18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
Instability of a coastal jet in a two-layer model ; application to the Ushant
front
Marc Pavec (1,2), Xavier Carton (1), Steven Herbette (1), Guillaume Roullet (1), Vincent Mariette (2)
(1) UBO/LPO, 6 av. Le Gorgeu, 29200 Brest, France ; (2) Actimar, 24 quai de la Douane, 29200 Brest, France ;
xcarton@univ-brest.fr
Abstract :
The instability of a shelf front, with the characteristics of the Ushant front (steep density gradient and narrow
jet), is investigated on the f-plane within the framework of a two-layer shallow-water model. Linear stability
analysis and nonlinear numerical simulations show that baroclinic waves, with 20 and 40 km wavelengths, are
the most unstable ones, with growth rates on the order of (1day)?1, which is comparable to what is observed on
SST images. The analysis of growing perturbations in a two-layer primitive equation model (MICOM) shows that
the latter are vertically shifted, which corroborates that the governing mechanism for the generation asymmetric
meanders and eddies in our simulations, is baroclinic instability. Then, in order to take into account tidal effects,
we add a time-periodic barotropic mean flow. Although we succeed observing some characteristics of parametric
resonance (frequency selection, stepwise growth), baroclinic instability remains the main mechanism to explain
the growth of meanders.
Key-words :
Frontal jet, baroclinic and parametric instability
1 Introduction
The baroclinic instability of coastal jets, and in particular of frontal currents (defined here as
currents associated with density fronts), has been the subject of recent studies (e.g. Boss et al.
(1996)). This instability leads to the growth of waves with length comparable to 2pi times the
internal radius of deformation. If the lower layer outcrops at the front, instability also occurs on
a separate band of shorter waves ; this mechanism was related to the interaction of Rossby and
Kelvin modes (Sakai (1989)). If the mean velocity of the jet varies periodically with time (due
to a variation of transport, for instance), parametric instability can also occur ; several studies
in a idealized framework evidenced its predominance near the curve of marginal baroclinic in-
stability (Pavec and Carton (2004), Pavec et al. (2005), Pavec et al. (2006)).
An interesting example of coastal jet which can a priori develop both baroclinic and parametric
instability is the Ushant front. This density front separates thermally stratified water masses
west of 5o20?W from tidally-mixed, homogeneous water masses east of this longitude. Ob-
servations, both from satellite and in-situ, provide information on this flow : the temperature
difference at the surface is 2 ? 3oC over 5km distance (the frontal jet width), the associated
currents are on average 0.2-0.3m/s, the internal deformation radius near in the stratified zone is
close to 6km, and this front undergoes a semi-diurnal tidal forcing with velocities that can reach
2m/s. Meanders with about 40km wavelength and filaments with about 20km cross-section are
observed on this front (see figure 1 (left)).
The objective of the present work is to determine if a simple two-layer shallow-water model,
initialized with a slightly perturbed narrow jet, can reproduce the observed characteristics of
the instability and if the interaction of the perturbation with a periodic barotropic mean flow
(mimicking the tidal forcing) can lead to parametric resonance.
1
18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
Figure 1: (left) SST in the Irish Sea on July 24, 2006 ; (right) average density structure of the front
2 Baroclinic instability of the frontal flow
A two-layer shallow-water model is chosen to allow strong isopycnal deviations (see figure
1 (right)). For this preliminary study, a f-plane hypothesis is chosen in reason of the small
latitudinal extent of the flow. The dynamical equations are
?tuj + (??uj ????uj)?f0vj = ??xPj (1)
?tvj + (??uj ????vj) + f0uj = ??yPj (2)
?thj +??uj ????hj + hj??? ???uj = 0 (3)
with j = 1,2 the upper, lower layer index. A consequence of these equations is the conservation
of potential vorticity qj = (?xvj ??yuj + f0)/hj. The mean flow is purely baroclinic U1(y) =
?U2(y), zonal (rotated by 90 degrees with respect to the ocean) and in geostrophic balance.
