arXiv subjects

Simen Å. Ellingsen

Publications and source records attributed to Simen Å. Ellingsen.

At least 19 recordsLinked to original sources

arXiv2026

Mapping surface height dynamics to subsurface flow physics in free-surface turbulent flow using a shallow recurrent decoder

Near-surface turbulent flows beneath a free surface are reconstructed from sparse measurements of the surface height variation, by a neural network algorithm known as the SHallow REcurrent Decoder (SHRED). The reconstruction of turbulent flow fields from limited, partial, or indirect measurements remains a grand challenge in science and engineering. The central goal in such applications is to leverage easy-to-measure proxy variables in order to estimate quantities which have not been, and perhaps cannot in practice be, measured. In the application considered here, the aim is to use a sparse number of surface height point measurements of a flow field, or drone video footage of surface features, in order to infer the turbulent flow field beneath the surface. SHRED is a deep learning architecture that learns a delay-coordinate embedding from a few surface height (point) sensors and maps it, via a shallow decoder trained in a compressed basis, to full subsurface fields, enabling fast, robust training from minimal data. We demonstrate the SHRED sensing architecture on two types of turbulent data from recent studies [Aarnes et al. J. Fluid Mech. 1007, A38 (2025) and Babiker et al. Phys. Rev. Fluids 11, 054802 (2026), respectively]: fully resolved DNS data and PIV laboratory data from a turbulent water tank. SHRED is capable of robustly mapping surface height fluctuations to full-state flow fields up to about one integral length scale deep, with as few as three surface measurements.

physics.flu-dyn↗

arXiv2026

Simultaneous free-surface profilometry and subsurface velocimetry with fringe projection and PIV

This work presents a novel combination of two well-established techniques: fringe projection profilometry (FPP) and particle image velocimetry (PIV). Despite seemingly conflicting requirements---FPP requires an opaque surface to project onto, while PIV requires a transparent fluid---both requirements are met by adding low concentrations (4--25 mg/L) of fluorescein dye to the water. This dye strongly absorbs the blue light projected onto the surface for FPP while remaining nearly transparent to the green PIV laser, achieving simultaneous opacity and transparency depending on wavelength. A set of three optical filters suppresses fluorescence-induced noise in the PIV images and specular reflections in the profilometry images, enabling clean simultaneous acquisition. Validated against point laser-induced fluorescence measurements of the surface, the method achieves a mean absolute error in surface elevation of 18 $μ$m at a dye concentration of 12 mg/L, above which further increases in concentration yield little improvement. PIV correlation values remain robust up to 20 mg/L. The technique resolves surface features differing in amplitude by two orders of magnitude and is demonstrated on flow past a cylinder interacting with surface waves and on droplet impacts on a quiescent surface.

physics.flu-dyn↗

arXiv2026

On the spatial statistics of free-surface turbulence and the complementarity of 'dimples' and 'scars'

The air--water interface governs the exchange of heat and gas between natural water bodies and their surrounding environment. Turbulence beneath the free surface imprints characteristic features: near-circular depressions (`dimples') and elongated indentations (`scars'). Recent studies have shown that these are linked to sub-surface flow features in a temporal sense. For instance, rapid increases in mean-square surface divergence due to upwelling events, precede dimple count surges. Yet the spatial structure of these connections remains unquantified. We employ spatial statistics to consider the spatial and temporal correlations between dimples and scars and two key velocity-derived fields, surface divergence $β=\partial_x u+\partial_y v$ and vertical vorticity $ω=\partial_x v-\partial_y u$. The dimples and scars are modelled as inhomogeneous Poisson point-processes, with intensity fields driven by the local variance of $β$ and $ω$. Parameters, including spatial support radius $r$ and time lag $τ$, are estimated by maximum likelihood against six DNS datasets, quantifying the spatial and temporal connection between dimples, scars, surface divergence and vorticity. Our results demonstrate a clear complementarity: Dimples show strong local connection to the vertical vorticity field but has weak spatial connection with surface divergence and a spatially ``global'' model is required for dimples to work as estimators of surface divergence; scars, in a similar but opposite manner, couple locally to surface divergence but globally to the vertical vorticity. The complementarity sheds new light on the way dimples and scars may be used to infer fluxes across the surface, e.g., in remote sensing contexts.

