Comments on 'Balance and the slow quasimanifold: some explicit results'

The concept of slow vortical dynamics and its role in theoretical understanding is central to geophysical fluid dynamics. It leads, for example, to “potential vorticity thinking” (Hoskins et al. 1985). Mathematically, one imagines an invariant manifold within the phase space of solutions, called the slow manifold (Leith 1980; Lorenz 1980), to which the dynamics are constrained. Whether this slow manifold truly exists has been a major subject of inquiry over the past 20 years. It has become clear that an exact slow manifold is an exceptional case, restricted to steady or perhaps temporally periodic flows (Warn 1997). Thus the concept of a “fuzzy slow manifold” (Warn and Ménard 1986) has been suggested. The idea is that nearly slow dynamics will occur in a stochastic layer about the putative slow manifold. The natural question then is, how thick is this layer? In a recent paper, Ford et al. (2000) argue that Lighthill emission—the spontaneous emission of freely propagating acoustic waves by unsteady vortical flows—is applicable to the problem of balance, with the Mach number Ma replaced by the Froude number F, and that it is a fundamental mechanism for this fuzziness. They consider the rotating shallow-water equations and find emission of inertia–gravity waves at O(F2). This is rather surprising at first sight, because several studies of balanced dynamics with the rotating shallow-water equations have gone beyond second order in F, and found only an exponentially small unbalanced component (Warn and Ménard 1986; Lorenz and Krishnamurthy 1987; Bokhove and Shepherd 1996; Wirosoetisno and Shepherd 2000). We have no technical objection to the analysis of Ford et al. (2000), but wish to point out that it depends crucially on R 1, where R is the Rossby number. This condition requires the ratio of the characteristic length scale of the flow L to the Rossby deformation radius LR to go to zero in the limit F → 0. This is the low Froude number scaling of Charney (1963), which, while originally designed for the Tropics, has been argued to be also relevant to mesoscale dynamics (Riley et al. 1981). If L/LR is fixed, however, then F → 0 implies R → 0, which is the standard quasigeostrophic scaling of Charney (1948; see, e.g., Pedlosky 1987). In this limit there is reason to expect the fuzziness of the slow manifold to be “exponentially thin,” and balance to be much more accurate than is consistent with (algebraic) Lighthill emission.

Spectral nonlocality, absolute equilibria and Kraichnan-Leith-Batchelor phenomenology in two-dimensional turbulent energy cascades

We study the degree to which Kraichnan–Leith–Batchelor (KLB) phenomenology describes two-dimensional energy cascades in α turbulence, governed by ∂θ/∂t+J(ψ,θ)=ν∇2θ+f, where θ=(−Δ)α/2ψ is generalized vorticity, and ψ^(k)=k−αθ^(k) in Fourier space. These models differ in spectral non-locality, and include surface quasigeostrophic flow (α=1), regular two-dimensional flow (α=2) and rotating shallow flow (α=3), which is the isotropic limit of a mantle convection model. We re-examine arguments for dual inverse energy and direct enstrophy cascades, including Fjørtoft analysis, which we extend to general α, and point out their limitations. Using an α-dependent eddy-damped quasinormal Markovian (EDQNM) closure, we seek self-similar inertial range solutions and study their characteristics. Our present focus is not on coherent structures, which the EDQNM filters out, but on any self-similar and approximately Gaussian turbulent component that may exist in the flow and be described by KLB phenomenology. For this, the EDQNM is an appropriate tool. Non-local triads contribute increasingly to the energy flux as α increases. More importantly, the energy cascade is downscale in the self-similar inertial range for 2.5<α<10. At α=2.5 and α=10, the KLB spectra correspond, respectively, to enstrophy and energy equipartition, and the triad energy transfers and flux vanish identically. Eddy turnover time and strain rate arguments suggest the inverse energy cascade should obey KLB phenomenology and be self-similar for α<4. However, downscale energy flux in the EDQNM self-similar inertial range for α>2.5 leads us to predict that any inverse cascade for α≥2.5 will not exhibit KLB phenomenology, and specifically the KLB energy spectrum. Numerical simulations confirm this: the inverse cascade energy spectrum for α≥2.5 is significantly steeper than the KLB prediction, while for α<2.5 we obtain the KLB spectrum.

