The generalized Coriolis term
\[ \begin{align} \mathbf{v}\times\etabi = \mathbf{v}\times\left(\mathbf{f} + \zetabi\right) = \mathbf{v}\times\left(\mathbf{f} + \nabla\times\mathbf{v}\right) \end{align} \]
requires special attention on a C-lattice. In particular, the following three questions must be answered:
In this chapter, these questions will first be considered in the light of the shallow water equations (SWEs).. The SWEs are without friction
\[ \begin{align} \frac{\partial\mathbf{v}}{\partial t} & \stackrel{\href{ch-12-important-approximations.html#eq:swe_0}{\text{Glg. (13.171)}}}{=} -g\nabla\left(h + b\right) - f\mathbf{k}\times\mathbf{v} \textcolor{red}{- \nabla k - \zeta\mathbf{k}\times\mathbf{v}} + \mu\mathbf{k},\tag{28.2}\label{eq:swe_gen_cori_0}\\ \frac{\partial h}{\partial t} & \stackrel{\href{ch-12-important-approximations.html#eq:swe_1}{\text{Glg. (13.172)}}}{=} -\nabla\cdot\left(h\mathbf{v}\right), \end{align} \]
where $\mathbf{v}$ is the velocity, $g$ is the gravitational acceleration, $h$ is the deflection of the surface from the rest position, $f$ is the Coriolis parameter, $\mathbf{k}$ is the vertical unit vector, $k \coloneqq \frac{1}{2}\mathbf{v}^2$ is the specific kinetic energy, $\zeta\coloneqq\mathbf{k}\cdot\left(\nabla\times\mathbf{v}\right)$ the relative vorticity and $\mu$ a Lagrange multiplier which, in the case of a curved set such as a sphere, ensures that the right-hand side of the equation is tangent to that set. The red terms are the momentum advection terms, which are also the nonlinear terms. Thus, the linearized momentum equation of the SWEs is
\[ \begin{align} \frac{\partial\mathbf{v}}{\partial t} = -g\nabla\left(h + b\right) - f\mathbf{k}\times\mathbf{v}. \end{align} \]
If you project Eq. (28.2) to a normal vector $\mathbf{n}$ of an edge, one obtains
\[ \begin{align} \frac{\partial v_n}{\partial t} = -g\frac{\partial\left(h + b\right)}{\partial n} - f\mathbf{n}\cdot\left(\mathbf{k}\times\mathbf{v}\right) - \frac{\partial k}{\partial n} - \zeta\mathbf{n}\cdot\left(\mathbf{k}\times\mathbf{v}\right), \end{align} \]
is there
\[ \begin{align} v_n \coloneqq \mathbf{n}\cdot\mathbf{v} \end{align} \]
the normal velocity component at the edge. $\partial/\partial n$ denote partial derivatives perpendicular to the edge. With Eq. (A.159) is obtained
\[ \begin{align} \mathbf{n}\cdot\left(\mathbf{k}\times\mathbf{v}\right) &= \mathbf{v}\cdot\left(\mathbf{n}\times\mathbf{k}\right) \stackrel{\href{ch-39-derivations-of-some-mathematical-formula.html#eq:cross_product_anticommutative}{\text{Glg. (A.149)}}}{=} -\mathbf{v}\cdot\left(\mathbf{k}\times\mathbf{n}\right) = -\mathbf{v}\cdot\mathbf{t} = -v_t,\tag{28.7}\label{eq:v_t_c-grid} \end{align} \]
where $\mathbf{t} \coloneqq \mathbf{k}\times\mathbf{n}$ is the tangential unit vector on the edge and
\[ \begin{align} v_t \coloneqq \mathbf{v}\cdot\mathbf{t} \end{align} \]
the tangential velocity component at the edge. Therefore applies
\[ \begin{align} \frac{\partial v_n}{\partial t} = -g\frac{\partial\left(h + b\right)}{\partial n} + \left(f + \zeta\right)v_t - \frac{\partial k}{\partial n}. \end{align} \]
The linear part of this is
\[ \begin{align} \frac{\partial v_n}{\partial t} = -g\frac{\partial\left(h + b\right)}{\partial n} + fv_t. \end{align} \]
The vorticity and divergence dynamics of the SWEs can be summarized in the equations (13.172) and (15.170):
\[ \begin{align} \frac{\partial h}{\partial t} + \nabla\cdot\left(h\mathbf{v}\right) = 0, & {} & \frac{\partial\left(hq\right)}{\partial t} + \nabla\cdot\left(hq\mathbf{v}\right) = 0 \end{align} \]
From the second equation (this is a form of the barotropic vorticity equation) one can
\[ \begin{align} \frac{d}{dt}\int_{4\pi}hqd\omega &= \int_{4\pi}\frac{\partial\left(hq\right)}{\partial t}d\omega = -\int_{4\pi}\nabla\cdot\left(hq\mathbf{v}\right)d\omega \end{align} \]
conclude. The integral over the divergence of the potential vorticity flux density over the unit sphere can be evaluated by adding a third dimension in the form of a thin spherical shell of thickness $\Delta r$. No potential vorticity flows through the vertical edges of this set, since in the SWEs $w = 0$. Thus follows
\[ \begin{align} \frac{d}{dt}\int_{4\pi}hqd\omega &= 0. \end{align} \]
The global integral of the incompressible potential vorticity $q$ is therefore constant.
