4.1 The Basis
101
The speed of frontal flows decreases exponentially (on a spatial scale of R) with
increasing distance from the surface outcrop of density surfaces. In the general case,
a counterflow establishes in the water column underneath a surface density front.
Density fronts in the coastal ocean attain typical widths of 1–20 km.
4.1.6 The 2.5d Shallow-Water Model
On the basis of vanishing gradients of variables in the y-direction and inclusion of
the Coriolis force, the horizontal momentum equations for a vertical ocean slice can
be written as:
∂u
∂t
+ u
∂u
∂ x
+ w
∂u
∂z
− f v = −
1
ρ o
∂( p + q)
∂ x
+ Diff(u)
(4.14)
∂v
∂t
+ u
∂v
∂ x
+ w
∂v
∂z
+ f u = Diff(v)
(4.15)
where f is the Coriolis parameter and Diff(ψ) is the diffusion operator, being specified in Sect. 3.15. The other model equations are the same as in Exercise 12.
4.1.7 Implementation of the Coriolis Force
In the 2.5d version of the Arakawa C-grid, the v-components of velocity are calculated at pressure grid points (see Fig. 3.3). Hence, interpolation of velocity values
is required for calculation of the Coriolis force. It can be shown that explicit formulation of the Coriolis force is numerically unstable. The nonhydrostatic solver of
the Navier-Stokes equations is already formulated in an implicit manner in terms
of dynamic pressure (see Sect. 3.4). Using another semi-implicit approach for the
Coriolis force would lead to an even more complex solver. To avoid this, the Coriolis
force is treated here by the local-rotation approach (described in Sect. 3.14 of K¨ ampf
(2009)). With the sole presence of the Coriolis force, this approach leads to the
finite-difference equations:
u
n+1
= cos(α)u
n
+ sin(α)v
n
,
v
n+1
= cos(α)v
n
− sin(α)u
n
where the rotation angle is α = 2 arcsin (0.5Δt f ). For sufficiently small numerical
time steps of Δt | f | << 1, this can be approximated by α ≈ Δt f . With inclusion of
other forces, this approach can be added to the first-guess of velocity components,
yielding:
101
The speed of frontal flows decreases exponentially (on a spatial scale of R) with
increasing distance from the surface outcrop of density surfaces. In the general case,
a counterflow establishes in the water column underneath a surface density front.
Density fronts in the coastal ocean attain typical widths of 1–20 km.
4.1.6 The 2.5d Shallow-Water Model
On the basis of vanishing gradients of variables in the y-direction and inclusion of
the Coriolis force, the horizontal momentum equations for a vertical ocean slice can
be written as:
∂u
∂t
+ u
∂u
∂ x
+ w
∂u
∂z
− f v = −
1
ρ o
∂( p + q)
∂ x
+ Diff(u)
(4.14)
∂v
∂t
+ u
∂v
∂ x
+ w
∂v
∂z
+ f u = Diff(v)
(4.15)
where f is the Coriolis parameter and Diff(ψ) is the diffusion operator, being specified in Sect. 3.15. The other model equations are the same as in Exercise 12.
4.1.7 Implementation of the Coriolis Force
In the 2.5d version of the Arakawa C-grid, the v-components of velocity are calculated at pressure grid points (see Fig. 3.3). Hence, interpolation of velocity values
is required for calculation of the Coriolis force. It can be shown that explicit formulation of the Coriolis force is numerically unstable. The nonhydrostatic solver of
the Navier-Stokes equations is already formulated in an implicit manner in terms
of dynamic pressure (see Sect. 3.4). Using another semi-implicit approach for the
Coriolis force would lead to an even more complex solver. To avoid this, the Coriolis
force is treated here by the local-rotation approach (described in Sect. 3.14 of K¨ ampf
(2009)). With the sole presence of the Coriolis force, this approach leads to the
finite-difference equations:
u
n+1
= cos(α)u
n
+ sin(α)v
n
,
v
n+1
= cos(α)v
n
− sin(α)u
n
where the rotation angle is α = 2 arcsin (0.5Δt f ). For sufficiently small numerical
time steps of Δt | f | << 1, this can be approximated by α ≈ Δt f . With inclusion of
other forces, this approach can be added to the first-guess of velocity components,
yielding: