An accurate momentum advection scheme for a z-level coordinate models

Abstract

In this paper, we focus on a conservative momentum advection discretisation in the presence of zlayers. While in the 2D case conservation of momentum is achieved automatically for an Eulerian advection scheme, special attention is required in the multi-layer case. We show here that an artificial vertical structure of the flow can be introduced solely by the presence of the z-layers, which we refer to as the staircase problem. To avoid this staircase problem, the z-layers have to be remapped in a specific way. The remapping procedure also deals with the case of an uneven number of layers adjacent to a column side, thus allowing one to simulate flooding and drying phenomena in a 3D model.

Accurate vertical profiles of turbulent flow in z‐layer models

“…With or without the remapping, the computation of advection over bottom steps requires attention, as, e.g., illustrated in Kleptsova et al [2010Kleptsova et al [ , 2012. Otherwise, the solution may deteriorate due to spurious form drag.…”

“…The advection terms can be approximated, e.g., using a conservative upwind scheme as in Kleptsova et al . [] or an Eulerian‐Lagrangian scheme [e.g., Casulli and Cheng , ; Ham et al ., ].…”

“…The discretized momentum and free-surface equations read: where i, j, and k are the indices corresponding to the x direction, y direction, and z direction, m is the bottom layer index, M n is the free-surface layer index at time level n, and Fu and Fv contain the explicitly discretized advection and horizontal diffusion terms. The advection terms can be approximated, e.g., using a conservative upwind scheme as in Kleptsova et al [2010] or an Eulerian-Lagrangian scheme [e.g., Casulli and Cheng, 1990;Ham et al, 2006]. where r x and r y are the Taylor coefficients, which we determine from the logarithmic law of the wall: Equations (25) and (26) follow when u 1 and v 1 are taken as the velocities in the bottom layer, i.e., at z 1 5Dz n m =2, in (7).…”

“…Even using a partial-cell or shaved-cell approach [see e.g., Adcroft et al, 1997;Pacanowski and Gnanadesikan, 1998], these boundaries cause problems. First, inadequate treatment of advection introduces implicit form drag along staircase boundaries [e.g., Beckmann and D€ oscher, 1997;Song and Chao, 2000;Chen, 2004;Ezer and Mellor, 2004;Kleptsova et al, 2010]. Second, thin layers-occurring when the bottom or free surface crosses a layer interface-cause discontinuities in velocity and shear stress, see Figure 1.…”

Multi-scale modelling of coastal, shelf and global ocean dynamics

“…They find that higher-order schemes on relatively coarse grids generally perform better than low-order schemes on fine grids. Kleptsova et al (2010) assess various advection schemes for z-coordinate, threedimensional models in which flooding and drying is taken into account. In this study, the ability to conserve momentum is regarded as the main criterion for selecting a suitable method.…”