In the eastern South Pacific Ocean (Figure 1), the simultaneous occurrence of colder waters near the coast (upwelling) and a southward warmer intrusion offshore (during El Niño) configures a scenario with strong SST gradients. The classical definition of the El Niño phenomenon is related to SST anomalies and the intrusion of warmer temperatures along the coastal eastern Pacific Ocean.

Figure 1. The study region (0˚S - 30˚S, 70˚W to 100˚W) is located in the eastern South Pacific Ocean.

But during 2002-2003, a non-canonical El Niño called El Niño Modoki took place, following reported contributions [1][2]. Also, some authors describe it as an Eastward Central Pacific El Niño [3][4].

Along the equator, during 2002-2003, El Niño Modoki is characterized by a warming of the NINO3 (150˚W - 90˚W; 5˚N - 5˚S) and NINO4 (150˚E - 150˚W; 5˚N - 5˚S) regions of the order of 1˚C. Otherwise, along the Perú coast, important SST anomalies were observed. During autumn 2002, SST anomalies of up to 2.5 ˚C appeared off Chicama (7.8˚S), diminishing towards the south and north. Also, it was possible to observe a reactivation of the upwelling in very coastal zones between 6˚S and 11˚S (Punta Falsa and Huacho). The 15˚C isotherm was located between 30 and 50 m, below its usual depth to the north of 9˚S [5]. In summer 2003 along the coast (3.5˚S - 12˚S), in January, the deepening of isotherms continued, but in February ascended because of the reactivation of coastal upwelling [6].

A challenge of El Niño Modoki is the fact that its generating mechanisms are still largely unknown [7]. Otherwise, it is well known that the Sea Surface Temperature is very dependent on radiation from the sun [8]. In the oceans, the SST variability is the dynamic result of the energy obtained from the sun and the current circulation in the deep and coastal ocean. A possible link between heat fluxes and warming during El Niño Modoki needs to be reviewed carefully.

In particular, the exchange of heat fluxes during warming El Niño events in the Pacific Ocean was not well discussed in the literature. But an important possibility associated with the warming effects during El Niño was indicated by Weare in 1983 [9]. Additionally, a study analysis [10] supports the view that surface heat flux variations have a role in the meridional warming of SST during ENSO development. This new conclusion differs from previous studies (e.g., [11]). Also, the comparison of East-West Short Wave Radiation (SWR) and SST [8] indicates that in the Eastern Pacific the SWR is higher during the initiation of El Niño, reaching higher SST.

Also, the influence of atmospheric heat fluxes has not been considered in numerical simulations in its real magnitude. But some signs of its possible influence were observed in some works. In the pioneering simulation of Philander and Siegel (1985) for El Niño 1982-1983 [12], constant surface heating was considered (short wave and long wave back radiation), obtaining high SST distributions with intrusion of warm waters off the South American coast. Moreover, Jia, Wells and Rowe (1990) [13] also simulated El Niño 1982-1983 but without heat input at the surface from the atmosphere. Their results show the persistence of cold waters off the South American coast (which was not understood), indicating implicitly that an important forcing was not included.

The aforementioned signals, associated with atmospheric heat fluxes, are a strong reason to explore the influence of atmospheric forcings, including solar radiation, on the evolution of sea surface temperature and currents in the eastern Pacific Ocean. The warming processes of the eastern Pacific Ocean during El Niño Modoki were not studied dynamically and need to be reviewed carefully. The present paper is a little step in this direction.

In the present study, the role of only time dependent regional atmospheric forcings will be investigated numerically, evaluating the time evolution of the resulting SST variability in space and time and the current fields in the domain. In this way, the importance of the regional atmospheric forcings could be really understood. The numerical study is performed covering the region between the equator and 30˚S and between 100˚W and 70˚W (Figure 1) with an unstructured finite element mesh (Figure 2).

The use of complicated models can obscure the basic physical processes at work in them, whereas the adequate formulation of simple models can show clearly the importance of particular processes. Here, a simple model ( 1 1 2 layer) of limited area domain is presented to describe the upper layer dynamics off Perú during El Niño Modoki 2002-2003. Numerically, the presented model has some technical advantages. The finite element Petrov-Galerkin formulation model uses a stabilizing operator in space and time to improve the classical Galerkin approach [14][15]. This methodology allows accurate solutions, precluding the use of artificial filters, mass diffusion and dampers, commonly used in several popular models. The numerical solutions are obtained strictly from the finite element formulation of the governing equations in the domain. An additional advantage is the treatment of the open boundaries which is based on weakly reflective open boundary conditions derived from the characteristic equations [16]. The effectiveness of those kinds of open boundary conditions for coastal circulation problems has been previously reported [17][18].

