Vertex splitting and connectivity augmentation in hypergraphs

We consider problems of splitting and connectivity augmentation in hypergraphs. In a hypergraph G = (V +s, E), to split two edges su, sv, is to replace them with a single edge uv. We are interested in doing this in such a way as to preserve a defined level of connectivity in V . The splitting technique is often used as a way of adding new edges into a graph or hypergraph, so as to augment the connectivity to some prescribed level. We begin by providing a short history of work done in this area. Then several preliminary results are given in a general form so that they may be used to tackle several problems. We then analyse the hypergraphs G = (V + s, E) for which there is no split preserving the local-edge-connectivity present in V. We provide two structural theorems, one of which implies a slight extension to Mader’s classical splitting theorem. We also provide a characterisation of the hypergraphs for which there is no such “good” split and a splitting result concerned with a specialisation of the local-connectivity function. We then use our splitting results to provide an upper bound on the smallest number of size-two edges we must add to any given hypergraph to ensure that in the resulting hypergraph we have λ(x, y) ≥ r(x, y) for all x, y in V, where r is an integer valued, symmetric requirement function on V*V. This is the so called “local-edge-connectivity augmentation problem” for hypergraphs. We also provide an extension to a Theorem of Szigeti, about augmenting to satisfy a requirement r, but using hyperedges. Next, in a result born of collaborative work with Zoltán Király from Budapest, we show that the local-connectivity augmentation problem is NP-complete for hypergraphs. Lastly we concern ourselves with an augmentation problem that includes a locational constraint. The premise is that we are given a hypergraph H = (V,E) with a bipartition P = {P1, P2} of V and asked to augment it with size-two edges, so that the result is k-edge-connected, and has no new edge contained in some P(i). We consider the splitting technique and describe the obstacles that prevent us forming “good” splits. From this we deduce results about which hypergraphs have a complete Pk-split. This leads to a minimax result on the optimal number of edges required and a polynomial algorithm to provide an optimal augmentation.

Transport in the low-latitude tropopause zone diagnosed using particle trajectories

Recent literature has described a “transition zone” between the average top of deep convection in the Tropics and the stratosphere. Here transport across this zone is investigated using an offline trajectory model. Particles were advected by the resolved winds from the European Centre for Medium-Range Weather Forecasts reanalyses. For each boreal winter clusters of particles were released in the upper troposphere over the four main regions of tropical deep convection (Indonesia, central Pacific, South America, and Africa). Most particles remain in the troposphere, descending on average for every cluster. The horizontal components of 5-day trajectories are strongly influenced by the El Niño–Southern Oscillation (ENSO), but the Lagrangian average descent does not have a clear ENSO signature. Tropopause crossing locations are first identified by recording events when trajectories from the same release regions cross the World Meteorological Organization lapse rate tropopause. Most crossing events occur 5–15 days after release, and 30-day trajectories are sufficiently long to estimate crossing number densities. In a further two experiments slight excursions across the lapse rate tropopause are differentiated from the drift deeper into the stratosphere by defining the “tropopause zone” as a layer bounded by the average potential temperature of the lapse rate tropopause and the profile temperature minimum. Transport upward across this zone is studied using forward trajectories released from the lower bound and back trajectories arriving at the upper bound. Histograms of particle potential temperature (θ) show marked differences between the transition zone, where there is a slow spread in θ values about a peak that shifts slowly upward, and the troposphere below 350 K. There forward trajectories experience slow radiative cooling interspersed with bursts of convective heating resulting in a well-mixed distribution. In contrast θ histograms for back trajectories arriving in the stratosphere have two distinct peaks just above 300 and 350 K, indicating the sharp change from rapid convective heating in the well-mixed troposphere to slow ascent in the transition zone. Although trajectories slowly cross the tropopause zone throughout the Tropics, all three experiments show that most trajectories reaching the stratosphere from the lower troposphere within 30 days do so over the west Pacific warm pool. This preferred location moves about 30°–50° farther east in an El Niño year (1982/83) and about 30° farther west in a La Niña year (1988/89). These results could have important implications for upper-troposphere–lower-stratosphere pollution and chemistry studies.

