Calculating and communicating ensemble-based volcanic ash dosage and concentration risk for aviation

During volcanic eruptions, aviation stakeholders require an assessment of the volcanic ash hazard. Operators and regulators are required to make fast decisions based on deterministic forecasts, which are subject to various sources of uncertainty. For a robust decision to be made, a measure of the uncertainty of the hazard should be considered but this can lead to added complexity preventing fast decision making. Here a proof-of-concept risk matrix approach is presented that combines uncertainty estimation and volcanic ash hazard forecasting into a simple warning system for aviation. To demonstrate the methodology, an ensemble of 600 dispersion model simulations is used to characterise uncertainty (due to eruption source parameters, meteorology and internal model parameters) in ash dosages and concentrations for a hypothetical Icelandic eruption. To simulate aircraft encounters with volcanic ash, trans-Atlantic air routes between New York (JFK) and London (LHR) are generated using time-optimal routing software. This approach has been developed in collaboration with operators, regulators and engine manufacturers; it demonstrates how an assessment of ash dosage and concentration risk can be used to make fast and robust flight-planning decisions even 23 when the model uncertainty spans several orders of magnitude. The results highlight the benefit of using an ensemble over a deterministic forecast and a new method for visualising dosage risk along flight paths. The risk matrix approach is applicable to other aviation hazards such as SO2 dosages, desert dust, aircraft icing and clear-air turbulence and is expected to aid flight-planning decisions by improving the communication of ensemble-based forecasts to aviation

Global W2,pW^{2,p} estimates for solutions to the linearized Monge--Amp\`ere equations

In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge-Amp\`ere equations under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin's global $W^{2,p}$ estimates for the Monge-Amp\`ere equations.Comment: v2: presentation improve

Balance conditions in variational data assimilation for a high-resolution forecast model

This paper explores the role of balance relationships for background error covariance modelling as the model's grid box decreases to convective-scales. Data assimilation (DA) analyses are examined from a simplified convective-scale model and DA system (called ABC-DA) with a grid box size of 1.5km in a 2D 540km (longitude), 15km (height) domain. The DA experiments are performed with background error covariance matrices B modelled and calibrated by switching on/off linear balance (LB) and hydrostatic balance (HB), and by observing a subset of the ABC variables, namely v, meridional wind, r', scaled density (a pressure-like variable), and b', buoyancy (a temperature-like variable). Calibration data are sourced from two methods of generating proxies of forecast errors. One uses forecasts from different latitude slices of a 3D parent model (here called the `latitude slice method'), and the other uses sets of differences between forecasts of different lengths but valid at the same time (the National Meteorological Center method). Root-mean-squared errors computed over the domain from identical twin DA experiments suggest that there is no combination of LB/HB switches that give the best analysis for all model quantities. It is frequently found though that the B-matrices modelled with both LB and HB do perform the best. A clearer picture emerges when the errors are examined at different spatial scales. In particular it is shown that switching on HB in B mostly has a neutral/positive effect on the DA accuracy at `large' scales, and switching off the HB has a neutral/positive effect at `small' scales. The division between `large' and `small' scales is between 10 and 100km. Furthermore, one hour forecast error correlations computed between control parameters find that correlations are small at large scales when balances are enforced, and at small scales when balances are not enforced (ideal control parameters have zero cross correlations). This points the way to modelling B with scale-dependent balances

A regime-dependent balanced control variable based on potential vorticity

In this paper it is argued that rotational wind is not the best choice of leading control variable for variational data assimilation, and an alternative is suggested and tested. A rotational wind parameter is used in most global variational assimilation systems as a pragmatic way of approximately representing the balanced component of the assimilation increments. In effect, rotational wind is treated as a proxy for potential vorticity, but one that it is potentially not a good choice in flow regimes characterised by small Burger number. This paper reports on an alternative set of control variables which are based around potential vorticity. This gives rise to a new formulation of the background error covariances for the Met Office's variational assimilation system, which leads to flow dependency. It uses similar balance relationships to traditional schemes, but recognises the existence of unbalanced rotational wind which is used with a new anti-balance relationship. The new scheme is described and its performance is evaluated and compared to a traditional scheme using a sample of diagnostics

Introduction: Mathematics applied to the climate system: outstanding challenges and recent progress

The societal need for reliable climate predictions and a proper assessment of their uncertainties is pressing. Uncertainties arise not only from initial conditions and forcing scenarios, but also from model formulation. Here, we identify and document three broad classes of problems, each representing what we regard to be an outstanding challenge in the area of mathematics applied to the climate system. First, there is the problem of the development and evaluation of simple physically based models of the global climate. Second, there is the problem of the development and evaluation of the components of complex models such as general circulation models. Third, there is the problem of the development and evaluation of appropriate statistical frameworks. We discuss these problems in turn, emphasizing the recent progress made by the papers presented in this Theme Issue. Many pressing challenges in climate science require closer collaboration between climate scientists, mathematicians and statisticians. We hope the papers contained in this Theme Issue will act as inspiration for such collaborations and for setting future research directions