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SUMMARY In a rapidly rotating Boussinesq fluid, buoyant anomalies radiate low-frequency inertial wave packets that disperse along the rotation axis. The wave packets lead to axially elongated vortices, which propagate negative (positive) kinetic helicity upwards (downwards) with respect to the rotation vector. The kinetic helicity carried by the inertial wave packets is near-maximal relative to the velocity and vorticity fields. In classical mean-field theory, kinetic helicity is often associated with the α-effect, which is thought to be an important ingredient for planetary dynamos. The modification of inertial wave packets in the presence of a transverse uniform magnetic field is investigated here, motivated by small-scale dynamics in planetary cores, where a large-scale magnetic field affects fluid motions. We study numerically the dispersion of wave packets from an isolated buoyant source and from a random layer of buoyant anomalies, while varying the Lehnert number Le—the ratio of the frequencies of Alfvén and inertial waves. We find that for Le < 0.1, the vortices are columnar and continue to segregate kinetic helicity so that it is negative (positive) above (below) the buoyant source. Importantly, the wave packets induce an α-effect, which remains strong and coherent for Earth-like values of the Lehnert number (Le < 0.1). The interaction of wave packets emitted by multiple neighbouring buoyant sources results in an α-effect that is stronger than the α-effect induced by wave packets launched from an isolated buoyant source, and we provide an analytical explanation for this. The coherence of the α-effect induced by the wave packets, for Earth-like values of the Lehnert number, lends support to the α2 dynamo model driven by helical waves.

SUMMARY Helicity, the inner product of velocity and vorticity, is considered an important ingredient for the maintenance of a dipolar magnetic field in the geodynamo. Outside the tangent cylinder—an imaginary cylinder which circumscribes the inner core—a spatial segregation of helicity has been observed in several simulations, being negative in the north and positive in the south. Such a segregation pattern is important for a dynamo that relies on the α-effect. However, the origin of this pattern in these simulations is poorly understood. In this paper, we use three strongly forced numerical dynamo solutions to study the various sources of helicity, including those due to buoyancy $({H_T})$, Coriolis, Lorentz and viscous forces. We find a strong spatial correlation between the segregation pattern of helicity and ${H_T}$ both in the instantaneous and the time-averaged results. Our results show that, outside the tangent cylinder, ${H_T}$ is dominated by the product $- {u_z}\partial T/\partial \varphi $, where ${u_z}$ is the vertical velocity component and T is the temperature perturbation. It is known that when inertial waves are launched from a localized buoyant anomaly, ${H_T}$ takes the same sign as the local helicity. We conjecture that this is the reason for the spatial correlation between ${H_T}$ and helicity in our simulation results. The flow in our simulations being strongly turbulent, this effect seems to be a statistical one and manifests itself most clearly in the averaged quantities.

In numerical simulations of planetary dynamos there is an abrupt transition in the dynamics of both the velocity and magnetic fields at a ‘local’ Rossby number of 0.1. For smaller Rossby numbers there are helical columnar structures aligned with the rotation axis, which efficiently maintain a dipolar field. However, when the thermal forcing is increased, these columns break down and the field becomes multi-polar. Similarly, in rotating turbulence experiments and simulations there is a sharp transition at a Rossby number of ${\sim}0.4$. Again, helical axial columnar structures are found for lower Rossby numbers, and there is strong evidence that these columns are created by inertial waves, at least on short time scales. We perform direct numerical simulations of the flow induced by a layer of buoyant anomalies subject to strong rotation, inspired by the equatorially biased heat flux in convective planetary dynamos. We assess the role of inertial waves in generating columnar structures. At high rotation rates (or weak forcing) we find columnar flow structures that segregate helicity either side of the buoyant layer, whose axial length scale increases linearly, as predicted by the theory of low-frequency inertial waves. As the rotation rate is weakened and the magnitude of the buoyant perturbations is increased, we identify a portion of the flow which is more strongly three-dimensional. We show that the flow in this region is turbulent, and has a Rossby number above a critical value $Ro^{crit}\sim 0.4$, consistent with previous findings in rotating turbulence. We suggest that the discrepancy between the transition value found here (and in rotating turbulence experiments), and that seen in the numerical dynamos ($Ro^{crit}\sim 0.1$), is a result of a different choice of the length scale used to define the local $Ro$. We show that when a proxy for the flow length scale perpendicular to the rotation axis is used in this definition, the numerical dynamo transition lies at $Ro^{crit}\sim 0.5$. Based on this we hypothesise that inertial waves, continually launched by buoyant anomalies, sustain the columnar structures in dynamo simulations, and that the transition documented in these simulations is due to the inability of inertial waves to propagate for $Ro>Ro^{crit}$.

The distribution of kinetic helicity in a dipolar planetary dynamo is central to the success of that dynamo. Motivated by the helicity distributions observed in numerical simulations of the Earth’s dynamo, we consider the relationship between the kinetic helicity, $h=\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{u}$, and the buoyancy field that acts as a source of helicity, where $\boldsymbol{u}$ is velocity. We show that, in the absence of a magnetic field, helicity evolves in accordance with the equation $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}t=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{F}+S_{h}$, where the flux, $\boldsymbol{F}$, represents the transport of helicity by inertial waves, and the helicity source, $S_{h}$, involves the product of the buoyancy and the velocity fields. In the numerical simulations it is observed that the helicity outside the tangent cylinder is predominantly negative in the north and positive in the south, a feature which the authors had previously attributed to the transport of helicity by waves (Davidson & Ranjan, Geophys. J. Intl, vol. 202, 2015, pp. 1646–1662). It is also observed that there is a strong spatial correlation between the distribution of $h$ and of $S_{h}$, with $S_{h}$ also predominantly negative in the north and positive in the south. This correlation tentatively suggests that it is the in situ generation of helicity by buoyancy that establishes the distribution of $h$ outside the tangent cylinder, rather than the dispersal of helicity by waves, as had been previously argued by the authors. However, although $h$ and $S_{h}$ are strongly correlated, there is no such correlation between $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}t$ and $S_{h}$, as might be expected if the distribution of $h$ were established by an in situ generation mechanism. We explain these various observations by showing that inertial waves interact with the buoyancy field in such a way as to induce a source $S_{h}$ which has the same sign as the helicity in the local wave flux, and that the sign of $h$ is simply determined by the direction of that flux. We conclude that the observed distributions of $h$ and $S_{h}$ outside the tangent cylinder are consistent with the transport of helicity by waves.

