Internship Report lpo 2015: Long-Term Variability in a 5 Layers Shallow-Water Model



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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. 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.m-1.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 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) + vH vi + i-1 - i/ h
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

No outcropping, perturbations on the W side

R6_004_400

1/6

0.04

400

//

R6_006_400

1/6

0.06

400

No outcropping, perturbations over all the basin

R6_008_400

1/6

0.08

400

Outcropping, perturbation over all the basin

R6_010_400

1/6

0.10

400

Outcropping, chaotic, some defined perturbations in NW corner

R6_012_400

1/6

0.12

400

Outcropping, chaotic

R6_006_200

1/6

0.06

200

//

R6_006_800

1/6

0.06

800

No Outcropping, very clear perturbations over all basin

R3_006_3200

1/3

0.06

3200

No outcropping, perturbations focused in mid and top parts

R3_008_3200

1/3

0.08

3200

//

R12_006_45

1/12

0.06

45

Outcropping, chaotic perturbations over all basin

R6_006_45_3mo

1/6

0.06

45

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 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 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.
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 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, 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


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




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.




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