In the present paper, some of the results are the following: the model forced by regional forcings is able to describe the time evolution of the SST field which compares favorably with observations. The results verified the connection between the forcings and the thermal response of the regional domain.

The article is organized as follows: Section 2 describes the model and its parametrizations. Section 3 presents the results of the modelling of ocean dynamics during 2002-2003 and Section 4 presents the findings of the study.

The used ocean model (a gravity reduced 1 1 2 layer model) has one active layer of density ρ (entire layer above the thermocline), overlaying a deep layer of density ρ l , where horizontal gradients are assumed to vanish [15]. A Cartesian coordinates system (x, y) oriented positively to the East and North directions respectively is considered. The non-permanent vertical integrated upper layer governing equations of motion, continuity and heat

The governing Equations (1), (2) and (3) describe the time evolution of the velocity u = ( u , v ) , the upper layer thicknesshand temperature Tof the upper active layer in each point of the spatial domain. The Nabla operator is represented by ∇ = ( ∂ / ∂ x , ∂ / ∂ y ) . The lower layer has a constant temperatureT^l. The upper layer describes the entire layer above the thermocline and due dynamical reasons can become small during upwelling motions and also is able to deepen during downwelling events.

In the above equations, Q_S is the heat input at the ocean surface, Q_I is the heat loss at the interface, τ s = ( τ x s , τ y s ) is the wind stress at the surface, τ i = ( τ x i , τ y i ) is the stress at the interface, k is the unit vector in the vertical direction andfis the coriolis parameter (evaluated in function of the spatial coordinatesx, y). The entrainment velocity is w_e, c_p is the specific heat and the parameter σ = 1 − γ where γ = ρ / ρ l and γ ¯ = γ / ( γ − σ ) . The upper layer density varies according to ρ = ρ l [ 1 − α ( T − T l ) ] where α is the coefficient of thermal expansion. The entrainment velocity w_e cools the upper layer and prevent the interface of surfacing in regions of intense upwelling [19],

whereH_e is the entrainment referential depth andt_e is a specified relaxation time.

To completely define the proposed ocean model we need to characterize the applied external sources of energy, the time dependent boundary conditions and the initial state from which the relevant hydrothermodynamical response will be obtained. The hydrodynamics of the model is forced by the wind stresses evaluated from the wind velocity fields using the quadratic relations τ x s = ρ a i r c D W x | W | , τ y s = ρ a i r c D W y | W | and c D = ( 0.509 + 0.065 | W | ) / 1000 [20]. The terms τ x i , τ y i , are parameterized equal to ρhruand ρhrvrespectively, wherer is a constant.

The thermodynamic of the model is forced by the short wave radiation Q_sw and the resultant outgoing longwave radiation Q_lw. The heat balance of the model is associated to the surface heat input Q_S defined as

where Q_Sen is the sensible heat and Q_Lat is the latent heat. The sensible heat is calculated from the bulk formula Q s e n = ρ a i r c p a i r C S ( T − T a i r ) | W | , where c_pair is the specific heat of the air, C_S an empirical coefficient, andTand T_air are the sea surface temperature and air temperature respectively. The latent heat is estimated from Q L a t = ρ a i r C L L ( q − q a i r ) | W | , whereL is the latent heat of vaporization, C_L an empirical coefficient and q, q_air are the saturation specific humidity of the sea surface and air respectively. For a constant lower layer temperatureT^l,the heat flux across the interface is given by:

in this way the gain or loss of heat across this interface, depends on the dynamical convergence or divergence of the flows ω in the upper layer, as expressed by the parameter k I = ω H .

Concerning the boundary conditions it is assumed that land type boundaries give rise to the classical non-slip conditions

and for upper thickness and for the temperature, homogeneous conditions are prescribed. On the sea side boundaries we prescribe a weakly reflective open boundary condition, based on the characteristic method. This is done assuming that on a normal direction (x_n) to the open boundary

where the invariant R ± = u n ± 2 c are valid, along the characteristics ( d x / d t ) ± = u n ± c normal to the open boundary. u_n represents the velocity component normal to the boundary and c = g σ h is the celerity wave. The weakly reflective conditions on the open boundary are defined by the in-going characteristic of the presented equations. For the temperature at the open boundaries, homogeneous condition is assumed.

Symmetric forms are always desirable because they possess better stability properties, an important feature to be used in the design of approximate solutions of advection dominated problems and shallow water wave problems [21].

