We present here a rapid review on recent laboratory investigations on geostrophic adjustment of density fronts. Several configurations were studied: warm core lens, cyclonic-anticyclonic PV patches and uniform PV front. The geostrophic adjustment is the first dynamical process which converts a significant fraction of the potential energy input of the atmosphere and the ocean into kinetic energy. According to the cases we studied we have shown that during this rapid adjustment toward a quasi-equilibrium state, an important part of the initial energy could be transferred to wave motion or dissipated by small-scale non-hydrostatic instabilities. A mean adjusted state is always reached after one or two inertial period. Even if a strong wave activity is present in the initial region of unbalance, the time-averaged mean flow could nevertheless be adjusted. We have shown that the wave modes frequency is concentrated around the inertial frequency. Besides, some anticyclonic structures may also exhibit sub-inertial o

MITKIN V.

STEGNER, Alexandre

ZEITLIN, Vladimir

I-Revues

18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
1
GEOSTROPHIC ADJUSTEMENT OF DENSITY FRONTS:
what do we learn from recent laboratory experiments ?
A. Stegner , V.Mitkin1, V. Zeitlin
Laboratoire de Météorologie Dynamique (CNRS)
24, Rue Lhomond 75005, Paris, France (stegner@lmd.ens.fr)
T. Pichon
 Unité de Mécanique (ENSTA)
Chemin de la hunière,91120 Palaiseau France
Abstract  :
 We present here a rapid review on recent laboratory investigations on geostrophic adjustment of density
fronts. Several configurations were studied: warm core lens, cyclonic-anticyclonic PV patches and
uniform PV front. The geostrophic adjustment is the first dynamical process which converts a significant
fraction of the potential energy input of the atmosphere and the ocean into kinetic energy. According to
the cases we studied we have shown that during this rapid adjustment toward a quasi-equilibrium state,
an important part of the initial energy could be transferred to wave motion or dissipated by small-scale
non-hydrostatic instabilities. A mean adjusted state is always reached after one or two inertial period.
Even if a strong wave activity is present in the initial region of unbalance, the time-averaged mean flow
could nevertheless be adjusted. We have shown that the wave modes frequency is concentrated around
the inertial frequency. Besides, some anticyclonic structures may also exhibit sub-inertial oscillations.
Key-words : Geostrophic adjustment, inertial gravity waves, instability
1 Introduction
. The geostrophic adjustment is the first dynamical process which converts a significant
fraction of the potential energy input of the atmosphere and the ocean into kinetic energy. The
first theoretical guidelines for solving geostrophic adjustment problems were proposed by
Rossby (1938) in the framework of rotating shallow water fluid, for a review see Blumen
(1972). It is assumed that the initial unbalanced state will evolve naturally toward a steady
balanced state, geostrophic or cyclogeostrophic, satisfying mass and lagrangian conservation of
potential vorticity (PV). This standard Rossby scenario conveniently avoids the time-dependent
problem and predicts the final state. The energy of this predicted adjusted state is always less
than the initial energy. Inertia-gravity waves, with zero PV, are then expected to carry the
missing part of the energy away from the adjusted state. However, the time-dependent problem
deals with the complex reversible-irreversible interactions between the waves and the mean
flow and the time needed to reach an adjusted state (if any) is not well defined. Previous
laboratory experiments have shown that the geostrophic adjustement is a rapid process
(Ungarish et al. (2001); Bouruet-Aubertot and Linden (2002); Rubino and Brandt (2003);
Thivolle-Cazat et al. (2005)). But very few studies investigated the characteristic time of this
process especially when strong wave activity is present. Besides, this standard scenario and
recent theoretical studies on nonlinear geostrophic adjustment (Kuo and Polvani (2000); Zeitlin,
et al. (2003); Plougonwen and Zeitlin (2005)) didn’t take into account the instabilities or the
dissipative processes that may affect the final adjusted state in a real three-dimensional fluid.
 In order to study both the small-scale dissipative process and the spectrum of the wave
emission during the geostrophic adjustment we performed several laboratory experiments in
rotating fluids based on lock released configurations. In a two-layer configuration, vertical
                                                           