Balance model for equatorial long waves

Geophysical fluid models often support both fast and slow motions. As the dynamics are often dominated by the slow motions, it is desirable to filter out the fast motions by constructing balance models. An example is the quasi geostrophic (QG) model, which is used widely in meteorology and oceanography for theoretical studies, in addition to practical applications such as model initialization and data assimilation. Although the QG model works quite well in the mid-latitudes, its usefulness diminishes as one approaches the equator. Thus far, attempts to derive similar balance models for the tropics have not been entirely successful as the models generally filter out Kelvin waves, which contribute significantly to tropical low-frequency variability. There is much theoretical interest in the dynamics of planetary-scale Kelvin waves, especially for atmospheric and oceanic data assimilation where observations are generally only of the mass field and thus do not constrain the wind field without some kind of diagnostic balance relation. As a result, estimates of Kelvin wave amplitudes can be poor. Our goal is to find a balance model that includes Kelvin waves for planetary-scale motions. Using asymptotic methods, we derive a balance model for the weakly nonlinear equatorial shallow-water equations. Specifically we adopt the ‘slaving’ method proposed by Warn et al. (Q. J. R. Meteorol. Soc., vol. 121, 1995, pp. 723–739), which avoids secular terms in the expansion and thus can in principle be carried out to any order. Different from previous approaches, our expansion is based on a long-wave scaling and the slow dynamics is described using the height field instead of potential vorticity. The leading-order model is equivalent to the truncated long-wave model considered previously (e.g. Heckley & Gill, Q. J. R. Meteorol. Soc., vol. 110, 1984, pp. 203–217), which retains Kelvin waves in addition to equatorial Rossby waves. Our method allows for the derivation of higher-order models which significantly improve the representation of Rossby waves in the isotropic limit. In addition, the ‘slaving’ method is applicable even when the weakly nonlinear assumption is relaxed, and the resulting nonlinear model encompasses the weakly nonlinear model. We also demonstrate that the method can be applied to more realistic stratified models, such as the Boussinesq model.

The Brownian traveller on manifolds

We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the at case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the at case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.

Rheology of discrete failure regimes of anisotropic sea ice

A rheological model of sea ice is presented that incorporates the orientational distribution of ice thickness in leads embedded in isotropic floe ice. Sea ice internal stress is determined by coulombic, ridging and tensile failure at orientations where corresponding failure criteria are satisfied at minimum stresses. Because sea ice traction increases in thinner leads and cohesion is finite, such failure line angles are determined by the orientational distribution of sea ice thickness relative to the imposed stresses. In contrast to the isotropic case, sea ice thickness anisotropy results in these failure lines becoming dependent on the stress magnitude. Although generally a given failure criteria type can be satisfied at many directions, only two at most are considered. The strain rate is determined by shearing along slip lines accompanied by dilatancy and closing or opening across orientations affected by ridging or tensile failure. The rheology is illustrated by a yield curve determined by combining coulombic and ridging failure for the case of two pairs of isotropically formed leads of different thicknesses rotated with regard to each other, which models two events of coulombic failure followed by dilatancy and refreezing. The yield curve consists of linear segments describing coulombic and ridging yield as failure switches from one lead to another as the stress grows. Because sliding along slip lines is accompanied by dilatancy, at typical Arctic sea ice deformation rates a one-day-long deformation event produces enough open water that these freshly formed slip lines are preferential places of ridging failure.

