, an analysis of the frequency dependence of the Casimir force by Ford is generalised to imperfectly reflecting mirrors. A paradox is pointed out, in that the effects of a perturbation of the reflecting properties of the mirrors in a finite frequency window can be calculated in two ways giving different results. It is concluded that optimistic conclusions reached by Ford based on one of these methods, which seems to allow radically changing and tailoring the Casimir force with engineered materials, can not be realised.
Part 2 presents several explicit calculations of the Casimir energy of different wedge and cylinder geometries. The Casimir energy of a perfectly conducting wedge intercut by a circularly cylindrical arc, either perfectly conducting or (magneto)dielectric, is calculated. The energy is found to include a singular and non-regularisable term due to the corners where the arc meets the wedge, whereas the finite part is an immediate generalisation of the previously known results for a circular cylinder. The energy of a magnetodielectric wedge obeying a criterion of isorefractivity (spatially uniform speed of light) superimposed coaxially on a perfectly conducting cylindrical shell is calculated. This is the first expression for the energy of a wedge which is not perfectly reflecting. Finally, the energy of the perfectly conducting wedge and arc (and, as a special case, cylinder) is extended to the case of non-zero temperatures. After a regularisation procedure making use of the Chowla-Selberg formula an analytical expression for the temperature-dependent energy at all temperatures is derived, and showed to coincide with previously calculated high-temperature asymptotics by Bordag, Nesterenko and Pirozhenko.
Part 3 considers numerical and analytical studies of the Casimir-Polder forces acting on particles prepared in a given eigenstate (or superposition of such) in an environment which is otherwise at thermal equilibrium. We first consider cold polar molecules outside a metallic halfspace. It is found that the force in the near-zone (non-retarded regime) is much weaker than what would result from a naïve perturbative calculation, and that in the far-zone (retarded regime) the force becomes spatially oscillatory. It is demonstrated how these spatial oscillations may be enhanced in a resonating planar cavity, although for polar molecules the resulting amplitude is still insufficient for observation. A cylindrical cavity, however, can achieve a better enhancement factor. The Casimir-Polder forces on Rydberg atoms near a surface are calculated; because of the very large transition dipole moments of Rydberg transitions, the force is enormous on an atomic scale. We show that the oscillating force on Rydberg atoms can be enhanced into the observable regime by use of a fine-tuned cylindrical cavity. A particle in an eigenstate which is in the non-retarded regime with respect to all its dominant transitions is shown to feel a Casimir-Polder force which is virtually independent of temperature from zero to room temperature and beyond. Both for cold polar molecules and Rydberg atoms, the temperature-independent regime extends to a few and hundreds of micrometers, respectively, and includes the separations generally accessed in experimentsPhD i energi- og prosessteknikkPhD in Energy and Process Engineerin
Experimental study of channel flow with sinusoidal walls using stereo-PIV
No full textEt eksperiment har blitt designet for å utforske resultatene til Brostrøm, Holum, Akselsen og Ellingsen i første omgang, og masteroppgaven kan bli sett på som en fortsettelse av dette arbeidet (Brostrøm, 2020, Holum, 2020 and Akselsen and Ellingsen, 2020). De studert plan Poiseuille flow med modifisert veggeometri bestående av kryssende sinusbølger både teoretisk og numerisk. Derimot finnes det ikke noen eksperimentell data på akkurat denne geometrien som forfatteren vet om.Dette eksperimentet ble derfor designet og brukt som en liten åpen vindtunnel med en testseksjon så lik som mulig det numeriske oppsettet til Brostrøm og Holum (Brostrøm, 2020, Holum, 2020).Den gjennomsnittelige kanalhøyden ble designet til å være 2 cm høy for at laminær, turbulent og overgangsstrømning skal kunne produsers som i de numeriske simuleringene. For å ta målingen bledet brukt et stereo-PIV-oppsett for å hente ut informasjon om all tre hastighetskomponenter i et plan stående normalt på strømningen. Topplata på kanalen var også utstyrt med trykkutganger for å kunne måle trykktappet i kanalen. Fire mål ble satt før og under jobbingen ettersom ny infomasjon kom frem. De fire målene var:
1. Lag et eksperimentelt oppsett som kan produsere strømningstilstandene i simuleringen til Brostrøm og Holum, inkludert lavt nok Reynoldstall og riktig overflategeometri.
