Layer-integrated equations

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 $z_k$ by considering the Leibniz rule

$\displaystyle \int_{z_{k-1}}^{z_k}\partial_x f\,dz = \partial_x\int_{z_{k-1}}^{z_k} f\,dz -f(z_k)\partial_xz_k +f(z_{k-1})\partial_xz_{k-1}$

(20)

for any function $f$ . For the vertical staggering of the layer notation see figure 7.

More details about the layer integration are given in Burchard and Petersen (1997).

With the further definitions of layer integrated transport,

$\displaystyle p_k:=\int_{z_{k-1}}^{z_k}u\,dz,\qquad q_k:=\int_{z_{k-1}}^{z_k}v\,dz,$

(21)

layer mean velocities,

$\displaystyle u_k:=\frac{p_k}{h_k},\qquad v_k:=\frac{q_k}{h_k},$

(22)

and layer averaged tracer concentrations and buoyancy,

$\displaystyle c^i_k:=\frac{1}{h_k}\int_{z_{k-1}}^{z_k}c^i\,dz,\qquad b_k:=\frac{1}{h_k}\int_{z_{k-1}}^{z_k}b\,dz,$

(23)

and the grid related vertical velocity,

$\displaystyle \bar w_k:=(w-\partial_tz-u\partial_xz-v\partial_yz)_{z=z_k},$

(24)

the continuity equation (3) has the layer-integrated form:

$\displaystyle \partial_t h_k + \partial_x p_k + \partial_y q_k + \bar w_k - \bar w_{k-1}=0.$

(25)

It should be noted that the grid related velocity is located on the layer interfaces. After this, the layer-integrated momentum equations read as:

$\begin{displaymath}\begin{array}{l} \partial_t p_k +\bar w_k \tilde u_k -\bar w_... ...tial^*_xb)_j \right)\Bigg\} = -gh_k\partial_x\zeta, \end{array}\end{displaymath}$

(26)

$\begin{displaymath}\begin{array}{l} \partial_t q_k +\bar w_k \tilde v_k -\bar w_... ...rtial^*_yb)_j \right)\Bigg\} = -gh_k\partial_y\zeta \end{array}\end{displaymath}$

(27)

with suitably chosen advective horizontal velocities $\tilde u_k$ and $\tilde v_k$ (see section 8.13.7) on page , the shear stresses

$\displaystyle \tau^x_k = \left(\nu_t \partial_z u \right)_k,$

(28)

and

$\displaystyle \tau^y_k = \left(\nu_t \partial_z v \right)_k,$

(29)

and the horizontal buoyancy gradients

$\displaystyle (\partial^*_xb)_k=\frac12(\partial_xb_{k+1}+\partial_x b_k) -\partial_xz_k\frac{b_{k+1}-b_k}{\frac12(h_{k+1}+h_k)}$

(30)

and

$\displaystyle (\partial^*_yb)_k=\frac12(\partial_yb_{k+1}+\partial_y b_k) -\partial_yz_k\frac{b_{k+1}-b_k}{\frac12(h_{k+1}+h_k)}.$

(31)

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 $w$ which is convenient in the discrete space if $w$ is evaluated at the layer centres (see Deleersnijder and Ruddick (1992)):

$\displaystyle w_k=\frac{1}{h_k}\left( \partial_t(h_kz_{k-1/2})+\partial_x(p_kz_{k-1/2})+\partial_y(q_kz_{k-1/2}) +\bar w_kz_k-\bar w_{k-1}z_{k-1}\right).$

(32)

It should be mentioned that $w$ only needs to be evaluated for post-processing reasons.

For the layer-integrated tracer concentrations, we obtain the following expression:

$\begin{displaymath}\begin{array}{l} \partial_t (h_k c^i_k) + \partial_x (p_kc^i_... ...rtial_y\left(A_k^Th_k\partial_yc^i_k\right) =Q^i_k. \end{array}\end{displaymath}$

(33)

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).