Time-dependent Velocity Field Research Articles

Superconvergence of Arbitrary Lagrangian--Eulerian Discontinuous Galerkin Methods for Linear Hyperbolic Equations

In this paper, we study the superconvergence properties of an arbitrary Lagrangian--Eulerian discontinuous Galerkin (ALE-DG) method with approximations to one-dimensional linear hyperbolic equations. The ALE-DG method is a mesh moving method; we need to deal with the new difficulties brought by the time-dependent test function space and grid velocity field. Since the time derivative cannot commute with the space projections for the ALE-DG method, we will introduce the material derivative in our analysis. With the help of the scaling argument and material derivative, we build a special interpolation function by constructing the correction functions and prove that the numerical solution is superclose to the interpolation function in the $L^2$-norm. The order of the superconvergence is 2k+1 when piecewise polynomials of degree at most $k$ are used. We also rigorously prove a (2k+1)th-order superconvergence rate for the domain and cell average and at the downwind points in the maximal and average norm. Furthermore, we prove that the function value approximation is superconvergent with a rate $k+2$ at the right Radau points and a superconvergence rate $k+1$ for the derivative approximation at all interior left Radau points. All theoretical findings are confirmed by numerical experiments.

Properties of Sediment Trap Catchment Areas in Fram Strait: Results From Lagrangian Modeling and Remote Sensing

Vertical particle fluxes are responsible for the transport of carbon and biogenic material from the surface to the deep ocean, hence understanding these particle fluxes is of climate relevance. Sediment traps operated in Fram Strait in the framework of the Arctic long-term observatory FRAM provide an estimate of vertical particle fluxes in a region of high CO2 uptake. Until now the source area (catchment area) of trapped particles is unclear; however, lateral advection of particles is supposed to play an important role. This study presents a Lagrangian method to backtrack the origin of particles for two Fram Strait moorings equipped with sediment traps in 200 m and 2300 m depth by using the time-dependent velocity field of a high-resolution, eddy-resolving ocean-sea ice model. Our study shows that the extent of the catchment area is larger the deeper the trap and the slower the settling velocity. Chlorophyll-a concentration as well as sea ice coverage of the catchment area are highest in the summer months. The high sea ice coverage in summer compared to winter can possibly be related to a weaker across-strait sea level pressure difference, which allows more sea ice to enter the then well-stratified central Fram Strait where moorings are located. Hence, highest vertical particle fluxes may be expected in late summer and autumn. Furthermore, a backward sea ice tracking approach shows that the origin and age of sea ice drifting through Fram Strait, partly responsible for vertical particle fluxes, varies strongly from year to year, pointing to a high variability in the composition of particles trapped in the moorings.

Low-Order Velocity Estimation and Acoustic Noise Generation by a Turbulent Wall Jet

A method for estimating the velocity fluctuations, hydrodynamic pressure fluctuations, and far-field acoustic pressure in a planar turbulent wall jet is examined. The Reynolds number based on nozzle height and exit velocity is 25,500. Streamwise-aligned, two-component particle image velocimetry and surface pressure measurements are obtained synchronously along the center line of the wall jet. Modified stochastic estimation is used to estimate a low-dimensional representation of the time-dependent velocity field, and performance metrics for the evaluation of the estimates are presented. The use of proper orthogonal decomposition in the estimation is evaluated against the estimation of individual velocity vectors from the measurements to investigate the filtering effect of using a subset of proper orthogonal decomposition modes in the velocity reconstruction. The velocity estimates are used to solve Poisson’s equation for the time-dependent hydrodynamic pressure fluctuations. Curle’s acoustic analogy is evaluated using the estimated velocity and pressure and compared with far-field acoustic measurements via a phased array of microphones. The estimated acoustic spectrum scaled by spanwise integration width is comparable to scaled measurements of the acoustic spectrum. The estimation method highlights the importance of the interaction between the pressure and velocity fields to the acoustic generation process.

Time-Dependent Flow seen through Approximate Observer Killing Fields.

