A comparison of numerical solutions to the Eady frontogenesis problem

Cullen, Michael J.

doi:10.1002/qj.335

Cited by 12 publications

(30 citation statements)

References 21 publications

(67 reference statements)

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“…Behind the forced frontal updraught and above the frontal surface, alternate regions of descent and ascent are evident, suggestive of a gravity wave propagating vertically away from the surface front, which acts to distort the isentropes above and behind the surface front. Similar gravity‐wave‐like structures were found by Cullen () in two‐dimensional simulations of the Eady frontogenesis problem and by Muir and Reeder () in a high‐resolution, two‐dimensional simulation of an idealized front. Overall, the structure of the noBL front is in agreement with previous idealized experiments; the absence of a PBL scheme results in unrealistically strong surface potential temperature gradients, limited convergence and hence unrealistically weak vertical motion.…”

Section: Structure and Evolution Of Cold Frontssupporting

confidence: 79%

Force balances and dynamical regimes of numerically simulated cold fronts within the boundary layer

Sinclair

Keyser

2015

Quart J Royal Meteoro Soc

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The governing dynamics of numerically simulated cold fronts, as they collapse towards a minimum cross-frontal scale, are identified by determining force balances and the Rossby and Ekman numbers in frontal regions. A hierarchy of numerical simulations of cold fronts is performed with the Weather Research and Forecasting model and individual terms in the horizontal momentum equations are calculated from the model output to construct force balances. An inviscid, three-dimensional, idealized front is simulated at three different horizontal grid spacings (100, 20 and 4 km) and then the experiments are repeated with a planetary boundary layer (PBL) parametrization scheme included. Additionally, a fullphysics simulation of the cold front originally analyzed by Sanders is conducted. The leading edge of the idealized, inviscid cold front is characterized by small Rossby numbers and hence balanced dynamics at all three model resolutions. When the PBL scheme is included, large Rossby numbers and unbalanced flow develop at the leading edge of the cold front, but only when the front is simulated at high resolution; at coarse resolution, balanced dynamics remain. The acceleration is in the cross-front direction and arises due to a localized pressure gradient force. Unlike the inviscid experiments or the coarse-resolution PBL experiments, the front in the high-resolution PBL experiment has a large cross-front thermal gradient, strong forced ascent and sharp surface pressure trough, which enables a large cross-front pressure gradient force to develop. The dynamics of the Sanders front agree qualitatively with the idealized PBL simulation, but much larger Rossby numbers occur at the leading edge of the Sanders front. This finding indicates that the dynamics of the Sanders front are more unbalanced than the dynamics of the idealized simulations.

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Section: Structure and Evolution Of Cold Frontssupporting

confidence: 79%

Force balances and dynamical regimes of numerically simulated cold fronts within the boundary layer

Sinclair

Keyser

2015

Quart J Royal Meteoro Soc

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“…A numerical model should not generate excessive imbalance, and, ideally, should not require artificial damping mechanisms to control imbalance. A thorough investigation of this issue requires an examination of how the model performs in the asymptotic limits of small Rossby or Froude number (Cullen, 2007(Cullen, , 2008; this is the subject of ongoing work. For present purposes we examine some simple diagnostics of imbalance in the barotropic instability test case, applied to the mimetic finite volume model and, for comparison, to ENDGame.…”

Section: Balancementioning

confidence: 99%

A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: comparison of hexagonal–icosahedral and cubed-sphere grids

Thuburn

Cotter

Dubos³

2014

Geosci. Model Dev.

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Abstract.A new algorithm is presented for the solution of the shallow water equations on quasi-uniform spherical grids. It combines a mimetic finite volume spatial discretization with a Crank-Nicolson time discretization of fast waves and an accurate and conservative forward-in-time advection scheme for mass and potential vorticity (PV). The algorithm is implemented and tested on two families of grids: hexagonal-icosahedral Voronoi grids, and modified equiangular cubed-sphere grids.Results of a variety of tests are presented, including convergence of the discrete scalar Laplacian and Coriolis operators, advection, solid body rotation, flow over an isolated mountain, and a barotropically unstable jet. The results confirm a number of desirable properties for which the scheme was designed: exact mass conservation, very good available energy and potential enstrophy conservation, consistent mass, PV and tracer transport, and good preservation of balance including vanishing ∇ × ∇, steady geostrophic modes, and accurate PV advection. The scheme is stable for large wave Courant numbers and advective Courant numbers up to about 1.In the most idealized tests the overall accuracy of the scheme appears to be limited by the accuracy of the Coriolis and other mimetic spatial operators, particularly on the cubed-sphere grid. On the hexagonal grid there is no evidence for damaging effects of computational Rossby modes, despite attempts to force them explicitly.

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“…Above the boundary layer, the second‐order approximation of the material derivative dominated below Ro = 0.1. The second‐order convergence rate was in agreement with that found for the dynamics‐only SG case during the early phases of baroclinic development (Cullen, ). These findings were in agreement with the estimates made in Eqs and .…”

Section: Resultsmentioning

confidence: 99%

“…Our new test involved running HPE models at the small‐ Ro limit, and comparing with a balanced model that included a boundary layer: the SGT model. Previous work focused on the convergence of the HPE solutions to SG solutions for the dynamics‐only case (Cullen, ; Visram et al ). For the first time, we determined the role of the boundary‐layer parametrization in modifying the dynamics‐only results.…”

Section: Discussionmentioning

confidence: 99%

“…The SGT model was first‐order accurate within the boundary layer, because it was strongly constrained by Ekman balance. However, it was second‐order accurate in the free troposphere, in agreement with the dynamics‐only case (Cullen, ; Visram et al ). Thus the convergence of HPE solutions to SGT model was dominated by the differences in the boundary layer.…”

Section: Discussionmentioning

confidence: 99%

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Validating weather and climate models at small Rossby numbers: including a boundary layer

Beare

Cullen

2016

Quart J Royal Meteoro Soc

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Ideally, the validation of weather and climate models requires that the predictions remain close to an exact solution of the governing equations. The complexity of weather and climate models means that it is not possible to compute exact solutions except in trivial cases. However, in the limit of small Rossby number, the exact solution of the Euler equations can be shown to be close to that of a semi-geostrophic model, which can be computed. Previous studies have used the small Rossby-number limit to validate numerical methods for a baroclinic wave without sub-grid physics. However, the method of coupling to the sub-grid physics plays an important role in the performance of weather and climate models. The aim of this paper is thus to extend the previous studies to include a boundary-layer parametrization. We use a balanced model that includes a known boundary-layer parametrization, the semi-geotriptic model. We then demonstrate that the semi-geotriptic model is the appropriate small Rossby-number limit of the solution of the Euler equations with the same boundary layer representation.The semi-geotriptic model is then used to expose weaknesses in the numerical methods for coupling the boundary layer to the rest of the model.

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A comparison of numerical solutions to the Eady frontogenesis problem

Cited by 12 publications

References 21 publications

Force balances and dynamical regimes of numerically simulated cold fronts within the boundary layer

Force balances and dynamical regimes of numerically simulated cold fronts within the boundary layer

A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: comparison of hexagonal–icosahedral and cubed-sphere grids

Validating weather and climate models at small Rossby numbers: including a boundary layer

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