Internship Report LPO 2015:
Long-Term Variability in a 2.5 Layers Shallow-Water
Model
Florent Aguesse,
University of Southampton,
Academy of Ocean and Earth Sciences
Email: fa2g13@soton.ac.uk
This spontaneous Internship was conducted as an end of 2nd year project at
the Laboratoire de Physique des Oceans, Universite de Bretagne
Occidentale, UFR Sciences, 6 av. Le Gorgeu, Brest, France, supervised by
Thierry Huck, CNRS Researcher at the LPO.
June-August 2015
Table of Contents
1) Introduction
2) Model Set up
3) Interannual Variability
4) Sensitivity Experiments
5) Interpretations with regards to Unstable Rossby Waves
6) Conclusion
1
1. Introduction
Using a 2.5 layers shallow water model from Jerome Sirven (Sirven et al, in
review) long-term baroclinic instabilities were investigated in a simplified
basin resembling the north Atlantic. Sirven et al. describes decadal
variability, centered on the intergyre region resulting from the outcropping
allowed by the model. Variability from seasonal factors appears negligible,
however, a stronger variability in the region of instability arises from long
term phenomena such as the AMOC and the NAO (Sirven et al, in review).
Using this paper as the basis for the research, the model was further
simplified in order to determine the most basic configuration under which
interannual instabilities could arise.
Following Sirven’s diagnostics, non linear Rossby waves were expected to
develop and propagate around the intergyre region following the Gulf
Stream path. Our results are quite different. Indeed, the intergyre region
defined by Sirven et al. (in review) as the main region for interannual
variability is here of lesser importance.
2. Model Set up
The model is Shallow water with 2.5 layers, the third layer being at rest and
of infinite depth. The diagnostics are made using variations on the top 2
layers. Coordinates are semi-spherical and extend from 15 to 55N, 70 to
10W. No coastal features are included in our version of the model, as
opposed to Sirven et al. original configuration. Furthermore, the wind was
made constant instead of climatological. No overturning circulation was
prescribed.
The model circulation follows a double gyre pattern with a strong western
boundary current as shown in fig. 1. This circulation holds for every
experiment made, with the volume transport varying by ±20 Sv depending
mostly on . A permanent gyre can be seen at 35N in the Western
Boundary Current.
This model allows the thickness of the first layer h1 to shrink towards 0 in
the north west corner of the basin if the wind forcing was sufficient,
resulting in the second layer outcropping. Under climatological winds
Sirven et al. observed outcropping, however, if the wind stress is low, no
outcropping will occur. The threshold lies in between 0.06 and 0.08 kg.m1.s-2 . The size of the outcropping region is proportional to the  value used.
The outcropping is maintained through the use of antidiffusive fluxes as
2
described by Liu et al. (1999) to keep the model stable (fig. 2). Thinning in
h1 is compensated by the thickening of h2, maintaining the thickness of the
two active layers at 1400m.
The momentum conservation equations are:
∂tvi +(curl vi +f) n  vi = - grad (Bi) + v ∆H vi + i-1 - i/ hi
Where: i= 1 (surface layer) or i=2 (mid layer), n is the vector normal to the
earth’s surface, Bi are the Bernoulli functions B1 = 2bh1 + bh2 +v12 /2 and
B2 = bh1 + bh2 + v22/2 where b= g∆/ is the reduced gravity. Because ∆ is
constant, we can take b= 0.004 m.s-2.
n  vi is the wedge product of the velocity (vi) with respect to the Earth’s
surface. Eddy viscosity was set as = constant throughout the basin.
Initial values for layer thicknesses were h1=450m, h2= 950m. The model ran
for 100 years simulation for wind stress values ranging from 0.02 to 0.12
kg.m-1.s-2 and 3 resolutions: 1/3, 1/6, 1/12. Yearly means for vorticity,
velocities, layer thicknesses were extracted and processed under Matlab,
yielding results detailed in section 2. Most results were evaluated over the
last 30 years of the run, due to the absence of seasonal and decadal
variability.