U1 = ?1f dP1dy = ?gf ddy(h1 + h2) (4)
U2 = ?1f dP2dy = ?gf ddy
parenleftBigg
h1 + h2 + ??? h2
parenrightBigg
(5)
leading to upper and lower layer thicknesses h1 and h2, in the steady state, defined by
h1 = H1 ? 11???/2??h2 tanh
parenleftbiggy
?
parenrightbigg
(6)
h2 = H2 + 12?h tanh
parenleftbiggy
?
parenrightbigg
(7)
To be consistent with the characteristics of the Ushant front, we choose H1 = 50m,H2 =
50m, ?h = 80m. Furthermore, ?? = ???T, with ? = 1.79 10?4 K?1. In the refer-
ence case, ??/? = 7 10?4 and ? = 5km. The domain is a zonally periodic square basin
2
18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
of 100km?100km.
In the linear analysis, equations (1-2-3) are linearized around the mean flow (4-5) with a normal
mode perturbation of the form (u?j,v?j,p?j,h?j) = (u?j0,v?j0,p?j0,h?j0)(y)eik(x?ct) and boundaries
conditions for the perturbation are no normal flow at the northern and southern boundaries. The
resulting matrix problem is solved with a generalized eigenvalue-eigenvector method.
The nonlinear equations are solved numerically with the MICOM model (Bleck and Boudra
(1981), Bleck and Boudra (1986), Herbette et al. (2004)) ; the horizontal grid size is 1km and
we set the Smagorinski coefficient of the biharmonic horizontal dissipation operator to 0.1.
2.1 Reference case
The values of the parameters given here-above lead to a baroclinic velocity on the order of
0.275m/s, quite compatible with measurements performed across the Ushant front (Mariette
(1983)), and fall in the range of baroclinic instability as suggested by linear stability analysis.
Numerical simulations, in which we initially add a small amplitude white-noise disturbance
to the mean flow, show growing meanders and formation of eddies (See lower layer potential
vorticity maps from day 11 to day 14 in figure 2). The growth of meanders and the wrapping
Figure 2: Potential vorticity maps in the lower layer from day 11 to day 14
up of their tip leads to eddy formation. Though initially, a wavelength close to 30km seems
to dominate, the front becomes irregular after a few days, indicating the presence of other
wavelengths. Indeed, a spectral analysis of the perturbation in zonal modes indicates that mode
2 grows nearly as fast as mode 3, but to a smaller amplitude, and that mode 1 becomes large
once the vortices have formed (see figure 3). The baroclinic nature of the instability is confirmed
3
18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
by the vertical phase shift between the layer perturbations of potential vorticity (shown in Pavec
(2007)).
Figure 3: Spectral analysis of potential vorticity (PV) in the reference case : the log of the meridionally
averaged PV of each mode is plotted with respect to time (in days)
2.2 Sensitivity study
This part shows how sensitive are the growth rates of modes 1-2-3 to both the width of the front
? and the density jump ??. If ?? is kept constant and ? increases, growth rates decrease (fig-
ures shown in Pavec (2007)) . Conversely, growth rates increase ; in both cases, the baroclinic
velocity varies. Now if the baroclinic velocity is kept constant and equal to about 0.275 m/s), ?
is proportional to ?? in a good approximation. Figure 4 shows that the growth rates decrease
with increasing ? in this case, and thus the width of the front is the dominant parameter in this
evolution.
A series of nonlinear simulations (for a front width varying between 2 and 20km) shows that
too narrow a front leads to a very regular spacing of meanders (with a dominant mode 3 of per-
turbation) while too wide a front leads to too slow a growth of these meanders. A front width of
5km leads to results closer to observations, in particular two dominant modes with 30 and 50km
wavelengths, comparable to the observed 20 and 40km spacing of meanders and filaments. Our
simulations also indicate that, as the front is less marked, the wavelength increases, a feature
indeed observed on the Ushant front in spring.
3 Parametric resonance
Since tidal flow influences the Ushant front, we study here the influence of an oscillatory,
barotropic mean flow on the stability of our idealized baroclinic current. Three different am-
plitudes, 0.02, 0.2 and 2m/s, and seven periods, ranging from 6 to 12 hours, are tested for this
oscillating barotropic mean flow. Theory (e.g. Pavec et al. (2005), Pavec et al. (2006)) indi-
cates that parametric instability of coastal currents is most effective near the curve of marginal
baroclinic instability. The resonant period is then given by the pulsation of the marginal baro-
clinic wave.