2. Finn ut ved hvilket Reynoldstall strømningen går fra laminær til turbulent ved å se på foreksempel hastighetsprofiler, turbulensintensitet og øyeblikksbilder av hastigheter.
3. Sammenlign strømningen kvalitativt med de numeriske resultatene til Brostrøm og Holum så godt som mulig (Brostrøm, 2020, Holum, 2020).
4. (Hvis mulig finn eller ta tilsvarende data for plan Poiseuille flow).
Det første målet ble gjennomført tilfredsstillende, med noen mulige forbedringer, og dette ga mulighet for å finne svar på de tre andre målene. Ett svar på det andre målet ble funnet ved hjelp av blant annet de nevnte hjelpemiddlene samt tidsgjennomsnittelige ”streamwise” hastigheter. Et konservativt estimat på området hvor strømningen skifter fra laminær til turbulent ble funnet til å grovt være mellom Re0= 1400 og Re0= 2400, noe som plasserer det forutsagte overgangs-Reynoldstallet på 2000 av Holum nesten midt i dette området (Holum, 2020). Så til det tredje målet. Sammeligninger mellom eksperimentelle resultater og numeriske resultater fra Brostrøm ogHolum ble gjennomført, men i små mengder siden det var lite tid igjen (Brostrøm, 2020, Holum,2020). Det fjerde målet ble lagt til sent ettersom iden ikke hadde prioritet fra start. Likevel ble ett røft forsøk ved å bruke glassplater plassert oppå de bølgete veggen gjennomført. Dessverre ble kvaliteten og kvantiteten av disse målingene for dårlige til å slutte noen konklusjoner. Fremtidig arbeid anbefales å lage flate plater med og uten overflateruhet for å tilføre eller avkrefte påstanden om at veggeometrien studert her senker overgangs-Reynoldstallet.An experiment has been designed to explore the findings of Brostrøm, Holum, Akselsen and Ellingsen first and foremost, and can be seen as a continuation of that work (Brostrøm, 2020, Holum, 2020 and Akselsen and Ellingsen, 2020). They studied plane Poiseuille flow with a modified wall geometry made up of crossing sinusoidal waves both theoretically and numerically. However experimental data on this specific geometric construction is not known to exist to the author. This experiment was therefore designed and put into operation as a small open return wind tunnel with a test section made as close to equal to the numerical setup of Brostrøm and Holum (Brostrøm,2020, Holum, 2020). The mean channel height was designed to be 2 cm, to allow for laminar, turbulent and transitional flow as in the numerical simulations. To take the measurements a stereo-PIV setup was used to gather all three velocity components in a spanwise plane. The top plate was also equipped with pressure ports to measure the pressure drop in the channel. Four goals was set before and during the making and measuring phase of the experiments as new information becameavailable. These four were:
1. Make an experimental set-up able to produce flow conditions as in the simulations of Brostrøm and Holum, including low enough Reynolds number and surface geometry.
2. Find out at which Reynolds number the flow transitions, by looking at for example velocityprofiles, turbulence intensity and instantaneous velocities.
3. Compare the flow qualitatively to the numerical results of Brostrøm and Holum as best aspossible (Brostrøm, 2020, Holum, 2020).
4. (If possible find or make comparable data for plane Pouiseuille flow).