Name
Resolution
in deg
Wind forcing in
kg.m-1.s-2
Viscosity in
m2.s-1
Notes on the
simulation
R6_002_400
1/6
0.02
400
R6_004_400
R6_006_400
1/6
1/6
0.04
0.06
400
400
R6_008_400
1/6
0.08
400
R6_010_400
1/6
0.10
400
R6_012_400
R6_006_200
R6_006_800
1/6
1/6
1/6
0.12
0.06
0.06
400
200
800
No outcropping,
perturbations on the W
side
//
No outcropping,
perturbations over all the
basin
Outcropping,
perturbation over all the
basin
Outcropping, chaotic,
some defined
perturbations in NW
corner
Outcropping, chaotic
//
No Outcropping, very
clear perturbations over
3
R3_006_3200
1/3
0.06
3200
R3_008_3200
1/3
0.08
3200
R12_006_45
1/12
0.06
45
0.06
45
R6_006_45_3mo 1/6
all basin
No outcropping,
perturbations focused in
mid and top parts
//
Outcropping, chaotic
perturbations over all
basin
3 months means. Clearly
defined eddies, stable
perturbations
3. Interannual Variability
The first series of experiments were made at 1/6 resolution for all  values
studied, in order to have some reference points for further experiments.
The viscosity parameter  was set at 400m2.s-1 , following Sirven et al.
prescriptions.
Observed large-scale anomalies have velocities and size matching Rossby
waves. Those Rossby waves propagating westwards were observed,
regardless of the  used (fig. 3 b)), emerging from the north-east corner of
the basin and propagating westwards. Those waves exhibited a strong tilt
and had a very repetitive behaviour. The tilt is an important feature as it
creates latitudinal switch is the sign of the anomaly, effectively enhancing it.
This lead us to believe that this behaviour was robust, however, the
presence of those waves at <0.06 lead us to consider them as not being an
inherent consequence of the outcropping mechanism.
Longitudinal sections were studied at high latitudes where the waves
seemed predominant showed typical phase lag between h1 and h2 (fig. 4).
This behaviour is characteristic of baroclinic instability as the wave
propagation in a given layer will positively feedback the propagation in the
other layer. The amplitude of the perturbations varies longitudinally,
indeed, when the outcropping region was small enough as to allow
perturbations to reach the western side, the amplitude of h1 and h2
anomalies grew before diminishing at around 50 W. This behaviour held
for all experiments at 1/6.
Considering the characteristics of those perturbations, Empirical
Orthogonal Functions analysis was performed, showing percentage of
variance explained close to 80% for <0.10 (fig.5). At ≥0.10, the
perturbations bore no more tilt and the EOF values shrank to 30% of the
4
variance explained. PC analysis mirrored this behaviour, with chaotic
variations arising at high .
Emitting the hypothesis that those perturbations were low frequency non
linear Rossby waves as described by Hochet et al. (2015), several analysis
were performed to determine their mode, as well as spatial and variability.
Zonal velocities, noted u1 and u2 for the top and mid layer respectively were
westward over most of the basin (fig.6). u1 <0 is a required condition for
baroclinic instability to arise, which is verified in our case.
Using Quasi-Geostrophic equations, the unstable regions were analytically
identified as lying in the north-west corner of the basin (fig. 7). This
matched the observations and the regions at which the imaginary part of
the phase speeds was ≠ ø. This is particularly striking for high temporal
resolution. However, for runs at lower temporal resolution, growth regions
were seen in the south-west corner of the basin, in the sub-tropical gyre, as
well as in the north east corner. This is opposite to the initial diagnostics
made using the phase speed calculations. Such was the case at all  values.
This is in agreement with results from Liu et al. 1999 and Cox (1987),
however, this does not match the regions where the anomalies were
observed. This discrepancy was described by Hochet et al. 2015, who shows
that analytical methods tend to yield results quite different from
observations. The reason for this behaviour has not been established and is
thought to be resulting from gaps in our analytical understanding of this
phenomenon. Non-linear terms and approximations made could be a
potential explanation for these discrepancies
4. Sensitivity Experiments:
Experiments were carried out for 3 values of horizontal viscosity  at 1/6:
200, 400(reference) and 800m2.s-1. On the reference experiment, at =0.12,
Rossby waves were destroyed before they could reach the eastern side of
the basin. This is likely due to the meso-scales turbulences (Huck et al.