We explored changes in seasonal distribution and behavior of waterbirds in the Strait of Georgia, Canada, in response to increased presence of a major avian predator, the Bald Eagle (Haliaeetus leucocephalus). Eagles were widespread and their increase through fall and winter coincided with migratory movements of waterbirds. Many species of waterbird used inshore waters in early fall when eagles were scarce. Diving birds moved away from inshore waters when eagles returned in late fall and winter, whereas dabbling ducks formed large flocks in inshore waters and spent proportionally more time being vigilant as winter progressed. Flock sizes and avoidance flight distances of scoters and dabblers, but not gulls, increased with proximity to eagles. Waterbirds did not alter vigilance with distance to eagles. We discuss our findings in context of management issues regarding apparent declines and importance of understanding indirect effects of predators on prey for wildlife monitoring.

We consider the dispersion of waves in a rapidly rotating, Boussinesq fluid which is threaded by a magnetic field and stirred by slowly gravitating buoyant blobs. Motivated by dynamics in the core of the Earth, we focus on the evolution of inertial-Alfven wave packets radiated from the buoyant anomalies. These waves resemble conventional low-frequency inertial waves, in the sense that energy disperses on the fast timescale of the background rotation rate, though they also exhibit slower Alfven-like propagation along magnetic field lines. When the magnetic field is uniform, inertial-Alfven waves automatically focus energy radiation onto the rotation axis, a property they share with conventional low-frequency inertial waves in the hydrodynamic case, and which ensures that they dominate the dispersion pattern. However, the situation changes significantly when the magnetic field, B, is non-uniform. In particular, we show any non-uniformity of B causes inertial-Alfven waves to evolve into a more general form of magnetic-Coriolis (MC) wave, and that these waves refract, dispersing somewhat offaxis. Moreover, if inertial-Alfven waves are launched near the equator they can be confined to low latitudes by a critical layer at which the axial group velocity drops to zero. Given that the magnetic field in a planetary core is inevitably non-uniform, we conclude that quasi-geostrophy is most likely achieved through a combination of weakly modified inertial waves and a form of slightly off-axis MC wave in which the inertial and Alfven frequencies are comparable.

Waterbirds are often used as indicators of ecosystem function across broad spatial and temporal scales. Resolving which species are declining and the ecological characteristics they have in common can offer insights into ecosystem changes and their underlying mechanisms. Using 20 years of citizen science data collected by the British Columbia Coastal Waterbird Survey, we examine species-specific trends in abundance of 50 species in the Salish Sea and 37 species along the outer Pacific Ocean coast that we considered to form the core wintering coastal bird community of British Columbia, Canada. Further, we explore whether ecological commonalities increase the likelihood of a species undergoing declines by testing the hypotheses that waterbird abundance trends are influenced by dietary specialization and migration distance to breeding grounds. Results suggest that most populations are stable (i.e., temporal trends are not significant) in both the Salish Sea (36 of 50 spp.) and Pacific coast (32 of 37 spp.) regions. Twelve species displayed significant decline trends in the Salish Sea, whereas two had significant increases. Along the Pacific coast, only three species displayed significant decline trends, and two significant increases. This result is corroborated by guild-specific trends indicating that waterbirds occupying the Salish Sea are faring significantly worse than those residing along the outer coastal regions, almost irrespective of dietary specialization or migration distance. Our results provide evidence that differential environmental pressures between the inner and outer coastal regions may be causing overall loss of wintering waterbirds within, or abundance shifts away from, the Salish Sea. Potential mechanisms responsible for these observed patterns are discussed, including environmental (e.g., climate) and human-induced (e.g., nutrient and chemical pollution) pressures. Collaborative, inter-disciplinary research priorities to help understand these mechanisms are suggested.

Many shorebirds (Order: Charadriiformes; Family: Charadriidae, Recurvirostridae, Scolopacidae, Haematopodidae, Jacanidae) are highly migratory, traversing thousands of kilometers between high latitude breeding and low latitude nonbreeding sites. In doing so, they are dependent on networks of coastal and interior wetland ecosystems. To aid in the effective conservation and management of their populations, and to assess the impact of threats facing shorebirds, standardized data on shorebird abundance are needed from multiple sites representing a gradient of conditions across the hemisphere. Such data would provide insight on whether fluctuations at one location represent real changes in abundance because of some localized threat, or whether other factors acting across broader scales such as the redistribution of predators, are responsible. We designed the Migratory Shorebird Project (MSP), now implemented in 11 countries along the Pacific Coast of the Americas, to characterize spatial and temporal variation in shorebird abundance, to understand which factors (habitat, threats) most influence their populations across the flyway, and to increase capacity for integrated research, monitoring, and conservation. We used bird inventory data from historical surveys and spatial data on habitat distribution along with a set of hypotheses about important threats to guide the design of the project and to identify data requirements to test hypotheses. We counted birds one time per year at 84 nonbreeding sites (~1400 sampling units) between 15 November and 15 February, when shorebirds were relatively stationary. In each of the first three annual counts from 2013/14 to 2015/16, the Migratory Shorebird Project counted ~1M shorebirds representing 44 species, including five species for which > 20% of the estimated biogeographic population was recorded annually, and nine additional species with > 5% recorded annually. The magnitude of variability in estimates of shorebird abundance was inversely correlated with survey effort with the most uncertainty, and lowest survey effort, in the South Temperate region followed by the Neotropical region (southern Mexico to northern Peru) and then the North Temperate region. Evaluation of variance highlighted both among-site and among-unit ("units" are nested within sites) variation in bird abundance and cover types as well as threats such as potential disturbance and predator abundance. Overall, shorebird density was significantly, and positively, associated with the area (ha) of intertidal mudflats, beaches, and aquaculture. Survey units with intermediate levels of bare ground and flooding had the highest shorebird density. As for threats, we found, contrary to our hypothesis, that shorebird abundance was significantly, and positively, associated, with the density of Peregrine Falcons (Falco peregrinus); however we found no relationship between shorebird abundance and the amount of urban development, our index to potential human disturbance, in the surrounding landscape. The Migratory Shorebird Project is providing an essential data foundation and network for increased knowledge of the factors affecting shorebirds across the Pacific Coast of the Americas and identifying what and where conservation actions could have the greatest impact.

