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  • View in gallery

    Mean potential temperature (°C) on the equator from (a) the OGCM and (b) Argo measurements. Mean zonal velocity (cm s−1) on the equator from (c) the OGCM and (d) observations (Johnson et al. 2002).

  • View in gallery

    Mean vertical profiles of zonal velocity (cm s−1) from the OGCM and TAO buoy observation on the equator for three locations: (a) 170°, (b) 140°, and (c) 110°W.

  • View in gallery

    Mean potential temperature over the Niño-3 region (5°N–5°S, 90°–150°W) for the upper 200 m (a) simulated by OGCM and (b) based on Argo measurements during 2003–07 (the overlapping period between model simulation and Argo float measurement).

  • View in gallery

    Comparison of adjoint sensitivity and forward perturbation-based sensitivity for a 6° × 1.5° box centered at 0°N, 162.75°W. The adjoint sensitivity over the region (solid line) is obtained by spatial summation of adjoint sensitivity covering grid points within that box [refer to Eq. (1), which displays similar summation over the Niño-3 region]. Crosses are obtained from perturbation experiments (see section 2 for details).

  • View in gallery

    Adjoint sensitivity of monthly average temperature in the upper 30 m of the Niño-3 region in December 2006 to monthly zonal wind stress in previous months during 2006.

  • View in gallery

    Adjoint sensitivity of the peak Niño-3 surface temperature in December to monthly local zonal wind stress in the region (5°N–5°S, 90°–150°W) during the 2-yr period before the peak. Simulations for six ENSO events are shown: one strong El Niño event, three intermediate El Niño events, and two La Niña events. A climatological run is shown for reference. The derived adjoint sensitivity from regression coefficients [Eq. (13)] is also plotted for a 1-yr period.

  • View in gallery

    As in Fig. 6, but for sensitivity of the peak Niño-3 surface temperature to remote zonal wind stress in the region 5°N–5°S, 145°E–155°W.

  • View in gallery

    Scatterplot between the subsurface vertical temperature gradient in the Niño-3 region (x axis) and maximum positive adjoint sensitivity of Niño-3 surface temperature to local zonal wind stress (y axis). The solid line shows the regression curve. The subsurface vertical temperature gradient is defined by Eq. (6).

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Adjoint Sensitivity of the Niño-3 Surface Temperature to Wind Forcing

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  • 1 Scripps Institution of Oceanography, La Jolla, California, and CSIRO Marine and Atmospheric Research, Hobart, Tasmania, Australia
  • | 2 Scripps Institution of Oceanography, La Jolla, California
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Abstract

The evolution of sea surface temperature (SST) over the eastern equatorial Pacific plays a significant role in the intense tropical air–sea interaction there and is of central importance to the El Niño–Southern Oscillation (ENSO) phenomenon. Effects of atmospheric fields (especially wind stress) and ocean state on the eastern equatorial Pacific SST variations are investigated using the Massachusetts Institute of Technology general circulation model (MITgcm) and its adjoint model, which can calculate the sensitivities of a cost function (in this case the averaged 0–30-m temperature in the Niño-3 region during an ENSO event peak) to previous atmospheric forcing fields and ocean state going backward in time. The sensitivity of the Niño-3 surface temperature to monthly zonal wind stress in preceding months can be understood by invoking mixed layer heat balance, ocean dynamics, and especially linear equatorial wave dynamics. The maximum positive sensitivity of the Niño-3 surface temperature to local wind forcing usually happens ~1–2 months before the peak of the ENSO event and is hypothesized to be associated with the Ekman pumping mechanism. In model experiments, its magnitude is closely related to the subsurface vertical temperature gradient, exhibiting strong event-to-event differences with strong (weak) positive sensitivity during La Niña (strong El Niño) events. The adjoint sensitivity to remote wind forcing in the central and western equatorial Pacific is consistent with the standard hypothesis that the remote wind forcing affects the Niño-3 surface temperature indirectly by exciting equatorial Kelvin and Rossby waves and modulating thermocline depth in the Niño-3 region. The current adjoint sensitivity study is consistent with a previous regression-based sensitivity study derived from perturbation experiments. Finally, implication for ENSO monitoring and prediction is also discussed.

Corresponding author address: Xuebin Zhang, Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Dr., MS 0230, La Jolla, CA 92093-0230. E-mail: Xuebin.Zhang@csiro.au