Urban tracer dispersion experiments during the second DAPPLE field campaign in London 2004

As part of the DAPPLE programme two large scale urban tracer experiments using multiple simultaneous releases of cyclic perfluoroalkanes from fixed location point sources was performed. The receptor concentrations along with relevant meteorological parameters measured are compared with a three screening dispersion models in order to best predict the decay of pollution sources with respect to distance. It is shown here that the simple dispersion models tested here can provide a reasonable upper bound estimate of the maximum concentrations measured with an empirical model derived from field observations and wind tunnel studies providing the best estimate. An indoor receptor was also used to assess indoor concentrations and their pertinence to commonly used evacuation procedures.

Accessing extremes of mid-latitudinal wave activity: methodology and application

A statistical methodology is proposed and tested for the analysis of extreme values of atmospheric wave activity at mid-latitudes. The adopted methods are the classical block-maximum and peak over threshold, respectively based on the generalized extreme value (GEV) distribution and the generalized Pareto distribution (GPD). Time-series of the ‘Wave Activity Index’ (WAI) and the ‘Baroclinic Activity Index’ (BAI) are computed from simulations of the General Circulation Model ECHAM4.6, which is run under perpetual January conditions. Both the GEV and the GPD analyses indicate that the extremes ofWAI and BAI areWeibull distributed, this corresponds to distributions with an upper bound. However, a remarkably large variability is found in the tails of such distributions; distinct simulations carried out under the same experimental setup provide sensibly different estimates of the 200-yr WAI return level. The consequences of this phenomenon in applications of the methodology to climate change studies are discussed. The atmospheric configurations characteristic of the maxima and minima of WAI and BAI are also examined.

Radiative forcing due to aviation water vapour emissions

Three emissions inventories have been used with a fully Lagrangian trajectory model to calculate the stratospheric accumulation of water vapour emissions from aircraft, and the resulting radiative forcing. The annual and global mean radiative forcing due to present-day aviation water vapour emissions has been found to be 0.9 [0.3 to 1.4] mW m^2. This is around a factor of three smaller than the value given in recent assessments, and the upper bound is much lower than a recently suggested 20 mW m^2 upper bound. This forcing is sensitive to the vertical distribution of emissions, and, to a lesser extent, interannual variability in meteorology. Large differences in the vertical distribution of emissions within the inventories have been identified, which result in the choice of inventory being the largest source of differences in the calculation of the radiative forcing due to the emissions. Analysis of Northern Hemisphere trajectories demonstrates that the assumption of an e-folding time is not always appropriate for stratospheric emissions. A linear model is more representative for emissions that enter the stratosphere far above the tropopause.

Brief communication. Rain effect on the load of tephra deposits

Accumulation of tephra fallout produced during explosive eruptions can cause roof collapses in areas near the volcano, when the weight of the deposit exceeds some threshold value that depends on the quality of buildings. The additional loading of water that remains trapped in the tephra deposits due to rainfall can contribute to increasing the loading of the deposits on the roofs. Here we propose a simple approach to estimate an upper bound for the contribution of rain to the load of pyroclastic deposits that is useful for hazard assessment purposes. As case study we present an application of the method in the area of Naples, Italy, for a reference eruption from Vesuvius volcano.

On the statistical modeling of persistence in total ozone anomalies

Geophysical time series sometimes exhibit serial correlations that are stronger than can be captured by the commonly used first‐order autoregressive model. In this study we demonstrate that a power law statistical model serves as a useful upper bound for the persistence of total ozone anomalies on monthly to interannual timescales. Such a model is usually characterized by the Hurst exponent. We show that the estimation of the Hurst exponent in time series of total ozone is sensitive to various choices made in the statistical analysis, especially whether and how the deterministic (including periodic) signals are filtered from the time series, and the frequency range over which the estimation is made. In particular, care must be taken to ensure that the estimate of the Hurst exponent accurately represents the low‐frequency limit of the spectrum, which is the part that is relevant to long‐term correlations and the uncertainty of estimated trends. Otherwise, spurious results can be obtained. Based on this analysis, and using an updated equivalent effective stratospheric chlorine (EESC) function, we predict that an increase in total ozone attributable to EESC should be detectable at the 95% confidence level by 2015 at the latest in southern midlatitudes, and by 2020–2025 at the latest over 30°–45°N, with the time to detection increasing rapidly with latitude north of this range.

