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.

Intensification of winter transatlantic aviation turbulence in response to climate change

Atmospheric turbulence causes most weather-related aircraft incidents1. Commercial aircraft encounter moderate-or-greater turbulence tens of thousands of times each year worldwide, injuring probably hundreds of passengers (occasionally fatally), costing airlines tens of millions of dollars and causing structural damage to planes1, 2, 3. Clear-air turbulence is especially difficult to avoid, because it cannot be seen by pilots or detected by satellites or on-board radar4, 5. Clear-air turbulence is linked to atmospheric jet streams6, 7, which are projected to be strengthened by anthropogenic climate change8. However, the response of clear-air turbulence to projected climate change has not previously been studied. Here we show using climate model simulations that clear-air turbulence changes significantly within the transatlantic flight corridor when the concentration of carbon dioxide in the atmosphere is doubled. At cruise altitudes within 50–75° N and 10–60° W in winter, most clear-air turbulence measures show a 10–40% increase in the median strength of turbulence and a 40–170% increase in the frequency of occurrence of moderate-or-greater turbulence. Our results suggest that climate change will lead to bumpier transatlantic flights by the middle of this century. Journey times may lengthen and fuel consumption and emissions may increase. Aviation is partly responsible for changing the climate9, but our findings show for the first time how climate change could affect aviation.

The composite-tendency RAW filter in semi-implicit integrations

The time discretization in weather and climate models introduces truncation errors that limit the accuracy of the simulations. Recent work has yielded a method for reducing the amplitude errors in leapfrog integrations from first-order to fifth-order. This improvement is achieved by replacing the Robert--Asselin filter with the RAW filter and using a linear combination of the unfiltered and filtered states to compute the tendency term. The purpose of the present paper is to apply the composite-tendency RAW-filtered leapfrog scheme to semi-implicit integrations. A theoretical analysis shows that the stability and accuracy are unaffected by the introduction of the implicitly treated mode. The scheme is tested in semi-implicit numerical integrations in both a simple nonlinear stiff system and a medium-complexity atmospheric general circulation model, and yields substantial improvements in both cases. We conclude that the composite-tendency RAW-filtered leapfrog scheme is suitable for use in semi-implicit integrations.

Sensitivity of a hydraulic model to changes in channel erosion during extreme flooding

Recent research into flood modelling has primarily concentrated on the simulation of inundation flow without considering the influences of channel morphology. River channels are often represented by a simplified geometry that is implicitly assumed to remain unchanged during flood simulations. However, field evidence demonstrates that significant morphological changes can occur during floods to mobilise the boundary sediments. Despite this, the effect of channel morphology on model results has been largely unexplored. To address this issue, the impact of channel cross-section geometry and channel long-profile variability on flood dynamics is examined using an ensemble of a 1D-2D hydraulic model (LISFLOOD-FP) of the 1:2102 year recurrence interval floods in Cockermouth, UK, within an uncertainty framework. A series of hypothetical scenarios of channel morphology were constructed based on a simple velocity based model of critical entrainment. A Monte-Carlo simulation framework was used to quantify the effects of channel morphology together with variations in the channel and floodplain roughness coefficients, grain size characteristics, and critical shear stress on measures of flood inundation. The results showed that the bed elevation modifications generated by the simplistic equations reflected a good approximation of the observed patterns of spatial erosion despite its overestimation of erosion depths. The effect of uncertainty on channel long-profile variability only affected the local flood dynamics and did not significantly affect the friction sensitivity and flood inundation mapping. The results imply that hydraulic models generally do not need to account for within event morphodynamic changes of the type and magnitude modelled, as these have a negligible impact that is smaller than other uncertainties, e.g. boundary conditions. Instead morphodynamic change needs to happen over a series of events to become large enough to change the hydrodynamics of floods in supply limited gravel-bed rivers like the one used in this research.

Stochastic climate theory and modeling

Stochastic methods are a crucial area in contemporary climate research and are increasingly being used in comprehensive weather and climate prediction models as well as reduced order climate models. Stochastic methods are used as subgrid-scale parameterizations (SSPs) as well as for model error representation, uncertainty quantification, data assimilation, and ensemble prediction. The need to use stochastic approaches in weather and climate models arises because we still cannot resolve all necessary processes and scales in comprehensive numerical weather and climate prediction models. In many practical applications one is mainly interested in the largest and potentially predictable scales and not necessarily in the small and fast scales. For instance, reduced order models can simulate and predict large-scale modes. Statistical mechanics and dynamical systems theory suggest that in reduced order models the impact of unresolved degrees of freedom can be represented by suitable combinations of deterministic and stochastic components and non-Markovian (memory) terms. Stochastic approaches in numerical weather and climate prediction models also lead to the reduction of model biases. Hence, there is a clear need for systematic stochastic approaches in weather and climate modeling. In this review, we present evidence for stochastic effects in laboratory experiments. Then we provide an overview of stochastic climate theory from an applied mathematics perspective. We also survey the current use of stochastic methods in comprehensive weather and climate prediction models and show that stochastic parameterizations have the potential to remedy many of the current biases in these comprehensive models.

