Numerical representation of geostrophic modes on arbitrarily structured C-grids

Copyright © 2009 Elsevier. NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational Physics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational Physics, Vol. 228, Issue 22 (2009), DOI: 10.1016/j.jcp.2009.08.006A C-grid staggering, in which the mass variable is stored at cell centers and the normal velocity component is stored at cell faces (or edges in two dimensions) is attractive for atmospheric modeling since it enables a relatively accurate representation of fast wave modes. However, the discretization of the Coriolis terms is non-trivial. For constant Coriolis parameter, the linearized shallow water equations support geostrophic modes: stationary solutions in geostrophic balance. A naive discretization of the Coriolis terms can cause geostrophic modes to become non-stationary, causing unphysical behaviour of numerical solutions. Recent work has shown how to discretize the Coriolis terms on a planar regular hexagonal grid to ensure that geostrophic modes are stationary while the Coriolis terms remain energy conserving. In this paper this result is extended to arbitrarily structured C-grids. An explicit formula is given for constructing an appropriate discretization of the Coriolis terms. The general formula is illustrated by showing that it recovers previously known results for the planar regular hexagonal C-grid and the spherical longitude–latitude C-grid. Numerical calculation confirms that the scheme does indeed give stationary geostrophic modes for the hexagonal–pentagonal and triangular geodesic C-grids on the sphere

Impacts of future urban expansion on urban heat island effects during heatwave events in the city of Melbourne in southeast Australia

The city of Melbourne in southeast Australia is planning to expand urban areas substantially by the year 2050 and this expansion has the potential to alter the Urban Heat Island (UHI), that is, higher temperatures in urban areas as compared to surrounding rural areas. Moreover, Melbourne has been experiencing more frequent heatwaves for last two decades, and the intensity and duration of heatwaves is expected to increase in the future, which could exacerbate the UHI. This study evaluates the potential impacts of future urban expansion on the urban meteorology in Melbourne city during four of the most severe heatwave events during the period 2000–2009. Urban expansion is implemented as high‐density urban with a high urban fraction of 0.9 to investigate the maximum possible impact. Simulations are carried out using the Weather Research and Forecasting model coupled with the Single‐Layer Urban Canopy Model with current land‐use and future urban expansion scenarios. Urban expansion increases the near‐surface (2‐m) UHI (UHI2) by 0.75 to 2.80 °C and the skin‐surface UHI (UHIsk) by 1.9 to 5.4 °C over the expanded urban areas during the night, with no changes in existing urban areas. No substantial changes in UHI2 and UHIsk occur during the day over both existing and expanded urban areas. This is largely driven by changes in the storage heat flux, with an increase in storage heat at night and a decrease during the day; that is, excess storage heat accumulated during the day is released at night, which causes a slower decrease of near‐surface temperature and increase in the UHI. Urban expansion did not affect human thermal comfort (HTC) in existing urban areas and there were no marked differences in HTC between existing and expanded urban areas

A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids

Copyright © 2010 Elsevier. NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational Physics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational Physics, Vol. 229, Issue 9 (2010), DOI: 10.1016/j.jcp.2009.12.007A numerical scheme applicable to arbitrarily-structured C-grids is presented for the nonlinear shallow-water equations. By discretizing the vector-invariant form of the momentum equation, the relationship between the nonlinear Coriolis force and the potential vorticity flux can be used to guarantee that mass, velocity and potential vorticity evolve in a consistent and compatible manner. Underpinning the consistency and compatibility of the discrete system is the construction of an auxiliary thickness equation that is staggered from the primary thickness equation and collocated with the vorticity field. The numerical scheme also exhibits conservation of total energy to within time-truncation error. Simulations of the standard shallow-water test cases confirm the analysis and show convergence rates between 1st1st- and 2nd2nd-order accuracy when discretizing the system with quasi-uniform spherical Voronoi diagrams. The numerical method is applicable to a wide class of meshes, including latitude–longitude grids, Voronoi diagrams, Delaunay triangulations and conformally-mapped cubed-sphere meshes