Hamiltonian geophysical fluid dynamics

Hamiltonian dynamics describes the evolution of conservative physical systems. Originally developed as a generalization of Newtonian mechanics, describing gravitationally driven motion from the simple pendulum to celestial mechanics, it also applies to such diverse areas of physics as quantum mechanics, quantum field theory, statistical mechanics, electromagnetism, and optics – in short, to any physical system for which dissipation is negligible. Dynamical meteorology consists of the fundamental laws of physics, including Newton’s second law. For many purposes, diabatic and viscous processes can be neglected and the equations are then conservative. (For example, in idealized modeling studies, dissipation is often only present for numerical reasons and is kept as small as possible.) In such cases dynamical meteorology obeys Hamiltonian dynamics. Even when nonconservative processes are not negligible, it often turns out that separate analysis of the conservative dynamics, which fully describes the nonlinear interactions, is essential for an understanding of the complete system, and the Hamiltonian description can play a useful role in this respect. Energy budgets and momentum transfer by waves are but two examples.

On wave action and phase in the non-canonical Hamiltonian formulation

The long time–evolution of disturbances to slowly–varying solutions of partial differential equations is subject to the adiabatic invariance of the wave action. Generally, this approximate conservation law is obtained under the assumption that the partial differential equations are derived from a variational principle or have a canonical Hamiltonian structure. Here, the wave action conservation is examined for equations that possess a non–canonical (Poisson) Hamiltonian structure. The linear evolution of disturbances in the form of slowly varying wavetrains is studied using a WKB expansion. The properties of the original Hamiltonian system strongly constrain the linear equations that are derived, and this is shown to lead to the adiabatic invariance of a wave action. The connection between this (approximate) invariance and the (exact) conservation laws of pseudo–energy and pseudomomentum that exist when the basic solution is exactly time and space independent is discussed. An evolution equation for the slowly varying phase of the wavetrain is also derived and related to Berry's phase.

A Hamiltonian weak-wave model for shallow-water flow

A reduced dynamical model is derived which describes the interaction of weak inertia–gravity waves with nonlinear vortical motion in the context of rotating shallow–water flow. The formal scaling assumptions are (i) that there is a separation in timescales between the vortical motion and the inertia–gravity waves, and (ii) that the divergence is weak compared to the vorticity. The model is Hamiltonian, and possesses conservation laws analogous to those in the shallow–water equations. Unlike the shallow–water equations, the energy invariant is quadratic. Nonlinear stability theorems are derived for this system, and its linear eigenvalue properties are investigated in the context of some simple basic flows.

On Hamiltonian balanced dynamics and the slowest invariant manifold

The concept of a slowest invariant manifold is investigated for the five-component model of Lorenz under conservative dynamics. It is shown that Lorenz's model is a two-degree-of-freedom canonical Hamiltonian system, consisting of a nonlinear vorticity-triad oscillator coupled to a linear gravity wave oscillator, whose solutions consist of regular and chaotic orbits. When either the Rossby number or the rotational Froude number is small, there is a formal separation of timescales, and one can speak of fast and slow motion. In the same regime, the coupling is weak, and the Kolmogorov–Arnold-Moser theorem is shown to apply. The chaotic orbits are inherently unbalanced and are confined to regions sandwiched between invariant tori consisting of quasi-periodic regular orbits. The regular orbits generally contain free fast motion, but a slowest invariant manifold may be geometrically defined as the set of all slow cores of invariant tori (defined by zero fast action) that are smoothly related to such cores in the uncoupled system. This slowest invariant manifold is not global; in fact, its structure is fractal; but it is of nearly full measure in the limit of weak coupling. It is also nonlinearly stable. As the coupling increases, the slowest invariant manifold shrinks until it disappears altogether. The results clarify previous definitions of a slowest invariant manifold and highlight the ambiguity in the definition of “slowness.” An asymptotic procedure, analogous to standard initialization techniques, is found to yield nonzero free fast motion even when the core solutions contain none. A hierarchy of Hamiltonian balanced models preserving the symmetries in the original low-order model is formulated; these models are compared with classic balanced models, asymptotically initialized solutions of the full system and the slowest invariant manifold defined by the core solutions. The analysis suggests that for sufficiently small Rossby or rotational Froude numbers, a stable slowest invariant manifold can be defined for this system, which has zero free gravity wave activity, but it cannot be defined everywhere. The implications of the results for more complex systems are discussed.

