#### 5.1. Eddy Kinetic Energy

[55] As observed in other upwelling system models (see for example *Marchesiello et al.* [2003] for the California Current System and *Penven et al.* [2001] for the Benguela Current System), in the absence of synoptic and interannual variability in the surface fluxes, this model is able to generate an intense mesoscale activity through oceanic instability processes. Looking at animations of the model vorticity or sea surface height, one can note that a large number of eddies are generated from the upwelling front and then propagate offshore. The nonseasonal and noninterannual (i.e. by employing the seasonal averages to obtain the anomalies) eddy kinetic energy (EKE) has been computed using the geostrophy from the sea surface height for both the model simulation and the altimeter observations (Figure 13).

[56] For ROMS, the EKE in the PCS is higher at the coast, with values ranging from 40 cm^{2}.s^{−2} to 80 cm^{2}.s^{−2} (Figure 13a). These values increase further as the equator is approached, for example the EKE attains 100 cm^{2}.s^{−2} close to the SECC. The EKE decreases offshore and remains at relatively low levels (below 10 cm^{2}.s^{−2}) in the southwestern part of the model domain. Altimeter EKE follows the same pattern: high values in the upwelling front and close to the equator, and lower values offshore (Figure 13b). There are 3 areas where the difference between model and altimeter EKE is significant: close to the equator, in the southern part of the domain, and in a specific location (17°S–75.5°W) where the EKE values are higher in the observations. In the tropical region, variability is mostly associated with coastal trapped waves of large-scale equatorial origin [*Brink et al.*, 1983]. Since all nonseasonal variations have been filtered in the large-scale oceanic data employed to force the model boundaries, an important source of variability is missing north of 5°S. South of 18°S and west of 76°W, modeled EKE is relatively low in comparison to observations. Three reasons can explain this discrepancy: eddies coming from the southern part of the Humboldt system are not represented in this simulation; the POC and its variations are not correctly reproduced; and since the observed values in EKE are quite low, the synoptic wind variations might have a relatively greater effect. The third area of model/data EKE discrepancy is located offshore of Nazca (∼17°S–75.5°W). In this area, altimeter data present a maximum in variability which is not seen in the model solution. The Pisco - San Juan upwelling plume can be an important source of eddies for the area and it would necessitate further invertigation.

[57] In the central PCS (i.e. in a band situated between latitudes 15°S and 7°S), the model is able to produce a level of EKE only 10% to 30% lower than the observations (Figure 13c). This accomplishment has been achieved in the absence of synoptic variability in the model boundary conditions (surface and lateral). Hence, for the central PCS, at least 70% to 90% of the nonseasonal, noninterannual surface EKE (associated with geostrophic currents) can be explained by locally generated turbulent processes. One can note the progress achieved since the work of *Stammer and Boening* [1992]. For the Atlantic Ocean, they found that the variability in the WOCE model (1/3° × 2/5° resolution) was systematically up to 4 times lower than GEOSAT observations. Even if the level of eddy kinetic energy in our model is slightly too low, Figure 13 gives confidence for the use of the model outputs to study, at least qualitatively, the mesoscale dynamics in the PCS with a relatively realistic level of energy.

#### 5.2. Eddy Length Scales and Kinetic Energy Spectra

[58] As an illustration of the turbulent character of the simulation, the vertical component of vorticity and the Okubo-Weiss parameter (λ) at 20 m depth are presented for 21 June of model year 8 on Figure 14. The Okubo-Weiss parameter takes the form:

where u and v are the horizontal velocity components, and x and y are the horizontal coordinates. The first 2 terms of equation (5) represent the deformation and the last term, the relative vorticity. Therefore, if λ is positive, the deformation dominates, otherwise if λ is negative, the relative vorticity dominates. The Okubo Weiss parameter thus helps to detect the boundary of the eddies. Eddies are characterized by a strong rotation in their center, and a strong deformation in the periphery; therefore they can be represented by patches of negative values of Okubo-Weiss parameter surrounded by positive rings.