4
18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
Figure 4: Growth rates (in seconds?1) with respect to the width of the front for a baroclinic mean velocity
set equal to 0.275 m/s
Here, for a realistic choice of flow parameters, nonlinear simulations lead to the following re-
sults : whatever the choice of oscillatory flow amplitude, the growth rates previously obtained
for baroclinic instability are not considerably altered. Nevertheless, for a finite range of periods
of oscillatory flow, the growth of the perturbation has a steplike evolution with time (see modes
k = 1,3,4 on figure 5) ; this effect is most marked for the 9h period whereas linear calculations
suggest a peak of resonance at the 7h period. Frequency selection and steplike growth are char-
acteristic of parametric resonance. But the influence of parametric resonance remains moderate
here because for realistic flow conditions, baroclinic instability is dominant.
4 Conclusions
The objective of this study was to analyze possible mechanisms leading to the growth of short
waves on a shelf front in an idealized model, with application to the Ushant front. In fact, for
the latter, satellite observations of SST, show meanders and filaments at 20 to 40 km spacing.
Numerical simulations, with a two-layer shallow-water model, of the evolution of a narrow
shelf jet in geostrophic balance, was able to reproduce the growth of meanders and eddies with
similar sizes as those that are observed for the Ushant front. The width of the front appears to
be the key parameter that governs its evolution. The analysis of the vertical phase shift of the
growing perturbations confirmed that the governing mechanism was baroclinic and frontal insta-
bility. Within the range of realistic parameters values, the presence of an oscillatory barotropic
mean flow was not able to trigger another type of instability, and baroclinic instability always
remained the dominant process. Nevertheless, we could observe some signs of parametric res-
onance: frequency selection and step like growth.
A first step towards more realism after this study should consider a meridional mean flow on
the beta-plane; indeed, the beta-effect would then favor the zonal growth of perturbations. In a
second step, realistic stratification, bathymetry and forcing should also be included.
References
Bleck, R., Boudra, B. 1981 Initial testing of a numerical ocean circulation model using a hybrid
5
18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
Figure 5: Growth of the first four modes (k=1,2,3,4) with respect to time (in days) for a barotropic
oscillatory flow of 2m/s with periods of 6, 7, 9 and 12 hours ; as in figure 3, the modal amplitude is the
logarithm of meridionally averaged PV.
6
18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
(quasi-isopycnic) vertical coordinate J. Phys. Oceanogr. 11 755-770.
Bleck, R., Boudra, B.1986 Wind-driven spin-up eddy resolving ocean models formulated in
isopycnic and isobaric coordinates. J. Geophys. Res. 91 (C6), 7611-7621.
Boss, E., Paldor, N., Thompson L. 1996 Stability of a potential geostrophic front: from quasi-
geostrophy to shallow-water, J. Fluid Mech. 315 65-84
Herbette, S., Morel, Y., Arhan, M. 2004 Subduction of a surface vortex under an outcropping
front J. Phys. Oceanogr. 34 1610-1627.
Mariette, V. 1983 Effets des ?changes atmosph?riques sur la structure thermique marine; appli-
cation ? des zones du large et ? une zone c?ti?re Th?se de Doctorat, Universit? de Bretagne
Occidentale
Pavec, M., Carton, X. 2004 Parametric instability of a two-layer wall jet Reg. Chaot. Dyn. 9
499-508
Pavec, M., Carton, X., Swaters, G. 2005 Baroclinic instability of frontal geostrophic currents
over a slope J. Phys. Oceanogr. 35 911-918
Pavec, M., Carton, X., Poulin, F. 2006 Parametric instability of boundary currents in a two-layer
QG model Dyn. Atmos. Ocean submitted
Pavec, M. 2007 Instabilit?s barocline et param?trique des courants de bord ; application au front
d?Ouessant Manuscrit soumis ? th?se de Doctorat, Universit? de Bretagne Occidentale
Sakai, S. 1989 Rossby-Kelvin instability : a new type of ageostrophic instability J. Fluid Mech.
202 149-176
7