Interaction of anthocyanins with human serum albumin: influence of pH and chemical structure on binding

The affinity of anthocyanins for human serum albumin (HSA) was determined by a fluorescence quenching method. The effects of pH and structure of anthocyanins on the binding constants were studied. The constants for binding of anthocyanins to HSA ranged from 1.08 x 10^5 M-1 to 13.16 x 10^5 M-1. A hydrophobic effect at acidic pH was shown by the relatively high positive entropy values under the conditions studied. Electrostatic interactions including hydrogen bonding contributed to the binding at pH 7.4. The effect of structure of anthocyanins on the affinity was pH dependent, particularly the effect of additional hydroxyl substituents. Hydroxyl substituents and glycosylation of anthocyanins decreased the affinity for binding to HSA at lower pH (especially pH 4), but increased the strength of binding at pH 7.4. In contrast, methylation of a hydroxyl group enhanced the binding at acidic pH, while this substitution reduced the strength of binding at pH 7.4. This paper has shown that changes in anthocyanin structure or reductions in pH, which may occur in the region of inflammatory sites, have an effect of the binding of anthocyanins to HSA.

Nonlinear wave-activity conservation laws and Hamiltonian structure for the two-dimensional anelastic equations

Exact, finite-amplitude, local wave-activity conservation laws are derived for disturbances to steady flows in the context of the two-dimensional anelastic equations. The conservation laws are expressed entirely in terms of Eulerian quantities, and have the property that, in the limit of a small-amplitude, slowly varying, monochromatic wave train, the wave-activity density A and flux F, when averaged over phase, satisfy F = cgA where cg is the group velocity of the waves. For nonparallel steady flows, the only conserved wave activity is a form of disturbance pseudoenergy; when the steady flow is parallel, there is in addition a conservation law for the disturbance pseudomomentum. The above results are obtained not only for isentropic background states (which give the so-called “deep form” of the anelastic equations), but also for arbitrary background potential-temperature profiles θ0(z) so long as the variation in θ0(z) over the depth of the fluid is small compared with θ0 itself. The Hamiltonian structure of the equations is established in both cases, and its symmetry properties discussed. An expression for available potential energy is also derived that, for the case of a stably stratified background state (i.e., dθ0/dz > 0), is locally positive definite; the expression is valid for fully three-dimensional flow. The counterparts to results for the two-dimensional Boussinesq equations are also noted.

A general method for finding extremal states of Hamiltonian dynamical systems, with applications to perfect fluids

In addition to the Hamiltonian functional itself, non-canonical Hamiltonian dynamical systems generally possess integral invariants known as ‘Casimir functionals’. In the case of the Euler equations for a perfect fluid, the Casimir functionals correspond to the vortex topology, whose invariance derives from the particle-relabelling symmetry of the underlying Lagrangian equations of motion. In a recent paper, Vallis, Carnevale & Young (1989) have presented algorithms for finding steady states of the Euler equations that represent extrema of energy subject to given vortex topology, and are therefore stable. The purpose of this note is to point out a very general method for modifying any Hamiltonian dynamical system into an algorithm that is analogous to those of Vallis etal. in that it will systematically increase or decrease the energy of the system while preserving all of the Casimir invariants. By incorporating momentum into the extremization procedure, the algorithm is able to find steadily-translating as well as steady stable states. The method is applied to a variety of perfect-fluid systems, including Euler flow as well as compressible and incompressible stratified flow.