This article illustrates and describes in detail a fine central European chasuble of the late c17 which, together with two dalmatics, ‘The Fetternear Vestments,’ were bequeathed to the Diocese of Aberdeen, in 1921 by the Leslie family, many of whom had been distinguished soldiers on the continent and especially in the Empire. After some contextual discussion of the alleged origins of the Leslie family and of their success in Imperial service, the article examines the traditional belief that the vestments, now at the Blairs Museum, Aberdeen, were made for Count James Leslie (c.1621-1694) partly out of Turkish textiles captured in 1683 at the Siege of Vienna. Detailed analysis of the embroidery on the chasuble, especially of the use of metal thread and ‘plate,’ demonstrates that the gold work is indeed of Turkish origin, the rest of the needle work central European, and thus makes the case that this extraordinary hybrid object is indeed a votive vestment made for the Catholic Leslies partly from captured Turkish work.

We derive scaling relationships for planetary dynamos based on a balance between energy production and Joule dissipation, and between the curl of the buoyancy and Coriolis forces. These scaling relationships are deduced for the particular case of dynamos driven by helical waves, but are shown to have a much broader applicability. They are consistent with the evidence of the numerical dynamos, yielding predictions consistent with published empirical scaling laws and also with the observed transition from dipolar to multipolar dynamos. A direct comparison with the observational evidence for the planets is hampered by the fact that we do not know what sets the smallest scale of the motion in the planets. Nevertheless, we use our scaling relationships to show that the traditional assumption that the Elsasser number is of order unity is inconsistent with the observation that the gas-giant dynamos are dipolar dynamos, as is the more recent suggestion that the strength of the dipole is independent of rotation rate and controlled by the buoyancy flux alone. On the other hand, we show that the observational data is consistent with the hypothesis that a dipolar dynamo saturates at the lowest permissible magnetic energy compatible with a given buoyancy flux.

In most numerical simulations of the Earth's core the dynamo resides outside the tangent cylinder and may be crudely classified as being of the α2 type. In this region the flow comprises a sea of thin columnar vortices aligned with the rotation axis, taking the form of alternating cyclones and anticyclones. The dynamo is thought to be driven by these columnar vortices within which the flow is observed to be highly helical, helicity being a crucial ingredient of planetary dynamos. As noted in Davidson, one of the mysteries of this dynamo cartoon is the origin of the helicity, which is observed to be positive in the south and negative in the north. While Ekman pumping at the mantle can induce helicity in some of the overly viscous numerical simulations, it is extremely unlikely to be a significant source within planets. In this paper we return to the suggestion of Davidson that the helicity observed in the less viscous simulations owes its existence to helical wave packets, launched in and around the equatorial plane where the buoyancy flux is observed to be strong. Here we show that such wave packets act as a potent source of planetary helicity, constituting a simple, robust mechanism that yields the correct sign for h north and south of the equator. Since such a mechanism does not rely on the presence of a mantle, it can operate within both the Earth and the gas giants. Moreover, our numerical simulations show that helical wave packets dispersing from the equator produce a random sea of thin, columnar cyclone/anticyclone pairs, very like those observed in the more strongly forced dynamo simulations. We examine the local dynamics of helical wave packets dispersing from the equatorial regions, as well as the overall nature of an α2-dynamo driven by such wave packets. Our local analysis predicts the mean emf induced by helical waves, an analysis that rests on a number of simple approximations which are consistent with our numerical experiments, while our global analysis yields exact integral relationships between the mean emf induced by the wave packets and the large-scale dipole and azimuthal field. Combining these local and integral equations yields a kinematic model for an α2-dynamo driven by helical waves. Order-of-magnitude estimates based on these equations suggest that such a dynamo is indeed feasible.

This article re-considers Alexander Seton, First Earl of Dunfermline (1555–1622) in his cultural context, particularly in the light of the recently-discovered inventory of his private library. This sophisticated collection of continental books, with strong holdings in art, architecture and Catholic apologetics, offers new information on the intense private Catholicism of a statesman who conformed outwardly to Protestantism. The inventory casts light also on his work of building and decoration at Pinkie House, Musselburgh. The article concludes by raising the possibility of some kind of continuance at Pluscarden Priory in Moray after the reformation and under Seton’s protection, and that Seton’s daughter Sophia may have built a freestanding Catholic chapel on her marriage to David Lindsay, first Lord Balcarres.

We consider a rapidly rotating, Boussinesq fluid stirred by buoyant anomalies. In such a system it is known that, in the absence of a magnetic field, inertial waves whose wave vectors lie normal to the rotation axis play a key role in establishing quasi-geostrophic motion. In particular, buoyant anomalies radiate low-frequency inertial wave packets which disperse along the rotation axis, leading to axially elongated columnar vortices. Here we focus on the influence of an ambient magnetic field on this process, motivated by the dynamics of planetary cores. We find that, once again, the waves responsible for establishing quasi-geostrophic structures have wave vectors normal to the rotation axis; however, these are not conventional inertial waves, but rather hybrid ‘inertial–Alfvén waves’. Their frequency equals that of an Alfvén wave but their axial group velocity is half that of the equivalent inertial wave. They have maximal kinetic, magnetic and cross-helicity, carry magnetic and kinetic energy in equal amounts, and are particularly potent in establishing columnar, helical vortices through the spontaneous emission of axially elongated wave packets. Although our hybrid inertial–Alfvén waves have been overlooked in dynamo literature to date, we speculate that they in fact play a central role in planetary dynamos.

