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.

Nonlinear stability and the saturation of instabilities to axisymmetric vortices

Nonlinear stability theorems are presented for axisymmetric vortices under the restriction that the disturbance is independent of either the azimuthal or the axial coordinate. These stability theorems are then used, in both cases, to derive rigorous upper bounds on the saturation amplitudes of instabilities. Explicit examples of such bounds are worked out for some canonical profiles. The results establish a minimum order for the dependence of saturation amplitude on supercriticality, and are thereby suggestive as to the nature of the bifurcation at the stability threshold.

On the interpretation of Andrews’ theorem

Andrews (1984) has shown that any flow satisfying Arnol'd's (1965, 1966) sufficient conditions for stability must be zonally-symmetric if the boundary conditions on the flow are zonally-symmetric. This result appears to place very strong restrictions on the kinds of flows that can be proved to be stable by Arnol'd's theorems. In this paper, Andrews’ theorem is re-examined, paying special attention to the case of an unbounded domain. It is shown that, in that case, Andrews’ theorem generally fails to apply, and Arnol'd-stable flows do exist that are not zonally-symmetric. The example of a circular vortex with a monotonic vorticity profile is a case in point. A proof of the finite-amplitude version of the Rayleigh stability theorem for circular vortices is also established; despite its similarity to the Arnol'd theorems it seems not to have been put on record before.

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

Idealised simulations of sting-jet cyclones

An idealised modelling study of sting-jet cyclones is presented. Sting jets are descending mesoscale jets that occur in some extratropical cyclones and produce localised regions of strong low-level winds in the frontal fracture region. Moist baroclinic lifecycle (LC1) simulations are performed with modifications to produce cyclones resembling observed sting-jet cyclones. A sting jet exists in the idealised control cyclone with similar characteristics to the sting jet in a simulation of windstorm Gudrun (a confirmed sting-jet case). Unlike in windstorm Gudrun, a low-level layer of strong moist static stability prohibits the descent of the strong winds from above the boundary layer to the surface in the idealised case. Conditional symmetric instability (CSI) exists in the cloud head and dissipates as the sting jet leaves the cloud head and descends. The descending, initially moist, sting-jet trajectories consistently have negative or near-zero saturated moist potential vorticity but moist static stability and inertial stability, consistent with CSI release; the moist static stability becomes negative during the period of most rapid descent, by which time the air is relatively dry implying conditional instability release is unlikely. Sensitivity experiments show that the existence of the sting jet is robust to changes in the initial state, and that the initial tropospheric static stability significantly impacts the descent rate of the sting jet. Inertial and conditional instability are probably being released in the experiment with the weakest initial static stability. This suggests that sting jets can arise through the release of all three instabilities associated with negative saturated moist potential vorticity. While evaporative cooling occurs along the sting-jet trajectories, a sensitivity experiment with evaporation effects turned off shows no significant change to the wind strength or descent rate of the sting jet implying that instability release is the dominant sting-jet driving mechanism.

Measurements of NO, NOy, N2O, and O3 during SPURT: implications for transport and chemistry in the lowermost stratosphere

We present measurements of NO, NOy, O3, and N2O within the lowermost stratosphere (LMS) over Europe obtained during the SPURT project. The measurements cover all seasons between November 2001 and July 2003. They span a broad band of latitudes from 30° N to 75° N and a potential temperature range from 290 to 380 K. The measurements represent a comprehensive data set of these tracers and reveal atmospheric transport processes that influence tracer distributions in the LMS. Median mixing ratios of stratospheric tracers in equivalent latitude-potential temperature coordinates show a clear seasonal cycle related to the Brewer-Dobson circulation, with highest values in spring and lowest values in autumn. Vertical tracer profiles show strong gradients at the extratropical tropopause, suggesting that vertical (cross-isentropic) mixing is reduced above the tropopause. Pronounced meridional gradients in the tracer mixing ratios are found on potential temperature surfaces in the LMS. This suggests strongly reduced mixing along isentropes. Concurrent large gradients in static stability in the vertical direction, and of PV in the meridional direction, suggest the presence of a mixing barrier. Seasonal cycles were found in the correlation slopes ΔO3/ΔN2O and ΔNOy/ΔN2O well above the tropopause. Absolute slope values are smallest in spring indicating chemically aged stratospheric air originating from high altitudes and latitudes. Larger values were measured in summer and autumn suggesting that a substantial fraction of air takes a "short-cut" from the tropical tropopause region into the extratropical LMS. The seasonal change in the composition of the LMS has direct implications for the ozone chemistry in this region. Comparisons of measured NO with the critical NO value at which net ozone production changes from negative to positive, imply ozone production up to 20 K above the local tropopause in spring, up to 30 K in summer, and up to 40 K in autumn. Above these heights, and in winter, net ozone production is negative.

