Now showing 1 - 10 of 15
  • Publication
    From Richardson to early numerical weather prediction
    (Cambridge University Press, 2010-12)
    The development of computer models for numerical simulation of the atmosphere and oceans is one of the great scientific triumphs of the past fifty years. These models have added enormously to our understanding of the complex processes in the atmosphere and oceans. The consequences for humankind of ongoing climate change will be far-reaching. Earth system models are the best means we have of predicting the future of our climate. The basic ideas of numerical forecasting and climate modeling were developed about a century ago, long before the first electronic computer was constructed. However, advances on several fronts were necessary before numerical prediction could be put into practice. A fuller understanding of atmospheric dynamics allowed the development of simplified systems of equations; regular observations of the free atmosphere provided the initial conditions; stable finite difference schemes were developed; and powerful electronic computers provided a practical means of carrying out the calculations required to predict the changes in the weather. In this chapter, we trace the history of computer forecasting from Richardson’s prodigious manual computation, through the ENIAC (Electronic Numerical Integrator and Computer) integrations to the early days of operational numerical weather prediction and climate modeling. The useful range of deterministic prediction is increasing by about one day each decade. We set the scene for the story of the remarkable progress in weather forecasting and in climate modeling over the past fifty years, which will be treated in subsequent chapters.
  • Publication
    Improving the Laplace transform integration method
    (Wiley, 2015-11-03) ;
    We consider the Laplace transform filtering integration scheme applied to the shallow water equations, and demonstrate how it can be formulated as a finite difference scheme in the time domain. In addition, we investigate a more accurate treatment of the non linear terms. The advantages of the resulting algorithms are demonstrated by means of numerical integrations.
      322Scopus© Citations 2
  • Publication
    The Fractal Boundary of the Power Tower Function
    (Associacao Ludus, 2017-08-23)
    We consider the function called the power tower function, defined by iterated exponentiation (or tetration) of the complex variable z. For real values x, it converges on the interval exp(−e)
  • Publication
    Pedro Nunes and the Retrogression of the Sun
    (Irish Mathematical Society, 2018)
    In northern latitudes we are used to the Sun rising in the East, following a smooth and even course through the southern sky and setting in the West. The idea that the compass bearing of the Sun might reverse seems fanciful. But that was precisely what Portuguese mathematician Pedro Nunes showed in 1537. Nunes made an amazing prediction: in certain circumstances, the shadow cast by the gnomon of a sun dial moves backwards.
  • Publication
    Weather and climate forecasting : chronicle of a revolution
    (World Meteorological Organization, 2010-06)
    Remarkable advances in weather forecasts during the past half-century have brought great benefits to humanity. Accurate forecasts save many lives, and early warnings mitigate the worst effects of extreme weather events, when they are available. Detailed, accurate forecasts are of huge economic value, with numerous studies showing that the benefits of forecasts outweigh the costs many times over. Advances in climate modeling over the past fifty years have also been outstanding. General circulation models have been developed and applied to examine the factors causing changes in our climate, and their likely timing and severity.
  • Publication
    A Lagrange-Laplace Integration Scheme for Weather Prediction and Climate Modelling
    (MDPI, 2022-09-27)
    A time integration scheme based on semi-Lagrangian advection and Laplace transform adjustment has been implemented in a baroclinic primitive equation model. The semi-Lagrangian scheme makes it possible to use large time steps. However, errors arising from the semi-implicit scheme increase with the time step size. In contrast, the errors using the Laplace transform adjustment remain relatively small for typical time steps used with semi-Lagrangian advection. Numerical experiments confirm the superior performance of the Laplace transform scheme relative to the semiimplicit reference model. The algorithmic complexity of the scheme is comparable to the reference model, making it computationally competitive, and indicating its potential for integrating weather and climate prediction models.
  • Publication
    Stokes's Fundamental Contributions to Fluid Dynamics
    (Oxford University Press, 2019-06-27)
    George Gabriel Stokes was one of the giants of hydrodynamics in the nineteenth century. He made fundamental mathematical contributions to fluid dynamics that had profound practical consequences. The basic equations formulated by him, the Navier-Stokes equations, are capable of describing fluid flows over a vast range of magnitudes. They play a central role in numerical weather prediction, in the simulation of blood flow in the body and in countless other important applications. In this chapter we put the primary focus on the two most important areas of Stokes’s work on fluid dynamics, the derivation of the Navier-Stokes equations and the theory of finite amplitude oscillatory water waves. Stokes became an undergraduate at Cambridge in 1837. He was coached by the ‘Senior Wrangler-maker’, William Hopkins and, in 1841, Stokes was Senior Wrangler and first Smith’s Prizeman. It was following a suggestion of Hopkins that Stokes took up the study of hydrodynamics, which was at that time a neglected area of study in Cambridge. Stokes was to make profound contributions to hydrodynamics, his most important being the rigorous establishment of the mathematical equations for fluid motions, and the theoretical explanation of a wide range of phenomena relating to wave motions in water.
  • Publication
    The two-child paradox : dichotomy and ambiguity
    (Irish Mathematical Society, 2011-07)
    Given that one of the children in a two-child family is a boy, what are the chances that the other is also a boy. The intuitive answer is 50 : 50. More careful investigation leads us to a 1-in-3 chance. We investigate circumstances under which these answers are correct. The imposition of further conditions yields some very surprising results.
  • Publication
    Laplace transform integration of the shallow-water equations. Part 1: Eulerian formulation and Kelvin waves
    (Wiley, 2011-04) ;
    A filtering integration scheme is developed, using a modification of the contour used to invert the Laplace transform (LT). It is shown to eliminate components with frequencies higher than a specified cut-off value. Thus it is valuable for integrations of the equations governing atmospheric flow. The scheme is implemented in a shallow water model with an Eulerian treatment of advection. It is compared to a reference model using the semi-implicit (SI) scheme. The LT scheme is shown to treat dynamically important Kelvin waves more accurately than the SI scheme.
      342Scopus© Citations 8
  • Publication
    Precession and recession of the rock'n'roller
    (IOP Publishing, 2009-09-30) ;
    We study the dynamics of a spherical rigid body that rocks and rolls on a plane under the effect of gravity. The distribution of mass is non-uniform and the centre of mass does not coincide with the geometric centre. The symmetric case, with moments of inertia I1 = I2 < I3, is integrable and themotion is completely regular. Three known conservation laws are the total energy E, Jellett’s quantity QJ and Routh’s quantity QR. When the inertial symmetry I1 = I2 is broken, even slightly, the character of the solutions is profoundly changed and new types of motion become possible. We derive the equations governing the general motion and present analytical and numerical evidence of the recession, or reversal of precession, that has been observed in physical experiments. We present an analysis of recession in terms of critical lines dividing the (QR,QJ ) plane into four dynamically disjoint zones. We prove that recession implies the lack of conservation of Jellett’s and Routh’s quantities, by identifying individual reversals as crossings of the orbit (QR(t ),QJ (t)) through the critical lines. Consequently, a method is found to produce a large number of initial conditions so that the system will exhibit recession.
      447Scopus© Citations 11