A strategy to extract turbulence structures from direct numerical simulation (DNS) data is described along with a systematic analysis of geometry and spatial distribution of the educed structures. A DNS dataset of decaying homogeneous isotropic turbulence at Reynolds number ${\mathit{Re}}_{\lambda } = 141$ is considered. A bandpass filtering procedure is shown to be effective in extracting enstrophy and dissipation structures with their smallest scales matching the filter width, $L$. The geometry of these educed structures is characterized and classified through the use of two non-dimensional quantities, ‘planarity’ and ‘filamentarity’, obtained using the Minkowski functionals. The planarity increases gradually by a small amount as $L$ is decreased, and its narrow variation suggests a nearly circular cross-section for the educed structures. The filamentarity increases significantly as $L$ decreases demonstrating that the educed structures become progressively more tubular. An analysis of the preferential alignment between the filtered strain and vorticity fields reveals that vortical structures of a given scale $L$ are most likely to align with the largest extensional strain at a scale 3–5 times larger than $L$. This is consistent with the classical energy cascade picture, in which vortices of a given scale are stretched by and absorb energy from structures of a somewhat larger scale. The spatial distribution of the educed structures shows that the enstrophy structures at the $5\eta $ scale (where $\eta $ is the Kolmogorov scale) are more concentrated near the ones that are 3–5 times larger, which gives further support to the classical picture. Finally, it is shown by analysing the volume fraction of the educed enstrophy structures that there is a tendency for them to cluster around a larger structure or clusters of larger structures.

We consider freely decaying, anisotropic, statistically axisymmetric, Saffman turbulence in which $E(k\ensuremath{\rightarrow} 0)\ensuremath{\sim} {k}^{2} $, where $E$ is the energy spectrum and $k$ the wavenumber. We note that such turbulence possesses two statistical invariants which are related to the form of the spectral tensor ${\Phi }_{ij} (\mathbi{k})$ at small $k$. These are ${M}_{\parallel } = {\Phi }_{\parallel } ({k}_{\parallel } = 0, {k}_{\perp } \ensuremath{\rightarrow} 0)$ and ${M}_{\perp } = 2{\Phi }_{\perp } ({k}_{\parallel } = 0, {k}_{\perp } \ensuremath{\rightarrow} 0)$, where the subscripts $\parallel $ and $\perp $ indicate quantities parallel and perpendicular to the axis of symmetry. Since ${M}_{\parallel } \ensuremath{\sim} { u}_{\parallel }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } $ and ${M}_{\perp } \ensuremath{\sim} { u}_{\perp }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } $, $u$ and $\ell $ being integral scales, self-similarity of the large scales (when it applies) demands ${ u}_{\parallel }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } = \text{constant} $ and ${ u}_{\perp }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } = \text{constant} $. This, in turn, requires that ${ u}_{\parallel }^{2} / { u}_{\perp }^{2} $ is constant, contrary to the popular belief that freely decaying turbulence should exhibit a ‘return to isotropy’. Numerical simulations performed in large periodic domains, with different types and levels of initial anisotropy, confirm that ${M}_{\parallel } $ and ${M}_{\perp } $ are indeed invariants and that, in the fully developed state, ${ u}_{\parallel }^{2} / { u}_{\perp }^{2} = \text{constant} $. Somewhat surprisingly, the same simulations also show that ${\ell }_{\parallel } / {\ell }_{\perp } $ is more or less constant in the fully developed state. Simple theoretical arguments are given which suggest that, when ${ u}_{\parallel }^{2} / { u}_{\perp }^{2} $ and ${\ell }_{\parallel } / {\ell }_{\perp } $ are both constant, the integral scales should evolve as ${ u}_{\perp }^{2} \ensuremath{\sim} { u}_{\parallel }^{2} \ensuremath{\sim} {t}^{\ensuremath{-} 6/ 5} $ and ${\ell }_{\perp } \ensuremath{\sim} {\ell }_{\parallel } \ensuremath{\sim} {t}^{2/ 5} $, irrespective of the level of anisotropy and of the presence of helicity. These decay laws, first proposed by Saffman (Phys. Fluids, vol. 10, 1967, p. 1349), are verified by the numerical simulations.

We investigate wind tunnel turbulence generated by both conventional and multi-scale grids. Measurements were made in a tunnel which has a large test-section, so that possible side wall effects are very small and the length assures that the turbulence has time to settle down to a homogeneous shear-free state. The conventional and multi-scale grids were all designed to produce turbulence with the same integral scale, so that a direct comparison could be made between the different flows. Our primary finding is that the behavior of the turbulence behind our multi-scale grids is virtually identical to that behind the equivalent conventional grid. In particular, all flows exhibit a power-law decay of energy, $u^2 \sim t^{-n}$, where $n$ is very close to the classical Saffman exponent of $n = 6/5$. Moreover, all spectra exhibit classical Kolmogorov scaling, with the spectra collapsing on the integral scales at small $k$, and on the Kolmogorov micro-scales at large $k$. Our results are at odds with some other experiments performed on similar multi-scale grids, where significantly higher energy decay exponents and turbulence levels have been reported., Comment: 19 pages, 18 figures

We discuss the long-range interactions that arise in homogeneous turbulence as a consequence of the Biot–Savart law. We note that, somewhat surprisingly, these long-range correlations are very weak in decaying, isotropic turbulence, and we argue that this should also be true for magnetohydrodynamic, rotating and stratified turbulence. If this is indeed the case, it is possible to make explicit predictions for the rate of decay of energy in these anisotropic systems, and it turns out that these predictions are consistent with the available numerical and experimental evidence.

High-resolution direct numerical simulations are used to investigate freely decaying two-dimensional turbulence. We focus on the interplay between coherent vortices and vortex filaments, the second of which give rise to an inertial range. We find that Batchelor's prediction for the inertial-range enstrophy spectrum Eω(k, t) ~ β2/3k−1, where β is the enstrophy dissipation rate, is reasonably well satisfied once the turbulence is fully developed, but that the assumptions which underpin the usual interpretation of his theory are not valid. For example, the lack of a quasi-equilibrium cascade means the enstrophy flux Πω(k) is highly non-uniform throughout the inertial range, thus the common assumption that β can act as a surrogate for Πω(k) becomes questionable. We present a variant of Batchelor's theory which accounts for the wavenumber-dependence of Πω; in particular we propose Eω(k, t) ~ Πω(k1)2/3k−1, where k1 is the wavenumber marking the start of the observed k−1 region of the enstrophy spectrum. This provides a better collapse of the data and, unlike Batchelor's original theory, can be justified on theoretical grounds. The basis for our proposal is the observation that the straining of the vortex filaments, which fuels the enstrophy flux through the inertial range, comes almost exclusively from the strain field of the coherent vortices, and this can be characterized by Πω(k1)1/3. Thus Eω(k) is a function of only k and Πω(k1) in the inertial range, and dimensional analysis then yields Eω ~ Πω(k1)2/3k−1. We also confirm the prediction by Davidson (Phys. Fluids, vol. 20, 2008, 025106) that in the inertial range Πω varies as Πω(k)/Πω(k1) = 1 − a−1 ln(k/k1), where a is a constant of order 1. This corresponds to ∂Eω/∂t ~ k−1. Surprisingly, the measured enstrophy fluxes imply that the dynamics of the inertial range as defined by the behaviour of Πω extend to wavenumbers much smaller than k1, but this is masked in Eω(k, t) by the presence of coherent vortices which also contribute to Eω in this region. In fact, we find that kEω(k, t) ≈ H(k) + A(t), or ∂Eω/∂t ~ k−1 in this extended low-k region, where H(k) is almost independent of time and represents the signature of the coherent vortices. In short, the inertial range defined by ∂Eω/∂t ~ k−1 or Πω(k) ~ ln(k) is much broader than the observed Eω ~ k−1 region.