physics.flu-dyn↗

arXiv2026

Experimental investigation relating free-surface features to sub-surface turbulence

Turbulent flows beneath a free surface play a central role in the Earth system, yet their coupling to observable surface features remains incompletely understood. Recent studies using Direct Numerical Simulations (DNS) have reported strong correlation between observable surface features and surface divergence as well as velocity statistics directly beneath, but were limited to Reynolds numbers ($Re$) far below those typical of natural flows, and do not carry the inherent challenges of measurement and flow fidelity that real flows present. We present a laboratory study in which free-surface topology and sub-surface turbulent velocity are measured simultaneously in a jet-stirred tank, extending these numerical results to the physical domain. Using a novel combination of particle-image velocimetry (PIV) and free-surface profilometry, we access $Re$ up to two orders of magnitude higher than in the DNS. A computer vision method developed for identifying turbulent imprints on the free surface is successfully applied to experimental data, enabling direct comparison with the DNS. The correlation between time series of mean-square surface divergence and surface features is found to persist as strongly at higher Reynolds numbers, despite the increased disparity of turbulent scales. Beyond the thin viscous layer, all surface-to-bulk correlations scale with the integral length scale across both experimental and numerical cases. The normalized cross-correlation between mean-square horizontal velocity divergence and surface area covered by structures decreases linearly with depth and remains significant even two integral scales beneath the surface, unlike point-to-point correlations which decay fast, illustrating how correlations are near-instantaneous but spatially non-local. These results demonstrate that visible surface features provide considerable... [truncated due to arXiv length constraint]

physics.flu-dyn↗

arXiv2026

Exact dispersion relation for linear surface waves on arbitrary vertical shear

We derive the formal solution to the dispersion relation for linear surface waves on a horizontal mean current with arbitrary vertical dependence. The problem is cast in a Green's function framework for the Rayleigh equation, neglecting viscosity but making no further approximations about the mean velocity profile. The solution is the dispersion relation in the form of a single, implicit equation relating -- and containing only -- the velocity profile, wave frequency, and wavenumber. By isolating curvature effects in a path-ordered exponential, we obtain a solution that serves as a natural starting point for systematic approximations. We demonstrate that our solution reduces to the expression found by Shrira (1993, J. Fluid Mech. 252, 565--584) in the deep-water limit, yields known asymptotic approximations, and recovers known analytical solutions in special cases.

physics.flu-dyn↗

arXiv2026

Turbulence-induced anti-Stokes flow: experiments and theory

We report experimental evidence of an Eulerian-mean flow, $\overline{u}(z)$, created by the interaction of surface waves and tailored ambient sub-surface turbulence, which partly cancels the Stokes drift, $u_s(z)$, and present supporting theory. Water-side turbulent velocity fields and Eulerian-mean flows were measured with particle image velocimetry before vs after the passage of a wave group, and with vs without the presence of regular waves. We compare different wavelengths, steepnesses and turbulent intensities. In all cases, a significant change in the Eulerian-mean current is observed, strongly focused near the surface, where it opposes the Stokes drift. The observations support the picture that when waves encounter ambient sub-surface turbulence, the flow undergoes a transition during which Eulerian-mean momentum is redistributed vertically (without changing the depth-integrated mass transport) until a new equilibrium state is reached, wherein the near-surface ratio between $|\mathrm{d}\overline{u}/\mathrm{d}z|$ and $|\mathrm{d}u_s/\mathrm{d} z|$ approximately equals the ratio between the streamwise and vertical Reynolds normal stresses. This accords with a simple statistical theory derived here and holds regardless of the absolute turbulence level, whereas stronger turbulence means faster growth of the Eulerian-mean current. We present a model based on Rapid Distortion Theory which describes the generation of the Eulerian-mean flow as a consequence of the action of the Stokes drift on the background turbulence. Predictions are in qualitative, and reasonable quantitative, agreement with experiments on wave groups, where equilibrium has not yet been reached. Our results could have substantial consequences for predicting the transport of water-borne material in the oceans.