For the upper layer dynamics problem treated in this paper a complete symmetrization is not achieved. Nevertheless, the hydrodynamic part of the equations governing the ocean circulation could be symmetrized. Furthermore, the whole set of unknown variables retains the same physical dimensions, which means that the transformed equations are adequately scaled and a new set of variables are defined as d = 2 g σ h and b = c 0 T / T u which have the dimensions of velocity. HereT^uisa referential upper layer temperature and c o = g σ H is a referential celerity. After the change of variables, the set of equations for the unknown vector Vcould be written as:

where

A space-time finite element partition is considered. The time interval [0, T_f] is divided into time sub-intervals I n = ( t n + 1 − t n ) = Δ t , for t ∈ [ 0 , T f ] , where the time levels t_n, t_n+1 belong to an ordered partition of time levels t 0 = 0 < t 1 < ⋯ < t n < t n + 1 < ⋯ < t F = T f .

The spatial domain Ω is partitioned into N sub-domains Ω_e. For each level of timen, the domain space-time integration is the Slab S n = Ω × I n with border Γ n = Γ × I n , so that the Slab is composed of N space-time elements S n e = Ω e × I n .

A finite-dimensional trial solution space U n h and a test function space U ^ j n are defined. For the presented model, continuous interpolation in space and time is adopted. The Petrov-Galerkin approximate solution for the ocean model problem is the vector V h ∈ U n h (of piecewise polynomials) which satisfies the following variational formulation for all V ^ h ∈ U ^ n h

where the residual vector R^h is

and the space-time operator G^h is

For an equation system, the Ψ ¯ e definition is even more complicated due to the presence of multiple wave components. For the ocean problem focused in this paper

where l e = ( ∂ ξ i / ∂ x ) max 2 + ( ∂ ξ i / ∂ y ) max 2 is the representative length of the triangular spatial element Ω_e, and ξ_i are the local co-ordinates. The first integral in the variational formulation (13) is the Galerkin term. The second integral represent the Petrov-Galerkin contribution, where Ψ ¯ e contain the intrinsic time scale free parameters [22]. The resulting system of algebraic equations is solved using the direct method of Gauss (L-U descomposition). The program code is written in Fortran.

Numerical experiments are performed to clarify the relative importance of regional forcings in the resulting time dependent evolution of the upper layer dynamics.

The study area extends in the north-south direction from equator to the 30˚S and in the east-west direction from the 70˚W to 100˚W. This region was represented by an unstructured mesh of 5684 surface triangular elements and 2994 nodes, as illustrated in Figure 2.

Near the coastal boundaries, a finer mesh was employed which gradually becomes coarser and coarser into the offshore region. The smaller elements are located along the coastline.

A vertically integrated 1 1 2 layer stabilized finite element model has been considered to understand the surface variability produced by only regional forcings during El Niño 2002-2003 in the eastern South Pacific Ocean. The model includes the hydrodynamics and thermodynamics of the upper ocean layer in a limited-area domain. The formulation uses a stabilizing operator in space and time to improve the classical Galerkin approach. This methodology allows accurate solutions.

The model solutions are forced by ten-day-averaged regional meteorological data, extracted from ECMWF. The time evolution results for the SST field compare well with the observed SST evolution during 2002-2003. The southward migration of the warm water tongue to the south, during the two phases of El Niño, was reproduced. During the warming phases, southward velocity components are induced and a coastal countercurrent is developed.

The source of the warming in the model region is the net incident heat flux and the promising results were possible due to a particular observed thermal feature of the region: the rate of change of the Sea Surface Temperature (∂T/dt) and the net incident heat flux (Q_n) are linked during 2002-2003.

The evolution of the integrated SST of the model in the study region follows the time variability of integrated observed SST, validating the time evolution of the thermal behavior in the region. The results show no significant accumulation or loss of heat in the system that can cause divergence of the time dependent solution.

It could be remark that it has been developed a 1 1 2 layer ocean model that is numerically adequated to simulate many aspects of the time evolution of the coastal ocean response off Peru. Additionally, the model has a simple structure (one dynamic layer) to understand the surface variability. In this way, it was possible to determine the real role of the regional forcings off Peru. Particularly, the importance of the influence of the heat fluxes in the development and decline of the warming phases during the El Niño event 2002-2003.

The presented model of the coastal ocean off Peru generates the SST variabilities during El Niño Modoki 2002-2003 by means of a surface atmosphere-ocean interaction.

This research was supported by “Fondo para la Innovación, Ciencia y Tecnología (FINCYT)”, Grant No. ECIP-1-P-033-14 and sponsored by the Instituto del Mar del Peru. Thanks to C. Arellano and C. Quispe for the data support and to J. Tam and D. Gutierrez for the valuable comments and coordination.