Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 2. pseudoenergy-based theory

This paper represents the second part of a study of semi-geostrophic (SG) geophysical fluid dynamics. SG dynamics shares certain attractive properties with the better known and more widely used quasi-geostrophic (QG) model, but is also a good prototype for balanced models that are more accurate than QG dynamics. The development of such balanced models is an area of great current interest. The goal of the present work is to extend a central body of QG theory, concerning the evolution of disturbances to prescribed basic states, to SG dynamics. Part 1 was based on the pseudomomentum; Part 2 is based on the pseudoenergy. A pseudoenergy invariant is a conserved quantity, of second order in disturbance amplitude relative to a prescribed steady basic state, which is related to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linear stability theorem analogous to Arnol'd's ‘first theorem’; and (ii) a small-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit. The results are analogous to their quasi-geostrophic forms, and reduce to those forms in the limit of small Rossby number. The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to β-plane compressible flow by Magnusdottir & Schubert. Novel features particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability criteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stable basic states. The interior semi-geostrophic dynamics has an underlying Hamiltonian structure, which guarantees that symmetries in the system correspond naturally to the system's invariants. This is an important motivation for the theoretical approach used in this study. The connection between symmetries and conservation laws is made explicit using Noether's theorem applied to the Eulerian form of the Hamiltonian description of the interior dynamics.

Nonlinear stability of Eady's model

A nonlinear stability theorem is established for Eady's model of baroclinic flow. In particular, the Eady basic state is shown to be nonlinearly stable (for arbitrary shear) provided (Δz)/(Δy) > 2(5)^1/2f/(πN),where Δz is the height of the domain, Δy the channel width, f the Coriolis parameter, and N the buoyancy frequency. When this criterion is satisfied, explicit bounds can be derived on the disturbance potential enstrophy, the disturbance energy, and the disturbance available potential energy on the rigid lids, which are expressed in terms of the initial disturbance fields. The disturbances are completely general (with nonzero potential vorticity) and are not assumed to be of small amplitude. The results may be regarded as an extension of Arnol'd's second nonlinear stability theorem to continuously stratified quasigeostrophic baroclinic flow.

The stability of a two-dimensional vorticity filament under uniform strain

The quantitative effects of uniform strain and background rotation on the stability of a strip of constant vorticity (a simple shear layer) are examined. The thickness of the strip decreases in time under the strain, so it is necessary to formulate the linear stability analysis for a time-dependent basic flow. The results show that even a strain rate γ (scaled with the vorticity of the strip) as small as 0.25 suppresses the conventional Rayleigh shear instability mechanism, in the sense that the r.m.s. wave steepness cannot amplify by more than a certain factor, and must eventually decay. For γ < 0.25 the amplification factor increases as γ decreases; however, it is only 3 when γ e 0.065. Numerical simulations confirm the predictions of linear theory at small steepness and predict a threshold value necessary for the formation of coherent vortices. The results help to explain the impression from numerous simulations of two-dimensional turbulence reported in the literature that filaments of vorticity infrequently roll up into vortices. The stabilization effect may be expected to extend to two- and three-dimensional quasi-geostrophic flows.

The kinetics of αIIbβ3 activation determines the size and stability of thrombi in mice: implications for antiplatelet therapy.

Two major pathways contribute to Ras-proximate-1-mediated integrin activation in stimulated platelets. Calcium and diacyglycerol-regulated guanine nucleotide exchange factor I (CalDAG-GEFI, RasGRP2) mediates the rapid but reversible activation of integrin αIIbβ3, while the adenosine diphosphate receptor P2Y12, the target for antiplatelet drugs like clopidogrel, facilitates delayed but sustained integrin activation. To establish CalDAG-GEFI as a target for antiplatelet therapy, we compared how each pathway contributes to thrombosis and hemostasis in mice. Ex vivo, thrombus formation at arterial or venous shear rates was markedly reduced in CalDAG-GEFI(-/-) blood, even in the presence of exogenous adenosine diphosphate and thromboxane A(2). In vivo, thrombosis was virtually abolished in arterioles and arteries of CalDAG-GEFI(-/-) mice, while small, hemostatically active thrombi formed in venules. Specific deletion of the C1-like domain of CalDAG-GEFI in circulating platelets also led to protection from thrombus formation at arterial flow conditions, while it only marginally increased blood loss in mice. In comparison, thrombi in the micro- and macrovasculature of clopidogrel-treated wild-type mice grew rapidly and frequently embolized but were hemostatically inactive. Together, these data suggest that inhibition of the catalytic or the C1 regulatory domain in CalDAG-GEFI will provide strong protection from athero-thrombotic complications while maintaining a better safety profile than P2Y12 inhibitors like clopidogrel.

Proactive R&D management and firm growth: a punctuated equilibrium model

The external environment is characterized by periods of relative stability interspersed with periods of extreme change, implying that high performing firms must practice exploration and exploitation in order to survive and thrive. In this paper, we posit that R&D expenditure volatility indicates the presence of proactive R&D management, and is evidence of a firm moving from exploitation to exploration over time. This is consistent with a punctuated equilibrium model of R&D investment where shocks are induced by reactions to external turbulence. Using an unbalanced panel of almost 11,000 firm-years from 1997 to 2006, we show that greater fluctuations in the firm's R&D expenditure over time are associated with higher firm growth. Developing a contextual view of the relationship between R&D expenditure volatility and firm growth, we find that this relationship is weaker among firms with higher levels of corporate diversification and negative among smaller firms and those in slow clockspeed industries.

