General Characteristics of Density-Turbidity Currents in the Ross Sea (Antarctica)
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where y, B2 are suitable constants, Rj is the Richardson number and v0 is the eddy
viscosity for unstratified fluids. This shows that it is not an easy task to fully compare our schematic assumption [Eq. (1)] with those discussed in the literature.
Intuitively, our model can be considered valid if the sédiment is so fine-grained
and light as to be dispersed immediately inside the thin turbulent bottom flow.
On general physical grounds, this situation would correspond to steady thermohaline currents generated by severe climatological conditions, entraining finegrained sédiment as its crossed the shelf or slope and found their way into the
deep océan along submarine canyons Crossing the shelf [7]. Note how this situation is truly different from that of a large amount of bottom sédiment carried by
shelf currents during a severe storm, i.e., a true turbidity current.
4 Bottom Current Evolution
For the current we assume that the longitudinal scale is naturally to be considered
large; the latéral confinement is such that the Coriolis effect can be neglected and
there is no entrainment of water at the upper surface of the flow [12], that is the
Richardson number is R; ~ 1. The équation of motion is thus:
3V/dt + (V-V)V = - Vp./p + vAV + py+F
(3)
y=(p-p0)g/p
p = (sin 0,0, -cos 0) ~ (0,0,-1),
where F is some external forcing for unit of mass, g dénotés the gravitational
accélération, V dénotés the velocity and p» dénotés the pressure minus the hydrostatic term.
One may now wonder as to the effect of any hydrodynamic instability on these
érosion-déposition dynamics. Natural modelling involves computing the bottom
stress due to both the quasi-steady current and its short space scale perturbation.
After some long computations [3], Eqs. (1) and (3) finally give
ô2h7ôt2=yohoô2h7ôx2-i-gEho/2d3h’2/dx3+v(â2/9x2d/dt)h’-(vn0h7ât)/4h2()+hoôF/dx, (4)
with
£=l/g[4y2opoho(l-e',/4r)]/[n2(To-Yopoho0)]^4/gTT2[Y2opoho]/[To-Yopoho0]|t/t._O(io)>
where the fundamental rôle of the “ignition” time t* = h02 / (n2v) will be clear in
the following.
The physical meaning of e as a dimensionless coefficient of non-linear effects
now emerges naturally. Equation (4) is a complex non-linear diffusive équation;
its treatment in some cases reveals some mathematical pathologies [2], namely
an explosive steepening of the signais [26]. A numerical solution, even if it
exists, cannot consequently be considered fully satisfactory. So let us first discuss some simpler cases: if e = 0, the dynamic effect of bottom entrainment vanishes quickly, so the bottom layer dynamics is driven only by the hydrological
227
where y, B2 are suitable constants, Rj is the Richardson number and v0 is the eddy
viscosity for unstratified fluids. This shows that it is not an easy task to fully compare our schematic assumption [Eq. (1)] with those discussed in the literature.
Intuitively, our model can be considered valid if the sédiment is so fine-grained
and light as to be dispersed immediately inside the thin turbulent bottom flow.
On general physical grounds, this situation would correspond to steady thermohaline currents generated by severe climatological conditions, entraining finegrained sédiment as its crossed the shelf or slope and found their way into the
deep océan along submarine canyons Crossing the shelf [7]. Note how this situation is truly different from that of a large amount of bottom sédiment carried by
shelf currents during a severe storm, i.e., a true turbidity current.
4 Bottom Current Evolution
For the current we assume that the longitudinal scale is naturally to be considered
large; the latéral confinement is such that the Coriolis effect can be neglected and
there is no entrainment of water at the upper surface of the flow [12], that is the
Richardson number is R; ~ 1. The équation of motion is thus:
3V/dt + (V-V)V = - Vp./p + vAV + py+F
(3)
y=(p-p0)g/p
p = (sin 0,0, -cos 0) ~ (0,0,-1),
where F is some external forcing for unit of mass, g dénotés the gravitational
accélération, V dénotés the velocity and p» dénotés the pressure minus the hydrostatic term.
One may now wonder as to the effect of any hydrodynamic instability on these
érosion-déposition dynamics. Natural modelling involves computing the bottom
stress due to both the quasi-steady current and its short space scale perturbation.
After some long computations [3], Eqs. (1) and (3) finally give
ô2h7ôt2=yohoô2h7ôx2-i-gEho/2d3h’2/dx3+v(â2/9x2d/dt)h’-(vn0h7ât)/4h2()+hoôF/dx, (4)
with
£=l/g[4y2opoho(l-e',/4r)]/[n2(To-Yopoho0)]^4/gTT2[Y2opoho]/[To-Yopoho0]|t/t._O(io)>
where the fundamental rôle of the “ignition” time t* = h02 / (n2v) will be clear in
the following.
The physical meaning of e as a dimensionless coefficient of non-linear effects
now emerges naturally. Equation (4) is a complex non-linear diffusive équation;
its treatment in some cases reveals some mathematical pathologies [2], namely
an explosive steepening of the signais [26]. A numerical solution, even if it
exists, cannot consequently be considered fully satisfactory. So let us first discuss some simpler cases: if e = 0, the dynamic effect of bottom entrainment vanishes quickly, so the bottom layer dynamics is driven only by the hydrological