Michael J. P. Cullen

Lagrangian solutions of semigeostrophic equations in physical space

The semigeostrophic equations are a simple model of large-scale atmosphere/ocean flows. Previous work by J.-D. Benamou and Y. Brenier, M. Cullen and W. Gangbo, and M. Cullen and H. Maroofi proves that the semigeostrophic equations can be solved in the cases, respectively, of 3-dimensional (3-d) incompressible flow between rigid boundaries, vertically averaged 3-d incompressible flow with a free surface, and fully compressible flow. However, all these results prove only the existence of weak solutions in dual variables, where the dual variables result from a change of variables introduced by Hoskins. This makes it difficult to relate the solutions to the full Euler or Navier--Stokes equations, or to those of other simple atmosphere/ocean models. We therefore seek to extend these results to prove existence of a solution in physical variables. We do this using the Lagrangian form of the equations in physical space. The proof is based on the recent results of L. Ambrosio on transport equations and ODE for B...

The breakdown of large-scale flows in rotating, stratified fluids

Flows under the influences of environmental rotation and stable density stratification often exhibit an approximate force balance and a consequently slow rate of evolution at large Reynolds number. Such flows are typically anisotropic in their velocity field. This regime is relevant to large-scale motions in the Earth’s atmosphere and ocean, as well as many other planetary and astrophysical systems. The Balance Equations are usually an accurate approximate model for this regime. However, they have solvability limits associated with a change of type in their time-integration operator. In this paper we derive these limiting conditions for the conservative Balance Equations in isentropic coordinates, show that the least familiar of these conditions coincides with loss of convexity of the streamfunction for horizontal velocity in the inertial reference frame, and identify these conditions with the general conditions for symmetric loss of stability for circular and parallel flows as well as for the three-dimensional loss of stability for elliptical flows. We then conjecture that the identified limits of balance coincide generally with the boundary between the distinctive nonlinear dynamical behaviors (i.e., their turbulent cascade and dissipation rates) associated with the large- and small-scale regimes in geophysical and astrophysical flows.

Large eddy simulation of the atmosphere on various scales

Numerical simulations of the atmosphere are routinely carried out on various scales for purposes ranging from weather forecasts for local areas a few hours ahead to forecasts of climate change over periods of hundreds of years. Almost without exception, these forecasts are made with space/time-averaged versions of the governing Navier–Stokes equations and laws of thermodynamics, together with additional terms representing internal and boundary forcing. The calculations are a form of large eddy modelling, because the subgrid-scale processes have to be modelled. In the global atmospheric models used for long-term predictions, the primary method is implicit large eddy modelling, using discretization to perform the averaging, supplemented by specialized subgrid models, where there is organized small-scale activity, such as in the lower boundary layer and near active convection. Smaller scale models used for local or short-range forecasts can use a much smaller averaging scale. This allows some of the specialized subgrid models to be dropped in favour of direct simulations. In research mode, the same models can be run as a conventional large eddy simulation only a few orders of magnitude away from a direct simulation. These simulations can then be used in the development of the subgrid models for coarser resolution models.

Diagnosis of boundary-layer circulations

Diagnoses of circulations in the vertical plane provide valuable insights into aspects of the dynamics of the climate system. Dynamical theories based on geostrophic balance have proved useful in deriving diagnostic equations for these circulations. For example, semi-geostrophic theory gives rise to the Sawyer–Eliassen equation (SEE) that predicts, among other things, circulations around mid-latitude fronts. A limitation of the SEE is the absence of a realistic boundary layer. However, the coupling provided by the boundary layer between the atmosphere and the surface is fundamental to the climate system. Here, we use a theory based on Ekman momentum balance to derive an SEE that includes a boundary layer (SEEBL). We consider a case study of a baroclinic low-level jet. The SEEBL solution shows significant benefits over Ekman pumping, including accommodating a boundary-layer depth that varies in space and structure, which accounts for buoyancy and momentum advection. The diagnosed low-level jet is stronger than that determined by Ekman balance. This is due to the inclusion of momentum advection. Momentum advection provides an additional mechanism for enhancement of the low-level jet that is distinct from inertial oscillations.

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.

How mathematical models can aid understanding of climate

IMA Conference on the Mathematics of the Climate System; Reading, United Kingdom, 13–15 September 2011. nAbout 40 researchers attended a conference by the Institute for Mathematics and its Applications (IMA) to discuss the mathematics of the climate system. The conference focused on the construction and use of mathematical and computational models. The entire hierarchy of models was considered, from the conceptual to the comprehensive. Conceptual models provide understandable paradigms for dynamical climate system behavior, enabling researchers to assess and interpret comprehensive models.

Classical Solutions to Semi-geostrophic System with Variable Coriolis Parameter

We prove the short time existence and uniqueness of smooth solutions (in

A Rigorous Treatment of Moist Convection in a Single Column

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