Sharply sheared interfaces determine the structure of turbulent motions both on large and small scales, as recent experiments and simulations have demonstrated. Shear layers form along contours defined by very large gradients in turbulence intensity and/or large gradients in the large scale velocity field. Inhomogeneous eddy motions adjacent to these layers stretch and distort the larger scale vorticity so as to counter the tendency of the interface to thicken by small scale eddy diffusion. These concepts are applied to the structure of high Reynolds number turbulence by focusing on the dynamics of the flow within and near the shear layers that are observed to exist between large eddies. The numerical simulations of Ishihara et al. (2009) show that on the edges of the layers and in their interiors thin viscous layers form, with thickness approximately equal to the Taylor microscale. Intermittent small-scale vortices within these viscous layers are stretched and amplified, as their thickness reduces to a limiting value determined by viscous diffusion, which is equal to the Kolmogorov micro length scale. But the root-mean -square values of vorticity and velocity, and also their flatness factors, are given by statistical averaging across these layers, taking into account the degree to which the thin layers are sufficiently convoluted to be ‘space-filling’. Through the blocking of the external scale eddies outside the layers as they impinge onto the interfaces, upscale and downscale energy transfer processes which are of comparable magnitude lead to distorted inertial-range motions with a wide range of length scales and characteristic self-similar power-law spectra. Here the skewness of the velocity derivatives becomes negative. The upscale motions are essential for maintaining the large structures and the thin shear layers between them, as two-dimensional experiments demonstrate. A significant implication of this ‘interface dynamics’ mechanism is that small scale turbulence is produced as much by larger eddy structures, as by interactions between progressively smaller, and more numerous, self-similar eddies – the ‘cascade’ mechanism proposed by L.F. Richardson. The latter mechanism, which is dynamically equivalent to the Kolmogorov–Obukhov statistical-physics models for homogeneously distributed fluctuations, implies that turbulence structure evolves more slowly than is observed. By contrast the model presented here for the statistics and dynamics of high Reynolds number turbulence is based on a quasi-deterministic zonal analysis within and adjacent to intermittently distributed thin shear layers, which occupy a finite fraction of the flow and cause the eddy motions to have different ranges of length scales in separate parts of the local flow fields. The structure does not change qualitatively as the Reynolds number varies, but is significantly changed if the turbulence is forced at small as well as large scales.

We examine the integral properties of freely decaying homogeneous magnetohydrodynamic (MHD) turbulence subject to an imposed magnetic field B0 at low-magnetic Reynolds number. We confirm that, like conventional isotropic turbulence, the fully developed state possesses a Loitsyansky-like integral invariant, in this case I// = − ∫ r⊥2 〈 u⊥·u′⊥〉 dr, where 〈u(x) ·u(x + rc)〉 = 〈u·u′〉 is the usual two-point velocity correlation and the subscript ⊥ indicates components perpendicular to the imposed field. The conservation of I// for fully developed turbulence places a fundamental restriction on the way in which the integral scales can develop, i.e. it implies u⊥2 ℓ⊥4 ℓ// ≈ constant where u⊥, ℓ⊥ and ℓ// are integral scales. This constraint can be used to estimate the evolution of u⊥(t; B0), ℓ⊥(t; B0) and ℓ//(t; B0), and these theoretical decay laws are shown to be in good agreement with numerical simulations.

Background Changes in regulatory standards that restrict use of identifiable health information can reduce patient recruitment to clinical trials and increase recruitment costs. Purpose To compare subject accrual rates and costs of three recruitment strategies that comply with new regulatory standards within the context of a clinical trial evaluating the impact of shared decision-making on colorectal cancer screening adherence. Methods Sequential cohorts of English-speaking, average-risk patients due for colorectal cancer screening were allocated to one of three recruitment strategies: (1) a provider-initiated electronic ‘opt-in’ referral (Click) method; (2) a provider-mediated ‘opt-in’ referral letter (Letter) method; and (3) an investigator-initiated direct contact ‘opt-out’ (Call) method. Results During distinct 6-month recruitment periods between March 2005 and April 2006, 100 potential subjects were identified using the Click method, 847 by the Letter method, and 758 by the Call method. After excluding ineligible prescreened patients, accrual rates were higher for the Call method (188 of 531 [35.4%]) than either the Click (12 of 72 [16.7%]; p = 0.002) or Letter (17 of 816 [2.1%]; p < 0.001) methods. The average cost per patient enrolled for the Call ($156) method was competitive with the Click ($129) and substantially lower than the Letter ($1967) methods; the Call method was least expensive if combined with automated patient identification ($99). Data extrapolation suggest it would take 2.4 years at an overall cost of $138,518 to recruit a target sample of 900 patients by the Call method, 40.5 years at $62,419 for the Click method and 27.9 years at $1,737,757 for the Letter method. Limitations The study was nonrandomized and findings may not be generalizable to other research settings. Conclusion The investigator-initiated direct contact ‘opt-out’ strategy is significantly more cost-effective and feasible than provider-initiated and provider-mediated ‘opt-in’ strategies for patient recruitment to clinical trials. Clinical Trials 2009; 6: 597—609. http://ctj.sagepub.com

