The BSH New Operational Circulation Model Using General Vertical Co-ordinates 
21 
hold. The transformed system has the spatial co 
ordinates x , у , and s (as vertical co-ordinate). Now, 
compared to the above, consider г from the opposite 
point of view, i.e. from the perspective of the general 
vertical co-ordinate system. Height z then is a 
function of (.r, y, s, t) , where time t enters as another 
independent variable. Furthermore, let v^, w be 
conventional horizontal and vertical components of 
velocity, respectively. Let us look at the vertical 
velocity as represented in the (.r, y, s, t) system. 
It reads 
Dz 3z -t-, ' 3z 
w= = — + ГУ z + s-— 
Dr 31 v s is 
where 
(1) 
V ■ - denotes the horizontal nabla operator applied on 
a surface of constant ,v. 
components of vertical motion of an object as seen 
from the viewpoint of a general frame of non 
horizontal vertically moving surfaces. These are, 
firstly, the apparent motion of the object due to the 
up-and-down motion of the reference level, secondly, 
the vertical motion due to horizontal motion of the 
object and inclination of the reference level and, 
thirdly, the relative motion due to motion along the 
vertical with respect to the rising or falling reference 
level. The latter component of vertical velocity, across 
the moving s surface, is called pseudo-vertical 
velocity. 
The above equation (1) provides a basic relation 
between the listed velocity components, and it holds 
true for any system with a generalised vertical co 
ordinate. To characterise any special system, one 
more relation is required. Here, no algebraic equation 
will be introduced but another relation between 
pseudo-vertical velocity and up-and-down velocity, 
cf. (Kleine, 2003) 
ds 
dt 
d_ 
ds 
Г 
dz/d s 
v.(»v,=) 
(2) 
This equation is to specify the flow-driven co 
ordinate surface motion. Non-negatively valued 
adjustable parameters (Uy, p, v) were chosen for the 
equation. As regards dimensions, y Dis 
nondimensional while p and v are diffusion 
coefficients [m 2 /s\. The terms associated with p and v 
ore used to regularise the co-ordinate surface 
dynamics as they work to relax the configuration, 
balancing the distortion caused by co-motion with the 
flow. In the suggested equation, the pseudo-vertical 
velocity is related to the up-and-down velocity of the 
co-ordinate surface, a vertical redistribution process, 
and horizontal diffusion-like smoothing. 
Equation (1) is a statement of a physical fact 
while equation (2) is a specifying ansatz - 
construction, essentially a model. In conjunction, they 
constitute a system of equations for the evolution of 
the vertical representation which may be solved for 
the up-and-down velocity and the pseudo-vertical 
velocity. 
It is easily demonstrated that the suggested layer 
model comprises all common forms of vertical 
representation which are, firstly, height, secondly, 
sigma, eta (re-scaled height), and thirdly, material 
(isopycnal) surfaces, which are all covered as special 
limit cases. For the proof see Kleine (2003). Our 
implemented setting is intermediate in order to 
achieve a reasonable compromise between co-motion 
and regularity. 
To focus on dz/ds and its evolution, consider the 
equation of continuity (for an incompressible fluid). 
In the context of a general vertical co-ordinate, it 
takes the form of 
= 0 
As a prognostic equation for differential layer 
thickness dz/Ss in flux form, it reads 
3 ( - r 
7 Ì V 
"7 
(-dz) 
— W- V\ 
1 z + \ 
1 
v — 
we recognize the 
ds 
S 1 
s 
ds) 
dt 
1 
1 141 
3 
+ 
( * dz y 
s — 
T, 
' - dz ^ 
v— 
ÒS 
3s 
\ / 
ds 
\ / 
= 0 
(3) 
Plug Eq.(2) into the continuity equation (3). The 
result is an evolution equation for differential layer 
thickness, controlled by horizontal flow, vertical 
rectification, and horizontal smoothing. 
(1 + 
dt Us- ds 
dz/ds 
П I -9z V-, ( dz 
(4) 
To completely specify the evolution of dz/Ss (as 
a spatially distributed and time-varying scalar 
quantity), an initial-boundary value problem should 
be formulated. For a budget equation in flux form, it 
is natural to prescribe normal flux, i.e. the flow 
component normal to the boundary. In our special 
setting, the boundary conditions (for the co-ordinate 
evolution problem) are as of the eta (re-scaled height) 
model, i.e., firstly, vanishing pseudo-vertical velocity 
s dz/ds = 0 at th e surface and bottom and, secondly, 
fixed height dz/dt = 0 at the bottom. 
4. Model Setup and First Results 
In the horizontal, the model equations of 
BSHcmod are defined for a computational grid of 
spherical co-ordinates. Grid spacing in the German 
Bight and the western Baltic Sea is approx. 900 m 
(AL= 50"; AtpD = 30"), and in the other parts of the 
North Sea and the Baltic approx. 5 km (AL = 5 Atp 
□ = 3'). 
The entire model, of which the equation of 
continuity is only one component, is formulated in the