Flow fields are usually visualized relative to a global observer, i.e., a single frame of reference. However, often no global frame can depict all flow features equally well. Likewise, objective criteria for detecting features such as vortices often use either a global reference frame, or compute a separate frame for each point in space and time. We propose the first general framework that enables choosing a smooth trade-off between these two extremes. Using global optimization to minimize specific differential geometric properties, we compute a time-dependent observer velocity field that describes the motion of a continuous field of observers adapted to the input flow. This requires developing the novel notion of an observed time derivative. While individual observers are restricted to rigid motions, overall we compute an approximate Killing field, corresponding to almost-rigid motion. This enables continuous transitions between different observers. Instead of focusing only on flow features, we furthermore develop a novel general notion of visualizing how all observers jointly perceive the input field. This in fact requires introducing the concept of an observation time, with respect to which a visualization is computed. We develop the corresponding notions of observed stream, path, streak, and time lines. For efficiency, these characteristic curves can be computed using standard approaches, by first transforming the input field accordingly. Finally, we prove that the input flow perceived by the observer field is objective. This makes derived flow features, such as vortices, objective as well.

Saturated impulse for fully clamped square plates under blast loading

Saturated impulse refers to the critical value, beyond which the deflection of a beam or plate under pulse loading will no longer increase with further applied load. The present paper investigates the saturated impulse of fully clamped square plates subjected to a blast loading that is assumed to be a uniformly distributed Linearly Rising Exponentially Decaying (LRED) pressure pulse. Considering both bending moment and membrane force, the transient dynamic plastic response is predicted by a rigid, perfectly plastic (R-PP) model characterized by travelling plastic hinge lines and time-dependent velocity field. The saturated duration, saturated impulse, saturated deflection, as well as the evolution of deformation mechanism are explored. By comparing three characteristic durations, i.e. the travelling duration of the plastic hinge lines, saturated duration and loading duration, the dynamic responses of a square plate are classified into three types such that a regime map on the loading parameters’ plane is constructed accordingly. The correlation is made between the results obtained from the elastic-plastic numerical simulation and the R-PP theoretical approximation. To facilitate engineering designs, a method of replacing the LRED pressure pulse by an equivalent rectangular pressure pulse is proposed to predict the maximum deflection of fully clamped square plates.

A combined DEM & FEM approach for modelling roll compaction process

Roll compaction is a continuous manufacturing process aiming to produce particulate granules from powders. A roll press typically consists of a screw feeding system, two rolls and a side sealing. Despite its conceptual simplicity, numerical modelling of the process is challenging due to the complexity involving two different mechanisms: feeding by the screw and powder compaction between the rolls.To represent the materials' behaviour both in the feeding zone and in the compaction area, a combined three-dimensional Discrete Elements Method (DEM) and Finite Elements Method (FEM) is developed in this work. The DEM, which is a more suitable method to describe the flow of granular material, is used to model the motion of particles in the feeding zone. As the granular material deforms under high pressure between rolls, FEM offers a more versatile approach to represent the powder behaviour and frictional conditions. In the proposed approach the DEM and FEM are treated as complementary methods, enabling us to take advantages of the strengths of both.In this proposed approach, the time dependent velocity field of the particles at the end of the screw feeder is evaluated as a continuous field using the coarse graining (CG) framework, which was used as input data for the FEM model. FEM is then used to simulate the powder compaction in between the rolls, and the resultant roll pressure and ribbon relative density are obtained.Our results show a direct correlation between the particle velocity driven by the screw conveyor and the roll pressure, both oscillating with the same period. This translates into an anisotropic ribbon with a density profile varying sinusoidally along its width, with a period equal to the duration of a screw turn.

