Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces Hs(Ω) and tildeHs(Ω), for s in R and an open Ω in R^n. We exhibit examples in one and two dimensions of sets Ω for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if Ω is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large.

Probabilistic parameterisation of the surface mass balance--elevation feedback in regional climate model simulations of the Greenland ice sheet

We present a new parameterisation that relates surface mass balance (SMB: the sum of surface accumulation and surface ablation) to changes in surface elevation of the Greenland ice sheet (GrIS) for the MAR (Modèle Atmosphérique Régional: Fettweis, 2007) regional climate model. The motivation is to dynamically adjust SMB as the GrIS evolves, allowing us to force ice sheet models with SMB simulated by MAR while incorporating the SMB–elevation feedback, without the substantial technical challenges of coupling ice sheet and climate models. This also allows us to assess the effect of elevation feedback uncertainty on the GrIS contribution to sea level, using multiple global climate and ice sheet models, without the need for additional, expensive MAR simulations. We estimate this relationship separately below and above the equilibrium line altitude (ELA, separating negative and positive SMB) and for regions north and south of 77� N, from a set of MAR simulations in which we alter the ice sheet surface elevation. These give four “SMB lapse rates”, gradients that relate SMB changes to elevation changes. We assess uncertainties within a Bayesian framework, estimating probability distributions for each gradient from which we present best estimates and credibility intervals (CI) that bound 95% of the probability. Below the ELA our gradient estimates are mostly positive, because SMB usually increases with elevation: 0.56 (95% CI: −0.22 to 1.33) kgm−3 a−1 for the north, and 1.91 (1.03 to 2.61) kgm−3 a−1 for the south. Above the ELA, the gradients are much smaller in magnitude: 0.09 (−0.03 to 0.23) kgm−3 a−1 in the north, and 0.07 (−0.07 to 0.59) kgm−3 a−1 in the south, because SMB can either increase or decrease in response to increased elevation. Our statistically founded approach allows us to make probabilistic assessments for the effect of elevation feedback uncertainty on sea level projections (Edwards et al., 2014).

Effect of uncertainty in surface mass balance--elevation feedback on projections of the future sea level contribution of the Greenland ice sheet

We apply a new parameterisation of the Greenland ice sheet (GrIS) feedback between surface mass balance (SMB: the sum of surface accumulation and surface ablation) and surface elevation in the MAR regional climate model (Edwards et al., 2014) to projections of future climate change using five ice sheet models (ISMs). The MAR (Modèle Atmosphérique Régional: Fettweis, 2007) climate projections are for 2000–2199, forced by the ECHAM5 and HadCM3 global climate models (GCMs) under the SRES A1B emissions scenario. The additional sea level contribution due to the SMB– elevation feedback averaged over five ISM projections for ECHAM5 and three for HadCM3 is 4.3% (best estimate; 95% credibility interval 1.8–6.9 %) at 2100, and 9.6% (best estimate; 95% credibility interval 3.6–16.0 %) at 2200. In all results the elevation feedback is significantly positive, amplifying the GrIS sea level contribution relative to the MAR projections in which the ice sheet topography is fixed: the lower bounds of our 95% credibility intervals (CIs) for sea level contributions are larger than the “no feedback” case for all ISMs and GCMs. Our method is novel in sea level projections because we propagate three types of modelling uncertainty – GCM and ISM structural uncertainties, and elevation feedback parameterisation uncertainty – along the causal chain, from SRES scenario to sea level, within a coherent experimental design and statistical framework. The relative contributions to uncertainty depend on the timescale of interest. At 2100, the GCM uncertainty is largest, but by 2200 both the ISM and parameterisation uncertainties are larger. We also perform a perturbed parameter ensemble with one ISM to estimate the shape of the projected sea level probability distribution; our results indicate that the probability density is slightly skewed towards higher sea level contributions.