Modeling Coulombic failure of sea ice with leads

[1] Sea ice failure under low-confinement compression is modeled with a linear Coulombic criterion that can describe either fractural failure or frictional granular yield along slip lines. To study the effect of anisotropy we consider a simplified anisotropic sea ice model where the sea ice thickness depends on orientation. Accommodation of arbitrary deformation requires failure along at least two intersecting slip lines, which are determined by finding two maxima of the yield criterion. Due to the anisotropy these slip lines generally differ from the standard, Coulombic slip lines that are symmetrically positioned around the compression direction, and therefore different tractions along these slip lines give rise to a nonsymmetric stress tensor. We assume that the skewsymmetric part of this tensor is counterbalanced by an additional elastic stress in the sea ice field that suppresses floe spin. We consider the case of two leads initially formed in an isotropic ice cover under compression, and address the question of whether these leads will remain active or new slip lines will form under a rotation of the principal compression direction. Decoupled and coupled models of leads are considered and it is shown that for this particular case they both predict lead reactivation in almost the same way. The coupled model must, however, be used in determining the stress as the decoupled model does not resolve the stress asymmetry properly when failure occurs in one lead and at a new slip line.

Continuum sea ice rheology determined from subcontinuum mechanics

[1] A method is presented to calculate the continuum-scale sea ice stress as an imposed, continuum-scale strain-rate is varied. The continuum-scale stress is calculated as the area-average of the stresses within the floes and leads in a region (the continuum element). The continuum-scale stress depends upon: the imposed strain rate; the subcontinuum scale, material rheology of sea ice; the chosen configuration of sea ice floes and leads; and a prescribed rule for determining the motion of the floes in response to the continuum-scale strain-rate. We calculated plastic yield curves and flow rules associated with subcontinuum scale, material sea ice rheologies with elliptic, linear and modified Coulombic elliptic plastic yield curves, and with square, diamond and irregular, convex polygon-shaped floes. For the case of a tiling of square floes, only for particular orientations of the leads have the principal axes of strain rate and calculated continuum-scale sea ice stress aligned, and these have been investigated analytically. The ensemble average of calculated sea ice stress for square floes with uniform orientation with respect to the principal axes of strain rate yielded alignment of average stress and strain-rate principal axes and an isotropic, continuum-scale sea ice rheology. We present a lemon-shaped yield curve with normal flow rule, derived from ensemble averages of sea ice stress, suitable for direct inclusion into the current generation of sea ice models. This continuum-scale sea ice rheology directly relates the size (strength) of the continuum-scale yield curve to the material compressive strength.

Anisotropic model for granulated sea ice dynamics

A continuum model describing sea ice as a layer of granulated thick ice, consisting of many rigid, brittle floes, intersected by long and narrow regions of thinner ice, known as leads, is developed. We consider the evolution of mesoscale leads, formed under extension, whose lengths span many floes, so that the surrounding ice is treated as a granular plastic. The leads are sufficiently small with respect to basin scales of sea ice deformation that they may be modelled using a continuum approach. The model includes evolution equations for the orientational distribution of leads, their thickness and width expressed through second-rank tensors and terms requiring closures. The closing assumptions are constructed for the case of negligibly small lead ice thickness and the canonical deformation types of pure and simple shear, pure divergence and pure convergence. We present a new continuum-scale sea ice rheology that depends upon the isotropic, material rheology of sea ice, the orientational distribution of lead properties and the thick ice thickness. A new model of lead and thick ice interaction is presented that successfully describes a number of effects: (i) because of its brittle nature, thick ice does not thin under extension and (ii) the consideration of the thick sea ice as a granular material determines finite lead opening under pure shear, when granular dilation is unimportant.

The Landau free energy of hard ellipses obtained from microscopic simulations

Systems of two-dimensional hard ellipses of varying aspect ratios and packing fractions are studied by Monte Carlo simulations in the generalised canonical ensemble. From this microscopic model, we extract a coarse-grained macroscopic Landau-de Gennes free energy as a function of packing fraction and orientational order parameter. We separate the free energy into the ideal orientational entropy of non-interacting two-dimensional spins and an excess free energy associated with excluded volume interactions. We further explore the isotropic-nematic phase transition using our empirical expression for the free energy and find that the nature of the phase transition is continuous for the aspect ratios we studied.