Quantitative visualization of temperature field and measurement of local heat transfer coefficient over heat exchanger elements in sinusoidal oscillating flow

The flow and heat transfer in the near field of simplified heat exchanger elements (two cylinders in a tandem arrangement and a longitudinal finned tube) immersed in oscillatory flow were experimentally investigated. The effects of cylinder spacing on heat transfer in the tandem arrangement were examined. The ambient flow was a sinusoidal planar oscillating flow. A synchronous PIV-PLIF (Particle Image Velocimetry- Planar Laser Induced Fluorescence) technique was used to simultaneously capture the time-dependent velocity and temperature fields. In the tandem cylinder configuration, a higher temperature drop in the gap region between the cylinders was observed for the larger cylinder spacing, which yielded to a higher heat transfer rate. For the finned tube configuration, both the velocity and temperature fields featured two recirculation zones at the bases of the attached longitudinal fins. These recirculation bubbles contained flow particles with low velocity and high temperature which led to a lower heat transfer rate in these zones. The heat transfer characteristics of heat exchanger elements in oscillating flows were found to be significantly different from those in steady flows. The results were compared to the heat transfer from a single circular cylinder under identical conditions.

A LAGRANGIAN GAUSS-NEWTON-KRYLOV SOLVER FOR MASS- AND INTENSITY-PRESERVING DIFFEOMORPHIC IMAGE REGISTRATION.

We present an efficient solver for diffeomorphic image registration problems in the framework of Large Deformations Diffeomorphic Metric Mappings (LDDMM). We use an optimal control formulation, in which the velocity field of a hyperbolic PDE needs to be found such that the distance between the final state of the system (the transformed/transported template image) and the observation (the reference image) is minimized. Our solver supports both stationary and non-stationary (i.e., transient or time-dependent) velocity fields. As transformation models, we consider both the transport equation (assuming intensities are preserved during the deformation) and the continuity equation (assuming mass-preservation). We consider the reduced form of the optimal control problem and solve the resulting unconstrained optimization problem using a discretize-then-optimize approach. A key contribution is the elimination of the PDE constraint using a Lagrangian hyperbolic PDE solver. Lagrangian methods rely on the concept of characteristic curves. We approximate these curves using a fourth-order Runge-Kutta method. We also present an efficient algorithm for computing the derivatives of the final state of the system with respect to the velocity field. This allows us to use fast Gauss-Newton based methods. We present quickly converging iterative linear solvers using spectral preconditioners that render the overall optimization efficient and scalable. Our method is embedded into the image registration framework FAIR and, thus, supports the most commonly used similarity measures and regularization functionals. We demonstrate the potential of our new approach using several synthetic and real world test problems with up to 14.7 million degrees of freedom.

Spatial resolution and velocity field improvement of 4D-flow MRI.

4D-flow MRI obtains a time-dependent 3D velocity field; however, its use for the calculation of higher-order parameters is limited by noise. We present an algorithm for denoising 4D-flow data. By integrating a velocity field and eliminating streamlines in noisy flow, depicted by high curvature, a denoised dataset may be extracted. This method, defined as the velocity field improvement (VFIT) algorithm, was validated in an analytical dataset and using in vivo data in comparison with a computation fluid dynamics (CFD) simulation. As a proof of principal, wall shear stress (WSS) measurements in the descending aorta were compared with those defined by CFD. The VFIT algorithm achieved a >100% noise reduction of a corrupted analytical dataset. In addition, 4D-flow data were cleaned to show improved spatial resolution and near wall velocity representation. WSS measures compared well with CFD data and bulk flow dynamics were retained (<2% difference in flow measurements). This study presents a method for denoising 4D-flow datasets with improved spatial resolution. Bulk flow dynamics are accurately conserved while velocity and velocity gradient fields are improved; this is important in the calculation of higher-order parameters such as WSS, which are shown to be more comparable to CFD measures. Magn Reson Med 78:1959-1968, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

INFERRING THE MAGNETOHYDRODYNAMIC STRUCTURE OF SOLAR FLARE SUPRA-ARCADE PLASMAS FROM A DATA-ASSIMILATED FIELD TRANSPORT MODEL