The excitation of ionospheric convection

This review presents recent observations of high-latitude ionospheric plasma convection, obtained using the EISCAT radar in the 'Polar' experiment mode. The paper is divided into two main parts. Firstly, the delay in the response of dayside high-latitude flows to changes in the interplanetary magnetic field is discussed. The results show the importance for the excitation of dayside convection of the transfer of magnetic flux from the dayside into the tail lobe. Consequently, ionospheric convection should be thought of as the sum of two intrinsically time-dependent flow patterns. The first of these patterns is directly driven by solar wind-magnetosphere coupling, dominates ionospheric flows on the dayside, is associated with an expanding polar cap area and is the F-region flow equivalent of the DP-2 E-region current system. The second of the two patterns is driven by the release of energy stored in the geomagnetic tail, dominates ionospheric flows on the nightside, is associated with a contracting polar cap and is equivalent to the DP-1, or substorm, current system. In the second half of the paper, various transient flow bursts observed in the vicinity of the dayside cusp are studied. These radar data, combined with simultaneous optical observations of transient dayside aurorae, strongly suggest that momentum is transferred across the magnetopause and into the ionosphere in a series of bursts, each associated with voltages of 30-80 kV. Similarities between these bursts and flux transfer events observed at the magnetopause are discussed.

The excitation of plasma convection in the high-latitude ionosphere

Recent observations of ionospheric flows by ground-based radars, in particular by the European Incoherent Scatter (EISCAT) facility using the “Polar” experiment, together with previous analyses of the response of geomagnetic disturbance to variations of the interplanetary magnetic field (IMF), suggest that convection in the high-latitude ionosphere should be considered to be the sum of two intrinsically time-dependent patterns, one driven by solar wind-magnetosphere coupling at the dayside magnetopause, the other by the release of energy in the geomagnetic tail (mainly by dayside and nightside reconnection, respectively). The flows driven by dayside coupling are largest on the dayside, where they usually dominate, are associated with an expanding polar cap area, and are excited and decay on ∼10-min time scales following southward and northward turnings of the IMF, respectively. The latter finding indicates that the production of new open flux at the dayside magnetopause excites magnetospheric and ionospheric flow only for a short interval, ∼10 min, such that the flow driven by this source subsequently decays on this time scale unless maintained by the production of more open flux tubes. Correspondingly, the flows excited by the release of energy in the tail, mainly during substorms, are largest on the nightside, are associated with a contracting polar cap boundary, and are excited on ∼1-hour time scales following a southward turn of the IMF. In general, the total ionospheric flow will be the sum of the flows produced by these two sources, such that due to their different response times to changes in the IMF, considerable variations in the flow pattern can occur for a given direction and strength of the IMF. Consequently, the ionospheric electric field cannot generally be regarded as arising from a simple mapping of the solar wind electric field along open flux tubes.

Using lexical diversity measures to operationalise language dominance in bilinguals

Treffers-Daller and Korybski propose to operationalize language dominance on the basis of measures of lexical diversity, as computed, in this particular study, on transcripts of stories told by Polish-English bilinguals in each of their languages They compute four different Indices of Language Dominance (ILD) on the basis of two different measures of lexical diversity, the Index of Guiraud (Guiraud, 1954) and HD-D (McCarthy & Jarvis, 2007). They compare simple indices, which are based on subtracting scores from one language from scores for another language, to more complex indices based on the formula Birdsong borrowed from the field of handedness, namely the ratio of (Difference in Scores) / (Sum of Scores). Positive scores on each of these Indices of Language Dominance mean that informants are more English-dominant and negative scores that they are more Polish-dominant. The authors address the difficulty of comparing scores across languages by carefully lemmatizing the data. Following Flege, Mackay and Piske (2002) they also look into the validity of these indices by investigating to what extent they can predict scores on other, independently measured variables. They use correlations and regression analysis for this, which has the advantage that the dominance indices are used as continuous variables and arbitrary cut-off points between balanced and dominant bilinguals need not be chosen. However, they also show how the computation of z-scores can help facilitate a discussion about the appropriateness of different cut-off points across different data sets and measurement scales in those cases where researchers consider it necessary to make categorial distinctions between balanced and dominant bilinguals. Treffers-Daller and Korybski correlate the ILD scores with four other variables, namely Length of Residence in the UK, attitudes towards English and life in the UK, frequency of usage of English at home and frequency of code-switching. They found that the indices correlated significantly with most of these variables, but there were clear differences between the Guiraud-based indices and the HDD-based indices. In a regression analysis three of the measures were also found to be a significant predictor of English language usage at home. They conclude that the correlations and the regression analyses lend strong support to the validity of their approach to language dominance.