An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems

Disturbances of arbitrary amplitude are superposed on a basic flow which is assumed to be steady and either (a) two-dimensional, homogeneous, and incompressible (rotating or non-rotating) or (b) stably stratified and quasi-geostrophic. Flow over shallow topography is allowed in either case. The basic flow, as well as the disturbance, is assumed to be subject neither to external forcing nor to dissipative processes like viscosity. An exact, local ‘wave-activity conservation theorem’ is derived in which the density A and flux F are second-order ‘wave properties’ or ‘disturbance properties’, meaning that they are O(a2) in magnitude as disturbance amplitude a [rightward arrow] 0, and that they are evaluable correct to O(a2) from linear theory, to O(a3) from second-order theory, and so on to higher orders in a. For a disturbance in the form of a single, slowly varying, non-stationary Rossby wavetrain, $\overline{F}/\overline{A}$ reduces approximately to the Rossby-wave group velocity, where (${}^{-}$) is an appropriate averaging operator. F and A have the formal appearance of Eulerian quantities, but generally involve a multivalued function the correct branch of which requires a certain amount of Lagrangian information for its determination. It is shown that, in a certain sense, the construction of conservable, quasi-Eulerian wave properties like A is unique and that the multivaluedness is inescapable in general. The connection with the concepts of pseudoenergy (quasi-energy), pseudomomentum (quasi-momentum), and ‘Eliassen-Palm wave activity’ is noted. The relationship of this and similar conservation theorems to dynamical fundamentals and to Arnol'd's nonlinear stability theorems is discussed in the light of recent advances in Hamiltonian dynamics. These show where such conservation theorems come from and how to construct them in other cases. An elementary proof of the Hamiltonian structure of two-dimensional Eulerian vortex dynamics is put on record, with explicit attention to the boundary conditions. The connection between Arnol'd's second stability theorem and the suppression of shear and self-tuning resonant instabilities by boundary constraints is discussed, and a finite-amplitude counterpart to Rayleigh's inflection-point theorem noted

Selected wheat seed defense proteins exhibit competitive binding to model microbial lipid interfaces

Puroindolines (Pins) and purothionins (Pths) are basic, amphiphilic, cysteine-rich wheat proteins that play a role in plant defense against microbial pathogens. We have examined the co-adsorption and sequential addition of Pins (Pin-a, Pin-b and a mutant form of Pin-b with Trp-44 to Arg-44 substitution) and β-purothionin (β-Pth) model anionic lipid layers, using a combination of surface pressure measurements, external reflection FTIR spectroscopy and neutron reflectometry. Results highlighted differences in the protein binding mechanisms, and in the competitive binding and penetration of lipid layers between respective Pins and β-Pth. Pin-a formed a blanket-like layer of protein below the lipid surface that resulted in the reduction or inhibition of β-Pth penetration of the lipid layer. Wild-type Pin-b participated in co-operative binding with β-Pth, whereas the mutant Pin-b did not bind to the lipid layer in the presence of β-Pth. The results provide further insight into the role of hydrophobic and cationic amino acid residues in antimicrobial activity.

Dopant-vacancy binding effects in Li-doped magnesium hydride

We use a combination of ab initio calculations and statistical mechanics to investigate the substitution of Li+ for Mg2+ in magnesium hydride (MgH2) accompanied by the formation of hydrogen vacancies with positive charge (with respect to the original ion at the site). We show that the binding energy between dopants and vacancy defects leads to a significant fraction of trapped vacancies and therefore a dramatic reduction in the number of free vacancies available for diffusion. The concentration of free vacancies initially increases with dopant concentration but reaches a maximum at around 1 mol % Li doping and slowly decreases with further doping. At the optimal level of doping, the corresponding concentration of free vacancies is much higher than the equilibrium concentrations of charged and neutral vacancies in pure MgH2 at typical hydrogen storage conditions. We also show that Li-doped MgH2 is thermodynamically metastable with respect to phase separation into pure magnesium and lithium hydrides at any significant Li concentration, even after considering the stabilization provided by dopant-vacancy interactions and configurational entropic effects. Our results suggest that lithium doping may enhance hydrogen diffusion hydride but only to a limited extent determined by an optimal dopant concentration and conditioned to the stability of the doped phase.