The first one was accomplished satisfactorily, with some possible improvements, and therefore allowed the opportunity to find answers to the three others. An answer to the second question was found using the tools mentioned in the question and the time-averaged streamwise velocities. The range was conservatively found to be roughly between Re0= 1400 and Re0= 2400, which places the predicted transitional Reynolds number of 2000 by Holum firmly in the middle of this region (Holum, 2020). Onto the third goal. Comparisons between the experimental results and numerical results of Brostrøm and Holum were done, but in to low quantity due to time constraints (Brostrøm, 2020, Holum, 2020). The fourth goal was added late as the idea did not have priority from the start. However a rough attempt using acrylic plates placed on top of the wavy walls was attempted. However the measurement quality and quantity was too bad to make any definite conclusions. Further work is recommended to focus on the making of flat plates with and without surface roughness to substantiate or disprove the claim that the wall geometry studied here do infact lower the transitional Reynolds number
Experimental study of channel flow with sinusoidal walls using stereo-PIV
No full textEt eksperiment har blitt designet for å utforske resultatene til Brostrøm, Holum, Akselsen og Ellingsen i første omgang, og masteroppgaven kan bli sett på som en fortsettelse av dette arbeidet (Brostrøm, 2020, Holum, 2020 and Akselsen and Ellingsen, 2020). De studert plan Poiseuille flow med modifisert veggeometri bestående av kryssende sinusbølger både teoretisk og numerisk. Derimot finnes det ikke noen eksperimentell data på akkurat denne geometrien som forfatteren vet om.Dette eksperimentet ble derfor designet og brukt som en liten åpen vindtunnel med en testseksjon så lik som mulig det numeriske oppsettet til Brostrøm og Holum (Brostrøm, 2020, Holum, 2020).Den gjennomsnittelige kanalhøyden ble designet til å være 2 cm høy for at laminær, turbulent og overgangsstrømning skal kunne produsers som i de numeriske simuleringene. For å ta målingen bledet brukt et stereo-PIV-oppsett for å hente ut informasjon om all tre hastighetskomponenter i et plan stående normalt på strømningen. Topplata på kanalen var også utstyrt med trykkutganger for å kunne måle trykktappet i kanalen. Fire mål ble satt før og under jobbingen ettersom ny infomasjon kom frem. De fire målene var:
1. Lag et eksperimentelt oppsett som kan produsere strømningstilstandene i simuleringen til Brostrøm og Holum, inkludert lavt nok Reynoldstall og riktig overflategeometri.
2. Finn ut ved hvilket Reynoldstall strømningen går fra laminær til turbulent ved å se på foreksempel hastighetsprofiler, turbulensintensitet og øyeblikksbilder av hastigheter.
3. Sammenlign strømningen kvalitativt med de numeriske resultatene til Brostrøm og Holum så godt som mulig (Brostrøm, 2020, Holum, 2020).
4. (Hvis mulig finn eller ta tilsvarende data for plan Poiseuille flow).
Det første målet ble gjennomført tilfredsstillende, med noen mulige forbedringer, og dette ga mulighet for å finne svar på de tre andre målene. Ett svar på det andre målet ble funnet ved hjelp av blant annet de nevnte hjelpemiddlene samt tidsgjennomsnittelige ”streamwise” hastigheter. Et konservativt estimat på området hvor strømningen skifter fra laminær til turbulent ble funnet til å grovt være mellom Re0= 1400 og Re0= 2400, noe som plasserer det forutsagte overgangs-Reynoldstallet på 2000 av Holum nesten midt i dette området (Holum, 2020). Så til det tredje målet. Sammeligninger mellom eksperimentelle resultater og numeriske resultater fra Brostrøm ogHolum ble gjennomført, men i små mengder siden det var lite tid igjen (Brostrøm, 2020, Holum,2020). Det fjerde målet ble lagt til sent ettersom iden ikke hadde prioritet fra start. Likevel ble ett røft forsøk ved å bruke glassplater plassert oppå de bølgete veggen gjennomført. Dessverre ble kvaliteten og kvantiteten av disse målingene for dårlige til å slutte noen konklusjoner. Fremtidig arbeid anbefales å lage flate plater med og uten overflateruhet for å tilføre eller avkrefte påstanden om at veggeometrien studert her senker overgangs-Reynoldstallet
Laser light deformation of microdroplets
No full textThis masters work describes the deformation of a droplet that is illuminated by a laser beam. The theory for linear fluid mechanical motion of the droplet is discussed. This is combined with Lorenz-Mie scattering. Droplet deformations resulting from the optical radiation pressure are computed.