ABSTRACT Supra-arcade fans are highly dynamic structures that form in the region above post-reconnection flare arcades. In these features the plasma density and temperature evolve on the scale of a few seconds, despite the much slower dynamics of the underlying arcade. Further, the motion of supra-arcade plasma plumes appears to be inconsistent with the low-beta conditions that are often assumed to exist in the solar corona. In order to understand the nature of these highly debated structures, it is, therefore, important to investigate the interplay of the magnetic field with the plasma. Here we present a technique for inferring the underlying magnetohydrodynamic processes that might lead to the types of motions seen in supra-arcade structures. Taking as a case study the 2011 October 22 event, we begin with extreme-ultraviolet observations and develop a time-dependent velocity field that is consistent with both continuity and local correlation tracking. We then assimilate this velocity field into a simplified magnetohydrodynamic simulation, which deals simultaneously with regions of high and low signal-to-noise ratio, thereby allowing the magnetic field to evolve self-consistently with the fluid. Ultimately, we extract the missing contributions from the momentum equation in order to estimate the relative strength of the various forcing terms. In this way we are able to make estimates of the plasma beta, as well as predict the spectral character and total power of Alfvén waves radiated from the supra-arcade region.

Theoretical study of time-dependent, ultrasound-induced acoustic streaming in microchannels.

Based on first- and second-order perturbation theory, we present a numerical study of the temporal buildup and decay of unsteady acoustic fields and acoustic streaming flows actuated by vibrating walls in the transverse cross-sectional plane of a long straight microchannel under adiabatic conditions and assuming temperature-independent material parameters. The unsteady streaming flow is obtained by averaging the time-dependent velocity field over one oscillation period, and as time increases, it is shown to converge towards the well-known steady time-averaged solution calculated in the frequency domain. Scaling analysis reveals that the acoustic resonance builds up much faster than the acoustic streaming, implying that the radiation force may dominate over the drag force from streaming even for small particles. However, our numerical time-dependent analysis indicates that pulsed actuation does not reduce streaming significantly due to its slow decay. Our analysis also shows that for an acoustic resonance with a quality factor Q, the amplitude of the oscillating second-order velocity component is Q times larger than the usual second-order steady time-averaged velocity component. Consequently, the well-known criterion v(1)≪c(s) for the validity of the perturbation expansion is replaced by the more restrictive criterion v(1)≪c(s)/Q. Our numerical model is available as supplemental material in the form of comsol model files and matlab scripts.

Time-dependent flow velocity measurement using two-dimensional color Doppler flow imaging and evaluation by Hagen-Poiseuille equation.

This paper aims to develop a technique to assess velocity flow profile and wall shear stress (WSS) spatial distribution across a vessel phantom representing an artery. Upon confirming the reliability of the technique, it was then used on a set of carotid arteries from a cohort of human subjects. We implemented color Doppler flow imaging (CDFI) for measurement of velocity profile in the artery cross section. Two dimensional instantaneous and time-dependent flow velocity and WSS vector fields were measured and their waveforms of peak velocities based on the technique were compared with WSS values generated by Hagen-Poiseuille equation. Seventy-five patients with intima-media thickening were prospectively enrolled and were divided into an IMT group. At the same time, another 75 healthy volunteers were enrolled as the control group. All the subjects were scanned and the DICOM files were imported into our in-house program. Next, we determine the velocity profile of carotid arteries in a set of 150 human subjects and compared them again. The peak velocities by the CDFI and Hagen-Poiseuille equation techniques were compared and statistically evaluated. The amounts of deviation for the two measured WSS profiles were performed and we demonstrated that they are not significantly different. At two different flow settings with peak flow velocity of 0.1, 0.5 (×10(-11)) m/s, the obtained WSS were 0.021 ± 0.04, 0.038 ± 0.05 m/s, respectively. For the patient population study, the mean WSS value calculated by Hagen-Poiseuille equation was 2.98 ± 0.15 dyne/cm(2), while it was 2.31 ± 0.14 dyne/cm(2) by our CDFI analysis program. The difference was not statistically significant (t = -1.057, P = 0.259). Similar to the Hagen-Poiseuille equation, a negative linear correlation was also found between the calculated WSS and intima-media thickness (P = 0.000). Using CDFI analysis, we found that the WSS distribution at the middle of the proximal plaque shoulder was larger than the top of the shoulder. CDFI can assess the velocity and WSS profile accurately and efficiently and may be used for clinical diagnosis of cardiovascular conditions.