Stability of an ice sheet on an elastic bed

The stability of stationary flow of a two-dimensional ice sheet is studied when the ice obeys a power flow law (Glen's flow law). The mass accumulation rate at the top is assumed to depend on elevation and span and the bed supporting the ice sheet consists of an elastic layer lying on a rigid surface. The normal perturbation of the free surface of the ice sheet is a singular eigenvalue problem. The singularity of the perturbation at the front of the ice sheet is considered using matched asymptotic expansions, and the eigenvalue problem is seen to reduce to that with fixed ice front. Numerical solution of the perturbation eigenvalue problem shows that the dependence of accumulation rate on elevation permits the existence of unstable solutions when the equilibrium line is higher than the bed at the ice divide. Alternatively, when the equilibrium line is lower than the bed, there are only stable solutions. Softening of the bed, expressed through a decrease of its elastic modulus, has a stabilising effect on the ice sheet.

Long-term cultivation-independent microbial diversity analysis demonstrates that bacterial communities infecting the adult cystic fibrosis lung show stability and resilience.

Whilst not true in all cases, the microbial communities that chronically infect the airways of patients with CF can vary little over a year despite antibiotic perturbation. The species present tended to vary more between than within subjects, suggesting that each CF airway infection is unique, with relatively stable and resilient bacterial communities. The inverse relationship between community richness and disease severity is similar to findings reported in other mucosal infections.

The effect of the tensor force on the predicted stability of superheavy nuclei

The effect of the tensor component of the Skyrme effective nucleon-nucleon interaction on the single-particle structure in superheavy elements is studied. A selection of the available Skyrme forces has been chosen and their predictions for the proton and neutron shell closures investigated. The inclusion of the tensor term with realistic coupling strength parameters leads to a small increase in the spin-orbit splitting between the proton 2f7/2 and 2f5/2 partners, opening the Z=114 shell gap over a wide range of nuclei. The Z=126 shell gap, predicted by these models in the absence of the tensor term, is found to be stongly dependent on neutron number with a Z=138 gap opening for large neutron numbers, having a consequent implication for the synthesis of neutron-rich superheavy elements. The predicted neutron shell structures remain largely unchanged by inclusion of the tensor component.

Biomineralisation by earthworms: an investigation into the stability and distribution of amorphous calcium carbonate

Background Many biominerals form from amorphous calcium carbonate (ACC), but this phase is highly unstable when synthesised in its pure form inorganically. Several species of earthworm secrete calcium carbonate granules which contain highly stable ACC. We analysed the milky fluid from which granules form and solid granules for amino acid (by liquid chromatography) and functional group (by Fourier transform infrared (FTIR) spectroscopy) compositions. Granule elemental composition was determined using inductively coupled plasma-optical emission spectroscopy (ICP-OES) and electron microprobe analysis (EMPA). Mass of ACC present in solid granules was quantified using FTIR and compared to granule elemental and amino acid compositions. Bulk analysis of granules was of powdered bulk material. Spatially resolved analysis was of thin sections of granules using synchrotron-based μ-FTIR and EMPA electron microprobe analysis. Results The milky fluid from which granules form is amino acid-rich (≤ 136 ± 3 nmol mg−1 (n = 3; ± std dev) per individual amino acid); the CaCO3 phase present is ACC. Even four years after production, granules contain ACC. No correlation exists between mass of ACC present and granule elemental composition. Granule amino acid concentrations correlate well with ACC content (r ≥ 0.7, p ≤ 0.05) consistent with a role for amino acids (or the proteins they make up) in ACC stabilisation. Intra-granule variation in ACC (RSD = 16%) and amino acid concentration (RSD = 22–35%) was high for granules produced by the same earthworm. Maps of ACC distribution produced using synchrotron-based μ-FTIR mapping of granule thin sections and the relative intensity of the ν2: ν4 peak ratio, cluster analysis and component regression using ACC and calcite standards showed similar spatial distributions of likely ACC-rich and calcite-rich areas. We could not identify organic peaks in the μ-FTIR spectra and thus could not determine whether ACC-rich domains also had relatively high amino acid concentrations. No correlation exists between ACC distribution and elemental concentrations determined by EMPA. Conclusions ACC present in earthworm CaCO3 granules is highly stable. Our results suggest a role for amino acids (or proteins) in this stability. We see no evidence for stabilisation of ACC by incorporation of inorganic components.

Stability of probiotic Lactobacillus plantarum in dry microcapsules under accelerated storage conditions

This study investigated the stability of freeze dried and fluid bed dried alginate microcapsules coated with chitosan containing model probiotic bacteria, Lactobacillus plantarum, during storage for up to 45 days at different water activities (0.11, 0.23, 0.40 and 0.70) and temperatures (4, 30 and 37 °C). The loss in cell viability was around 0.8 log in the case of fluid bed drying and around 1.3 in the case of freeze drying, with the former method resulting in dried capsules of smaller size (~ 1 mm vs 1.3 mm), more irregular shape, and with a rougher surface. In both cases, the water activity and water content were less than 0.25 and 10% w/w, respectively, which favours high storage stability. The storage stability studies demonstrated that as the water activity and temperature decreased the survival of the dried encapsulated cells increased. Considerably better survival was observed for fluid bed dried encapsulated cells compared to freeze dried encapsulated cells and freeze dried free cells with 10% sucrose (control), and in some cases, e.g. at 4 and 30 °C at water activities of 0.11, 0.23 and 0.40, there was more than 1 log difference after 45 days, with concentrations higher than 108 CFU/g after 45 days of storage. The results indicate that fluid bed drying is an effective and efficient manufacturing method to produce probiotic containing capsules with enhanced storage stability.