114
4 2.5D Vertical Slice Modelling
over time under continuous offshore Ekman transport, the sea level at the open
boundary is kept at its initial value throughout the simulation.
The surface-mixed layer is initially 25 m thick and of a density of 1,027 kg/m
3 .
A pycnocline is situated at the base of this layer across which density changes by
1 kg/m
3 . This corresponds to a local stability frequency of N = 4.4 × 10
−2 s
−1 .
Density is assumed to increase with depth underneath the pycnocline with a stability
frequency of N = 6.2 × 10
−3 s
−1 . The Coriolis parameter is set to f = −1 ×
10
−4 s
−1 , representing mid-latitudes in the Southern Hemisphere. Recall that, in the
Southern Hemisphere, the net wind-driven water movement in the surface Ekman
layer is at right angle and to left with respect to the wind direction.
Horizontal eddy viscosity and eddy viscosity are set to a constant value of A h =
K h = 1 m
2 /s. Vertical eddy viscosity and diffusivity, assumed to be equal, are diagnosed from an advanced turbulence closure scheme, described in the next section.
A value of K z = A z = 0.05 m
2 /s is prescribed near the surface to mimic the immediate effect of wind mixing. Ambient background values of A z,min = K z,min =
1×10
−4 m
2 /s are used. The bottom friction parameter (assuming a quadratic bottomfriction law) is set to r = 0.003.
The model is forced via prescription of a alongshore wind stress of the form:
τ
wind
y
= τ o sin (2π t/T )
(4.21)
where t is time, and the period T is chosen as 10 days, being characteristic for
synoptic weather events. Wind-stress components are defined at the same locations
as u and v.
Two different scenarios are considered, both running over a total of 5 days with
data output every 3 hrs. The first scenario uses τ o = + 0.2 Pa corresponding to
upwelling favorable coastal winds, whereas the second scenario uses τ o = −0.2 Pa,
which should lead to coastal downwelling. The mean wind-stress magnitude over
the 5-day simulation period is 2 |τ o | /π = 0.128 Pa. The surface stress felt by the
ocean is calculated from Eq. (2.5).
This application uses a free surface and a numerical time step of Δt = 12 s.
The pressure accuracy of the S.O.R. iteration is set to = 1 × 10
−4 Pa. Zerogradient conditions are employed for the alongshore velocity component v near
“dry” grid cells. This condition, called full-slip boundary condition, disables lateral
friction, which otherwise can create problems for variable bottom topography in
z-coordinate models.
4.5.3 Advanced Turbulence Closure
Vertical eddy diffusivity and eddy viscosity can be diagnosed from an advanced
turbulence closure proposed by Kochergin (1987). When applied to a vertical ocean
slice and under the assumption that eddy viscosity equals eddy diffusivity, this turbulence scheme reads:
K z = A z = (c 2 Δz)
2
(∂u/∂z)
2
+ (∂v/∂z)
2
− N 2 + A z,min
(4.22)
4 2.5D Vertical Slice Modelling
over time under continuous offshore Ekman transport, the sea level at the open
boundary is kept at its initial value throughout the simulation.
The surface-mixed layer is initially 25 m thick and of a density of 1,027 kg/m
3 .
A pycnocline is situated at the base of this layer across which density changes by
1 kg/m
3 . This corresponds to a local stability frequency of N = 4.4 × 10
−2 s
−1 .
Density is assumed to increase with depth underneath the pycnocline with a stability
frequency of N = 6.2 × 10
−3 s
−1 . The Coriolis parameter is set to f = −1 ×
10
−4 s
−1 , representing mid-latitudes in the Southern Hemisphere. Recall that, in the
Southern Hemisphere, the net wind-driven water movement in the surface Ekman
layer is at right angle and to left with respect to the wind direction.
Horizontal eddy viscosity and eddy viscosity are set to a constant value of A h =
K h = 1 m
2 /s. Vertical eddy viscosity and diffusivity, assumed to be equal, are diagnosed from an advanced turbulence closure scheme, described in the next section.
A value of K z = A z = 0.05 m
2 /s is prescribed near the surface to mimic the immediate effect of wind mixing. Ambient background values of A z,min = K z,min =
1×10
−4 m
2 /s are used. The bottom friction parameter (assuming a quadratic bottomfriction law) is set to r = 0.003.
The model is forced via prescription of a alongshore wind stress of the form:
τ
wind
y
= τ o sin (2π t/T )
(4.21)
where t is time, and the period T is chosen as 10 days, being characteristic for
synoptic weather events. Wind-stress components are defined at the same locations
as u and v.
Two different scenarios are considered, both running over a total of 5 days with
data output every 3 hrs. The first scenario uses τ o = + 0.2 Pa corresponding to
upwelling favorable coastal winds, whereas the second scenario uses τ o = −0.2 Pa,
which should lead to coastal downwelling. The mean wind-stress magnitude over
the 5-day simulation period is 2 |τ o | /π = 0.128 Pa. The surface stress felt by the
ocean is calculated from Eq. (2.5).
This application uses a free surface and a numerical time step of Δt = 12 s.
The pressure accuracy of the S.O.R. iteration is set to = 1 × 10
−4 Pa. Zerogradient conditions are employed for the alongshore velocity component v near
“dry” grid cells. This condition, called full-slip boundary condition, disables lateral
friction, which otherwise can create problems for variable bottom topography in
z-coordinate models.
4.5.3 Advanced Turbulence Closure
Vertical eddy diffusivity and eddy viscosity can be diagnosed from an advanced
turbulence closure proposed by Kochergin (1987). When applied to a vertical ocean
slice and under the assumption that eddy viscosity equals eddy diffusivity, this turbulence scheme reads:
K z = A z = (c 2 Δz)
2
(∂u/∂z)
2
+ (∂v/∂z)
2
− N 2 + A z,min
(4.22)