Design of delay-tolerant linear dispersion codes

In cooperative communication networks, owing to the nodes' arbitrary geographical locations and individual oscillators, the system is fundamentally asynchronous. This will damage some of the key properties of the space-time codes and can lead to substantial performance degradation. In this paper, we study the design of linear dispersion codes (LDCs) for such asynchronous cooperative communication networks. Firstly, the concept of conventional LDCs is extended to the delay-tolerant version and new design criteria are discussed. Then we propose a new design method to yield delay-tolerant LDCs that reach the optimal Jensen's upper bound on ergodic capacity as well as minimum average pairwise error probability. The proposed design employs stochastic gradient algorithm to approach a local optimum. Moreover, it is improved by using simulated annealing type optimization to increase the likelihood of the global optimum. The proposed method allows for flexible number of nodes, receive antennas, modulated symbols and flexible length of codewords. Simulation results confirm the performance of the newly-proposed delay-tolerant LDCs.

Power allocation strategies for distributed space-time codes in two-way relay networks

We study a two-way relay network (TWRN), where distributed space-time codes are constructed across multiple relay terminals in an amplify-and-forward mode. Each relay transmits a scaled linear combination of its received symbols and their conjugates,with the scaling factor chosen based on automatic gain control. We consider equal power allocation (EPA) across the relays, as well as the optimal power allocation (OPA) strategy given access to instantaneous channel state information (CSI). For EPA, we derive an upper bound on the pairwise-error-probability (PEP), from which we prove that full diversity is achieved in TWRNs. This result is in contrast to one-way relay networks, in which case a maximum diversity order of only unity can be obtained. When instantaneous CSI is available at the relays, we show that the OPA which minimizes the conditional PEP of the worse link can be cast as a generalized linear fractional program, which can be solved efficiently using the Dinkelback-type procedure.We also prove that, if the sum-power of the relay terminals is constrained, then the OPA will activate at most two relays.

Scenario and modelling uncertainty in global mean temperature change derived from emission-driven global climate models

We compare future changes in global mean temperature in response to different future scenarios which, for the first time, arise from emission-driven rather than concentration-driven perturbed parameter ensemble of a global climate model (GCM). These new GCM simulations sample uncertainties in atmospheric feedbacks, land carbon cycle, ocean physics and aerosol sulphur cycle processes. We find broader ranges of projected temperature responses arising when considering emission rather than concentration-driven simulations (with 10–90th percentile ranges of 1.7 K for the aggressive mitigation scenario, up to 3.9 K for the high-end, business as usual scenario). A small minority of simulations resulting from combinations of strong atmospheric feedbacks and carbon cycle responses show temperature increases in excess of 9 K (RCP8.5) and even under aggressive mitigation (RCP2.6) temperatures in excess of 4 K. While the simulations point to much larger temperature ranges for emission-driven experiments, they do not change existing expectations (based on previous concentration-driven experiments) on the timescales over which different sources of uncertainty are important. The new simulations sample a range of future atmospheric concentrations for each emission scenario. Both in the case of SRES A1B and the Representative Concentration Pathways (RCPs), the concentration scenarios used to drive GCM ensembles, lies towards the lower end of our simulated distribution. This design decision (a legacy of previous assessments) is likely to lead concentration-driven experiments to under-sample strong feedback responses in future projections. Our ensemble of emission-driven simulations span the global temperature response of the CMIP5 emission-driven simulations, except at the low end. Combinations of low climate sensitivity and low carbon cycle feedbacks lead to a number of CMIP5 responses to lie below our ensemble range. The ensemble simulates a number of high-end responses which lie above the CMIP5 carbon cycle range. These high-end simulations can be linked to sampling a number of stronger carbon cycle feedbacks and to sampling climate sensitivities above 4.5 K. This latter aspect highlights the priority in identifying real-world climate-sensitivity constraints which, if achieved, would lead to reductions on the upper bound of projected global mean temperature change. The ensembles of simulations presented here provides a framework to explore relationships between present-day observables and future changes, while the large spread of future-projected changes highlights the ongoing need for such work.

Averaging, slaving and balance dynamics in a simple atmospheric model

We report numerical results from a study of balance dynamics using a simple model of atmospheric motion that is designed to help address the question of why balance dynamics is so stable. The non-autonomous Hamiltonian model has a chaotic slow degree of freedom (representing vortical modes) coupled to one or two linear fast oscillators (representing inertia-gravity waves). The system is said to be balanced when the fast and slow degrees of freedom are separated. We find adiabatic invariants that drift slowly in time. This drift is consistent with a random-walk behaviour at a speed which qualitatively scales, even for modest time scale separations, as the upper bound given by Neishtadt’s and Nekhoroshev’s theorems. Moreover, a similar type of scaling is observed for solutions obtained using a singular perturbation (‘slaving’) technique in resonant cases where Nekhoroshev’s theorem does not apply. We present evidence that the smaller Lyapunov exponents of the system scale exponentially as well. The results suggest that the observed stability of nearly-slow motion is a consequence of the approximate adiabatic invariance of the fast motion.

Nonlinear symmetric stability of planetary atmospheres

The energy–Casimir method is applied to the problem of symmetric stability in the context of a compressible, hydrostatic planetary atmosphere with a general equation of state. Formal stability criteria for symmetric disturbances to a zonally symmetric baroclinic flow are obtained. In the special case of a perfect gas the results of Stevens (1983) are recovered. Finite-amplitude stability conditions are also obtained that provide an upper bound on a certain positive-definite measure of disturbance amplitude.

Nonlinear saturation of baroclinic instability: part I: the two-layer model

A rigorous bound is derived which limits the finite-amplitude growth of arbitrary nonzonal disturbances to an unstable baroclinic zonal flow within the context of the two-layer model. The bound is valid for conservative (unforced) flow, as well as for forced-dissipative flow that when the dissipation is proportional to the potential vorticity. The method used to derive the bound relies on the existence of a nonlinear Liapunov (normed) stability theorem for subcritical flows, which is a finite-amplitude generalization of the Charney-Stern theorem. For the special case of the Philips model of baroclinic instability, and in the limit of infinitesimal initial nonzonal disturbance amplitude, an improved form of the bound is possible which states that the potential enstrophy of the nonzonal flow cannot exceed ϵβ2, where ϵ = (U − Ucrit)/Ucrit is the (relative) supereriticality. This upper bound turns out to be extremely similar to the maximum predicted by the weakly nonlinear theory. For unforced flow with ϵ < 1, the bound demonstrates that the nonzonal flow cannot contain all of the potential enstrophy in the system; hence in this range of initial supercriticality the total flow must remain, in a certain sense, “close” to a zonal state.

Time development of small disturbances to plane Couette Flow

The question of linear sheared-disturbance evolution in constant-shear parallel flow is here reexamined with regard to the temporary-amplification phenomenon noted first by Orr in 1907. The results apply directly to Rossby waves on a beta-plane, and are also relevant to the Eady model of baroclinic instability. It is shown that an isotropic initial distribution of standing waves maintains a constant energy level throughout the shearing process, the amplification of some waves being precisely balanced by the decay of the others. An expression is obtained for the energy of a distribution of disturbances whose wavevectors lie within a given angular wedge and an upper bound derived. It is concluded that the case for ubiquitous amplification made in recent studies may have been somewhat overstated: while carefully-chosen individual Fourier components can amplify considerably before they decay. a general distribution will tend to exhibit little or no amplification.

Seasonal to interannual Arctic sea-ice predictability in current GCMs

We establish the first inter-model comparison of seasonal to interannual predictability of present-day Arctic climate by performing coordinated sets of idealized ensemble predictions with four state-of-the-art global climate models. For Arctic sea-ice extent and volume, there is potential predictive skill for lead times of up to three years, and potential prediction errors have similar growth rates and magnitudes across the models. Spatial patterns of potential prediction errors differ substantially between the models, but some features are robust. Sea-ice concentration errors are largest in the marginal ice zone, and in winter they are almost zero away from the ice edge. Sea-ice thickness errors are amplified along the coasts of the Arctic Ocean, an effect that is dominated by sea-ice advection. These results give an upper bound on the ability of current global climate models to predict important aspects of Arctic climate.