Radiative forcing and climate metrics for ozone precursor emissions: the impact of multi-model averaging

Multi-model ensembles are frequently used to assess understanding of the response of ozone and methane lifetime to changes in emissions of ozone precursors such as NOx, VOCs (volatile organic compounds) and CO. When these ozone changes are used to calculate radiative forcing (RF) (and climate metrics such as the global warming potential (GWP) and global temperature-change potential (GTP)) there is a methodological choice, determined partly by the available computing resources, as to whether the mean ozone (and methane) concentration changes are input to the radiation code, or whether each model's ozone and methane changes are used as input, with the average RF computed from the individual model RFs. We use data from the Task Force on Hemispheric Transport of Air Pollution source–receptor global chemical transport model ensemble to assess the impact of this choice for emission changes in four regions (East Asia, Europe, North America and South Asia). We conclude that using the multi-model mean ozone and methane responses is accurate for calculating the mean RF, with differences up to 0.6% for CO, 0.7% for VOCs and 2% for NOx. Differences of up to 60% for NOx 7% for VOCs and 3% for CO are introduced into the 20 year GWP. The differences for the 20 year GTP are smaller than for the GWP for NOx, and similar for the other species. However, estimates of the standard deviation calculated from the ensemble-mean input fields (where the standard deviation at each point on the model grid is added to or subtracted from the mean field) are almost always substantially larger in RF, GWP and GTP metrics than the true standard deviation, and can be larger than the model range for short-lived ozone RF, and for the 20 and 100 year GWP and 100 year GTP. The order of averaging has most impact on the metrics for NOx, as the net values for these quantities is the residual of the sum of terms of opposing signs. For example, the standard deviation for the 20 year GWP is 2–3 times larger using the ensemble-mean fields than using the individual models to calculate the RF. The source of this effect is largely due to the construction of the input ozone fields, which overestimate the true ensemble spread. Hence, while the average of multi-model fields are normally appropriate for calculating mean RF, GWP and GTP, they are not a reliable method for calculating the uncertainty in these fields, and in general overestimate the uncertainty.

Metrics for linking emissions of gases and aerosols to global precipitation changes

Recent advances in understanding have made it possible to relate global precipitation changes directly to emissions of particular gases and aerosols that influence climate. Using these advances, new indices are developed here called the Global Precipitation-change Potential for pulse (GPP_P) and sustained (GPP_S) emissions, which measure the precipitation change per unit mass of emissions. The GPP can be used as a metric to compare the effects of different emissions. This is akin to the global warming potential (GWP) and the global temperature-change potential (GTP) which are used to place emissions on a common scale. Hence the GPP provides an additional perspective of the relative or absolute effects of emissions. It is however recognised that precipitation changes are predicted to be highly variable in size and sign between different regions and this limits the usefulness of a purely global metric. The GPP_P and GPP_S formulation consists of two terms, one dependent on the surface temperature change and the other dependent on the atmospheric component of the radiative forcing. For some forcing agents, and notably for CO2, these two terms oppose each other – as the forcing and temperature perturbations have different timescales, even the sign of the absolute GPP_P and GPP_S varies with time, and the opposing terms can make values sensitive to uncertainties in input parameters. This makes the choice of CO2 as a reference gas problematic, especially for the GPP_S at time horizons less than about 60 years. In addition, few studies have presented results for the surface/atmosphere partitioning of different forcings, leading to more uncertainty in quantifying the GPP than the GWP or GTP. Values of the GPP_P and GPP_S for five long- and short-lived forcing agents (CO2, CH4, N2O, sulphate and black carbon – BC) are presented, using illustrative values of required parameters. The resulting precipitation changes are given as the change at a specific time horizon (and hence they are end-point metrics) but it is noted that the GPPS can also be interpreted as the time-integrated effect of a pulse emission. Using CO2 as a references gas, the GPP_P and GPP_S for the non-CO2 species are larger than the corresponding GTP values. For BC emissions, the atmospheric forcing is sufficiently strong that the GPP_S is opposite in sign to the GTP_S. The sensitivity of these values to a number of input parameters is explored. The GPP can also be used to evaluate the contribution of different emissions to precipitation change during or after a period of emissions. As an illustration, the precipitation changes resulting from emissions in 2008 (using the GPP_P) and emissions sustained at 2008 levels (using the GPP_S) are presented. These indicate that for periods of 20 years (after the 2008 emissions) and 50 years (for sustained emissions at 2008 levels) methane is the dominant driver of positive precipitation changes due to those emissions. For sustained emissions, the sum of the effect of the five species included here does not become positive until after 50 years, by which time the global surface temperature increase exceeds 1 K.