3.11 Exercise 6: Kelvin-Helmholtz Instability
51
3.11.2 Task Description
Consider a vertical ocean slice of 500 m in length and 100 m in depth with cyclic
lateral boundaries (see below) and grid spacings of Δx = 5 m and Δz = 2 m.
The density interface of two superimposed layers of different densities is located
at a depth of 50 m. The density of the top layer is set to ρ 1 = 1,027.0 kg/m
3 .
The bottom layer has a density of ρ 2 = 1,029.0 kg/m
3 , yielding a density difference of Δρ = 2 kg/m
3 . Density diffusion is included in the model. Horizontal
and vertical density diffusivities are set to small uniform values of K h = K z = 1 ×
10
−4 m
2
/s.
The initial lateral flow in the top layer has an average speed of 1 m/s, whereas
the lower layer is at rest. Randomly generated values of ±0.01 m/s are added to
the upper-layer flow. This technique promotes the initial growth of disturbances at
random locations. The random-number generator was taken from Press et al. (1989).
With a difference in layer speeds of Δu = 1 m/s, Eq. (3.62) suggests that the transition mixing zone attains a thickness of around 13 m. Equation (3.56) predicts that
mixing is initiated by disturbances of <18 m in wavelength. These length scales are
barely resolved with the grid spacings chosen. The reader is encouraged to use a
finer spatial resolution for more accurate results.
This task employs the rigid-lid version of the vertical ocean-slice model. The
total simulation time is 100 min with outputs every minute. The time step is set to
Δt = 1 s. The pressure accuracy for the S.O.R. iteration is set to = 1 × 10
−3 Pa.
Will the model adequately simulate vertical mixing inherent with the KelvinHelmholtz instability mechanism?
3.11.3 Cyclic Boundary Conditions
Cyclic boundary conditions mean that opposite lateral boundaries are connected
with each other, such that flow escaping through one boundary enters through the
opposite one. With our numerical grid, cyclic boundary conditions are implemented
with:
ψ k=0 = ψ k=nx
ψ k=nx+1 = ψ k=1
where ψ stands for any variable. The use of cyclic boundary conditions is justified
for dynamical disturbances of a wavelength much shorter than the length of the
model domain and occupying the entire model domain. Cyclic boundary conditions
correspond then to a model domain of infinite length. A rule of thumb is that the
model domain should extend at least ten wavelengths of disturbances. Otherwise,
some stretching bias will occur.
51
3.11.2 Task Description
Consider a vertical ocean slice of 500 m in length and 100 m in depth with cyclic
lateral boundaries (see below) and grid spacings of Δx = 5 m and Δz = 2 m.
The density interface of two superimposed layers of different densities is located
at a depth of 50 m. The density of the top layer is set to ρ 1 = 1,027.0 kg/m
3 .
The bottom layer has a density of ρ 2 = 1,029.0 kg/m
3 , yielding a density difference of Δρ = 2 kg/m
3 . Density diffusion is included in the model. Horizontal
and vertical density diffusivities are set to small uniform values of K h = K z = 1 ×
10
−4 m
2
/s.
The initial lateral flow in the top layer has an average speed of 1 m/s, whereas
the lower layer is at rest. Randomly generated values of ±0.01 m/s are added to
the upper-layer flow. This technique promotes the initial growth of disturbances at
random locations. The random-number generator was taken from Press et al. (1989).
With a difference in layer speeds of Δu = 1 m/s, Eq. (3.62) suggests that the transition mixing zone attains a thickness of around 13 m. Equation (3.56) predicts that
mixing is initiated by disturbances of <18 m in wavelength. These length scales are
barely resolved with the grid spacings chosen. The reader is encouraged to use a
finer spatial resolution for more accurate results.
This task employs the rigid-lid version of the vertical ocean-slice model. The
total simulation time is 100 min with outputs every minute. The time step is set to
Δt = 1 s. The pressure accuracy for the S.O.R. iteration is set to = 1 × 10
−3 Pa.
Will the model adequately simulate vertical mixing inherent with the KelvinHelmholtz instability mechanism?
3.11.3 Cyclic Boundary Conditions
Cyclic boundary conditions mean that opposite lateral boundaries are connected
with each other, such that flow escaping through one boundary enters through the
opposite one. With our numerical grid, cyclic boundary conditions are implemented
with:
ψ k=0 = ψ k=nx
ψ k=nx+1 = ψ k=1
where ψ stands for any variable. The use of cyclic boundary conditions is justified
for dynamical disturbances of a wavelength much shorter than the length of the
model domain and occupying the entire model domain. Cyclic boundary conditions
correspond then to a model domain of infinite length. A rule of thumb is that the
model domain should extend at least ten wavelengths of disturbances. Otherwise,
some stretching bias will occur.