There are two different ways to derive the layer-integrated equations. Burchard and Petersen (1997) transform first the equations into general vertical coordinate form (see Deleersnijder and Ruddick (1992)) and afterwards integrate the transformed equations over constant intervals in the transformed space. Lander et al. (1994) integrate the equations in the Cartesian space over surfaces by considering the Leibniz rule
for any function . For the vertical staggering of the layer notation see figure 8.
More details about the layer integration are given in Burchard and Petersen (1997).
With the further definitions of layer integrated transport,
layer mean velocities,
and layer averaged tracer concentrations and buoyancy,
and the grid related vertical velocity,
the continuity equation (3) has the layer-integrated form:
It should be noted that the grid related velocity is located on the layer interfaces. After this, the layer-integrated momentum equations read as:
with suitably chosen advective horizontal velocities and (see section 8.13.7) on page , the shear stresses
and the horizontal buoyancy gradients
The layer integration of the pressure gradient force is discussed in detail by Burchard and Petersen (1997).
A conservative formulation can be derived for the recalculation of the physical vertical velocity which is convenient in the discrete space if is evaluated at the layer centres (see Deleersnijder and Ruddick (1992)):
It should be mentioned that only needs to be evaluated for post-processing reasons.
For the layer-integrated tracer concentrations, we obtain the following expression:
It should be noted that the "horizontal" diffusion does no longer occur along geopotential surfaces but along horizontal coordinate lines. The properly transformed formulation would include some cross-diagonal terms which may lead to numerical instabilities due to violation of monotonicity. For an in-depth discussion of this problem, see Beckers et al. (1998) and Beckers et al. (2000).