Active region upflows

Context. Observations of many active regions show a slow systematic outflow/upflow from their edges lasting from hours to days. At present no physical explanation has been proven, while several suggestions have been put forward. Aims. This paper investigates one possible method for maintaining these upflows assuming that convective motions drive the magnetic field to initiate them through magnetic reconnection. Methods. We use Helioseismic and Magnetic Imager (HMI) data to provide an initial potential three dimensional magnetic field of the active region NOAA 11123 on 2010 November 13 where the characteristic upflow velocities are observed. A simple one-dimensional hydrostatic atmospheric model covering the region from the photosphere to the corona is derived. Local Correlation Tracking of the magnetic features in the HMI data is used to derive a proxy for the time dependent velocity field. The time dependent evolution of the system is solved using a resistive three-dimensional MagnetoHydro-Dynamic code. Results. The magnetic field contains several null points located well above the photosphere, with their fan planes dividing the magnetic field into independent open and closed flux domains. The stressing of the interfaces between the different flux domains is expected to provide locations where magnetic reconnection can take place and drive systematic flows. In this case, the region between the closed and open flux is identified as the region where observations find the systematic upflows. Conclusions. In the present experiment, the driving only initiates magneto-acoustic waves, without driving any systematic upflows at any of the flux interfaces.

Reply to comment by Behzad Ataie‐Ashtiani on “Effects of tidal fluctuations on mixing and spreading in coastal aquifers: Homogeneous case”

We would like to thank Prof. Behzad Ataie-Ashtiani for his explanatory comment on our paper [Ataie-Ashtiani, 2015]. He notes that in confined aquifers with a vertical seaside boundary, a tidal overheight does not develop. This is in contrast to an unconfined aquifer for which a tidal overheight may develop when the tidal amplitude is significant relative to the saturated thickness, in particular in the presences of a sloping land surface. As Prof. Ataie-Ashtiani points out, the absence of a tidal overheight in the case of a confined aquifer makes that the difference in the model outcomes for a specified head and a specified flux at the inland boundary will be less pronounced than for an unconfined aquifer. Whilst we agree with this observation in principle, we do wish to add that Prof. Ataie-Ashtiani's assertion that our results are applicable to confined aquifers only may be perhaps too restrictive. We have in fact shown that the effect of tides on the development of the intruded wedge and the mixing zone depends critically on the aquifer storativity, with the highest values of this parameter having the greatest impact on the simulated salinity distribution. We attributed this to the fact that the interaction between the tidally driven circulation and local dispersion related to the spatially nonuniform time-dependent velocity field that strongly influences the mixing behavior. There is no tidally induced mixing and spreading when the flow response to the hydraulic perturbation is instantaneous. As the storativity of unconfined aquifers is much larger than that of confined aquifers, our findings have important implications for the modeling of unconfined systems. It is indeed true that our results are not directly transferable to aquifers in which a tidal overheight develops, or where complex flow cells due to a sloping beach develop, but indeed our models can be approximative of unconfined aquifers with a near-constant transmissivity. Flow conditions similar to those that we considered can potentially exist in unconfined aquifers that have a large saturated thickness relative to the tidal amplitude, or for those with an inland extent far enough so that small-scale complications arising from the beach morphology can be considered negligible. We would further like to point out here that while the differences between models with a specified head or specified freshwater flux at the inland boundary were small enough to make that our general findings and conclusions were independent of the inland boundary condition, differences could be observed. Figure 1 shows the numerical results for the salt mass fraction distribution and the diagnostics defined in Pool et al. [2014], considering a prescribed flux and a prescribed head along the inland boundary and for different storativity values. In the three tested scenarios, the horizontal component of the center of mass for the prescribed head models penetrates further inland than those to the prescribed flux (Figure 1c, top). Moreover, based on visual inspection of the salinity contours, a constant head landward boundary condition slightly increases the effects of tidal fluctuations on the interface, which is consistent with the results described by Ataie-Ashtiani et al. [1999]. However, although small differences are observed between the results, the salt mass fraction distributions for the inland prescribed flux models are quite similar to that obtained from the prescribed head models (Figure 1a) and the diagnosis follow an identical trend as a function of the tidal mixing number (Figures 1b and 1c). (a) Normalized salinity contours (25, 50, and 75%) for the numerical models under tidal conditions imposing the head (dashed lines) and a freshwater flux (solid lines) at the landward boundary for three different values of the storativity. (b) Numerical results for the width of the mixing zone, longitudinal mixing length, and (c) the components of the center of mass with respect to the nontidal situation, in terms of the dimensionless constant . Note that where the open circle is not visible it is overprinted by the solid circle. This difference in behavior is attributed to the fact that the freshwater inflow in the case of a prescribed head boundary is variable with time, whereas it is constant with time if the flux is specified. This effect is expected to be more pronounced for studies where the inland boundary is closer to the coast, and when the effect of tidal fluctuations on the horizontal head gradient increases. As such, and in line with Prof. Ataie-Ashtiani's observations, there is further scope for investigations of smaller-scale systems, both confined and unconfined, but these were not the focus of our Pool et al. [2014] study. The data used in this paper can be obtained upon request from the corresponding author.

