50
3 Basics of Nonhydrostatic Modelling
∂u
∂t
+ u
∂u
∂ x
+ w
∂u
∂z
= −
1
ρ o
∂ P
∂ x
(3.57)
∂w
∂t
+ u
∂w
∂ x
+ w
∂w
∂z
= −
1
ρ o
∂ P
∂z
−
ρ
ρ o
g
(3.58)
∂ρ
∂t
+ u
∂ρ
∂ x
+ w
∂ρ
∂z
= 0
(3.59)
∂u
∂ x
+
∂w
∂z
= 0
(3.60)
where density diffusion and frictional effects are ignored, to zero-order approximation. Also, the existence of the sea surface is ignored, based on the assumption of
a fluid of infinite vertical extent. For a constant stability frequency N and an initial
horizontal flow of a speed that varies linearly with depth (∂u/∂z = constant), it can
then be shown that instability occurs locally, if the so-called Richardson number
falls below a threshold value of 1/4 (Richardson, 1920). The Richardson number is
hereby defined by:
Ri =
N
2
(∂u/∂z) 2
(3.61)
Accordingly, instability in a two-layer shear fluid, such as that depicted in
Fig. 3.23, can be expected to produce a final transition layer of thickness Δh with
Ri ≈ 0.25 throughout this layer. Using Eq. (3.61), the thickness of the resultant
transition layer can be estimated from the equation:
−g/ρ o Δρ/Δh
(Δu/Δh)
2
= 0.25 or Δh = 0.25
ρ o
|Δρ|
(Δu)
2
g
(3.62)
where Δρ and Δu, respectively, are differences in density and flow speed across this
layer. We will use the latter equation for verification of simulation results. Note that
the vertical scale from Eq. (3.62) is of the same order of magnitude as the horizontal
wavelength described by Eq. (3.56). Hence, horizontal and vertical length scales
of turbulent vortices inherent with the Kelvin-Helmholtz instability mechanism are
closely related to each other, which implies a nonhydrostatic nature of the dynamics
at play.
3.11 Exercise 6: Kelvin-Helmholtz Instability
3.11.1 Aim
The aim of this exercise is to simulate mixing at the density interface of a two-layer
vertical shear fluid.
3 Basics of Nonhydrostatic Modelling
∂u
∂t
+ u
∂u
∂ x
+ w
∂u
∂z
= −
1
ρ o
∂ P
∂ x
(3.57)
∂w
∂t
+ u
∂w
∂ x
+ w
∂w
∂z
= −
1
ρ o
∂ P
∂z
−
ρ
ρ o
g
(3.58)
∂ρ
∂t
+ u
∂ρ
∂ x
+ w
∂ρ
∂z
= 0
(3.59)
∂u
∂ x
+
∂w
∂z
= 0
(3.60)
where density diffusion and frictional effects are ignored, to zero-order approximation. Also, the existence of the sea surface is ignored, based on the assumption of
a fluid of infinite vertical extent. For a constant stability frequency N and an initial
horizontal flow of a speed that varies linearly with depth (∂u/∂z = constant), it can
then be shown that instability occurs locally, if the so-called Richardson number
falls below a threshold value of 1/4 (Richardson, 1920). The Richardson number is
hereby defined by:
Ri =
N
2
(∂u/∂z) 2
(3.61)
Accordingly, instability in a two-layer shear fluid, such as that depicted in
Fig. 3.23, can be expected to produce a final transition layer of thickness Δh with
Ri ≈ 0.25 throughout this layer. Using Eq. (3.61), the thickness of the resultant
transition layer can be estimated from the equation:
−g/ρ o Δρ/Δh
(Δu/Δh)
2
= 0.25 or Δh = 0.25
ρ o
|Δρ|
(Δu)
2
g
(3.62)
where Δρ and Δu, respectively, are differences in density and flow speed across this
layer. We will use the latter equation for verification of simulation results. Note that
the vertical scale from Eq. (3.62) is of the same order of magnitude as the horizontal
wavelength described by Eq. (3.56). Hence, horizontal and vertical length scales
of turbulent vortices inherent with the Kelvin-Helmholtz instability mechanism are
closely related to each other, which implies a nonhydrostatic nature of the dynamics
at play.
3.11 Exercise 6: Kelvin-Helmholtz Instability
3.11.1 Aim
The aim of this exercise is to simulate mixing at the density interface of a two-layer
vertical shear fluid.