physics.flu-dyn↗

arXiv2025

Rediscovering shallow water equations from experimental data

New data-driven methods have advanced the discovery of governing equations from observations, enabling parsimonious models for complex systems. Here, we 'rediscover' a shallow-water equation closely related to Korteweg--de Vries (KdV) using only video recordings of solitons in a simple flume. Two fundamentally different approaches -- weak-form sparse identification of nonlinear dynamics (WSINDy) and a novel Fourier-multiplier method -- recover the same PDE, demonstrating that the equation is inherent in the data and robust to the choice of method. Both identify the same terms with comparable magnitudes and errors. To validate the models, we solve the discovered equations forward in time and compare them with additional experimental cases that were not used in the discovery. Based on the results, we discuss absolute and cumulative errors, as well as the strengths and limitations of the two discovery approaches. Together, these results demonstrate the potential of equation discovery from everyday experiments ('GoPro physics') and highlight shallow-water waves as an ideal test bed for developing and benchmarking new methods.

physics.flu-dyn↗

arXiv2024

Vortex structures under dimples and scars in turbulent free-surface flows

Turbulence beneath a free surface leaves characteristic long-lived signatures on the surface, such as upwelling 'boils', near-circular 'dimples' and elongated 'scars', easily identifiable by eye, e.g., in riverine flows. In this paper, we use Direct Numerical Simulations to explore the connection between these surface signatures and the underlying vortical structures. We investigate dimples, known to be imprints of surface-attached vortices, and scars, which have yet to be extensively studied, by analysing the conditional probabilities that a point beneath a signature is within a vortex core as well as the inclination angles of sub-signature vorticity. The analysis shows that the likelihood of vortex presence beneath a dimple decreases from the surface down through the viscous and blockage layers in a near-Gaussian manner, influenced by the dimple's size and the bulk turbulence. When expressed as a function of depth over the Taylor microscale $λ_T$, this probability is independent of Reynolds and Weber number. Conversely, the probability of finding a vortex beneath a scar increases sharply from the surface to a peak at the edge of the viscous layer, at a depth of approximately $λ_T/4$. Distributions of vortical orientation also show a clear pattern: a strong preference for vertical alignment below dimples and an equally strong preference for horizontal alignment below scars. Our findings suggest that scars can be defined as imprints of horizontal vortices approximately a quarter of the Taylor microscale beneath the surface, analogous to how dimples can be defined as imprints of surface-attached vertical vortex tubes.

physics.flu-dyn↗

arXiv2024

The Effect of Horizontal Shear on Extracting Water Currents From Surface Wave Data

The dispersive motion of surface waves is now routinely used to remotely measure the currents close beneath the surface of oceans and other natural flows. The current manifests as wavelength-dependent Doppler shifts in the spatiotemporal wave spectrum, which is obtained by performing a Fourier transform of an observed wave field, a procedure which assumes that the current is horizontally uniform within the field of view. This assumption is frequently not satisfied. We here analyze the effects of the presence of horizontal shear in the velocity field, with special emphasis on biases: ``measured'' velocities which differ systematically in magnitude and/or direction compared to the average current within the field of view. We generate random synthetic wave data according to a spectrum prescribed at the domain center while varying the current and its gradient, and the mean direction $θ_0$ and angular spread of the waves' propagation direction, from which we extract Doppler shift velocities (DSVs) using one of the most common numerical schemes. Assuming currents vary by up to 10\% of a characteristic wave phase velocity across the field of view, we find that strong biases can occur for highly directional wave spectra, strongly dependent on $θ_0$ and the current direction relative to that of its magnitude gradient. We discuss how biases arise in the DSV extraction procedure and suggest practical steps to discover the presence of biases.