Loitsyanky's integral I = − ∫ r2〈u ⋅u′〉dr is known to be approximately conserved in certain types of fully developed, isotropic turbulence, and its near conservation controls the rate of decay of kinetic energy. Landau suggested that this integral is related to the angular momentum H = ∫ (x × u)dV of some large volume V of the turbulence, according to the expression I = 〈H2〉/V. He also suggested that the approximate conservation of I is related to the principle of conservation of angular momentum. However, Landau's analysis can be criticized because, formally, it applies only to inhomogeneous turbulence evolving in a closed domain. So how are we to interpret the near conservation of I? And what is its relationship, if any, to angular momentum conservation? We show that the key to extending Landau's analysis to strictly homogeneous turbulence is to rewrite Loitsyansky's integral in terms of the vector potential of the velocity field, i.e. I = 6 ∫〈A ⋅ A′〉dr, where ∇ × A = u. This yields I = 6〈[∫VAdV]2〉/V for any large spherical volume V of radius R. Crucially, J = 3∫VAdV can be rewritten as the weighted integral of the angular momentum density throughout all space. This fundamentally changes the way in which we interpret the dynamical behaviour of I. For example, we show that the conservation of 〈J2〉/V, and hence of I, which occurs when the long-range correlations are weak, is a direct consequence of the decorrelation of the flux of angular momentum out through a spherical control surface S and the local angular momentum in the vicinity of S. Thus, within the framework of strictly homogeneous turbulence, we provide the first self-consistent interpretation of Loitsyanky's integral in terms of angular momentum conservation. We also show that essentially the same ideas carry over to certain types of anisotropic turbulence, such as magnetohydrodynamic (MHD), rotating and stratified turbulence. This is important because conservation of angular momentum, which manifests itself in the form of a Loitsyansky-like invariant, places a fundamental restriction on the way in which the integral scales can evolve in such turbulence. This, in turn, controls the rate of decay of energy. We illustrate this by deriving new decay laws for MHD and stratified turbulence. The MHD decay laws are consistent with the available numerical evidence, but further study is required to verify, or otherwise, the predictions for stratified turbulence.

In the limit of vanishing viscosity, ν→0, Kolmogorov's two-thirds, 〈(Δυ)2〉~ε2/3r2/3, and five-thirds, E~ε2/3k−5/3, laws are formally equivalent. (Here 〈(Δυ)2〉 is the second-order structure function, ε the dissipation rate, r the separation in physical space, E the three-dimensional energy spectrum, and k the wavenumber.) However, for the Reynolds numbers encountered in terrestrial experiments, or numerical simulations, it is invariably easier to observe the five-thirds law. We ask why this should be. To this end, we create artificial fields of isotropic turbulence composed of a random sea of Gaussian eddies whose size and energy distribution can be controlled. We choose the energy of eddies of scale, s, to vary as s2/3, in accordance with Kolmogorov's 1941 law, and vary the range of scales, γ=smax/smin, in any one realization from γ=25 to γ=800. This is equivalent to varying the Reynolds number in an experiment from Rλ=60 to Rλ=600. We find that, while there is some evidence of a five-thirds law for γ>50; (Rλ>100), the two-thirds law only starts to become apparent when γ approaches 200 (Rλ~240). The reason for this discrepancy is that the second-order structure function is a poor filter, mixing information about energy and enstrophy, and from scales larger and smaller than r. In particular, in the inertial range, 〈(Δυ)2〉 takes the form of a mixed power law, a1 + a2r2 + a3r2/3, where a2r2 tracks the variation in enstrophy and a3r2/3 the variation in energy. These findings are shown to be consistent with experimental data where the ‘pollution’ of the r2/3 law by the enstrophy contribution, a2r2, is clearly evident. We show that higher-order structure functions (of even order) suffer from a similar deficiency.

We consider freely decaying two-dimensional isotropic turbulence. It is usually assumed that, in such turbulence, the energy spectrum at small wavenumber, k, takes the form E(k → 0) ∼ Ik3, where I is the two-dimensional version of Loitsyansky's integral. However, a second possibility is E(k → 0) ∼ Lk, where the pre-factor, L, is the two-dimensional analogue of Saffman's integral. We show that, as in three dimensions, L is an invariant and that E ∼ Lk spectra arise whenever the eddies possess a significant amount of linear impulse. The conservation of L is shown to be a direct consequence of the principle of conservation of linear momentum. We also show that isotropic turbulence dominated by a cloud of randomly located monopole vortices has a singular energy spectrum of the form E(k → 0) ∼ Jk−1, where J, like L, is an invariant. However, while E ∼ Jk−1 necessarily implies the existence of a sea of monopoles, the converse need not be true: a sea of monopoles whose spatial locations are not entirely random, but constrained in some way, need not give a E ∼ Jk−1 spectra. The constraint imposed by the conservation of energy is particularly important, ruling out E ∼ Jk−1 spectra for certain classes of initial conditions. Finally, we provide simple explicit examples of random vorticity fields with E ∼ Ik3, E ∼ Lk and E ∼ Jk−1 spectra.

Prandtl's secondary mean motions of the second kind are driven by the variation of Reynolds stresses near resistive boundaries. In the flows considered here the turbulence is generated away from the boundary in the absence of a mean flow and then impacts onto a rigid surface placed into the flow at t=0. The initial development of the distorted flow is obtained using the linear approximation and the statistical analysis of rapid distortion theory, following Hunt & Graham (1978) assuming homogeneous stationary high-Reynolds-number turbulence with an integral length scale L ∞ and r.m.s. velocity v' ∞ . First, the effects of axisymmetric anisotropy and of different forms of the spectra are analysed for turbulence impinging onto a plane surface lying at an angle a to the unit vector e of the axis of symmetry of the energy spectrum tensor Φ ij (k). R is defined as the ratio of the largest to smallest variances of the velocity components. The surface blocking leads to gradients of Reynolds shear stresses normal to the surface in the source layer B (s) with thickness of order L ∞ and thence to a mean velocity U(t)∼-tv' 2 ∞ sin 2α(1-1/R)/L ∞ along the slope in the opposite direction of the projection of e onto the plane (i.e. in the direction (eΛn)Λn where n is the normal into the flow). U is greatest near the surface where y > 1. Anisotropic free-stream turbulence also induces mean motions on undulating surfaces with the same wavelength A as that of the undulating surface, but the directions of these mean motions can be towards or away from the troughs/crests depending on the orientation of the anisotropy of the free stream. Flow visualization experiments conducted in a mixing box with oscillating anisotropic and isotropic grids demonstrated the existence of these mean flows and that they reach a steady state with an intensity and length scale comparable to those predicted. These results are also consistent with numerical simulation of Krettenauer & Schumann (1992) of convective turbulence over an undulating surface.