2015). Indeed, large changes were observed when modifying the 
parameter: when this parameter was increased to 800, all turbulences
smaller than a grid point were destroyed, further amplifying the long-term
anomalies. However, when  was decreased to 200, the perturbations were
propagating chaotically, without tilt.
Experiments were also made at higher temporal, then spatial resolution,
yielding different results. As expected, for 1/12 resolution, meso-scale
turbulences destroyed the Rossby waves, in the same manner as in runs
5
with low  values or high  values. As a result, the waves lost their tilted
horizontal structure, however, anomalies could still be seen, exhibiting the
same behaviour as simulations made at 1/6 with =0.12. At means
calculated every 3 months, the perturbations were more clearly defined and
eddies were standing out along the northern border of the basin.
Important changes were seen at 1/3. Indeed, reversed perturbations
flowing eastwards are seen south of the sub-polar gyre, in addition to
Rossby waves propagating westwards north of the gyre.
In both cases, the first 2 EOF explain only 10 to 20% of the variance
observed, depending on the  value.
As for temporal resolution increase, means were made every 3 months,
hence increasing by 4 the temporal resolution. This translates into more
stable Rossby waves crossing the basin, with the first 2 EOFs explaining
85% of the variance observed. This tends to prove that the first 2 modes are
driving the perturbations.
Over every experiments, the total Kinetic Energy was monitored at high
frequency in order to assess the impact of eddies. Following what was
expected, the KE increases with increasing , increasing resolution and
decreasing  (fig. 8). As most of the KE in the ocean is stored in meso-scale
eddies, this was a good indicator. Variations in KE were sinusoidal when the
anomalies were tilted and lost clear periodicity when  or the resolution
reached critical levels.
In the next section, we will attempt to explain the origin of the observed
anomalies and explore the possibility of Green modes being at work here.
Indeed, Green modes are the result of coupling between the Non-Doppler
shift mode, resembling the first mode and the Advective mode, impacted by
the mean flow, hence creating instability in planetary waves. We shall also
highlight discrepancies between our expectations from theory and the work
from Hochet et al and Sirven et al.
5. Interpretation with regards to Unstable Rossby Waves
As mentioned in section 2, our results were quite surprising, due to the
apparent robustness of the Rossby waves to varying , but their high
sensitivity to other parameters such as  or the resolution. We will here
attempt to classify these perturbations and identify their propagating
mode.
6
At first order, layer thicknesses standard deviations were computed, in
order to determine regions of highest perturbations’ influence. As  values
increase, the region most impacted by the waves drifts towards the eastern
part of the basin (fig. 9). 2 reasons may explain this: the outcropping region
emerges at high , preventing propagation of the waves in the top layer,
hence reducing their impact on the second layer. Secondly, as  increases,
the waves form faster than at low .
Rossby waves develop in presence of forcing and are an inherent oceanic
behaviour. They are the restoring mechanism acting against isopycnal
slopes, naturally contained in the mean state.Those waves can be stable or
unstable. In the case where Rossby waves are stable, they propagate
westwards with an unchanging wavelength and amplitude due to initial
spin up or changing winds. In our case, Rossby waves show varying
wavelength with longitude and their amplitude changes (fig.4). In our cases,
instabilities arise in all cases. This can be seen from the imaginary part of
the phase speed solution (fig. 10). As mentioned previously, those
instabilities are fed through a number of processes. The tilt of the
instabilities creates meridional alternating positive and negative anomalies,
hence promoting perturbations westwards propagation (Quentin Jamet,
LPO, Personal comment).
On vertical level, modes are defined by the sign of u1 and u2 with respect to
each other. If they are of the same sign, the first mode dominates and
perturbations are unaffected by the mean flow. If the two have different
sign, the second mode might be accelerated by the mean flow to the point
where it couples with the first mode and creates instabilities. In our case, u1
is westwards in the vast majority of the basin. However, as seen in fig. 6,
there are regions where the second mode is promoted.