physics.ao-ph↗

arXiv2023

Biases from spectral leakage in remote sensing of near-surface currents

Remotely measuring subsurface water currents from imagery of the wave field has become a much-used technique. We study the biases and errors in such measurements due to spectral leakage, and suggest mitigating procedures. Deviations between peak values in the three-dimensional wave spectrum and the known dispersion relation in quiescent water are extracted and interpreted as current-induced Doppler shifts, from which the sub-surface current is inferred. The use of discrete Fourier transforms, however, introduces spectral leakage between nearby frequency bins. Analysing synthetically generated wave data adhering to realistic input spectra we show that although no current is in fact present, spurious currents can be "measured" which can amount to a significant fraction of the phase speed at the spectral peak. We analyse the effects of data tapering, method of Doppler shift extraction, limited wavenumber and frequency resolution, peakedness and angular width of the input spectrum, and average misalignment between waves and Doppler shift velocity direction. The narrower the input wave spectrum in frequency and/or direction, the greater the biases become. The use of a window function reduces the severity in nearly all cases, yet mitigates the effects of limited resolution more effectively in space than in time. When a current is present the absolute biases remain essentially unchanged, when waves and currents are roughly aligned tapering much alleviates the concomitant biases, whereas in the case of a cross-current, biases remain significant even for tapered data.

physics.ao-ph↗

arXiv2022

Current Mapping from the Wave Spectrum

In this chapter we review methods by which near--surface ocean currents can be measured remotely using images of the water surface, as obtained by X-band radar in particular. The presence of a current changes the dispersive behavior of surface waves, so our challenge is to solve the inverse problem: to infer the spatially-varying current from measurements of the wavy surface. We examine how remote sensing of currents is achieved in practice by analyzing the wave spectrum, as may be measured for example by X-band radar. A set of consecutive backscatter images recorded as a function of time is Fourier-transformed to produce the spectrum, which gives information concerning the propagation of waves whose dispersion is altered by currents. X-band radar images measure the wave field over multiple square kilometers, and analyzing various spatial subsets of the images allows a map of the spatial variation of the currents to be reconstructed. Several algorithms for obtaining empirical dispersion relations from the measured spectrum and extracting the currents are reviewed: the least squares and iterative least squares method, the normalized scalar product method, and the polar current shell method. We go on to describe how the same methods and algorithms can be extended to also allowing the depth-dependence of the current to be determined. Reasonable agreement between radar-derived currents and in situ measurements has been demonstrated in multiple field measurements. However, more validation is necessary especially in the context of depth-varying flows. Understanding the extent to which Stokes drift is measured as part of the radar-derived current is not well-understood yet potentially important.

physics.flu-dyn↗

arXiv2021

Designing vortices in pipe flow with topography-driven Langmuir circulation

We present direct numerical simulation of a mechanism for creating longitudinal vortices in pipe flow, compared with a simple model theory. By furnishing the pipe wall with a pattern of crossing waves secondary flow in the form of spanwise vortex pairs is created. The mechanism `CL1' is kinematic and known from oceanography as a driver of Langmuir circulation. CL1 is strongest when the `wall wave' vectors make an accute angle with the axis, $φ=10^\circ$ - $20^\circ$ (a `contracted eggcarton'), changes sign near $45^\circ$ and is weak and opposite beyond this angle. A competing, dynamic mechanism driving secondary flow in the opposite sense is also observed created by the azimuthally varying friction. Whereas at smaller angles `CL1' prevails, the dynamic effect dominates when $φ\gtrsim 45^\circ$ reversing the flow. Curiously, circulation strength is a faster-than-linearly increasing function of Reynolds number for the contracted case. We explore an analogy with Prandtl's secondary motion of the second kind in turbulence. A transport equation for average streamwise vorticity is derived, and we analyse it for three different crossing angles, $φ=18.6^\circ, 45^\circ$ and $60^\circ$. Mean-vorticity production is organised in a ring-like structure with the two rings contributing to rotating flow in opposite senses. For the larger $φ$ the inner ring decides the main swirling motion, whereas for $φ=18.6^\circ$ outer-ring production dominates. For the larger angles the outer ring is mainly driven by advection of vorticity and the inner by deformation (stretching) whereas for $φ=18.6^\circ$ both contribute approximately equally to production in the outer ring.

physics.flu-dyn↗

arXiv2020

Langmuir-type vortices in boundary layers driven by a criss-cross wavy wall topography