Abstract

The evolution of sea surface temperature (SST) over the eastern equatorial Pacific plays a significant role in the intense tropical air–sea interaction there and is of central importance to the El Niño–Southern Oscillation (ENSO) phenomenon. Effects of atmospheric fields (especially wind stress) and ocean state on the eastern equatorial Pacific SST variations are investigated using the Massachusetts Institute of Technology general circulation model (MITgcm) and its adjoint model, which can calculate the sensitivities of a cost function (in this case the averaged 0–30-m temperature in the Niño-3 region during an ENSO event peak) to previous atmospheric forcing fields and ocean state going backward in time. The sensitivity of the Niño-3 surface temperature to monthly zonal wind stress in preceding months can be understood by invoking mixed layer heat balance, ocean dynamics, and especially linear equatorial wave dynamics. The maximum positive sensitivity of the Niño-3 surface temperature to local wind forcing usually happens ~1–2 months before the peak of the ENSO event and is hypothesized to be associated with the Ekman pumping mechanism. In model experiments, its magnitude is closely related to the subsurface vertical temperature gradient, exhibiting strong event-to-event differences with strong (weak) positive sensitivity during La Niña (strong El Niño) events. The adjoint sensitivity to remote wind forcing in the central and western equatorial Pacific is consistent with the standard hypothesis that the remote wind forcing affects the Niño-3 surface temperature indirectly by exciting equatorial Kelvin and Rossby waves and modulating thermocline depth in the Niño-3 region. The current adjoint sensitivity study is consistent with a previous regression-based sensitivity study derived from perturbation experiments. Finally, implication for ENSO monitoring and prediction is also discussed.

Corresponding author address: Xuebin Zhang, Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Dr., MS 0230, La Jolla, CA 92093-0230. E-mail: Xuebin.Zhang@csiro.au

1. Introduction

The El Niño–Southern Oscillation (ENSO) phenomenon, one of the most prominent interannual climate variations on the planet, has attracted attention from both the research community and the general public because it cannot only affect our climate system significantly, but it also has tremendous environmental and socioeconomic impacts over the globe (e.g., McPhaden et al. 2006; Philander 2006). The variations of sea surface temperature (SST) in the eastern equatorial Pacific play a significant role in the evolution of ENSO events. Vertical heat advection is the primary mechanism by which ENSO-related SST anomalies are generated in the eastern equatorial Pacific. Because the mean thermocline depth is shallow in this region, thermocline displacement is the primary source of SST anomalies on ENSO time scales (e.g., Zebiak and Cane 1987; Battisti and Hirst 1989). Interannual variations of thermocline depth in the eastern equatorial Pacific are mainly forced by remote wind forcing in the western and central equatorial Pacific (e.g., Zhang and McPhaden 2006, hereafter ZM06). Because the remote wind forcing plays a dominant role in ENSO-related SST variations in the eastern equatorial Pacific (e.g., Rasmusson and Carpenter 1982; Harrison and Larkin 1998; Wang and Picaut 2004), local wind forcing in the eastern equatorial Pacific and associated physical processes were often neglected in previous studies. However, local wind forcing may also vary on ENSO time scales and can have impacts on individual ENSO events (e.g., Rasmusson and Carpenter 1982; Harrison 1989; Harrison and Larkin 1998; Vintzileos et al. 2005; Kessler and McPhaden 1995; McPhaden and Yu 1999). The empirical analysis by ZM06 pointed out that local wind variations can have nonnegligible effects on interannual SST variations in the eastern equatorial Pacific, mainly through modifying vertical upwelling velocity. Through regression analysis, they found that a spatially uniform zonal wind stress (τx) anomaly of 0.01 N m−2 over the Niño-3 region (5°N–5°S, 90°–150°W) leads to approximately a 1°C SST anomaly there. Wind stress anomalies on the order of 0.01 N m−2 occurred in the eastern equatorial Pacific during the 1982–83 and 1997–98 El Niño events, accounting for about 1/3 of the maximum SST anomaly during these events. ZM06’s strictly empirical analysis was later verified by their numerical sensitivity tests (Zhang and McPhaden 2008, hereafter ZM08). By perturbing (masking) wind stress anomaly in an ocean general circulation model (OGCM) and comparing results between perturbed and nonperturbed runs, ZM08 were able to verify ZM06’s regression-based empirical analysis. They found that in the Niño-3 region a zonal wind stress anomaly of 0.01 N m−2 leads to about 0.9°C SST anomaly when the former leads the latter by 2 months, and air–sea heat fluxes tend to damp interannual SST anomalies generated by other physical processes at a rate of about 40 W m−2 °C−1.

The main purpose of this paper is to explore the sensitivity of the Niño-3 surface temperature to atmospheric fields and ocean state in the past with a very different approach by using another OGCM and its adjoint model, which allows sensitivity to be derived efficiently in a single adjoint model run. We focus on the sensitivity of the Niño-3 surface temperature to zonal wind stress, which may provide new insights into remote and local wind effect on the Niño-3 surface temperature change. Adjoint sensitivity analysis has been used in the oceanic and meteorological fields for more than 20 years (e.g., Le Dimet and Talagrand 1986; Thacker and Long 1988; Marotzke et al. 1999; Lee et al. 2001; Fukumori et al. 2004; Hill et al. 2004; Bugnion and Hill 2006; Bugnion et al. 2006; Losch and Heimbach 2007; Moore et al. 2009; Veneziani et al. 2009), but many applications remain unexplored. The adjoint model produces the partial derivatives of a cost function (in this case temperature) with respect to previous model states and forcing, linearized around the original (“reference”) forward model solution. The sensitivity gives the linearized Green’s function for the cost function since the perturbation to the cost function can be estimated by summing the sensitivity multiplied by the perturbed forcing or model state. For sufficiently small perturbations, the estimated cost function perturbation will match the perturbation calculated from rerunning the nonlinear model with perturbed initial conditions and forcing, but the range of perturbation size over which this linearity holds may be limited. In spite of the many applications of the adjoint, comparisons between adjoint and regression analysis are not common. This paper serves as an example of both the use of adjoint sensitivity analysis as a dynamical tool and its complementarity to regression analysis. The paper is organized as follows: in section 2, we introduce the OGCM and its adjoint model, model configuration, numerical experiment design, and model validation. Detailed discussion of adjoint sensitivity is given in section 3. In section 4, we reconcile current adjoint sensitivity study with previous regression-based sensitivity study. Finally, a summary and discussion are given in section 5.