1 Current affiliation : Institute for Problems in Mechanics RAS, Moscow, Russia (mitkin@ipmnet.ru).
18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
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boundaries (i.e. locks) are used to fix initial height (or density) steps in the upper layer. For such
cases when there is no relative motion in the layers the initial PV field is precisely controlled by
the layer thickness. If the release of the vertical walls is rapid enough, we could then follow the
geostrophic adjustment of a well defined initial condition corresponding to discontinuous
profiles of constant PV. The simplicity of the initial condition makes these experiments easily
reproducible.
2 Experimental configurations
We study the adjustment process in four experimental configurations. The term warm-core
lens is generally used for mesoscale vortices, which contain a finite volume of warm and light
water, at the ocean surface. A simple experimental configuration leads to similar dynamical
structure (Stegner et al. 2004). A fixed volume of buoyant water is initially confined within a
bottomless cylinder of radius Rc on the top of a dense rotating fluid Fig.1 (a).
We used the term "PV patches" for localised positive or negative potential vorticity (PV)
anomalies of constant values within a uniform layer. The "PV patch" model is the generalisation
of the Rankine vortex (cylindrical vorticity patch) for a rotating shallow-water layer. It is the
simplest description of potential vorticity front with no outcropping. The corresponding
experimental configurations for anticyclonic and cyclonic "PV patches" are shown respectively
in Fig.1 (b) and Fig.1 (c).
The geostrophic adjustment of a motionless horizontal density gradient generally leads to a
baroclinic tilted front corresponding to a simplified model of synoptic atmospheric fronts. We
use a three-layer setup to study the adjustment of an uniform PV front. Two upper fluid layers
having different density ρ1 and ρ 2, but the same thickness H1, are initially separated by a
bottomless cylinder. If the upper layers have exactly the same thickness the initial PV
distribution is uniform and have the same value inside (r<Rc) and outside (r>Rc) the cylinder
(Fig.1 (d)).
FIG. 1 – Initial experimental configuration for a “warm-core” lens (a), an anticyclonic (b)
and a cyclonic (c) PV-patch and a uniform PV front (d).
For all the cases, according to the small thickness parameter δ=H1/H2~0.1 and the weak
motion in the lower layer we assume that two upper layer follows the reduced-gravity rotating
shallow-water (RSW) equations. The final adjusted state predicted by the standard Rossby
18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
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scenario is then fixed by the Burger number: Bu=(Rd/Rc)
2 where 
€ 
Rd = g * H1 / 2Ω0  is the
internal deformation radius, g*=(ρ2-ρ1)/ρ2 the reduced-gravity and the relative elevation
parameter λ=η/H1 (configurations (b) and (c)).
Laser induced visualization (LIV) technique was used to measure with precision the upper
fluid layer thickness along a line while standard particle image velocimetry (PIV) was used to
measure the horizontal velocity field. Using simultaneously theses two technique was the first
attempt for direct and quantitative measurements of the potential vorticity field in rotating
shallow-water layer experiments (Stegner, 2007).
3 Mean adjusted state
According to all the cases we studied, a mean adjusted state is reached after approximately
one or two inertial period Tf. The rapidity of the geostrophic adjustment does not depends on the
size (i.e. Bu) or the amplitude (i.e. λ) of the initial unbalance state. The so-called mean state is
obtained from a simple time-averaging over Tf, in order to filter out the fast wave motion. We
say that this averaged state reaches an equilibrium (i.e. get adjusted) when it's temporal
evolution remain small in comparison with the characteristic wave frequency. Hence, even if a
strong wave activity is present in the initial region of unbalance, the mean flow could
nevertheless be adjusted. This experimental observation is in good agreement with the standard
hypothesis of dynamical splitting between the fast and the slow component of motion. In the
limit of small Rossby numbers, the asymptotic analysis shows that the slow component of
motion doesn't feel the fast one (Zeitlin et al. 2003).
FIG. 2 –Mean averaged height (a,b) and velocity profiles (c,d)  for anticyclonic
and cyclonic PV-patches at various times. The solid lines correspond to the final states predicted by
the standard adjustment scenario based on the lagrangian PVconservation (Stegner, 2007).
18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
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Theses two-layers or three-layers experiments show that the PV conservation remain
robust even if the initial state does not satisfy the assumptions of the rotating shallow-water
(RSW) model. Indeed, in almost all the cases, three-dimensional and non-hydrostatic motions
(shocks or gravity current head) could occur in the early stage of adjustment. Nevertheless, the
prediction of the RSW model based on the PV conservation gives a correct estimation of the
mean adjusted state (Fig. 2). A very good agreement is found for the cases of cyclonic PV patch
when there is no outcropping Fig. 2 (a,c). However, for the anticyclonic counterpart a
significant dissipation of the kinetic energy occurs Fig.2 (d). Moreover, the PV conservation
could be locally broken in the case of outcropping fronts when the initial PV profile exhibits a
singularity (i.e. the layer thickness vanish at a given position).
4 Inertial and sub-inertial wave activity
The relaxation of any unbalanced initial state in a rotating shallow-water model will
always leads to the emission of Poincarre waves. According to our experiments and previous