We investigate a mechanism to manipulate wall-bounded flows whereby wave-like undulations of the wall topography drives the creation of bespoke longitudinal vortices. A resonant interaction between the ambient vorticity of the undisturbed shear flow and the undulation of streamlines enforced by the wall topography serves to slightly rotate the spanwise vorticity of the mean flow into the streamwise direction, creating a swirling motion, in the form of regular streamwise rolls. The process is kinematic and essentially identical to the `direct drive' CL1 mechanism for Langmuir circulation (LC) proposed by Craik (1970). Boundary layers are modelled by selecting suitable primary flow profiles. A simple, easily integrable expression for the cross-plane stream function is found in two asymptotic regimes: the resonant onset of the essentially inviscid instability at early times, and the fully developed steady state viscous flow. Linear-order solutions for flow over undulating boundaries are obtained, fully analytical in the special case of a power-law profile. These solutions allow us to quickly map out the circulation response to boundary design parameters. The study is supplemented with direct numerical simulations which verify the manifestation of boundary induced Langmuir vortices in laminar flows with no-slip boundaries. Simulations show good qualitative agreement with theory. Quantitatively, the comparisons rest on a displacement length closure parameter adopted in the perturbation theory. While wall-driven LC appear to become unstable in turbulent flows, we propose that the mechanism can promote swirling motion in boundary layers, a flow feature which has been reported to reduce drag in some situations.

physics.flu-dyn↗

arXiv2019

An improved method for determining near-surface currents from wave dispersion measurements

A new inversion method for determining near-surface shear currents from a measured wave spectrum is introduced. The method is straightforward to implement and starts from the existing state-of-the-art technique of assigning effective depths to measured wavenumber-dependent Doppler shift velocities. A polynomial fit is performed, with the coefficients scaled based on a simple derived relation to produce a current profile that is an improved estimate of the true profile. The method involves no user-input parameters, with the optimal parameters involved in the polynomial fit being chosen based on a simple criterion involving the measured Doppler shift data only. The method is tested on experimental data obtained from a laboratory where current profiles of variable depth dependence could be created and measured by particle image velocimetry, which served as "truth" measurements. Applying the new inversion method to experimentally measured Doppler shifts resulted in a $>3\times$ improvement in accuracy relative to the state-of-the-art for current profiles with significant near-surface curvature. The experiments are dynamically similar to typical oceanographic flows such as wind-drift profiles and our laboratory thus makes a suitable and eminently useful scale model of the real-life setting. Our results show that the new method can achieve improved accuracy in reconstructing near-surface shear profiles from wave measurements by a simple extension of methods which are currently in use, incurring little extra complexity and effort. A novel adaptation of the normalized scalar product method has been implemented, able to extract Doppler shift velocities as a function of wavenumber from the measured wave spectrum.

physics.ao-ph↗

arXiv2019

Observation of surface wave patterns modified by sub-surface shear currents

We report experimental observations of two canonical surface wave patterns --- ship waves and ring waves --- skewed by sub-surface shear, thus confirming effects predicted by recent theory. Observed ring waves on a still surface with sub-surface shear current are strikingly asymmetric, an effect of strongly anisotropic wave dispersion. Ship waves for motion across a sub--surface current on a still surface exhibit striking asymmetry about the ship's line of motion, and large differences in wake angle and transverse wavelength for upstream vs downstream motion are demonstrated, all of which in good agreement with theoretical predictions. Neither of these phenomena can occur on a depth-uniform current. A quantitative comparison of measured vs predicted average phase shift for a ring wave is grossly mispredicted by no-shear theory, but in good agreement with predictions for the measured shear current. A clear difference in wave frequency within the ring wave packet is observed in the upstream vs downstream direction for all shear flows, while it conforms with theory for quiescent water for propagation normal to the shear current, as expected. Peak values of the measured 2-dimensional Fourier spectrum for ship waves are shown to agree well with the predicted criterion of stationary ship waves, with the exception of some cases where results are imperfect due to the limited wave-number resolution, transient effects and/or experimental noise. Experiments were performed on controlled shear currents created in two different ways, with a curved mesh, and beneath a blocked stagnant-surface flow. Velocity profiles were measured with particle image velocimetry, and surface waves with a synthetic schlieren method. Our observations lend strong empirical support to recent predictions that wave forces on vessels and structures can be greatly affected by shear in estuarine and tidal waters.

physics.flu-dyn↗

arXiv2019

A framework for modelling linear surface waves on shear currents in slowly varying waters