This paper discusses the decay of freely-evolving, two-dimensional turbulence. We present numerical simulations in which particular care is taken to avoid unphysical pollution from periodic boundary conditions. Batchelor's classical theory, which assumes that the kinetic energy is the only invariant of the flow, predicts that the integral scale grows as l∼t. In line with the results of Herring et al. [Evolution of Decaying Two-dimensional Turbulence and Self-similarity, Trends in Mathematics, Birkhauser Verlag Basel, Switzerland, 1999], and many others, our DNS results show that l grows approximately as t1/2. Bartello and Warn [J. Fluids Mech. 326 (1996) 357–372], and McWilliams [J. Fluids Mech. 146 (1984) 21–43], suggested that, in the limit Re→∞, two-dimensional turbulence possesses a second invariant: the peak in vorticity. It is now widely accepted that this explains the failure of Batchelor's theory. However, as yet, there is no satisfactory explanation for the t1/2 growth in l. Periodic boundary conditions impose mirror image long-range correlations of velocity and vorticity. Lilly [J. Fluid Mech. 45 (2) (1971) 395–415], and Davidson [Turbulence: An Introduction for Scientists and Engineers, Oxford University Press, 2004], note that these correlations have the potential to influence the dynamical behavior of the turbulence. Varying the energy-containing length-scale of the turbulence relative to the size of the periodic domain allows this effect to be investigated. To this end, we introduce the box-ratio, ldomain/lturbulence, as a measure of the number of energy containing eddies in our simulations. Over a wide range of box-ratios we show that l grows approximately as t1/2, the enstrophy decays as ∼t−1 (at least for large Re), and the l-normalised vorticity correlations more or less collapse onto a single, self-similar curve. We provide one possible explanation for the observed t1/2 growth of l.

In this paper we examine the evolution of a freely evolving vortex in a rotating conducting fluid. The work is motivated by the study of small-scale motion in the fluid core of the Earth where magnetic and Coriolis forces dominate over nonlinear and viscous forces. The flow is incompressible and the magnetic Reynolds number is small. Two configurations are considered: (i) the rotation axis is perpendicular to a homogeneous imposed magnetic field; (ii) the two are parallel. Large time asymptotic expressions suggest that, in the context of configuration (i), the global kinetic energy of the vortex decays algebraically on a time scale of the Joule damping time. Configuration (ii) exhibits a strong channelling of energy along the axis which leads to a slow-down of the Joule dissipation. Several initial conditions are studied by varying the initial orientation of the vortex. The characteristic shapes of the resulting flows are examined in detail and it is shown that, for Earth-like parameters, the flow evolves and decays on a time scale of days. The second part of this paper examines flows subject to continuous forcing. In particular, we discuss radiation patterns in forced rotating conducting flows. It is observed that Taylor columns are developed while magnetic effects (such as pseudodiffusion and the fracturing of vortices) shape the flow to form a series of plate-like cells orientated parallel to the magnetic and rotation axes.

To clarify the function of gravity in the shallow-water theory of the interfacial instability in aluminium reduction cells, we analyse the existing long-wave theory of the instability in the limit of vanishing gravitational acceleration $g$ . The flow then has an inner and outer structure, with gravity remaining essential within thin layers coating the cell walls. In those thin wall layers, the growing disturbance takes the form of a trapped magnetogravity wave propagating horizontally on the internal interface, and the growth rate $\sigma$ is determined by the coupling of that edge wave to the large-scale flow in the core of the cell; that coupling is expressed as an oblique-derivative problem for the core flow. Although $\sigma$ is asymptotically independent of $g$ , gravity is essential to the long-wave instability because a correlation imposed by the magneto-gravity waves is essential for the disturbance to extract power from the mean state.

In this paper we examine the evolution of an isolated vortex in a rotating conducting fluid which is threaded by a uniform magnetic field. Magnetic and rotation forces dominate over nonlinear and viscous effects and the flow is incompressible. The study is formulated in terms of an unbounded initial value problem with emphasis on the asymptotic solutions at large time. When the homogeneous imposed magnetic field is normal to the rotation axis it is observed that inertial waves, induced by the prescribed initial condition, transport energy on planes perpendicular to the magnetic field to form a series of counter-rotating travelling eddies. Transport along magnetic field lines occurs by pseudo-diffusion, common at low magnetic Reynolds numbers, and by wave propagation from those inertial waves that survive the strong magnetic damping.

We study the stability of a perfectly conducting body in a magnetostatic equilibrium. The body is immersed in a fluid which is threaded by a three-dimensional magnetic field. The fluid may be perfectly conducting, non-conducting or have finite conductivity. We generalise the classical stability criterion of Bernstein et al. (Proc. Roy. Soc. London Ser. A 244 (1958) 17–40; I.B. Bernstein, The variational principle for problems of ideal magnetohydrodynamic stability, in: A.A. Galeev, R.N. Sudan (Eds.), Basic Plasma Physics: Selected Chapters, North-Holland, Amsterdam, 1989, pp. 199–227) and show that the body is stable to small isomagnetic perturbations if and only if the magnetic energy has a minimum at the equilibrium. For an equilibrium of a body in potential magnetic field, we obtain a sufficient condition for genuine nonlinear stability.

In 1996, St Pierre (1996) reported numerical simulations of a buoyant blob migrating across the earth's outer core and subject to the combined effects of rotation and an azimuthal magnetic field. He noted that the blob rapidly fragments into a series of plate-like structures. Quite independently, Davidson (1995, 1997) discovered a similar behaviour in the context of low- R m turbulence (without a Coriolis force) and showed that this phenomenon has its roots in the destruction of angular momentum by the Lorentz force. The purpose of this paper is to pull together these earlier studies and, in particular, to determine whether or not St. Pierre's platelets are also a consequence of the destruction of angular momentum. We confirm that this is indeed the case.