Depending on the wavenumber and associated growth rate i, Rossby
waves can follow different propagation modes. At the scale of the internal
Rossby radius of deformation, two main modes are generally at work:
Charney mode and Green mode. As detailed by Hochet et al., the dominating
mode is a function of several parameters. Representing the growth rate as a
function of u1 and u2 allows us to identify the necessary development
conditions for each mode (fig. 11). In our case, Green mode seems to be
favoured, also Charney mode cannot be set apart from Green mode. Indeed,
the relationship between the wavenumber kx (in rad-1) and the growth rate
of the waves showed that growth was selectively significant only for high
wavenumbers. This would mean that Charney mode Rossby waves are
7
being destroyed by the Laplacian operator, due to the relationship being
with k2, thus promoting exclusively long-term and large-scale variability
through Green mode, due to Green mode waves being the expression of the
least damped mode generated by Rossby waves.
During early calculations, the long wave approximation: c=/k was used to
infer the regions of growth where I ≠0. Eigenvalues problem was solved in
order to obtain c and from there . Under the long wave approximation,
results correlated well with the obsertions as the regions were located in
the North-Western part of the basin, precisely where Rossby waves seemed
to grow from (fig. 2). Further calculations involved the potential vorticity
and  in the calculations, yielding different results, with a shift of the
growth regions towards the south-east corner of the basin. Similar
discrepancies were observed by Hochet et al. and Liu et al., neither
providing a satisfactory explanation for this behaviour.
It is clear, however, that due to ’s influence the positive growth rate is
limited to low k. Could it be then that the growth regions observed in the
NW corner are in fact, Charney mode perturbations?
We verified the conditions for high growth rate as derived by Hochet et al.
in the case of 2 layers of equal thicknesses under constant zonal flows. We
found that in our case, the boundaries of instability were different,
particularly for u2. Indeed, Hochet et al. prescribe boundaries of ±0.05 m/s,
while we had growth where u2 either <-0.05 or >0. It still appears safe to
assume that the perturbations observed are indeed Green mode Rossby
waves.
6. Conclusion:
Low-frequency variability in an idealised Shallow water 2.5 layers model
was studied in order to investigate non-linear Rossby waves processes. Due
to the modifications brought to Sirven’s model, our focus shifted from the
Gulf Stream’s variability to low-frequency Rossby waves. The absence of
any kind of seasonality or AMOC type of process means our variability is
perfectly periodic, as opposed to Sirven et al.’s initial research. On the
subject, Sévellec et al. (2015) also investigated the influence of Atlantic
Multidecadal Oscillation in presence of vertical AMOC fluxes and provides
thorough insight on the influence of the AMOC. Moreover, no NW corner
was present, potentially impacting the outcropping regions and influence.
Experiments were conducted at varying resolutions and wind-stress. In all
case, non-linear Rossby waves developed and propagated westwards,
8
following Green mode propagation scheme. Experiments with variability in
wind energy input would be valuable. Further investigation of the vertical
mode in all experiments should also be done. It should be established which
parameters amongst velocity, vorticity,  and  has the most influence on
the variability observed, by setting them all at their mean values and letting
one at a time follow its variations. The double gyre circulation pattern
might be a key parameter for the observed instabilities (Ben Jelloul & Huck
2003).
References
Ben Jelloul, M., and T. Huck, 2003: Basin modes interactions and
selection by the mean flow in a reduced-gravity quasigeostrophic model.
Cox, M. D., 1987: An eddy resolving numerical model of the ventilated
thermocline: Time dependence.
Hochet, Antoine, 2015: Étude des courants océaniques transitoires de
grande échelle: structure verticale, interaction avec la topographie et
le courant moyen, stabilité. PhD manuscript, Université de Bretagne
Occidentale, Brest, France.
Hochet, A., T. Huck, A. Colin de Verdière, 2015: Large scale baroclinic
instability of the mean oceanic circulation: a local approach.