[59] The snapshot of subsurface vorticity (Figure 14a) shows a succession of cyclones and anticyclones which are generated at the upwelling front. They appear more energetic at the shore, in agreement with Figure 13. In the case of geostrophic turbulence, if the spatial variations in the Rossby radius of deformation are not negligible (e.g. close to the equator), *Theiss* [2004] has established the presence of a critical latitude (ϕ_{c}). On the poleward side of ϕ_{c}, the turbulence is isotropic; while on the equatorward side, alternating zonal jets dominate the flow (see Figure 8 in *Theiss* [2004]). In our simulation, a distinct transition occurs between a turbulent flow, south of 3°S; and the presence of large scale equatorial currents in the north (Figure 14a). In the PCS, eddies are clearly separated in the image of the Okubo Weiss parameter (Figure 14b). Looking more closely on Figure 14b, it appears that the eddies seem, in general, larger in the northern part of the domain.

[60] To verify this impression, the characteristic eddy length scales were calculated in a more rigorous manner. *Stammer and Boening* [1992] analysed oceanic eddy characteristics from altimeter measurements. From autocorrelation functions derived from GEOSAT data in 10° × 10° subdomains in the Atlantic Ocean, they computed the lag of the first zero crossing (*L*_{0}), the linear integral scales (*L*_{1}) and the quadratic integral scales (*L*_{2}). They found a linear relation between *L*_{0} and the first Rossby radius of deformation outside the equatorial region (10°S–10°N). *Stammer* [1997] extended the same analysis to the global Ocean. In our case, the autocorrelation functions have been calculated from surface current anomalies in 2° × 2° boxes centered on each point of a 0.5° resolution grid. They have been averaged from year 3 to year 10:

where is the autocorrelation function for the box centered at longitude *X*_{0} and latitude *Y*_{0}, *u*′ and *v*′ are surface current horizontal anomaly components, and *d* is a distance. *L*_{0} is presented on Figure 15. This representation shows larger eddies in the northern part of the model domain (almost by a factor of 2) than in the south. We can also note a progressive increase in eddy sizes when moving from the coast to offshore.

[61] To validate this approach, another method was used to calculate the eddy characteristics. From the Okubo Weiss parameter described above, it is also possible to systematically follow the eddies. An eddy tracking algorithm has been developed following this procedure:

[62] 1. Compute the Okubo-Weiss parameter at 20 m depth. Spatially smooth it by applying a Hanning filter several times in order to keep only the significant structures.

[63] 2. Detect the local minimums (i.e. every grid point which presents a value that is below its eight surrounding points). These should correspond to eddy centers.

[64] 3. For each local minimum, compute several relevant eddy properties: the position of the center, the radius (i.e. the minimum distance between each local minimum and where the Okubo Weiss parameter changes its sign) and the mean relative vorticity (i.e. the relative vorticy averaged over the eddy radius).

[65] 4. Compute a generalized nondimensional distance (*X*) between the eddies detected during two successive time steps. For each eddy (*e*_{1}) of the first time step and for each eddy (*e*_{2}) of the second time step, is defined as an Euclidean distance in a nondimensional property space:

where Δ*X* is the spatial distance between *e*_{1} and *e*_{2}, Δ*R* is the variation of diameter between *e*_{1} and *e*_{2}, and Δξ is the variation of vorticity between *e*_{1} and *e*_{2}. *X*_{0} is a characteristic length scale (100 km), *R*_{0} is a characteristic radius (50 km) and ξ_{0} is a characteristic vorticity (10^{−6} s^{−1}).

[66] 5. Select the eddy pair (*e*_{1}, *e*_{2}) that minimize to be the same eddy that is tracked from the first time step to the second time step.

[67] *X* measures the degree of dissimilarity between two eddies. Hence, it helps to discriminate between cyclones and anticyclones and between large and small eddies. Other eddy properties could be added into *X* calculation such as potential vorticity, temperature or salinity.