We present a theoretical and numerical framework -- which we dub the Direct Integration Method (DIM) -- for simple, efficient and accurate evaluation of surface wave models allowing presence of a current of arbitrary depth dependence, and where bathymetry and ambient currents may vary slowly in horizontal directions. On horizontally constant water depth and shear current the DIM numerically evaluates the dispersion relation of linear surface waves to arbitrary accuracy, and we argue that for this purpose it is superior to two existing numerical procedures: the piecewise-linear approximation and a method due to \textit{Dong \& Kirby} [2012]. The DIM moreover yields the full linearized flow field at little extra cost. We implement the DIM numerically with iterations of standard numerical methods. The wide applicability of the DIM in an oceanographic setting in four aspects is shown. Firstly, we show how the DIM allows practical implementation of the wave action conservation equation recently derived by \textit{Quinn et al.} [2017]. Secondly, we demonstrate how the DIM handles with ease cases where existing methods struggle, i.e.\ velocity profiles $\mathbf{U}(z)$ changing direction with vertical coordinate $z$, and strongly sheared profiles. Thirdly, we use the DIM to calculate and analyse the full linear flow field beneath a 2D ring wave upon a near--surface wind--driven exponential shear current, revealing striking qualitative differences compared to no shear. Finally we demonstrate that the DIM can be a real competitor to analytical dispersion relation approximations such as that of \textit{Kirby \& Chen} [1989] even for wave/ocean modelling.

physics.flu-dyn↗

arXiv2018

Weakly non-linear transient waves on a shear current: Ring waves and skewed Langmuir rolls

We investigate the weakly nonlinear dynamics of transient gravity waves at infinite depth under the influence of a shear current varying linearly with depth. An analytical solution is permitted via integration of the Euler equations. Although similar problems were investigated in the 1960's and 70's for special cases of resonance, this is to our knowledge the first general wave interaction (mode coupling) solution derived to second order with a shear current present. Wave interactions are integrable in a spectral convolution to yield the second order dynamics of initial value problems. To second order, irrotational wave dynamics interacts with the background vorticity field in a way that creates new vortex structures. A notable example is the large parallel vortices which drive Langmuir circulation as oblique plane waves interact with an ocean current. We also investigate the effect on wave pairs which are misaligned with the shear current. In contrast to a conjecture by Leibovich (1983) we find similar, but skewed, vortex structures in every case except when the mean wave direction is perpendicular to the direction of the current. Similar nonlinear wave-shear interactions are found to also generate near-field vortex structures in the Cauchy-Poisson problem with an initial surface elevation. These interactions create further groups of dispersive ring waves in addition to those present in linear theory. The second order solution is derived in a general manner which accommodates any initial condition through mode coupling over a continuous wave spectrum. It is therefore applicable to a range of problems including special cases of resonance. As a by--product of the general theory, a simple expression for the Stokes drift due to a monochromatic wave propagating at oblique angle with a current of uniform vorticity is derived, for the first time to our knowledge.

physics.flu-dyn↗

arXiv2017

Approximate dispersion relations for waves on arbitrary shear flows

An approximate dispersion relation is derived and presented for linear surface waves atop a shear current whose magnitude and direction can vary arbitrarily with depth. The approximation, derived to first order of deviation from potential flow, is shown to produce good approximations at all wavelengths for a wide range of naturally occuring shear flows as well as widely used model flows. The relation reduces in many cases to a 3D generalization of the much used approximation by Skop [1987], developed further by Kirby & Chen [1989], but is shown to be more robust, succeeding in situations where the Kirby & Chen model fails. The two approximations incur the same numerical cost and difficulty. While the Kirby & Chen approximation is excellent for a wide range of currents, the exact criteria for its applicability have not been known. We explain the apparently serendipitous success of the latter and derive proper conditions of applicability for both approximate dispersion relations. Our new model has a greater range of applicability. A second order approximation is also derived. It greatly improves accuracy, which is shown to be important in difficult cases. It has an advantage over the corresponding 2nd order expression proposed by Kirby \& Chen that its criterion of accuracy is explicitly known, which is not currently the case for the latter to our knowledge. Our 2nd order term is also arguably significantly simpler to implement, and more physically transparent, than its sibling due to Kirby & Chen.

physics.flu-dyn↗