arXiv subjects

Paul D. Williams

Publications and source records attributed to Paul D. Williams.

5 recordsLinked to original sources

arXiv2026

Integral constraints on the linear instability of stratified flow with planar shear at an arbitrary angle to the vertical

Integral constraints on the linear instability of stratified parallel flow with planar shear at an arbitrary angle to the vertical are derived using the analytical approach of Miles and Howard, for perturbations with 2D spatial structure, which are thought to be the most unstable. The general stability formulation reproduces the Miles-Howard stability criterion for vertical shear, but yields no stability condition for non-vertical shear, confirming expectations from earlier studies. This study also extends Howard's semicircle theorem to non-vertical planar shear, and derives a new expression for the upper bound of the instability growth rate (extending that obtained by Howard), which is consistent with published numerical results.

physics.flu-dyn↗

arXiv2026

Detecting clear-air turbulence via beam broadening in a Rayleigh-scattering lidar system

The volume of clear-air turbulence (CAT) in the atmosphere at flight cruising altitudes is increasing rapidly, posing a growing problem for civil aviation and resulting in reduced confidence in aviation safety. There are limited remote detection capabilities for CAT, since clear air produces no measurable radar return. Lidar has been proposed as a viable detection methodology, and several systems have been demonstrated. However, these systems have to date demonstrated limited detection ranges of less than 15 km. In this work, we propose a novel lidar-based CAT detection methodology that uses Rayleigh scattering and relies on a differential detector measurement to quantify beam spread and thereby estimate the eddy dissipation rate (EDR), which is the international aircraft-independent metric for quantifying aviation turbulence strength. Additionally, we present experimental results demonstrating the validity of the optical efficiency model used in the detection simulations. We show that, under modest assumptions, a size, weight, and power (SWAP) constrained system that implements this method can detect moderate CAT at ranges in excess of 30 km, equating to two minutes of flight time for typical commercial aviation cruising speeds, which represents a substantial range improvement over prior approaches. This is an important advance because-for the first time-it potentially allows the cabin to be secured before the turbulence is encountered, reducing the injury risk to passengers and flight attendants.

physics.optics↗

arXiv2026

A Generalized Richardson Number Diagnostic for Turbulence in the Free Atmosphere

A new Richardson number formulation, Ri_new, is introduced to improve the diagnosis of turbulence in the stratified free atmosphere. The formulation is derived from the turbulent kinetic energy budget and accounts for both vertical wind shear and horizontal shear (deformation and divergence), weighted by the ratio of horizontal to vertical eddy viscosities (K_mh/K_mv). This extends the classical Richardson number Ri_old, which accounts only for vertical shear. The diagnostics Ri_new , Ri_old ,and the widely used Turbulence Index 1 (TI1), computed from ERA5 reanalysis, are evaluated using more than 247 million automated turbulence reports from commercial aircraft (2017--2024). Across various turbulence intensity thresholds, Ri_new consistently outperforms the other diagnostics, resulting in higher AUC values and improved probability of detection at operationally relevant false-alarm rates. The highest skill is obtained for K_mh/K_mv approximately 5000. Seasonal and regional evaluations indicate that the added value of Ri_new is largest where turbulence generation involves both vertical and horizontal shear, such as over the contiguous United States and during summer. Ri_new remains the best-performing diagnostic in all regions and seasons. Spatial case studies show that Ri_new identifies 83--98% of observed moderate-or-greater turbulence events compared with 54--85% for Ri_old. This substantial improvement in detection comes with a much smaller increase in false alarms, confirming that Ri_new provides a more physically realistic representation of turbulence-prone regions. These results demonstrate that incorporating horizontal wind shear into the Richardson number yields a physically consistent and statistically robust improvement in turbulence diagnostics, with relevance for research and operational applications.

physics.ao-ph↗

arXiv2017

Stochastic Parameterization: Towards a new view of Weather and Climate Models

The last decade has seen the success of stochastic parameterizations in short-term, medium-range and seasonal forecasts: operational weather centers now routinely use stochastic parameterization schemes to better represent model inadequacy and improve the quantification of forecast uncertainty. Developed initially for numerical weather prediction, the inclusion of stochastic parameterizations not only provides better estimates of uncertainty, but it is also extremely promising for reducing longstanding climate biases and relevant for determining the climate response to external forcing. This article highlights recent developments from different research groups which show that the stochastic representation of unresolved processes in the atmosphere, oceans, land surface and cryosphere of comprehensive weather and climate models (a) gives rise to more reliable probabilistic forecasts of weather and climate and (b) reduces systematic model bias. We make a case that the use of mathematically stringent methods for the derivation of stochastic dynamic equations will lead to substantial improvements in our ability to accurately simulate weather and climate at all scales. Recent work in mathematics, statistical mechanics and turbulence is reviewed, its relevance for the climate problem demonstrated, and future research directions outlined.

physics.ao-ph↗

arXiv2014

Stochastic Climate Theory and Modelling

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 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 modelling. 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 perspectives. 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.

physics.ao-ph↗