Optimal steady motions for oriented vehicles

Motivated by the motion planning problem for oriented vehicles travelling in a 3-Dimensional space; Euclidean space E3, the sphere S3 and Hyperboloid H3. For such problems the orientation of the vehicle is naturally represented by an orthonormal frame over a point in the underlying manifold. The orthonormal frame bundles of the space forms R3,S3 and H3 correspond with their isometry groups and are the Euclidean group of motion SE(3), the rotation group SO(4) and the Lorentzian group SO(1; 3) respectively. Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. For constant twist motions or helical motions, the corresponding curves g(t) 2 SE(3) are given in closed form by using the well known Rodrigues’ formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw/helical motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

Investigations into the formation and characterization of phospholipid microemulsions. IV. Pseudo-ternary phase diagrams of systems containing water-lecithin-alcohol and oil; The influence of oil

Phase studies have been performed for quaternary systems composed of egg lecithin, cosurfactant, water and oil. The lecithin used was the commercially available egg lecithin Ovothin 200 (which comprises ≥ 92% phosphatidylcholine). The cosurfactants employed were propanol and butanol, and these were used at lecithin/cosurfactant mixing ratios (Km) of 1:1 and 1.94:1 (weight basis). Six polar oils were investigated, including the alkanoic acids, octanoic and oleic, their corresponding ethyl esters and the medium and long chain triglycerides, Miglyol 812 and soybean oil. All oils, irrespective of the alcohol and the Km used, gave rise to systems that produced a stable isotropic region along the surfactant/oil axis (designated as a reverse microemulsion system). In addition, the systems incorporating propanol at both Km and butanol at a Km of 1.94: 1, generally gave rise to a liquid crystalline region and, in some cases, a second isotropic non-birefingent area (designated as a normal microemulsion system). The phase behaviour observed was largely dependent upon the alcohol and Km used and the size and the polarity of the oil present.

Computer simulations of the structure of colloidal ferrofluids

The structure of a ferrofluid under the influence of an external magnetic field is expected to become anisotropic due to the alignment of the dipoles into the direction of the external field, and subsequently to the formation of particle chains due to the attractive head to tail orientations of the ferrofluid particles. Knowledge about the structure of a colloidal ferrofluid can be inferred from scattering data via the measurement of structure factors. We have used molecular-dynamics simulations to investigate the structure of both monodispersed and polydispersed ferrofluids. The results for the isotropic structure factor for monodispersed samples are similar to previous data by Camp and Patey that were obtained using an alternative Monte Carlo simulation technique, but in a different parameter region. Here we look in addition at bidispersed samples and compute the anisotropic structure factor by projecting the q vector onto the XY and XZ planes separately, when the magnetic field was applied along the z axis. We observe that the XY- plane structure factor as well as the pair distribution functions are quite different from those obtained for the XZ plane. Further, the two- dimensional structure factor patterns are investigated for both monodispersed and bidispersed samples under different conditions. In addition, we look at the scaling exponents of structure factors. Our results should be of value to interpret scattering data on ferrofluids obtained under the influence of an external field.

