An orthogonal terrain-following coordinate and its preliminary tests using 2-D idealized advection experiments

We have designed an orthogonal curvilinear terrain-following coordinate (the orthogonal σ coordinate, or the OS coordinate) to reduce the advection errors in the classic σ coordinate. First, we rotate the basis vectors of the z coordinate in a specific way in order to obtain the orthogonal, terrain-following basis vectors of the OS coordinate, and then add a rotation parameter b to each rotation angle to create the smoother vertical levels of the OS coordinate with increasing height. Second, we solve the corresponding definition of each OS coordinate through its basis vectors; and then solve the 3-D coordinate surfaces of the OS coordinate numerically, therefore the computational grids created by the OS coordinate are not exactly orthogonal and its orthogonality is dependent on the accuracy of a numerical method. Third, through choosing a proper b, we can significantly smooth the vertical levels of the OS coordinate over a steep terrain, and, more importantly, we can create the orthogonal, terrain-following computational grids in the vertical through the orthogonal basis vectors of the OS coordinate, which can reduce the advection errors better than the corresponding hybrid σ coordinate. However, the convergence of the grid lines in the OS coordinate over orography restricts the time step and increases the numerical errors. We demonstrate the advantages and the drawbacks of the OS coordinate relative to the hybrid σ coordinate using two sets of 2-D linear advection experiments.

Using Depth-Normalized Coordinates to Examine Mass Transport Residual Circulation in Estuaries with Large Tidal Amplitude Relative to the Mean Depth

Residual (subtidal) circulation profiles in estuaries with a large tidal amplitude-to-depth ratio often are quite complex and do not resemble the traditional estuarine gravitational circulation profile. This paper describes how a depth-normalized σ-coordinate system allows for a more physical interpretation of residual circulation profiles than does a fixed vertical coordinate system in an estuary with a tidal amplitude comparable to the mean depth. Depth-normalized coordinates permit the approximation of Lagrangian residuals, performance of empirical orthogonal function (EOF) analysis, estimation of terms in the along-stream momentum equations throughout depth, and computation of a tidally averaged momentum balance. The residual mass transport velocity has an enhanced two-layer exchange flow relative to an Eulerian mean because of the Stokes wave transport velocity directed upstream at all depths. While the observed σ-coordinate profiles resemble gravitational circulation, and pressure and friction are the dominant terms in the tidally varying and tidally averaged momentum equations, the two-layer shear velocity from an EOF analysis does not correlate with the along-stream density gradient. To directly compare to theoretical profiles, an extension of a pressure–friction balance in σ coordinates is solved. While the barotropic riverine residual matches theory, the mean longitudinal density gradient and mean vertical mixing cannot explain the magnitude of the observed two-layer shear residual. In addition, residual shear circulation in this system is strongly driven by asymmetries during the tidal cycle, particularly straining and advection of the salinity field, creating intratidal variation in stratification, vertical mixing, and shear.

An orthogonal curvilinear terrain-following coordinate for atmospheric models

We have designed an orthogonal curvilinear terrain-following coordinate (the orthogonal σ coordinate, or the OS coordinate) to overcome two well-known problems in the classic σ coordinate, namely, pressure gradient force (PGF) errors and advection errors. First, in the design of basis vectors, we rotate the basis vectors of the z coordinate in a particular way in order to reduce the PGF errors and add a special rotation parameter b to each rotation angel in order to reduce the advection errors. Second, the corresponding definition of each OS coordinate is solved through its basis vectors. Third, the scalar equations of the OS coordinate are solved by expanding the vector equation using the basis vectors. Since the computational form of PGF has only one term in each momentum equation of the OS coordinate, the PGF errors will be significantly reduced, according to Li et al. (2012). When a proper b is chosen, the σ levels over a steep terrain can be significantly smoothed, therefore alleviating the advection errors in the OS coordinate. This is demonstrated by a series of 2-D linear advection experiments under a unified framework.