22
3 Basics of Nonhydrostatic Modelling
Fig. 3.1 Illustration of vertical levels in z-coordinate models and in σ -coordinate models
3.2 2D Vertical-Slice Modelling
3.2.1 Configuration
We consider a vertical ocean slice with the z-axis pointing upward and the x-axis
being the horizontal coordinate (Fig. 3.2). The free sea surface is located at z =
η. The sea floor can be found at z = −h o . The assumption of this chapter is
absence of both motion and gradients of variables perpendicular to this slice. This
implies neglect of the Coriolis force. A modification to the latter are so-called
2.5-dimensional ocean-slice models which allow for flow normal to the slice and
inclusion of the Coriolis force, but this flow is also assumed to have vanishing
gradients in the y-direction. Chapter 4 presents applications of such models. The
momentum equations for the two-dimensional ocean slice are given by:
∂u
∂t
+ Adv(u) = −
1
ρ o
∂ P
∂ x
+
∂
∂ x
A h
∂u
∂ x
+
∂
∂z
A z
∂u
∂z
∂w
∂t
+ Adv(w) = −
1
ρ o
∂ P
∂z
−
ρ
ρ o
g +
∂
∂ x
A h
∂w
∂ x
+
∂
∂z
A z
∂w
∂z
(3.1)
∂ρ
∂t
+ Adv(ρ
) =
∂
∂ x
K h
∂ρ
∂ x
+
∂
∂z
K z
∂ρ
∂z
The advection operator is given by:
Adv(ψ) = u
∂ψ
∂ x
+ w
∂ψ
∂z