Investigation of milk proteins binding to the oral mucosa

High protein dairy beverages are considered to be mouth drying. The drying sensation may be due to the product protein content; however the mechanism of this mouth drying is uncertain. This study investigated the potential adhesion of milk proteins to porcine oral mucosal tissues and their resistance to wash out with simulated saliva was monitored using fluorescence microscopy. Cadein was found to be more adhesive to porcine mucosa then lactogloubulin. Some investigation into the reason for this difference in mucoadhesion was conducted by thiol-content analysis, rheology and zeta-potential measurements. The higher viscosity of casein solution and smaller zeta-potential is believed to be responsible for its better retention on mucosal surfaces. These findings suggest that casein and whey protein are both capable of binding and eliciting mouth drying in high protein dairy beverages.

Variable binding and coreference in sentence comprehension: evidence from eye movements

The hypothesis that pronouns can be resolved via either the syntax or the discourse representation has played an important role in linguistic accounts of pronoun interpretation (e.g. Grodzinsky & Reinhart, 1993). We report the results of an eye-movement monitoring study investigating the relative timing of syntactically-mediated variable binding and discourse-based coreference assignment during pronoun resolution. We examined whether ambiguous pronouns are preferentially resolved via either the variable binding or coreference route, and in particular tested the hypothesis that variable binding should always be computed before coreference assignment. Participants’ eye movements were monitored while they read sentences containing a pronoun and two potential antecedents, a c-commanding quantified noun phrase and a non c-commanding proper name. Gender congruence between the pronoun and either of the two potential antecedents was manipulated as an experimental diagnostic for dependency formation. In two experiments, we found that participants’ reading times were reliably longer when the linearly closest antecedent mismatched in gender with the pronoun. These findings fail to support the hypothesis that variable binding is computed before coreference assignment, and instead suggest that antecedent recency plays an important role in affecting the extent to which a variable binding antecedent is considered. We discuss these results in relation to models of memory retrieval during sentence comprehension, and interpret the antecedent recency preference as an example of forgetting over time.

Analysis of tyrosine phosphorylation and phosphotyrosine-binding proteins in germinating seeds from Scots pine

Protein tyrosine phosphorylation in angiosperms has been implicated in various physiological processes, including seed development and germination. In conifers, the role of tyrosine phosphorylation and the mechanisms of its regulation are yet to be investigated. In this study, we examined the profile of protein tyrosine phosphorylation in Scots pine seeds at different stages of germination. We detected extensive protein tyrosine phosphorylation in extracts from Scots pine (Pinus sylvestris L.) dormant seeds. In addition, the pattern of tyrosine phosphorylation was found to change significantly during seed germination, especially at earlier stages of post-imbibition which coincides with the initiation of cell division, and during the period of intensive elongation of hypocotyls. To better understand the molecular mechanisms of phosphotyrosine signaling, we employed affinity purification and mass spectrometry for the identification of pTyr-binding proteins from the extracts of Scots pine seedlings. Using this approach, we purified two proteins of 10 and 43 kDa, which interacted specifically with pTyr-Sepharose and were identified by mass spectrometry as P. sylvestris defensin 1 (PsDef1) and aldose 1-epimerase (EC:5.1.3.3), respectively. Additionally, we demonstrated that both endogenous and recombinant PsDef1 specifically interact with pTyr-Sepharose, but not Tyr-beads. As the affinity purification approach did not reveal the presence of proteins with known pTyr binding domains (SH2, PTB and C2), we suggest that plants may have evolved a different mode of pTyr recognition, which yet remains to be uncovered.