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Scientific Report 2007-2009
Condensed matter physics and biophysics
C18. Stochastic convective plumes dynamics in stratified sea
The study of the deep sea convection processes is very
important for climate comprehension. Their dynamics
are very complex and far from being fully understood.
Every year they occur in some specific sites of
the world, allowing the deep water formation and oxygenation;
their position is responsible of the general sea
circulation and affects the earth climate. In particular
the Mediterranean Sea Circulation is driven by a lot of
these sites (MEDOC area, Adriatic Sea, Aegean Sea, and
so on): each of them is characterized by strong yearly
winds blowing over them and by not large sea stratification.
During winter, this is slowly eroded by the wind
stress in a finite sea region, so that an about circular
isopycnal, geostrophic, cyclonic ”doming”, ∼ 50 − 100
km large, quite homogeneous in its central area, appears;
the stratification (O(10 −3 − 10 −4 s −1 )) is vertically
decaying and horizontally growing from the center
to the boundary. At once, as a last violent wind proceeds
on the late winter, a lot of ∼ 1km wide vertical down
flows (∼ 3 − 10cm/s), so called ”plumes”, alternating
with slower upward velocities, are visible at a depth of
100 − 550 m, for a period t ≃ 2h < f −1 (f is the Coriolis
parameter); a decorrelation between plumes over a 2
Km horizontal range has been observed. These plumes
are thought to be efficient water mixing agents as a large
rotating ”chimney” is forming on longer times.
The problem of the physical processes involved in formation
and dynamics of these small plumes has been
studied experimentally, numerically and theoretically for
a long time. Laboratory experiments on a rotating tank
cooled on a finite region of its upper surface, so as numerical
and theoretical analyses have defined scale relations
for the initial plume formation phase, relating
the convective layer depth, its horizontal and vertical
velocity and the reduced gravity to the time-space average
surface buoyancy flux (due to surface cooling and
evaporation caused by the wind), the time and the sea
stratification. But the real plumes dynamics in the sea
have to be still investigated.
In the last few years, my contribution to the comprehension
of these processes has been given through the
development of a stochastic three-dimensional analytical
model describing the initial unsteady phase of convective
plumes generation and dynamics (for times ≤ f −1 ).
The hypothesis is that this kind of convection is not a
collective phenomenon generated by bulk fluctuations,
neither initially constrained by rotation. The analysis of
field data suggests the above-mentioned observed turbulent
plumes are likely generated by non uniformities of
the cooling effects. So it is possible this kind of convection
is due to external surface heterogeneous buoyancy
forcing, in such a way that every plume is independent
of the others.
The process is mathematically described by the
complete set of the non viscous Navier Stokes equations
(in Boussinesq and quasi-hydrostatic approximation)
coupled to the non diffusive mass conservation equation
in a rotating frame, by disregarding the wind stress.
Very small sea stratification has been introduced as
a perturbation. Still sea initial condition is given;
the space and time stochastic buoyancy horizontal
variability, driven by the transverse winds, is the source
of a convective process. By recognizing two space scales
(a small plume scale and a collective perturbed region
scale) acting together, a multiple space scale method
allows to decouple, in a stream function formulation,
the vertical transverse plane, over which the plumes set
is generated, from the winds direction line, along which
the plume is deviated from the Coriolis force on longer
times. Two time scales have been recognized; on the
short time scale the plumes generation and first evolution
can be described in a Lagrangian representation
on a 2D plane; on the long time scale, shear horizontal
instability allows a set of 3D small plumes to be defined:
an enhanced region of perturbation can be recognized,
due to stratification, driving to a different regime of
scale laws. A kind of ’transformation’ of the buoyancy
allows the effect of the entrainment-detrainment to
be analyzed. After all we have generation of a set of
independent quasi periodical small scale plumes, whose
distance is given by the horizontal correlation length
in the surface buoyancy. Their evolution is described
by an equation scalable with the penetration depth; it
is ruled by time power ’one plume laws’ depending on
the statistics of the external events, their frequency,
and by space-time buoyancy fluctuations power laws.
The convective motion is driven by the mean horizontal
in homogeneity of the surface buoyancy flux and its
space-time variability. For short times a linear stability
theory shows that the fastest growing in time internal
perturbations take a very long time to grow. The
analysis of the stability of the model, perturbed by
vertical internal fluctuations on longer times, shows
a weak intermittent behavior: but plume evolution
and scaling laws are ruled by random external forcing
leading to a higher time power behavior; this depends
on the probability of the event, which hides slower
internal randomness; if the air-sea interaction statistics
is such that it is impossible to define it, no self-similar
behavior is possible. Large internal fluctuations have
a mixing and turbulence generation effect. Numerical
simulation of the quasi-hydrostatic and not hydrostatic
model shows that the not hydrostatic effect is not
important.
References
1. V.Bouché, Int. Jour. Pure Appl. Math 4, 555 (2008).
Authors
V. Bouché
Sapienza Università di Roma 71 Dipartimento di Fisica