The Unified Method: I Non-Linearizable Problems on the Half-Line

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving four scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant, and on the computation of the large $k$ asymptotics of the eigenfunctions defining the relevant spectral functions.Comment: 39 page

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

River water level height measurements obtained from river cameras near Tewkesbury

This dataset contains extracted water level observations (WLOs) from four river cameras located on rivers Severn and Avon near Tewkesbury, UK, and owned by Farson Digital Ltd The data is extracted from hourly daylight river camera images between 21st Nov – 5th Dec 2012. The dataset consists of spreadsheets containing extracted water level data for each camera along with necessary metadata, all available Farson Digital Ltd river camera images between 21st Nov – 5th Dec 2012, and 3D point measurements of the location of each camera and locations of all measured points in the field-of-view for each camera

Ensemble data assimilation in the presence of cloud

In numerical weather prediction (NWP) data assimilation (DA) methods are used to combine available observations with numerical model estimates. This is done by minimising measures of error on both observations and model estimates with more weight given to data that can be more trusted. For any DA method an estimate of the initial forecast error covariance matrix is required. For convective scale data assimilation, however, the properties of the error covariances are not well understood. An effective way to investigate covariance properties in the presence of convection is to use an ensemble-based method for which an estimate of the error covariance is readily available at each time step. In this work, we investigate the performance of the ensemble square root filter (EnSRF) in the presence of cloud growth applied to an idealised 1D convective column model of the atmosphere. We show that the EnSRF performs well in capturing cloud growth, but the ensemble does not cope well with discontinuities introduced into the system by parameterised rain. The state estimates lose accuracy, and more importantly the ensemble is unable to capture the spread (variance) of the estimates correctly. We also find, counter-intuitively, that by reducing the spatial frequency of observations and/or the accuracy of the observations, the ensemble is able to capture the states and their variability successfully across all regimes

Advances in the study of boundary value problems for nonlinear integrable PDEs

In this review I summarise some of the most significant advances of the last decade in the analysis and solution of boundary value problems for integrable partial differential equations in two independent variables. These equations arise widely in mathematical physics, and in order to model realistic applications, it is essential to consider bounded domain and inhomogeneous boundary conditions. I focus specifically on a general and widely applicable approach, usually referred to as the Unified Transform or Fokas Transform, that provides a substantial generalisation of the classical Inverse Scattering Transform. This approach preserves the conceptual efficiency and aesthetic appeal of the more classical transform approaches, but presents a distinctive and important difference. While the Inverse Scattering Transform follows the "separation of variables" philosophy, albeit in a nonlinear setting, the Unified Transform is a based on the idea of synthesis, rather than separation, of variables. I will outline the main ideas in the case of linear evolution equations, and then illustrate their generalisation to certain nonlinear cases of particular significance

The approach of physiatrists to low back pain across Europe

Low back pain (LBP) is the most common type of musculoskeletal pain, thus it is one of the most commonly encountered conditions in Physical and Rehabilitation Medicine. The physicians who are primarily responsible for the nonsurgical management of LBP are physiatrists

White book on physical and rehabilitation medicine (PRM) in Europe : Chapter 3. A primary medical specialty: The fundamentals of PRM

In the context of the White Book of Physical and Rehabilitation Medicine (PRM) in Europe, this paper deals with the core concepts at the base of the PRM specialty. These are the essential constituents that make PRM a primary medical specialty, different from all the other medical specialties, and PRM physician the primary medical specialist among the rehabilitation professionals. The core concepts that will be discussed in this Section include: - PRM is a person/functioning oriented specialty, and this makes the specialty different from the organ/disease oriented, or treatment/age specific medical specialties - PRM physicians have medical responsibilities, like all the other medical specialists, but with an additional specificity of making a functional assessment - Like the other specialists, PRM physicians provide direct treatments, but they also work leading the multi-professional rehabilitation team, that works in a collaborative way with other professionals and medical specialists - Due to its function oriented approach, PRM has a multimodal approach including a wide variety of treatment tools (frequently provided by other rehabilitation professionals) and manages all persons' morbidities (health conditions), since it focuses on decreasing impairments and activity limitations to allow the best possible participation of patients - As PRM bases its work on functioning, it has a transversal role to other specialties: it overlaps with several of them, sharing part of their knowledge, but it is also totally independent from all of them, since it is based on a different and transversal body of knowledge - PRM is focused on the person and neither on the disease nor on the setting; in fact, PRM is not only transversal to specialties, but also to the settings of care, and PRM physicians should know these different realities: persons with disabilities and those with long-term health conditions in fact move inside the national health systems between various facilities to obtain the best possible functioning and participation through an appropriate rehabilitation process

White book on physical and rehabilitation medicine (PRM) in Europe. Chapter 7.The clinical field of competence: PRM in practice

In the context of the White Book on Physical and Rehabilitation Medicine (PRM) in Europe this paper deals with the scope and competencies of PRM starting from its definition as the "medicine of functioning." PRM uses the rehabilitative health strategy as its core strategy together with the curative strategy. According to the complexity of disabling health conditions, PRM also refers to prevention and maintenance and provides information to the patients and other caregivers. The rehabilitation process according to the so-called rehabilitation cycle including an assessment and definition of the (individual) rehabilitation goals, assignment to the rehabilitation program evaluation of individual outcomes. PRM physicians treat a wide spectrum of diseases and take a transversal across most of the medical specialties. They also focus on many functional problems such as immobilization, spasticity, pain syndromes, communication disorders, and others. The diagnosis in PRM is the interaction between the medical diagnosis and a PRM-specific functional assessment. The latter is based on the ICF conceptual framework, and obtained through functional evaluations and scales: these are classified according to their main focus on impairments, activity limitations or participation restrictions; environmental and personal factors are included as barriers or facilitators. Interventions in PRM are either provided directly by PRM physicians or within the PRM team. They include a wide range of treatments, including medicines, physical therapies, exercises, education and many others. Standardized PRM programs are available for many diseases and functional problems. In most cases rehabilitation is performed in multi-professional teams working in a collaborative way, as well as with other disciplines under the leadership of a PRM physician and it is a patient-centered approach. Outcomes of PRM interventions and programs, showed reduction of impairments in body functions, activity limitations, and impacting on participation restrictions, and also reduction in costs as well as decrease in mortality for certain groups of patients