The influence of anisotropic F region ion velocity distributions on ionospheric ion outflows into the magnetosphere

The contribution to the field-aligned ionospheric ion momentum equation, due to coupling between pressure anisotropy and the inhomogeneous geomagnetic field, is investigated. We term this contribution the “hydrodynamic mirror force” and investigate its dependence on the ion drift and the resulting deformations of the ion velocity distribution function from an isotropic form. It is shown that this extra upforce increases rapidly with ion drift relative to the neutral gas but is not highly dependent on the ion-neutral collision model employed. An example of a burst of flow observed by EISCAT, thought to be the ionospheric signature of a flux transfer event at the magnetopause, is studied in detail and it is shown that the nonthermal plasma which results is subject to a hydrodynamic mirror force which is roughly 10% of the gravitational downforce. In addition, predictions by the coupled University College London-Sheffield University model of the ionosphere and thermosphere show that the hydrodynamic mirror force in the auroral oval is up to 3% of the gravitational force for Kp of about 3, rising to 10% following a sudden increase in cross-cap potential. The spatial distribution of the upforce shows peaks in the cusp region and in the post-midnight auroral oval, similar to that of observed low-energy heavy ion flows from the ionosphere into the magnetosphere. We suggest the hydrodynamic mirror force may modulate these outflows by controlling the supply of heavy ions to regions of ion acceleration and that future simulations of the effects of Joule heating on ion outflows should make allowance for it.

Scattered power from non-thermal, F-region plasma observed by EISCAT—evidence for coherent echoes?

Three rapid, poleward bursts of plasma flow, observed by the U.K.-POLAR EISCAT experiment, are studied in detail. In all three cases the large ion velocities (> 1 kms−1) are shown to drive the ion velocity distribution into a non-Maxwellian form, identified by the characteristic shape of the observed spectra and the fact that analysis of the spectra with the assumption of a Maxwellian distribution leads to excessive rises in apparent ion temperature, and an anticorrelation of apparent electron and ion temperatures. For all three periods the total scattered power is shown to rise with apparent ion temperature by up to 6 dB more than is expected for an isotropic Maxwellian plasma of constant density and by an even larger factor than that expected for non-thermal plasma. The anomalous increases in power are only observed at the lower altitudes (< 300 km). At greater altitudes the rise in power is roughly consistent with that simulated numerically for homogeneous, anisotropic, non-Maxwellian plasma of constant density, viewed using the U.K.-POLAR aspect angle. The spectra at times of anomalously high power are found to be asymmetric, showing an enhancement near the downward Doppler-shifted ion-acoustic frequency. Although it is not possible to eliminate completely rapid plasma density fluctuations as a cause of these power increases, such effects cannot explain the observed spectra and the correlation of power and apparent ion temperature without an unlikely set of coincidences. The observations are made along a beam direction which is as much as 16.5° from orthogonality with the geomagnetic field. Nevertheless, some form of coherent-like echo contamination of the incoherent scatter spectrum is the most satisfactory explanation of these data.

The dynamics of Baroclinic Zonal jets

Multiple alternating zonal jets are a ubiquitous feature of planetary atmospheres and oceans. However, most studies to date have focused on the special case of barotropic jets. Here, the dynamics of freely evolving baroclinic jets are investigated using a two-layer quasigeostrophic annulus model with sloping topography. In a suite of 15 numerical simulations, the baroclinic Rossby radius and baroclinic Rhines scale are sampled by varying the stratification and root-mean-square eddy velocity, respectively. Small-scale eddies in the initial state evolve through geostrophic turbulence and accelerate zonally as they grow in horizontal scale, first isotropically and then anisotropically. This process leads ultimately to the formation of jets, which take about 2500 rotation periods to equilibrate. The kinetic energy spectrum of the equilibrated baroclinic zonal flow steepens from a −3 power law at small scales to a −5 power law near the jet scale. The conditions most favorable for producing multiple alternating baroclinic jets are large baroclinic Rossby radius (i.e., strong stratification) and small baroclinic Rhines scale (i.e., weak root-mean-square eddy velocity). The baroclinic jet width is diagnosed objectively and found to be 2.2–2.8 times larger than the baroclinic Rhines scale, with a best estimate of 2.5 times larger. This finding suggests that Rossby wave motions must be moving at speeds of approximately 6 times the turbulent eddy velocity in order to be capable of arresting the isotropic inverse energy cascade.