The vorticity flow term of the shallow water equations is
\[ \begin{align} -\left(f + \zeta\right)\mathbf{k}\times\mathbf{v}. \end{align} \]
In this section we want to find a discretization that satisfies the Thuburn condition and thus according to Eq. (27.141) also has a geostrophic mode. If you project this onto the normal vector $\mathbf{n}$ of an edge, you get according to Eq. (28.7)
\[ \begin{align} -\left[\left(f + \zeta\right)\mathbf{k}\times\mathbf{v}\right]\cdot\mathbf{n} &= \left(f + \zeta\right)v_t. \end{align} \]
This term is now linearized in the form
\[ \begin{align} \left(f + \zeta\right)v_t \to f_0v_t + \zeta U_t, \end{align} \]
here is
\[ \begin{align} U_t \coloneqq \left(\mathbf{k}\times\mathbf{n}\right)\cdot U\mathbf{i} \end{align} \]
the projection of a homogeneous zonal base current $U$ on the tangential direction. One now orientates oneself on the notations from Chap. 27. There, for the term $fv_t$ according to Eq. (27.134) the discretization
\[ \begin{align} f_0v_t \to \frac{f_0}{\sqrt{3}}\left(\newtilde{u}_2^{(3)} - \newtilde{u}_3^{(2)}\right) \end{align} \]
derived. According to Eq. (27.137) the Thuburn condition Eq. (27.102). For the term $\zeta U_t$ one sets with
\[ \begin{align} U_{t,1} &= 0,\tag{28.19}\label{eq:vorticity_flux_c-grid_ansatz_0_0}\\ U_{t,2} &= -\zeta_2\frac{\sqrt{3}}{2}U,\tag{28.20}\label{eq:vorticity_flux_c-grid_ansatz_0_1}\\ U_{t,3} &= -\zeta_3\frac{\sqrt{3}}{2}U\tag{28.21}\label{eq:vorticity_flux_c-grid_ansatz_0_2} \end{align} \]
at the edges in the $x_i-$direction initially
\[ \begin{align} \left(\zeta U_t\right)_1 &= 0,\\ \left(\zeta U_t\right)_2 &= -\zeta_2\frac{\sqrt{3}}{2}U,\\ \left(\zeta U_t\right)_3 &= -\zeta_3\frac{\sqrt{3}}{2}U \end{align} \]
to. These terms must also satisfy the Thuburn condition. One expects to be checked
\[ \begin{align} \newtilde{\left(\zeta U_t\right)_1}^{(1)} + \newtilde{\left(\zeta U_t\right)_2}^{(2)} + \newtilde{\left(\zeta U_t\right)_3}^{(3)} &= -\frac{\sqrt{3}}{2}U\left(\newtilde{\zeta_2}^{(2)} + \newtilde{\zeta_3}^{(3)}\right) \stackrel{\text{i. A.}}{\not=} 0. \end{align} \]
Inspired by Eq. (27.134) one modifies the approach of the equations (28.19) - (28.21)
\[ \begin{align} \left(\zeta U_t\right)_1 &= \frac{1}{\sqrt{3}}\left(-\zeta_2\frac{U}{2} + \zeta_3\frac{U}{2}\right),\\ \left(\zeta U_t\right)_2 &= \frac{1}{\sqrt{3}}\left(-\zeta_3\frac{U}{2} + \zeta_1U\right),\\ \left(\zeta U_t\right)_3 &= \frac{1}{\sqrt{3}}\left(-\zeta_1U + \zeta_2\frac{U}{2}\right). \end{align} \]
From this it follows
\[ \begin{align} \newtilde{\left(\zeta U_t\right)_1}^{(1)} + \newtilde{\left(\zeta U_t\right)_2}^{(2)} + \newtilde{\left(\zeta U_t\right)_3}^{(3)} &\propto \newtilde{\zeta_3 - \zeta_2}^{(1)} + \newtilde{2\zeta_1 - \zeta_3}^{(2)} + \newtilde{\zeta_2 - 2\zeta_1}^{(3)}\nonumber\\ = -\newtilde{\zeta_2}^{(1)} + \newtilde{\zeta_3}^{(1)} - \newtilde{\zeta_3}^{(2)} + 2\newtilde{\zeta_1}^{(2)} - 2\newtilde{\zeta_1}^{(3)} + \newtilde{\zeta_2}^{(3)} &\stackrel{\text{i. A.}}{\not=} 0. \end{align} \]
This approach also does not satisfy the Thuburn condition. Rather, a preparatory averaging of the vorticities is required, which leads to the approach
\[ \begin{align} \left(\zeta U_t\right)_1 &= \frac{1}{\sqrt{3}}\left(-\newoverline{\zeta_2}^{(3)}\frac{U}{2} + \newoverline{\zeta_3}^{(2)}\frac{U}{2}\right),\\ \left(\zeta U_t\right)_2 &= \frac{1}{\sqrt{3}}\left(-\newoverline{\zeta_3}^{(1)}\frac{U}{2} + \newoverline{\zeta_1}^{(3)}U\right),\\ \left(\zeta U_t\right)_3 &= \frac{1}{\sqrt{3}}\left(-\newoverline{\zeta_1}^{(2)}U + \newoverline{\zeta_2}^{(1)}\frac{U}{2}\right). \end{align} \]
This satisfies the Thuburn condition since averaging operators commute. An operator other than the simple averaging operator $\newoverline{\zeta_i}^{(j)}$ can also be used. With $\zeta \to \eta = \zeta + f$ the complete vorticity flow term on the regular C lattice can be expressed as follows.
It is not trivial how to extend the vorticity flow term to the deformed grid. To do this, we first start from the f-plane in order to be able to assume a homogeneous vorticity $f_0$. Rather than attempting to generalize the Thuburn condition to the deformed grid, the closely related statement is used as a basic requirement that a geostrophic mode should exist. Furthermore, the discretization should be energy-conserving.