Huck, T., O. Arzel, F. Sévellec, 2015: Multidecadal variability of the
overturning circulation in presence of eddy turbulence. \JPO{45} (1)
157-173, doi: http://dx.doi.org/10.1175/JPO-D-14-0114.1 .
Liu, Z., 1999: Planetary wave modes in the thermocline:
Non-Doppler-shift mode, advective mode and Green mode.
Sévellec, F, and T. Huck, 2015: Theoretical investigation of the
Atlantic multidecadal oscillation. Journal of Physical Oceanography,
Sirven, J., S. Février, C. Herbaut, 2015: Low frequency variability of
the separated western boundary current in response to a seasonal wind
stress in a 2.5 layer model with outcropping.
Huck, T., O. Arzel, F. Sévellec, 2015: Multidecadal variability of the
overturning circulation in presence of eddy turbulence
9
Sévellec, F, and T. Huck, 2015: Theoretical investigation of the
Atlantic multidecadal oscillation. Journal of Physical Oceanography,
List of figures
Fig. 1: Streamfunctions
Fig. 2: Liu et al. flux antidiffusive flows
Fig. 3 a): depth anomalies as a function of resolution
Fig. 3 b): Depth anomalies as a function of 
Fig. 3 c): Depth anomalies as a function of 
Fig. 4: h1 and h2 at given lats
Fig. 5: EOF and PCs
Fig. 6: regions were u1 and u2 <0
Fig. 7: growth regions
Fig. 8: KE graphs
Fig. 9: STD graphs.
Fig. 10: Phase speeds
Fig. 11: Hochet et al. Charney+ Green mode wrt u1 et u2
10
Figure 1:
Streamfunctions for the 3
spatial resolutions
studied: from top to
bottom R3_006_3200,
R6_006_400,
R12_006_45. The values
are actually in Sv.
11
Figure 2: Principle of Antidiffusive flow in a model in order to allow
outcropping. From Liu et al. 1999
12
Figure 3 a) Depth
anomalies at the last
year of each
simulation. From top
to bottom:
R3_006_3200,
R6_006_400
R12_006_45
13
Figure 3 b)
On this page,
top to bottom:
R6_002_400,
R6_004_400,
R6_006_400
Continued next
page
14
On this page: top to
bottom,
R6_008_400,
R6_010_400,
R6_012_400
15
Figure 3 c) On
this page, top
to bottom:
R6_006_200
R6_006_800
16
Figure 4:
Longitudinal profiles
of h1 and h2
anomalies. From top
to bottom:
R3_006_3200,
R6_006_400
R12_006_45
17
Figure 5:
EOF and PC
analysis for
top:
R6_006_400
bottom:
R3_006_3200
These 2
illustrate well
the situation
for well
defined
perturbations
and poorly
defined
perturbations.
18
Figure 6: Regions where Mean zonal velocities flows are <0 in
R6_008_400. Very little variability was observed over all experiments.
These regions indicate regions of potential instability growth.
19
Figure 7: Map of
growth regions as
defined by i≠0 as
calculated from
Quasi- geostrophic
eigenvalues
problem under the
long wave
approximation for
R6_006_400 and
R6_006_400_3mo
20
Figure 8: Kinetic
Energy over time for:
top to bottom:
R3_008_3200
R6_006_400
R12_006_45
See next page for
influence of  and  on
R=1/6 and =400
21
22
Figure 9: Standard
deviation for h1 and h2 for
from top to bottom:
R6_002_400
R6_004_400
R6_006_400
Continued next page
23
R6_008_400
R6_010_400
R6_012_400
24
Figure 10: Real
and imaginary
part of the long
Rossby waves
phase speed for
R6_006_400
The black
contours on the
real part
indicate regions
where iC≠0.
Note the
differences in
growth region
with Fig.7.
25
Figure 11: Growth time of large scale mode in a 2.5 layers Shallow Water
model as a function of u1 and u2 at lat=30N for h1=h2=200m. From Hochet
et al. These 2015.
26