[Figure][1] On Saturday 2nd October, at around 16.00 h, the fluid mechanics community lost one of its most valued members: Tim Nickels. Tim published widely in the fields of vortex dynamics and turbulence, but it is for his beautiful experiments that he will be best remembered. He was

We investigate the linear stability of inviscid flows which are subject to a conservative body force. This includes a broad range of familiar conservative systems, such as ideal MHD, natural convection, flows driven by electrostatic forces and axisymmetric, swirling, recirculating flow. We provide a simple, unified, linear stability criterion valid for any conservative system. In particular, we establish a principle of maximum action of the formformula herewhere η is the Lagrangian displacement,e is a measure of the disturbance energy, T and V are the kinetic and potential energies, and L is the Lagrangian. Here d represents a variation of the type normally associated with Hamilton's principle, in which the particle trajectories are perturbed in such a way that the time of flight for each particle remains the same. (In practice this may be achieved by advecting the streamlines of the base flow in a frozen-in manner.) A simple test for stability is that e is positive definite and this is achieved if L(η) is a maximum at equilibrium. This captures many familiar criteria, such as Rayleigh's circulation criterion, the Rayleigh–Taylor criterion for stratified fluids, Bernstein's principle for magnetostatics, Frieman & Rotenberg's stability test for ideal MHD equilibria, and Arnold's variational principle applied to Euler flows and to ideal MHD. There are three advantages to our test: (i) d2T(η) has a particularly simple quadratic form so the test is easy to apply; (ii) the test is universal and applies to any conservative system; and (iii) unlike other energy principles, such as the energy-Casimir method or the Kelvin–Arnold variational principle, there is no need to identify all of the integral invariants of the flow as a precursor to performing the stability analysis. We end by looking at the particular case of MHD equilibria. Here we note that when u and B are co-linear there exists a broad range of stable steady flows. Moreover, their stability may be assessed by examining the stability of an equivalent magnetostatic equilibrium. When u and B are non-parallel, however, the flow invariably violates the energy criterion and so could, but need not, be unstable. In such cases we identify one mode in which the Lagrangian displacement grows linearly in time. This is reminiscent of the short-wavelength instability of non-Beltrami Euler flows.

▪ Abstract Magnetic fields can be used to melt, pump, stir, and stabilize liquid metals. This provides a nonintrusive means of controlling the flow of metal in commercial casting and refining operations. The quest for greater efficiency and more control in the production of steel, aluminum, and high-performance superalloys has led to a revolution in the application of magnetohydrodynamics (MHD) to process metallurgy. Three typical applications are described here, chosen partially on the basis of their general interest to fluid dynamicists, and partially because of their considerable industrial importance. We look first at magnetic stirring, where a rotating magnetic field is used to agitate and homogenize the liquid zone of a partially-solidified ingot. This is a study in Ekman pumping. Next, we consider magnetic damping, where an intense, static magnetic field is used to suppress fluid motion. In particular, we look at the damping of jets, vortices, and turbulence. We conclude with a discussion of the magnetic destabilization of liquid-liquid interfaces. This is of particular importance in aluminum production.

We are concerned here with forced steady recirculating flows which are laminar, two-dimensional and have a high Reynolds number. The body force is considered to be prescribed and independent of the flow, a situation which arises frequently in magnetohydrodynamics. Such flows are subject to a strong constraint. Specifically, the body force generates kinetic energy throughout the flow field, yet dissipation is confined to narrow singular regions such as boundary layers. If the flow is to achieve a steady state, then the kinetic energy which is continually generated within the bulk of the flow must find its way to the dissipative regions. Now the distribution of u2/2 is governed by a transport equation, in which the only cross-stream transport of energy is diffusion, v∇2 (u2/2). It follows that there are only two possible candidates for the transport of energy to the dissipative regions: the energy could be diffused to the shear layers, or else it could be convected to the shear layers through entrainment of the streamlines. We investigate both options and show that neither is a likely candidate at high Reynolds number. We then describe numerical experiments for a model problem designed to resolve these issues. We show that, at least for our model problem, no stable steady solution exists at high Reynolds number. Rather, as soon as the Reynolds number exceeds a modest value of around 10, the flow becomes unstable via a supercritical Hopf bifurcation.

We investigate the stability of interfacial waves in conducting fluids under the influence of a vertical current density, paying particular attention to aluminium reduction cells in which such instabilities are commonly observed. We develop a wave equation for the interface in which the Lorentz force is expressed explicitly in terms of the fluid motion. Our wave equation differs from previous models, most notably that developed by Urata (1985), in that earlier formulations rested on a more complex, implicit coupling between the fluid motion and the Lorentz force. Our formulation furnishes a number of quite general stability results without the need to resort to Fourier analysis. (The need for Fourier analysis typifies previous studies.) Moreover, our equation supports both travelling and standing waves. We investigate each in turn.We obtain three new results. First, we show that travelling waves may become unstable in the presence of a uniform, vertical magnetic field. (Our travelling waves are quite different to those discovered by previous investigators (Sneyd 1985 and Moreau & Ziegler 1986) which require more complex magnetic fields to become unstable.) Second, in line with previous studies we confirm that standing waves may also become unstable. In this context we derive a simple energy criterion which shows which types of motion may extract energy from the background magnetic field. This indicates that a rotating, tilted interface is particularly prone to instability, and indeed such a motion is often seen in practice. Finally, we use Gershgorin's theorem to produce a sufficient condition for the stability of standing waves in a finite domain. This allows us to place a lower bound on the critical value of the background magnetic field at which an instability first appears, without solving the governing equations of motion.

SUMMARY: Rhabdomyolysis, myoglobinuria and acute renal failure are rare complication of surgery. Long operative time, increased body mass, lateral decubitus positioning and extracellular volume depletion may predispose to this condition. The authors describe the case of a 70-year-old man with renal cell carcinoma who underwent a laparoscopic right radical nephrectomy in the lateral decubitus position. His postoperative course was complicated by acute renal failure due to rhabdomyolysis. Heightened awareness, early recognition and treatment of this condition are important, particularly as laparoscopic nephrectomy is becoming a common procedure for living donor transplantation.