Etudes et évaluation de processus océaniques par des hiérarchies ...

4.7. ON THE BASIC STRUCTURE OF OCEANIC GRAVITY CURRENTS 133
Ocean Dynamics (2009) 59:551–563 553
tel-00545911, version 1 - 13 Dec 2010
2 Idealised oceanic gravity current
2.1 The physical problem considered
In the experiments presented here, I use an idealised
geometry, considering an infinitely long gravity current
on an inclined plane with constant slope, and I do not
allow for variations in the long-stream direction. A similar
geometry was investigated by Ezer and Weatherly
(1990). I thus consider only the dynamics of a vertical
slice perpendicular to the geostrophic flow direction
of the gravity current. Please note that such simplified
geometry inhibits large-scale instability and the
formation of cyclones and other large-scale features.
This is beneficial to our goal of studying the small-scale
fluxes in gravity currents. The simplified geometry also
filters out small-scale processes and instabilities in the
y direction. Instead of a gravity current descending in
space, along the direction of propagation, it descends
in time (this strategy was also used by MacCready 1994
and others). Such descent is also investigated when the
gravity current is initially homogeneous in the longslope
direction, as in laboratory experiments where
the dense fluid is injected axisymmetrically on a cone
structure (Shapiro and Zatsepin 1997; Sutherland et al.
2004); semi-analytical calculations of the instability of a
rotating gravity current (Meacham and Stephens 2001).
The results can be compared to the observations of
the transverse structure of a oceanic gravity current by
Umlauf et al. (2007). The local dynamics will not differ
from gravity currents descending in space as long as the
large-scale descent is slow in space and time compared
to the local turbulent dynamics, which is definitely
the case when the dynamics is close to a geostrophic
equilibrium. Such type of model is referred to as 2.5-
dimensional, as it is three-dimensional, but the variables
have no dependence on the stream-wise direction
(the y direction in this case). I like to emphasise that
none of the three components of the velocity vector are
trivial and that the main (geostrophic) transport is in
the y direction.
In observations and laboratory and numerical experiments,
large-scale instabilities are observed for a
wide range of parameter values. All the studies, that I
am aware of, include a geostrophic adjustment process,
which also destabilises the current. In laboratory experiments
where the gravity current was injected close
to a geostrophically adjusted state, these large-scale
instabilities developed only very slowly (Wirth and
Sommeria, unpublished manuscript), long time after
the 2.5-dimensional dynamics, studied here, had started
evolving. A geostrophically adjusted state also forms
the starting point of investigations concerning the stability
of oceanic gravity currents and the dynamics of
streamtube models.
The gravity currents considered here are on an inclined
plane of a constant slope of one degree, the
initial profile of its vertical extension above the inclined
ocean floor z = h(x) = max{H − x 2 /λ, 0} has a
parabolic shape with H = 200 m and λ = 5. · 10 5 m
leading to a gravity current that is 200 m high and
L = 20 km large. The values used in the experiment
are typical for oceanic gravity currents (see, e.g. Smith
1977; Price and Baringer 1994; Killworth 2001). If the
gravity current is initially geostrophically adjusted, the
velocity components are given by:
u G = 0; v G = g′ (∂ x h + tan α)
; w G = 0. (1)
f
This geostrophic velocity, also called the Nof speed
(Nof 1983), is due to the balance between the Coriolis
force and the buoyancy force in the downslope
direction. Outside the gravity current, the water is initially
at rest. Multiplying Eq. 1 by the thickness and
integrating across the gravity current shows that the
average geostrophic speed of the gravity current is
given by v G = (g ′ /f) tan α. The Coriolis parameter is
f = 1.0313 · 10 −4 s −1 corresponding to the earth rotation
at mid-latitude. The reduced gravity is g ′ = g(ρ gc −
ρ 0 )/ρ 0 = T · 2. · 10 −4 K −1 9.8065 m s −2 , where T is
the temperature difference between the ambient fluid
and the gravity current, a constant thermal expansion
coefficient of 2 · 10 −4 K −1 and a gravitational acceleration
of 9.8065 m s −2 are used. The temperature difference
between the gravity current and the surrounding
water ranges from 0.25 to 1.5 K. The values of the
average geostrophic speed are given in Table 1. The
vertical friction coefficient is ν v = 10 −3 m 2 s −1 , leading
to an Ekman layer thickness δ = √ 2ν v /f ≈ 4.4 m.
The dynamics depends on the six independent parameters
(α, g ′ , f, H, L, ν v ); four independent nondimensional
numbers can be obtained: (1) the slope
of the inclined plane tan α, (2) the Richardson number
Ri = g ′ H/v G 2 = H f 2 /(g ′ tan 2 α), (3) the Ekman number
Ek = (δ/H) 2 = 2ν v /( f H 2 ) = 4.84 · 10 −4 comparing
the frictional to the Coriolis force and (4) the ratio
L/L D , the width of the gravity current divided by
the Rossby radius L D = √ g ′ H/f. In the experiments
presented here, I choose to systematically vary the
Richardson number, identified as a key parameter (see,
e.g. Price and Baringer 1994; Killworth 2001), by varying
the density (temperature) difference. The Froude
number, F = Ri −1/2 , gives the geostrophic speed divided
by the speed of the shallow water gravity wave.