The dynamical core that predicts the three‐dimensional vorticity rather than the momentum, which is called Vector‐Vorticity Model (VVM), is implemented on a cubed sphere. Its horizontal coordinate system is not restricted to orthogonal, while the vertical coordinate is orthogonal to the horizontal surface. Accordingly, all the governing equations of the VVM, which are originally developed with Cartesian coordinates, are rewritten in terms of general curvilinear coordinates. The local coordinates on each cube surface are constructed with the gnomonic equiangular projection. Using global channel domains, the VVM on the cubed sphere has been evaluated by (1) advecting a passive tracer with a bell‐shaped initial perturbation along an east‐west latitude circle and along a north‐south meridional circle and (2) simulating the evolution of barotropic and baroclinic instabilities. The simulated results with the cubed‐sphere grids are compared to analytic solutions or those with the regular longitude‐latitude grids. The convergence with increasing spatial resolution is also quantified using standard error norms. The comparison shows that the solutions with the cubed‐sphere grids are quite reasonable for both linear and nonlinear problems when high resolutions are used. With coarse resolution, degeneracy appears in the solutions of the nonlinear problems such as spurious wave growth; however, it is effectively reduced with increased resolution. Based on the encouraging results in this study, we intend to use this model as the cloud‐resolving component in a global Quasi‐Three‐Dimensional Multiscale Modeling Framework.
Boundary layer formulations in orthogonal curvilinear coordinates for flow over wind-generated surface waves
The development of the governing equations for fluid flow in a surface-following coordinate system is essential to investigate the fluid flow near an interface deformed by propagating waves. In this paper, the governing equations of fluid flow, including conservation of mass, momentum and energy balance, are derived in an orthogonal curvilinear coordinate system relevant to surface water waves. All equations are further decomposed to extract mean, wave-induced and turbulent components. The complete transformed equations include explicit extra geometric terms. For example, turbulent stress and production terms include the effects of coordinate curvature on the structure of fluid flow. Furthermore, the governing equations of motion were further simplified by considering the flow over periodic quasi-linear surface waves wherein the wavelength of the disturbance is large compared to the wave amplitude. The quasi-linear analysis is employed to express the boundary layer equations in the orthogonal wave-following curvilinear coordinates with the corresponding decomposed equations for the mean, wave and turbulent fields. Finally, the vorticity equations are also derived in the orthogonal curvilinear coordinates in order to express the corresponding velocity–vorticity formulations. The equations developed in this paper proved to be useful in the analysis and interpretation of experimental data of fluid flow over wind-generated surface waves. Experimental results are presented in a companion paper.
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- Award ID(s):
- 1634051
- PAR ID:
- 10208693
- Date Published:
- Journal Name:
- Journal of fluid mechanics
- Volume:
- 888
- Issue:
- A11
- ISSN:
- 1469-7645
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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