162
6 Rotational Effects
6.15.5 Additional Exercise for the Reader
Repeat this exercise for an ocean uniform in density (ρ 1 = ρ 2 = 1028 kg m
−3
) to
explore whether the barotropic instability process alone can produce a similar form
of frontal instability. The two-layer version of the shallow-water equations can be
adopted for this task, but the reader should avoid division by zero. . .
6.16 Density-Driven Flows
6.16.1 Background
Density-driven f ows are bodies of dense water that cascade downward on the continental slope to a depth where they meet ambient water of the same density. At
this equilibrium density level, these f ows tend to fl w along topographic contours
of the continental slope. Detachment from the seafloo and injection into the ambient ocean is also possible. Numerical modelling of gravity plumes started with the
“streamtube model” of Smith (1975). This model considers a laterally integrated
streamtube with a variable cross-sectional area and, under the assumption of stationarity, it produces the path and laterally averaged properties (density contrast,
velocity) of the plume on a given slope.
With an advanced method describing the dynamics of such reduced-gravity
plumes, Jungclaus and Backhaus (1994) employed the shallow-water equations for
a two-layer flui with an upper layer at rest. Without exchange of flui across the
“skin” of the plume, the dynamic governing this model can be written as:
∂u 2
∂t
+ u 2
∂u 2
∂ x
+ v 2
∂u 2
∂ y
− f v 2 = −g
∂η 2
∂ x
−
τ
bot
x
h 2 ρ 2
∂v 2
∂t
+ u 2
∂v 2
∂ x
+ v 2
∂v 2
∂ y
+ f u 2 = −g
∂η 2
∂ y
−
τ
bot
y
h 2 ρ 2
(6.72)
∂η 2
∂t
+
∂(u 2 h 2 )
∂ x
+
∂(v 2 h 2 )
∂ y
= 0
where g
= (ρ 2 − ρ 1 )/ρ 2 g is reduced gravity, and (τ
bot
x , τ
bot
y ) represents the frictional stress at the seafloo . These equations are known as the reduced-gravity plume
model. Interface displacements η 2 are define with respect to a certain reference
level. When formulated in finit differences, the CFL stability criterion in such a
model is given by:
Δt ≤
min(Δx, Δy)
√
2g h max
where h max is maximum plume thickness.
6 Rotational Effects
6.15.5 Additional Exercise for the Reader
Repeat this exercise for an ocean uniform in density (ρ 1 = ρ 2 = 1028 kg m
−3
) to
explore whether the barotropic instability process alone can produce a similar form
of frontal instability. The two-layer version of the shallow-water equations can be
adopted for this task, but the reader should avoid division by zero. . .
6.16 Density-Driven Flows
6.16.1 Background
Density-driven f ows are bodies of dense water that cascade downward on the continental slope to a depth where they meet ambient water of the same density. At
this equilibrium density level, these f ows tend to fl w along topographic contours
of the continental slope. Detachment from the seafloo and injection into the ambient ocean is also possible. Numerical modelling of gravity plumes started with the
“streamtube model” of Smith (1975). This model considers a laterally integrated
streamtube with a variable cross-sectional area and, under the assumption of stationarity, it produces the path and laterally averaged properties (density contrast,
velocity) of the plume on a given slope.
With an advanced method describing the dynamics of such reduced-gravity
plumes, Jungclaus and Backhaus (1994) employed the shallow-water equations for
a two-layer flui with an upper layer at rest. Without exchange of flui across the
“skin” of the plume, the dynamic governing this model can be written as:
∂u 2
∂t
+ u 2
∂u 2
∂ x
+ v 2
∂u 2
∂ y
− f v 2 = −g
∂η 2
∂ x
−
τ
bot
x
h 2 ρ 2
∂v 2
∂t
+ u 2
∂v 2
∂ x
+ v 2
∂v 2
∂ y
+ f u 2 = −g
∂η 2
∂ y
−
τ
bot
y
h 2 ρ 2
(6.72)
∂η 2
∂t
+
∂(u 2 h 2 )
∂ x
+
∂(v 2 h 2 )
∂ y
= 0
where g
= (ρ 2 − ρ 1 )/ρ 2 g is reduced gravity, and (τ
bot
x , τ
bot
y ) represents the frictional stress at the seafloo . These equations are known as the reduced-gravity plume
model. Interface displacements η 2 are define with respect to a certain reference
level. When formulated in finit differences, the CFL stability criterion in such a
model is given by:
Δt ≤
min(Δx, Δy)
√
2g h max
where h max is maximum plume thickness.