Semi‐geostrophic solutions for flow over a ridge

Quart J Royal Meteoro Soc

2008

Self Cite

ABSTRACT:The semi-geostrophic solution to the Eady frontogenesis problem is believed to represent an asymptotic limit of the full Euler equations as the Rossby number tends to zero. This is investigated using numerical solutions. The exercise is non-trivial because the semi-geostrophic solution is discontinuous, and so may not be found as the limit of the Eulerian form of the equations. It is shown that solutions of the two-dimensional compressible Euler equations converge to a singular limit with geostrophic balance in the cross-front direction. The convergence is at the expected second-order rate in Rossby number up to the formation of the discontinuity and at a first-order rate for some time afterwards. Numerical robustness can be assured and the degree of convergence improved if a quasi-monotone form of interpolation is used in the semi-Lagrangian advection. Direct integrations of a semi-geostrophic model using standard numerical methods require significant computational compromises, and give a somewhat smoothed version of this limit.

Section: Introductionmentioning

confidence: 87%

Section: Semi-geostrophic Modelmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A comparison of numerical solutions to the Eady frontogenesis problem

Quart J Royal Meteoro Soc

2008

Self Cite

A framework for evaluating model error using asymptotic convergence in the Eady model

Visram

Cotter

Quart J Royal Meteoro Soc

2013

Self Cite

Operational weather forecasting requires the accurate simulation of atmospheric motions on scales ranging from the synoptic down to tens of kilometres. Weather fronts, characteristic of midlatitude weather systems, are generated through baroclinic instability on the large scale but are anisotropic features in which temperature and winds can vary rapidly on the short scale. We present a framework for evaluating model error in terms of asymptotic convergence using the Eady model. Through rescaling the problem, we are able to approach solutions of a balanced model, given by the semi‐geostrophic equations, using the non‐hydrostatic, incompressible Euler–Boussinesq Eady equations. Using this approach, we are able to validate the numerical implementation and assess the long‐term performance in terms of solution lifecycles. We present results using a finite‐difference method with semi‐implicit time‐stepping and semi‐Lagrangian transport, and show that we are able to proceed past the point of frontal collapse and recover the theoretical convergence rate. We propose that numerical diffusion of potential vorticity after collapse, as a result of insufficient Lagrangian conservation, is detrimental to the long‐term evolution of the solution.

Modelling atmospheric flows

2007

Acta Numerica

This article demonstrates how numerical methods for atmospheric models can be validated by showing that they give the theoretically predicted rate of convergence to relevant asymptotic limit solutions. This procedure is necessary because the exact solution of the Navier-Stokes equations cannot be resolved by production models. The limit solutions chosen are those most important for weather and climate prediction. While the best numerical algorithms for this purpose largely reflect current practice, some important limit solutions cannot be captured by existing methods. The use of Lagrangian rather than Eulerian averaging may be required in these cases.