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-grid. In particular, the following three questions must be answered:
Over which polygon must Stokes' theorem be evaluated to calculate the vorticity?
How must the tangential wind component on an edge be reconstructed from the normal wind components on the surrounding edges?
By what vorticity must the tangential wind component ultimately be multiplied to obtain a component of $\mathbf{v}\times\etabi$?
In this chapter, these questions will first be considered in the light of the shallow water equations (SWEs). Without friction, the SWEs read
\[ \begin{align} \frac{\partial\mathbf{v}}{\partial t} & \stackrel{\href{ch-12-important-approximations.html#eq:swe_0}{\text{Eq. (13.171)}}}{=} -g\nabla\left(h + b\right) - f\mathbf{k}\times\mathbf{v} \textcolor{red}{- \nabla k - \zeta\mathbf{k}\times\mathbf{v}} + \mu\mathbf{k},\tag{28.2}\label{eq:swe_gen_cori_0}\\ \frac{\partial h}{\partial t} & \stackrel{\href{ch-12-important-approximations.html#eq:swe_1}{\text{Eq. (13.172)}}}{=} -\nabla\cdot\left(h\mathbf{v}\right), \end{align} \]
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 reads
\[ \begin{align} \frac{\partial\mathbf{v}}{\partial t} = -g\nabla\left(h + b\right) - f\mathbf{k}\times\mathbf{v}. \end{align} \]
Projecting Eq. (28.2) onto a normal vector $\mathbf{n}$ of an edge, one obtains
\[ \begin{align} \frac{\partial v_n}{\partial t} = -g\frac{\partial\left(h + b\right)}{\partial n} - f\mathbf{n}\cdot\left(\mathbf{k}\times\mathbf{v}\right) - \frac{\partial k}{\partial n} - \zeta\mathbf{n}\cdot\left(\mathbf{k}\times\mathbf{v}\right), \end{align} \]
where
\[ \begin{align} v_n \coloneqq \mathbf{n}\cdot\mathbf{v} \end{align} \]
is the normal velocity component at the edge. $\partial/\partial n$ denotes partial derivatives perpendicular to the edge. With Eq. (A.161) one obtains
\[ \begin{align} \mathbf{n}\cdot\left(\mathbf{k}\times\mathbf{v}\right) &= \mathbf{v}\cdot\left(\mathbf{n}\times\mathbf{k}\right) \stackrel{\href{ch-39-derivations-of-some-mathematical-formula.html#eq:cross_product_anticommutative}{\text{Eq. (A.151)}}}{=} -\mathbf{v}\cdot\left(\mathbf{k}\times\mathbf{n}\right) = -\mathbf{v}\cdot\mathbf{t} = -v_t,\tag{28.7}\label{eq:v_t_c-grid} \end{align} \]
where $\mathbf{t} \coloneqq \mathbf{k}\times\mathbf{n}$ is the tangential unit vector on the edge and
\[ \begin{align} v_t \coloneqq \mathbf{v}\cdot\mathbf{t} \end{align} \]
is the tangential velocity component at the edge. Thus one has
\[ \begin{align} \frac{\partial v_n}{\partial t} = -g\frac{\partial\left(h + b\right)}{\partial n} + \left(f + \zeta\right)v_t - \frac{\partial k}{\partial n}. \end{align} \]
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 Eqs. (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 conclude
\[ \begin{align} \frac{d}{dt}\int_{4\pi}hqd\omega &= \int_{4\pi}\frac{\partial\left(hq\right)}{\partial t}d\omega = -\int_{4\pi}\nabla\cdot\left(hq\mathbf{v}\right)d\omega \end{align} \]
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 boundaries of this set, since $w = 0$ holds in the SWEs. It 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 flux term of the shallow water equations reads
\[ \begin{align} -\left(f + \zeta\right)\mathbf{k}\times\mathbf{v}. \end{align} \]
In this section, a discretization is to be found that satisfies the Thuburn condition and thus, according to Eq. (27.141), also possesses a geostrophic mode. Projecting this onto the normal vector $\mathbf{n}$ of an edge, one obtains, 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