Geo-acoustic inversion of marine sediments using vector sensor measurements

In this paper, a narrow-band geo-acoustic inversion scheme based on the acoustic vector field is presented. Measurements of acoustic particle velocity made with a vector sensor and short line array of hydrophones are studied with regards to the interference pattern generated by single 1–4 kHz source. This source was lowered from a research vessel positioned within three water depths (depth 20-m) of the receivers, and projected a series of 100-ms long continuous wave (cw) pulses as it was raised toward the surface. The interference pattern measured at the receivers is assessed by a non-dimensional property of the acoustic particle velocity, the degree of circularity [Dall’Osto and Dahl, J. Acoust. Soc. Am. 134, 109 (2013)], computed directly at the vector sensor and approximated along the line-array. This degree of circularity is highly sensitive to the phase of interfering multipaths, which depends on both the source-receiver geometry and the geo-acoustic properties of sea-bed. Source positions measured by GPS and depth sensors are used to model the time-dependent particle velocity field as the cw tone establishes a steady-state response. A best-fit model to the field data provides an estimate of the geo-acoustic properties of the sandy sediment at the experimental site.

Scale estimation for turbulent flows in porous media

The results of an experimental study to determine the smallest, dissipative scales of the turbulent flow in a randomly packed porous bed are presented and discussed. Particle Image Velocimetry was used to obtain time dependent two-dimensional velocity fields within the bed, which were then used to obtain turbulent statistical measures within the flow field. Results are presented for representative pores for a range of pore Reynolds numbers, Repore, from 839 to 3964, which include integral length and velocity scales and estimates of the Kolmogorov scales of length, velocity and time. The results show an asymptotic region at sufficiently high Repore, on the order of 2800, where scaling is normalized using the bed pore scales. It is also shown that the turbulent Reynolds number for this flow in the asymptotic limit is on the order of 0.07Repore, for the range of pores studied.

Pulsatile flow characterization in a vessel phantom with elastic wall using ultrasonic particle image velocimetry technique: the impact of vessel stiffness on flow dynamics.