Pulsating equilibria: stability through migration

This paper is to present a model of spatial equilibrium using a nonlinear generalization of Markov-chain type model, and to show the dynamic stability of a unique equilibrium. Even at an equilibrium, people continue to migrate among regions as well as among agent-types, and yet their overall distribution remain unchanged. The model is also adapted to suggest a theory of traffic distribution in a city.

On the role of the ocean in projected atmospheric stability changes in the Atlantic polar low region

The occurrence of destructive mesoscale ‘polar low’ cyclones in the subpolar North Atlantic is projected to decline under anthropogenic change, due to an increase in atmospheric static stability. This letter reports on the role of changes in ocean circulation in shaping the atmospheric stability. In particular, the Atlantic Meridional Overturning Circulation (AMOC) is projected to weaken in response to anthropogenic forcing, leading to a local minimum in warming in this region. The reduced warming is restricted to the lower troposphere, hence contributing to the increase in static stability. Linear correlation analysis of the CMIP3 climate model ensemble suggests that around half of the model uncertainty in the projected stability response arises from the varied response of the AMOC between models.

Impact of strong deep ocean stratification on the glacial carbon cycle

During the Last Glacial Maximum, the climate was substantially colder and the carbon cycle was clearly different from the late Holocene. According to proxy data deep oceanic δ13C was very low, and the atmospheric CO2 concentration also reduced. Several mechanisms have been proposed to explain these changes, but none can fully explain the data, especially the very low deep ocean δ13C values. Oceanic core data show that the deep ocean was very cold and salty, which would lead to enhanced deep ocean stratification. We show that such an enhanced stratification in the coupled climate model CLIMBER-2 helps get very low deep oceanic δ13C values. Indeed the simulated δ13C reaches values as low as −0.8‰ in line with proxy data evidences. Moreover it increases the oceanic carbon reservoir leading to a small, yet robust, atmospheric CO2 drop of approximately 10 ppm.

The Mid-Brunhes Event and West Antarctic ice sheet stability

The complex cyclical nature of Pleistocene climate, driven by the evolving orbital configuration of the Earth, is well known but not well understood. A major climatic transition took place at the Mid-Brunhes Event (MBE), ca. 430 ka ago after which the amplitude of the ca.100 ka climate oscillations increased, with substantially warmer interglacials, including periods warmer than present. Recent modelling has indicated that while the timing of these warmer-than-present transient (WPT) events is consistent with southern warming due to a deglaciation-forced slowdown of the Atlantic Meridional Overturning Circulation, the magnitude of warming requires a local amplification, for which a candidate is the feedback of significant West Antarctic Ice Sheet (WAIS) retreat. We here extend this argument, based on the absence of WPTs in the early ice core record (450–800 ka ago), to hypothesize that the MBE could be a manifestation of decreased WAIS stability, triggered by ongoing subglacial erosion.

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.

Political instability and the persistence of religion in Greece: the policy implications of the cultural defence paradigm

This paper examines the implications of ‘cultural defence’ in the nature of democracy and the stability of the political system in Greece. It focuses on the Greek Orthodox Church’s maintenance of power and political relevance by virtue of its strong link to national identity. We argue that the inhibition of secularization in Greece as a result of cultural defence has significant policy implications, especially in times of crises, when the role of nationalism as a cohesive factor against perceived threats is intensified. The paper further explores three policy/politics areas: (1) political orientation; (2) religious pluralism; and (3) education.