Coriolis and Advection: flux form

&namdyn_adv    !   formulation of the momentum advection
   ln_dynadv_vec = .true.  !  vector form (T) or flux form (F)
   nn_dynkeg     = 0       ! scheme for grad(KE): =0   C2  ;  =1   Hollingsworth correction
   ln_dynadv_cen2= .false. !  flux form - 2nd order centered scheme
   ln_dynadv_ubs = .false. !  flux form - 3rd order UBS      scheme
   ln_dynzad_zts = .false. !  Use (T) sub timestepping for vertical momentum advection

Options are defined through the namdyn_adv namelist variables. In the flux form (as in the vector invariant form), the Coriolis and momentum advection terms are evaluated using a leapfrog scheme, $ i.e.$ the velocity appearing in their expressions is centred in time (now velocity). At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied following Chap.8.

Coriolis plus curvature metric terms (dynvor.F90)

In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term. This altered Coriolis parameter is thus discretised at $ f$-points. It is given by:

f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i} -... u ^{j+1/2}\delta _{j+1/2} \left[ {e_{1u} } \right] }  \right)

Any of the (C.13), (6.6) and (C.15) schemes can be used to compute the product of the Coriolis parameter and the vorticity. However, the energy-conserving scheme (C.15) has exclusively been used to date. This term is evaluated using a leapfrog scheme, $ i.e.$ the velocity is centred in time (now velocity).

Flux form Advection term (dynadv.F90)

The discrete expression of the advection term is given by :

\begin{equation*}\left\{ \begin{aligned}\frac{1}{e_{1u} e_{2u} e_{3u}} \left( ...
...2w}\;w}^{j+1/2}  v_{vw} \right] \right)  \end{aligned} \right.\end{equation*}

Two advection schemes are available: a $ 2^{nd}$ order centered finite difference scheme, CEN2, or a $ 3^{rd}$ order upstream biased scheme, UBS. The latter is described in Shchepetkin and McWilliams [2005]. The schemes are selected using the namelist logicals ln_dynadv_cen2 and ln_dynadv_ubs. In flux form, the schemes differ by the choice of a space and time interpolation to define the value of $ u$ and $ v$ at the centre of each face of $ u$- and $ v$-cells, $ i.e.$ at the $ T$-, $ f$-, and $ uw$-points for $ u$ and at the $ f$-, $ T$- and $ vw$-points for $ v$.

$ 2^{nd}$ order centred scheme (cen2) (ln_dynadv_cen2=true)

In the centered $ 2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points :

\begin{equation*}\left\{ \begin{aligned}u_T^{cen2} &=\overline u^{i } \quad & u_...
...d & v_{vw}^{cen2} &=\overline v ^{k+1/2}  \end{aligned} \right.\end{equation*}

The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive ($ i.e.$ it may create false extrema). It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to produce a sensible solution. The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, so $ u$ and $ v$ are the now velocities.

Upstream Biased Scheme (UBS) (ln_dynadv_ubs=true)

The UBS advection scheme is an upstream biased third order scheme based on an upstream-biased parabolic interpolation. For example, the evaluation of $ u_T^{ubs} $ is done as follows:

$\displaystyle u_T^{ubs} =\overline u ^i-\;\frac{1}{6} \begin{cases}u''_{i-1/2}&...
... u''_{i+1/2}& \text{if $ \overline{e_{2u} e_{3u}  u}^i < 0$ } \end{cases}$ (6.16)

where $ u''_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results in a dissipatively dominant ($ i.e.$ hyper-diffusive) truncation error [Shchepetkin and McWilliams, 2005]. The overall performance of the advection scheme is similar to that reported in Farrow and Stevens [1995]. It is a relatively good compromise between accuracy and smoothness. It is not a positive scheme, meaning that false extrema are permitted. But the amplitudes of the false extrema are significantly reduced over those in the centred second order method. As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum ($ i.e.$ setting both ln_dynldf_lap and ln_dynldf_bilap to false), and it is recommended to do so.

The UBS scheme is not used in all directions. In the vertical, the centred $ 2^{nd}$ order evaluation of the advection is preferred, $ i.e.$ $ u_{uw}^{ubs}$ and $ u_{vw}^{ubs}$ in (6.16) are used. UBS is diffusive and is associated with vertical mixing of momentum.

For stability reasons, the first term in (6.17), which corresponds to a second order centred scheme, is evaluated using the now velocity (centred in time), while the second term, which is the diffusion part of the scheme, is evaluated using the before velocity (forward in time). This is discussed by Webb et al. [1998] in the context of the Quick advection scheme.

Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by one coefficient. Replacing $ 1/6$ by $ 1/8$ in (6.17) leads to the QUICK advection scheme [Webb et al., 1998]. This option is not available through a namelist parameter, since the $ 1/6$ coefficient is hard coded. Nevertheless it is quite easy to make the substitution in the dynadv_ubs.F90 module and obtain a QUICK scheme.

Note also that in the current version of dynadv_ubs.F90, there is also the possibility of using a $ 4^{th}$ order evaluation of the advective velocity as in ROMS. This is an error and should be suppressed soon.

Gurvan Madec and the NEMO Team
NEMO European Consortium2017-02-17