42 
Using £2 = (0,0,£2 V ) together with V^ = (0,0,V v ^) the p -equation is dimensionally 
uncoupled in the last two terms: 
V 2 P = -PoV - Vvv + V ■ F - V v pV v </> - V h ■ (2Clxp 0 v) h . (4) 
Now, to explain different approximations, a “transport velocity“ v with V • v = 0 is introduced 
and the time derivative of the divergence is retained: 
v V = -/7 0 -^V-v-/7 0 V-Vvv+V-F-V v /7-V v ^ v -V A -(2axp 0 v) h . (5) 
The individual approximation is then (after introducing some sort of co-ordinate system with 
vertical axis r) determined by the definitions of v , v and F : 
V=(v h ,v v ) 
v =<TV v v) 
non-hydrostatic, non-linear 
V =(v*,v v ) 
o 
o' 
o 
II 
F = (0,0,0) 
non-hydrostatic, linear 
V=(v h ,v v ) 
O 
o' 
O 
II 
F = (0,0,0) 
hydrostatic, linear 
V=(v h ,v v ) 
r 
V =(V* -jv fc -v h dr) 
—h 
F = (F h ,0) 
hydrostatic, non-linear 
r 
V =(V*-jv A •v h dr) 
—h 
r 
v =(v*-jv h -v h dr) 
—h 
F = (F h ,0) 
Boussinesq, non-linear 
v =(v*-fv A •v h dr) 
o 
o' 
o 
II 
F = (0,0,0) 
Boussinesq, linear. 
—h 
The first non-hydrostatic case is just a reformulation without any additional assumptions. In 
both non-hydrostatic approximations and for the Boussinesq equations, V • v = 0, which 
removes the first term from equation 5, regaining equation 4. In the hydrostatic cases, 
dv d 
—- = 0 is assumed. Thus the first term in equation 5 takes the form of -p 0 —V h ■v h and 
dt dt 
the horizontal part of the p -equation is completely uncoupled from the vertical. 
The hydrostatic formulation and the Boussinesq equations have in common that the vertical 
velocity component is computed diagnostically from the mass balance equation, and p is 
determined by integration of the third momentum equation: 
V 
Phs = Pr, + ¡pVvfidr 
Pb=P v + 
■ v h dr"dr' + 
V __ V 
Po {(Vvv) v dr + JpV v (f)dr . 
r r 
In the equation for p B , as compared to hydrostatic pressure, the first additional term is the 
most important one because it allows dispersion (frequency dispersion). The second 
additional term is neglected also in non-linear Boussinesq equations. 
Therefore, it may be relatively easy to integrate dispersion effects with the same accuracy 
that Boussinesq models have into models such as those used at the BSH, just by storing 
V h -v h at different time levels. By contrast, in the non-hydrostatic case, the third momentum 
equation would have to be solved in prognostic form, as well as the Poisson equation 4 with 
suitable boundary conditions (Marshall et al. 1997a,b).