studies (Bouruet-Aubertot and Linden, 2002; Rubino and Brandt, 2003; Thivolle-Cazat et al.,
2005) the energy released to the wave modes during the adjustment is mainly concentrated
around the inertial frequency. A significant wave activity remains for a long time (several
inertial period) inside both the cyclonic and the anticyclonic structures even if the mean steady
state is already adjusted Fig.2.
FIG. 3 – Spatio-temporal evolution of the upper layer relative oscillations for the cases of a anticyclonic
(a) and a cyclonic (b) PV-patch.
The most striking results is a strong cyclone-anticyclone asymmetry in the wave
frequency. According to the spatio-temporal plots, the oscillation is faster for the positive PV
anomaly (Fig 3 (b)) in comparison with the negative PV anomaly (Fig 3 (a)). In the rotating
shallow-water configuration, the apparition of sub-inertial modes (ω/f<1) corresponds to
trapped modes in other words, these modes must have an evanescent structure outside the PV-
patch. If the relative vorticity is strong enough, a finite number of trapped modes could appear
in anticyclonic vorticity region only (Plougonwen and Zeitlin, 2005). The present experiment
shows, for the first time in laboratory, the existence of sub-inertial modes within an anticyclonic
vortex.
18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
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5 Small-scale instabilities
For the warm core lens configuration, detailed analysis of the velocity field evolution show
that strong and localized dissipation occurs in the very initial stage of adjustment (t<2Tf) while
the flow experiences only a weak dissipation afterwards. This rapid dissipation, which occurs at
the edge of the anticyclonic lenses, induces a significant deficit in the kinetic energy of the
adjusted flow up to 50% or 80% (Stegner et al., 2004). Dye visualization reveals that transient
and rapid three-dimensional instabilities occur in the very first stage of adjustement. These
three-dimensional instabilities localized in time (less than one inertial period) and space (the
edge of the anticyclonic lens) provide an efficient mechanism of turbulent dissipation which
cascades energy toward small scales in the frontal region.
For the uniform PV front configuration, small and intense cyclones are formed in a very
short time (~0.5Tf) during the adjustement of a large scale anticyclonic front. By means of LIV,
we visualize an horizontal cross-section of the sharp density gradient just below the free surface.
The dynamical evolution of this sharp gradient is shown in figure 4. After the release of the
bottomless cylinder the upper front experiences a rapid radial contraction. Due to the angular
momentum conservation, this radial contraction generates at the same time a strong azimuthal
shear flow. During this very initial stage, small scale perturbation growths at the edge of the
front. Unlike the warm-core lens configuration, this instability leads to the formation of intense
cyclones. The rapid formation of these structures, which are much smaller than the deformation
radius, were not predicted by the standard scenario of adjustment and they could hardly be
captured by standard numerical simulations which have limited spatial resolution. The
laboratory experiment shows here a new mechanism of formation of small and intense cyclones
within a large-scale synoptic front.
FIG. 4 – Small-scale instabilities during the geostrophic adjustment of a uniform PV front (Bu=0.14) at
t=0 (a), t=0.25Tf (b), t=0.5Tf(c) and t=0.72Tf (d). Local image processing of edge front detection are
shown in (b) and (c), the black pixels corresponds to high values of the intensity gradient.
6 Conclusions
18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
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Recent measurements techniques were used to investigate the geostrophic adjustment of
density fronts. Several configurations were studied: warm core lens, cyclonic-anticyclonic PV
patches and uniform PV front. According to the cases we studied we have shown that during
this rapid adjustment toward a quasi-equilibrium state, an important part of the initial energy
could be transferred to wave motion or dissipated by small-scale non-hydrostatic instabilities. In
some specific case these transient instabilities could lead to the formation of intense and small-
scale cyclones within a large-scale synoptic front. Nevertheless, a mean adjusted state is always
reached after one or two inertial period. Outside the outcropping region, the PV conservation is
robust and the standard Rossby adjustment predict correctly the mean adjusted state. Moreover,
even if a strong wave activity is present in the initial region of unbalance, the time-averaged
mean flow could nevertheless be adjusted. This experimental observation is in good agreement
with the standard hypothesis of dynamical splitting between the fast and the slow component of
motion Besides, we have shown that the wave modes frequency is concentrated around the
inertial frequency, and some anticyclonic structures exhibit sub-inertial oscillations
corresponding to trapped modes.
References
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upwelling fronts. Part I: Laboratory experiments. Dyn. Atmos. Ocean. 36, pp 153-173.
Kuo, A. C. and Polvani, L. M. (2000) Nonlinear geostrophic adjustment, cyclone/anticyclone
asymmetry, and potential vorticity rearrangement. Phys. Fluids 12, 1087-1100.
Plougonwen, R. and Zeitlin, V. 2005 Lagrangian approach to geostrophic adjustment of frontal
anomalies in a stratified fluid, Geophys. Astrophys. Fluid Dyn., 99 (2), 101-135.
Rossby, C. G. 1937 On the mutual adjustment of pressure and velocity distribution in simple
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Geostrophic adjustement of density fronts: what do we learn from recent laboratory experiments ?

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