Serum protein layers on parylene-C and silicon oxide: effect on cell adhesion

Among the range of materials used in bioengineering, parylene-C has been used in combination with silicon oxide and in presence of the serum proteins, in cell patterning. However, the structural properties of adsorbed serum proteins on these substrates still remain elusive. In this study, we use an optical biosensing technique to decipher the properties of fibronectin (Fn) and serum albumin adsorbed on parylene-C and silicon oxide substrates. Our results show the formation of layers with distinct structural and adhesive properties. Thin, dense layers are formed on parylene-C, whereas thicker, more diffuse layers are formed on silicon oxide. These results suggest that Fn acquires a compact structure on parylene-C and a more extended structure on silicon oxide. Nonetheless, parylene-C and silicon oxide substrates coated with Fn host cell populations that exhibit focal adhesion complexes and good cell attachment. Albumin adopts a deformed structure on parylene-C and a globular structure on silicon oxide, and does not support significant cell attachment on either surface. Interestingly, the co-incubation of Fn and albumin at the ratio found in serum, results in the preferential adsorption of albumin on parylene-C and Fn on silicon oxide. This finding is supported by the exclusive formation of focal adhesion complexes in differentiated mouse embryonic stem cells (CGR8), cultured on Fn/albumin coated silicon oxide, but not on parylene-C. The detailed information provided in this study on the distinct properties of layers of serum proteins on substrates such as parylene-C and silicon oxide is highly significant in developing methods for cell patterning.

Satellite-supported flood forecasting in river networks: a real case study

Satellite-based (e.g., Synthetic Aperture Radar [SAR]) water level observations (WLOs) of the floodplain can be sequentially assimilated into a hydrodynamic model to decrease forecast uncertainty. This has the potential to keep the forecast on track, so providing an Earth Observation (EO) based flood forecast system. However, the operational applicability of such a system for floods developed over river networks requires further testing. One of the promising techniques for assimilation in this field is the family of ensemble Kalman (EnKF) filters. These filters use a limited-size ensemble representation of the forecast error covariance matrix. This representation tends to develop spurious correlations as the forecast-assimilation cycle proceeds, which is a further complication for dealing with floods in either urban areas or river junctions in rural environments. Here we evaluate the assimilation of WLOs obtained from a sequence of real SAR overpasses (the X-band COSMO-Skymed constellation) in a case study. We show that a direct application of a global Ensemble Transform Kalman Filter (ETKF) suffers from filter divergence caused by spurious correlations. However, a spatially-based filter localization provides a substantial moderation in the development of the forecast error covariance matrix, directly improving the forecast and also making it possible to further benefit from a simultaneous online inflow error estimation and correction. Additionally, we propose and evaluate a novel along-network metric for filter localization, which is physically-meaningful for the flood over a network problem. Using this metric, we further evaluate the simultaneous estimation of channel friction and spatially-variable channel bathymetry, for which the filter seems able to converge simultaneously to sensible values. Results also indicate that friction is a second order effect in flood inundation models applied to gradually varied flow in large rivers. The study is not conclusive regarding whether in an operational situation the simultaneous estimation of friction and bathymetry helps the current forecast. Overall, the results indicate the feasibility of stand-alone EO-based operational flood forecasting.

The dynamics of Baroclinic Zonal jets

Multiple alternating zonal jets are a ubiquitous feature of planetary atmospheres and oceans. However, most studies to date have focused on the special case of barotropic jets. Here, the dynamics of freely evolving baroclinic jets are investigated using a two-layer quasigeostrophic annulus model with sloping topography. In a suite of 15 numerical simulations, the baroclinic Rossby radius and baroclinic Rhines scale are sampled by varying the stratification and root-mean-square eddy velocity, respectively. Small-scale eddies in the initial state evolve through geostrophic turbulence and accelerate zonally as they grow in horizontal scale, first isotropically and then anisotropically. This process leads ultimately to the formation of jets, which take about 2500 rotation periods to equilibrate. The kinetic energy spectrum of the equilibrated baroclinic zonal flow steepens from a −3 power law at small scales to a −5 power law near the jet scale. The conditions most favorable for producing multiple alternating baroclinic jets are large baroclinic Rossby radius (i.e., strong stratification) and small baroclinic Rhines scale (i.e., weak root-mean-square eddy velocity). The baroclinic jet width is diagnosed objectively and found to be 2.2–2.8 times larger than the baroclinic Rhines scale, with a best estimate of 2.5 times larger. This finding suggests that Rossby wave motions must be moving at speeds of approximately 6 times the turbulent eddy velocity in order to be capable of arresting the isotropic inverse energy cascade.

