3.7 Exercise 4: Density-Driven Flows
41
criteria for both external gravity waves (Eq. 3.35) and advection (Eq. 3.43). Parameters of the S.O.R. scheme are kept at the same values as in the previous exercise.
3.7.3 Theory
Baroclinic pressure gradients associated with horizontal density differences will
produce a bottom-arrested density-driven flow. The resultant speed of the flow can
be estimated from energy conservation principles. In this process, potential energy
available from the initial density field is converted into kinetic energy of the flow.
This energy conversion can be quantified by the Bernoulli equation. Provided that
the plume density does not change, the Bernoulli equation reads:
0.5u
2
+ g
h = g
h o
(3.47)
where g
is reduced gravity, and h o and h, respectively, are initial and final plume
thicknesses. The Bernoulli equation is named after Daniel Bernoulli (Bernoulli,
1738) and his father Johann. The first true Bernoulli equation, however, was derived
by Euler (1755). The latter equation can be rewritten as:
u =
2g (h o − h)
(3.48)
3.7.4 Results
As anticipated, the initial density anomaly produces a density-driven flow spreading
toward the right-hand side of the model domain (Fig. 3.15). This flow forms an
isolated plume head. Counter-clockwise vortices forming in the lee of this head
induce vigorous mixing.
With g
= 0.0095 m s
−2
, h o = 100 m, and h ≈ 40 m, the Bernoulli equation
(Eq. 3.48) suggests a plume speed of u = 1.1 m/s. In good agreement with theory,
maximum speeds of the simulated plume vary by ±0.2 m/s around this value.
It should be highlighted that the model appears capable of simulating turbulence
initiated by vertical shear of the horizontal flow and the breaking of internal waves.
Exercise 6 will explore this feature in greater detail.
3.7.5 Can Reduced-Gravity Plumes Jump?
The author decided to repeat the latter experiment with inclusion of variable bottom
topography with a ramp and a vertical bar (Fig. 3.16). Will the reduced-gravity
plume make it over these obstacles?
Results show that the density-driven current is energetic enough to pass the ramp.
It shoots upward as it meets the vertical bar where it forms a counter-rotating vortex
41
criteria for both external gravity waves (Eq. 3.35) and advection (Eq. 3.43). Parameters of the S.O.R. scheme are kept at the same values as in the previous exercise.
3.7.3 Theory
Baroclinic pressure gradients associated with horizontal density differences will
produce a bottom-arrested density-driven flow. The resultant speed of the flow can
be estimated from energy conservation principles. In this process, potential energy
available from the initial density field is converted into kinetic energy of the flow.
This energy conversion can be quantified by the Bernoulli equation. Provided that
the plume density does not change, the Bernoulli equation reads:
0.5u
2
+ g
h = g
h o
(3.47)
where g
is reduced gravity, and h o and h, respectively, are initial and final plume
thicknesses. The Bernoulli equation is named after Daniel Bernoulli (Bernoulli,
1738) and his father Johann. The first true Bernoulli equation, however, was derived
by Euler (1755). The latter equation can be rewritten as:
u =
2g (h o − h)
(3.48)
3.7.4 Results
As anticipated, the initial density anomaly produces a density-driven flow spreading
toward the right-hand side of the model domain (Fig. 3.15). This flow forms an
isolated plume head. Counter-clockwise vortices forming in the lee of this head
induce vigorous mixing.
With g
= 0.0095 m s
−2
, h o = 100 m, and h ≈ 40 m, the Bernoulli equation
(Eq. 3.48) suggests a plume speed of u = 1.1 m/s. In good agreement with theory,
maximum speeds of the simulated plume vary by ±0.2 m/s around this value.
It should be highlighted that the model appears capable of simulating turbulence
initiated by vertical shear of the horizontal flow and the breaking of internal waves.
Exercise 6 will explore this feature in greater detail.
3.7.5 Can Reduced-Gravity Plumes Jump?
The author decided to repeat the latter experiment with inclusion of variable bottom
topography with a ramp and a vertical bar (Fig. 3.16). Will the reduced-gravity
plume make it over these obstacles?
Results show that the density-driven current is energetic enough to pass the ramp.
It shoots upward as it meets the vertical bar where it forms a counter-rotating vortex