Vertical velocity

The vertical velocity serves as a diagnostic variable for local volume conservation1, but is a physically important quantity, especially when a steep slope is present (Zhang et al. 2004). To solve the vertical velocity, we apply a finite-volume method to a typical prism, as depicted in Figure 6, assuming that \(w\) is constant within an element \(i\), and obtain -

\[\begin{equation} \label{eq01} \hat{S}_{k+1} \left( \overline{u}_{k+1}^{n+1} n_{k+1}^{x} + \overline{v}_{k+1}^{n+1} n_{k+1}^{y} + w_{i, k+1}^{n+1} n_{k+1}^{z} \right) - \hat{S}_{k} \left( \overline{u}_{k}^{n+1} n_{k}^{x} + \overline{v}_{k}^{n+1} n_{k}^{y} + w_{i, k}^{n+1} n_{k}^{z} \right) + \sum_{m=1}^{3} \hat{P}_{js(i, m)} \left( \hat{q}_{js(i, m), k}^{n+1} + \hat{q}_{js(i, m), k+1}^{n+1} \right)/2 = 0, (k=k^b, \cdots, N_z - 1) \end{equation}\]

where \(\hat{S}\) and \(\hat{P}\) are the areas of the prism surfaces (Figure 6), (\(n^x, n^y, n^z\)), are the normal vector (pointing upward), \(\overline{u}\) and \(\overline{v}\) the averaged horizontal velocities at the top and bottom surfaces, and \(\hat{q}\) is the outward normal velocity at each side center. The vertical velocity is then solved from the bottom to the surface, in conjunction with the bottom boundary condition \((u, v, q)\cdot\pmb{n}=0\). In the case of earthquake module (imm≠0), the bed velocity is prescribed. A compact form for Eqs \(\ref{eq01}\) is \(\sum_{j\epsilon S^+} \left| Q_j\right| = \sum_{j\epsilon S^-} \left| Q_j\right|\), where \(Q_j\) is the facial fluxes outward of a prism \(i\). This conservation will be utilized in the transport equation as the foundation for mass conservation and constancy.

References Zhang, Y., Baptista, A.M. and Myers, E.P. (2004) "A cross-scale model for 3D baroclinic circulation in estuary-plume-shelf systems: I. Formulation and skill assessment". Cont. Shelf Res., 24: 2187-2214.

  1. Although other definitions of volume/mass exist, we define volume/mass in the finite-volume sense throughout this paper and measure conservation based on this definition. ↩