Note: A balloon-borne accelerometer technique for measuring atmospheric turbulence

A weather balloon and its suspended instrument package behave like a pendulum with a moving pivot. This dynamical system is exploited here for the detection of atmospheric turbulence. By adding an accelerometer to the instrument package, the size of the swings induced by atmospheric turbulence can be measured. In test flights, strong turbulence has induced accelerations greater than 5g, where g = 9.81 m s−2. Calibration of the accelerometer data with a vertically orientated lidar has allowed eddy dissipation rate values of between 10−3 and 10−2 m2 s−3 to be derived from the accelerometer data. The novel use of a whole weather balloon and its adapted instrument package can be used as a new instrument to make standardized in situ measurements of turbulence.

Is there a Rhythm Of The Rain? An analysis of weather in popular music

Weather is frequently used in music to frame events and emotions, yet quantitative analyses are rare. From a collated base set of 759 weather-related songs, 419 were analysed based on listings from a karaoke database. This article analyses the 20 weather types described, frequency of occurrence, genre, keys, mimicry, lyrics and songwriters. Vocals were the principal means of communicating weather: sunshine was the most common, followed by rain, with weather depictions linked to the emotions of the song. Bob Dylan, John Lennon and Paul McCartney wrote the most weather-related songs, partly following their experiences at the time of writing.

Matching roots to their environment

Background Plants form the base of the terrestrial food chain and provide medicines, fuel, fibre and industrial materials to humans. Vascular land plants rely on their roots to acquire the water and mineral elements necessary for their survival in nature or their yield and nutritional quality in agriculture. Major biogeochemical fluxes of all elements occur through plant roots, and the roots of agricultural crops have a significant role to play in soil sustainability, carbon sequestration, reducing emissions of greenhouse gasses, and in preventing the eutrophication of water bodies associated with the application of mineral fertilisers. ● Scope This article provides the context for a Special Issue of Annals of Botany on ‘Matching Roots to Their Environment’. It first examines how land plants and their roots evolved, describes how the ecology of roots and their rhizospheres contributes to the acquisition of soil resources, and discusses the influence of plant roots on biogeochemical cycles. It then describes the role of roots in overcoming the constraints to crop production imposed by hostile or infertile soils, illustrates root phenotypes that improve the acquisition of mineral elements and water, and discusses high-throughput methods to screen for these traits in the laboratory, glasshouse and field. Finally, it considers whether knowledge of adaptations improving the acquisition of resources in natural environments can be used to develop root systems for sustainable agriculture in the future.

An investigation of the role of business analysts in IS development

Business analysis has developed since the early 1990s as an IS discipline that is concerned with understanding business problems, defining requirements and evaluating relevant solutions. However, this discipline has had limited recognition within the academic community and little research has been conducted into the practices and standards employed by business analysts. This paper reports on a study into business analysis that considered the activities conducted and the outcomes experienced on IS projects. Senior business analysts were interviewed in order to gain insights into the business analyst role and the techniques and approaches applied when conducting this work. The Context, Content, Process, Outcomes framework was adopted as a basis for developing the interview questions. The data collected was analysed using the template analysis technique and the template was based upon this framework. Additional themes concerning aspects of business analysis that may contribute to IS success emerged during data analysis. These included the key business analysis activities and the skills business analysts require to perform these activities. The organisational attitude was also identified as a key factor in enabling the use and contribution of business analysis.

Efficient incorporation of channel cross-section geometry uncertainty into regional and global scale flood inundation models

This paper investigates the challenge of representing structural differences in river channel cross-section geometry for regional to global scale river hydraulic models and the effect this can have on simulations of wave dynamics. Classically, channel geometry is defined using data, yet at larger scales the necessary information and model structures do not exist to take this approach. We therefore propose a fundamentally different approach where the structural uncertainty in channel geometry is represented using a simple parameterization, which could then be estimated through calibration or data assimilation. This paper first outlines the development of a computationally efficient numerical scheme to represent generalised channel shapes using a single parameter, which is then validated using a simple straight channel test case and shown to predict wetted perimeter to within 2% for the channels tested. An application to the River Severn, UK is also presented, along with an analysis of model sensitivity to channel shape, depth and friction. The channel shape parameter was shown to improve model simulations of river level, particularly for more physically plausible channel roughness and depth parameter ranges. Calibrating channel Manning’s coefficient in a rectangular channel provided similar water level simulation accuracy in terms of Nash-Sutcliffe efficiency to a model where friction and shape or depth were calibrated. However, the calibrated Manning coefficient in the rectangular channel model was ~2/3 greater than the likely physically realistic value for this reach and this erroneously slowed wave propagation times through the reach by several hours. Therefore, for large scale models applied in data sparse areas, calibrating channel depth and/or shape may be preferable to assuming a rectangular geometry and calibrating friction alone.