[68] This algorithm has been applied to the model outputs, and the resulting mean eddy diameters (*D*_{e}) are presented on Figure 15b. South of 3°S, *D*_{e} is in agreement with *L*_{0}. It is minimum at the coast, and increases towards the equator. North of 3°S (i.e. the critical latitude observed on Figure 14), *D*_{e} undergoes a marked decrease. In this area, *L*_{0} and *D*_{e} are in total disagreement. This strengthens the point that there is a sharp bifurcation in eddy dynamics in the PCS around 3°S. Outside an equatorial band, *L*_{0} appears to be a reliable measure for the mean eddy diameters.

[69] To further explore the turbulent aspects in the PCS, one-dimensional kinetic energy spectra for the surface currents are computed (Figure 16). They are spatially averaged on 2° wide latitude bands, centered at the latitudes: 17.5°S, 15°S, 12.5°S, 10°S, 7.5°S and 5°S. They are also temporally averaged from year 3 to year 10. In the case of geostrophic turbulence, the predicted energy spectra should be characterized by a *k*^{−3} power law for the direct enstrophy cascade (at wave numbers that are larger than the forcing wave numbers), and a *k*^{−5/3} inverse energy cascade spectrum (at wave numbers that are smaller than the forcing wave numbers) [*Charney*, 1971]. In the presence of β effect, the inverse cascade towards the large scales is halted when the scales are large enough for Rossby waves to be able to radiate turbulent energy [*Rhines*, 1975].

[70] For each latitude band, the modeled energy spectra obey almost rigorously the theoretical predictions, presenting well-developed energy and enstrophy inertial ranges. The energy injection length scales can be deducted from the marked change in slope in the spectra (e.g. the intersections between the *k*^{−3} and *k*^{−5/3} lines on Figure 16). One can note that the injection length scales are larger in the northern part of the model domain. For each spectrum on Figure 16 a cutoff scale is defined where the curve departs from the *k*^{−5/3} power low (i.e. where the red cascade is halted). Although it slightly increases towards the north, this cutoff scale does not present important variations around a mean value of about 400 km. The Rhines scale takes the form: , where U is a characteristic root mean square velocity [*Rhines*, 1975]. Using the model surface root mean square velocities we obtained a Rhines scale of the order of 70 km. Thus, It has not been possible to relate the model cutoff scale to the theoretical cutoff Rhines scale.

[71] Since the injection length scale increases towards the equator, while the cutoff scale varies by a much smaller amount; at 5°S, the energy cascade inertial region almost totally vanishes. This behavior is consistent with the findings of *Theiss* [2004]: in the quasigeostrophic framework, the critical latitude which marks the transition from an isotropic turbulent flow to alternating zonal currents is the latitude where the Rossby radius of deformation equals the cutoff Rhines scale. In our case, the injection length scale reaches the cutoff scale around 3°S. This might be an explanation for the sharp transition observed on Figure 14. However, at 3°S we lie within the equatorial band of twice the equatorial deformation radius (∼230 km) [*Chelton et al.*, 1998], degrading the validity of the formalism used by *Theiss* [2004].

[72] There are two major differences between our model and theoretical results. Firstly, idealised experiments show an equatorward energy cascade through the movement of eddies towards the equator. By tracking the eddies one by one, it has been possible to derive their average displacements. In out model, south of 12°S, eddies are moving straight offshore, while, between 5°S and 12°S, they present a southwestward movement. Secondly, while *Theiss* [2004] has detected a majority of cyclones, in our experiment, in the PCS, there is evidence for a larger population of anticylones.

[73] To summarize, the eddy length scale *L*_{0} and the mean eddy diameters (*D*_{e}) (both averaged in the first 1000 km from coast), the energy injection length scales, and the Rossby radius of deformation are presented together in Figure 17. South of 3°S, *D*_{e} and *L*_{0} are in relative agreement. They increase towards the north from about 40 km to more than 100 km. As explained by *Stammer and Boening* [1992], outside of the equatorial region, the eddy characteristic length scale follows the variations of the Rossby radius of deformation. The relation between the Rossby radius of deformation and the energy injection length scale is obvious. Because the first Rossby radius of deformation is the fastest growing mode for the baroclinic instability, this relation supports the agreement that baroclinic instability is the predominent eddy generation mechanism in the PCS.