Specific beam profiles are discussed in the literature for the purposes of optical droplet deformation, namely the cases of linear and circular polarized plane waves, a Gaussian beam and the Bessel beam. The general case of an arbitrary beam is not, to the author's knowledge, given in the published literature. Such a framework is developed from first principles and presented in this work.
A Mathematica script was written to compute deformations. These are calculated, fitting nicely to those found by Ellingsen in a recent article. The case of two plane waves from opposite directions is discussed here for the first time, and droplet shapes produced.
The size droplets considered is between the geometrical limit and the Rayleigh limit. The numerical load increases quickly with increasing values of the droplet radius and wave vector. The hardest to compute coefficients die off the most quickly with time
Laser light deformation of microdroplets
No full textThis masters work describes the deformation of a droplet that is illuminated by a laser beam. The theory for linear fluid mechanical motion of the droplet is discussed. This is combined with Lorenz-Mie scattering. Droplet deformations resulting from the optical radiation pressure are computed.
Specific beam profiles are discussed in the literature for the purposes of optical droplet deformation, namely the cases of linear and circular polarized plane waves, a Gaussian beam and the Bessel beam. The general case of an arbitrary beam is not, to the author's knowledge, given in the published literature. Such a framework is developed from first principles and presented in this work.
A Mathematica script was written to compute deformations. These are calculated, fitting nicely to those found by Ellingsen in a recent article. The case of two plane waves from opposite directions is discussed here for the first time, and droplet shapes produced.
The size droplets considered is between the geometrical limit and the Rayleigh limit. The numerical load increases quickly with increasing values of the droplet radius and wave vector. The hardest to compute coefficients die off the most quickly with time
Replication Data for: Sub-surface turbulence or non-breaking capillary waves: which dominates air-water gas transfer?
No full textThis dataset contains the data necessary to reproduce the plots in the publication 'Sub-surface turbulence or non-breaking capillary waves: which dominates air-water gas transfer?'. It contains results obtained by particle image velocimetry, planar laser-induced fluorescence, and O2 concentration measurements for flows with different combinations of sub-surface turbulence and surface wave properties
Dispersive Wave Focusing on a Shear Current: Part 2—Nonlinear Effects
No full textContinuing our recent work [Ellingsen et al., Water Waves (2024)] we investigate the influence of vertically sheared currents on the surface elevation as well as the kinematics of dispersively focusing wave groups up to second order in steepness. The groups are assumed long crested in deep water which may travel at oblique angles with the current, which has a depth-dependent profile in both magnitude and direction. A strong but realistic shear current affects the wave surface elevation only slightly but the wave-induced horizontal velocity beneath the point of focus is very significantly affected, and new phenomena occur at second order. Firstly, a shear current causes wave-induced superharmonic velocity to be nonzero, contributing significantly for moderate wave steepness. At linear order, following (opposing) shear causes horizontal velocities to be amplified (reduced); for crest-focused wave groups, the superharmonic contribution reduces the influence of shear, whereas for trough-focused waves the velocity change from linear and second-order waves add, causing a substantially larger shear-induced effect. Secondly, the sub-harmonic mean flow is not strictly a return flow, but can follow the direction of wave propagation at the depths nearest the surface. Thirdly, unlike the case without shear where the subharmonic mean flow vanishes in the limit of zero bandwidth, it can now tend to a finite value in the narrowband limit. The criterion for this to happen is that the shear current has nonzero curvature.publishedVersio
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