This study aims to experimentally investigate the impact of vessel stiffness on the flow dynamics of pulsatile vascular flow. Vessel phantoms with elastic walls were fabricated using polyvinyl alcohol cryogel to result in stiffness ranging from 60.9 to 310.3 kPa and tested with pulsatile flows using a flow circulation set-up. Two-dimensional instantaneous and time-dependent flow velocity and shear rate vector fields were measured using ultrasonic particle image velocimetry (EchoPIV). The waveforms of peak velocities measured by EchoPIV were compared with the ultrasonic pulse Doppler spectrum, and the measuring accuracy was validated. The cyclic vessel wall motion and flow pressure were obtained as well. The results showed that vessel stiffening influenced the waveforms resulting from vessel wall distension and flow pressure, and the fields of flow velocity and shear rate. The stiffer vessel had smaller inner diameter variation, larger pulse pressure and median pressure. The velocity and shear rate maximized at peak systole for all vessels. The results showed a decrease in wall shear stress for a stiffer vessel, which can initiate the atherosclerotic process. Our study elucidates the impact of vessel stiffness on several flow dynamic parameters, and also demonstrates the EchoPIV technique to be a useful and powerful tool in cardiovascular research.

Irreducible Representations of Oscillatory and Swirling Flows in Active Soft Matter

Recent experiments imaging fluid flow around swimming microorganisms have revealed complex time-dependent velocity fields that differ qualitatively from the stresslet flow commonly employed in theoretical descriptions of active matter. Here we obtain the most general flow around a finite sized active particle by expanding the surface stress in irreducible Cartesian tensors. This expansion, whose first term is the stresslet, must include, respectively, third-rank polar and axial tensors to minimally capture crucial features of the active oscillatory flow around translating Chlamydomonas and the active swirling flow around rotating Volvox. The representation provides explicit expressions for the irreducible symmetric, antisymmetric, and isotropic parts of the continuum active stress. Antisymmetric active stresses do not conserve orbital angular momentum and our work thus shows that spin angular momentum is necessary to restore angular momentum conservation in continuum hydrodynamic descriptions of active soft matter.

Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and &amp;quot;Nearly Invariant&amp;quot; tori

Abstract. In this paper we consider fluid transport in two-dimensional flows from the dynamical systems point of view, with the focus on elliptic behaviour and aperiodic and finite time dependence. We give an overview of previous work on general nonautonomous and finite time vector fields with the purpose of bringing to the attention of those working on fluid transport from the dynamical systems point of view a body of work that is extremely relevant, but appears not to be so well known. We then focus on the Kolmogorov–Arnold–Moser (KAM) theorem and the Nekhoroshev theorem. While there is no finite time or aperiodically time-dependent version of the KAM theorem, the Nekhoroshev theorem, by its very nature, is a finite time result, but for a "very long" (i.e. exponentially long with respect to the size of the perturbation) time interval and provides a rigorous quantification of "nearly invariant tori" over this very long timescale. We discuss an aperiodically time-dependent version of the Nekhoroshev theorem due to Giorgilli and Zehnder (1992) (recently refined by Bounemoura, 2013 and Fortunati and Wiggins, 2013) which is directly relevant to fluid transport problems. We give a detailed discussion of issues associated with the applicability of the KAM and Nekhoroshev theorems in specific flows. Finally, we consider a specific example of an aperiodically time-dependent flow where we show that the results of the Nekhoroshev theorem hold.

Modelling the trajectory of the corpses of mountaineers who disappeared in 1926 on Aletschgletscher, Switzerland

AbstractIn this paper we reconstruct the space–time trajectory beneath the surface of Aletschgletscher, Switzerland, of the corpses of three mountaineers that disappeared in March 1926 and reappeared at the glacier surface in June 2012. Our method integrates the time-dependent velocity field of an existing full-Stokes glacier model, starting at the point where the corpses were found at the glacier surface. Our main result is that we were able to localize the immersion location where the brothers presumably died. As a second result, the upstream end point of the computed trajectory emerges very close to the glacier surface in 1926, giving a new and global validation of the glacier model in space and time. Testing the sensitivity of the immersion location obtained with respect to the model and other uncertainties indicates an area of 0.6% of the entire glacier area where the accident could have occurred. Our result suggests that death was not caused by an avalanche or a fall into a crevasse; instead, it is likely that the mountaineers became disoriented in prolonged severe weather conditions and froze to death.