\[ \begin{align} \left(f + \zeta\right)v_t \to f_0v_t + \zeta U_t, \end{align} \]
where
\[ \begin{align} U_t \coloneqq \left(\mathbf{k}\times\mathbf{n}\right)\cdot U\mathbf{i} \end{align} \]
is the projection of a homogeneous zonal background flow $U$ onto the tangential direction. One now follows the notation from Chap. 27. There, the discretization
\[ \begin{align} f_0v_t \to \frac{f_0}{\sqrt{3}}\left(\newtilde{u}_2^{(3)} - \newtilde{u}_3^{(2)}\right) \end{align} \]
was derived for the term $fv_t$ according to Eq. (27.134). According to Eq. (27.137), this satisfies the Thuburn condition Eq. (27.102). For the term $\zeta U_t$, 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, one initially sets
\[ \begin{align} \left(\zeta U_t\right)_1 &= 0,\\ \left(\zeta U_t\right)_2 &= -\zeta_2\frac{\sqrt{3}}{2}U,\\ \left(\zeta U_t\right)_3 &= -\zeta_3\frac{\sqrt{3}}{2}U \end{align} \]
These terms must also satisfy the Thuburn condition. To check this, one computes
\[ \begin{align} \newtilde{\left(\zeta U_t\right)_1}^{(1)} + \newtilde{\left(\zeta U_t\right)_2}^{(2)} + \newtilde{\left(\zeta U_t\right)_3}^{(3)} &= -\frac{\sqrt{3}}{2}U\left(\newtilde{\zeta_2}^{(2)} + \newtilde{\zeta_3}^{(3)}\right) \stackrel{\text{in general}}{\not=} 0. \end{align} \]
Inspired by Eq. (27.134), one modifies the ansatz of Eqs. (28.19) - (28.21) to
\[ \begin{align} \left(\zeta U_t\right)_1 &= \frac{1}{\sqrt{3}}\left(-\zeta_2\frac{U}{2} + \zeta_3\frac{U}{2}\right),\\ \left(\zeta U_t\right)_2 &= \frac{1}{\sqrt{3}}\left(-\zeta_3\frac{U}{2} + \zeta_1U\right),\\ \left(\zeta U_t\right)_3 &= \frac{1}{\sqrt{3}}\left(-\zeta_1U + \zeta_2\frac{U}{2}\right). \end{align} \]
From this it follows
\[ \begin{align} \newtilde{\left(\zeta U_t\right)_1}^{(1)} + \newtilde{\left(\zeta U_t\right)_2}^{(2)} + \newtilde{\left(\zeta U_t\right)_3}^{(3)} &\propto \newtilde{\zeta_3 - \zeta_2}^{(1)} + \newtilde{2\zeta_1 - \zeta_3}^{(2)} + \newtilde{\zeta_2 - 2\zeta_1}^{(3)}\nonumber\\ = -\newtilde{\zeta_2}^{(1)} + \newtilde{\zeta_3}^{(1)} - \newtilde{\zeta_3}^{(2)} + 2\newtilde{\zeta_1}^{(2)} - 2\newtilde{\zeta_1}^{(3)} + \newtilde{\zeta_2}^{(3)} &\stackrel{\text{in general}}{\not=} 0. \end{align} \]
This ansatz also does not satisfy the Thuburn condition. Rather, a preparatory averaging of the vorticities is required, which leads to the ansatz
\[ \begin{align} \left(\zeta U_t\right)_1 &= \frac{1}{\sqrt{3}}\left(-\newoverline{\zeta_2}^{(3)}\frac{U}{2} + \newoverline{\zeta_3}^{(2)}\frac{U}{2}\right),\\ \left(\zeta U_t\right)_2 &= \frac{1}{\sqrt{3}}\left(-\newoverline{\zeta_3}^{(1)}\frac{U}{2} + \newoverline{\zeta_1}^{(3)}U\right),\\ \left(\zeta U_t\right)_3 &= \frac{1}{\sqrt{3}}\left(-\newoverline{\zeta_1}^{(2)}U + \newoverline{\zeta_2}^{(1)}\frac{U}{2}\right). \end{align} \]
This satisfies the Thuburn condition, since averaging operators commute. An operator other than the simple averaging operator $\newoverline{\zeta_i}^{(j)}$ may also be used. With $\zeta \to \eta = \zeta + f$, the complete vorticity flux term on the regular C-grid can be expressed in this way.
It is not trivial how the vorticity flux term is to be extended to the deformed grid. For this, one first starts from the f-plane in order to be able to assume a homogeneous vorticity $f_0$. Instead of attempting to generalize the Thuburn condition to the deformed grid, the closely related requirement that a geostrophic mode should exist is used as the basic condition. Furthermore, the discretization should be energy-conserving.