2. Model and methodology

a. Model configuration and evaluation

The OGCM used in this study is the MIT general circulation model (MITgcm), which is developed at the Massachusetts Institute of Technology (Marshall et al. 1997; more information can be found online at http://mitgcm.org/). It has been widely applied for many applications with various temporal and spatial scales, including the Estimating the Circulation and Climate of the Ocean (ECCO) Project (see online at http://ecco.mit.edu). The MITgcm and its adjoint model have been applied to the tropical Pacific basin for hindcasting of ocean state (Hoteit et al. 2005, 2008, 2010). Our current model setup is mainly based on Hoteit et al.’s previous work with several minor modifications, so only key attributes are discussed here. The model domain covers the tropical Pacific basin (26°S–26°N, 104°E–68°W), with constant 1/3° resolution in both zonal and meridional directions. The model has realistic bottom topography taken from the 5-minute gridded elevations/bathymetry for the world (ETOPO5; see online at http://www.ngdc.noaa.gov/mgg/global/etopo5.HTML). There are 51 vertical layers with fine resolution for the upper ocean (10 uniform layers for the upper 100 m). The daily surface forcing fields (including heat flux, wind stress, and freshwater flux) are all derived from the European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Reanalysis (see online at http://data-portal.ecmwf.int/data/d/interim_daily), which are available at 1.5° × 1.5° horizontal resolution. The current model domain has realistic open boundaries at 26°S, 26°N, and 104°W. An artificial island is added at the southwest corner grid for the ease of implementation of open boundary conditions. Two global 1° × 1° data assimilation products are used to provide initial and open boundary conditions (e.g., temperature, salinity, and horizontal velocity). For the model runs before 2004, global 1° × 1° ECCO product (Köhl et al. 2007) is used, while the global 1° × 1° Ocean Comprehensive Atlas (OCCA) product (Forget 2010) is used after 2004. Sponge layers are applied to the region within 3° of the open boundaries.

The tropical Pacific regional model has been tuned and validated with in situ observations and other modeling results, as the current configuration is based on our group’s previous work (Hoteit et al. 2005, 2008, 2010). The model can represent equatorial ocean dynamics and ENSO-related physical processes quite well (details can be found in above references). Here we only show comparison of several important features between model and in situ observation as support for carrying out this study based on this regional model. The model simulates the subsurface thermal structure on the equator quite well (Figs. 1a,b), as compared to Argo float measurements (Roemmich et al. 1998; Roemmich and Gilson 2009; see more information online at http://www.argo.ucsd.edu/). There is a strong zonal temperature gradient in the upper ocean, with the warm pool in the west and the cold tongue in the east. Below the surface mixed layer, there is a thermocline defined by a sharp vertical temperature gradient. The thermocline is not flat zonally with deeper (shallower) depth in the west (east). Two weaknesses of the model simulation can be found: 1) the simulated thermocline is not as tight as observed by Argo floats, and 2) cold bias exists in the cold tongue region. These are common problems for many OGCMs (e.g., Stockdale et al. 1998). Compared with observation-based analysis (Johnson et al. 2002), the model also simulates the mean vertical structure of zonal velocity along the equator: a westward South Equatorial Current (SEC) at the surface and an eastward Equatorial Undercurrent (EUC) with an eastward-shoaling core (Figs. 1c,d). The mean vertical profile of the EUC, another sensitive measurement of model fidelity, compares well with the Tropical Atmosphere Ocean (TAO) buoys measurements (McPhaden et al. 1998; see online at http://www.pmel.noaa.gov/tao/) at three representative locations on the equator (Fig. 2). The mean potential temperature over the Niño-3 region (5°N–5°S, 90°–150°W) for the upper 200 m simulated by the OGCM clearly catches the temporal evolution of thermal structure (such as the mixed layer, thermocline, and the vertical temperature gradient) as compared with Argo float measurements (Fig. 3). An almost constant cold bias (about 1°C) nonetheless exists in the simulated mixed layer temperature in the Niño-3 region. We also checked the upper-ocean heat balance (both mean and interannual variability) from model simulation and found similar results to Lee et al. (2004) and Kim et al. (2007) who used a global configuration of the same MITgcm model. So this tropical Pacific model based on MITgcm can be confidently applied for our current study.

Fig. 1.
Fig. 1.

Mean potential temperature (°C) on the equator from (a) the OGCM and (b) Argo measurements. Mean zonal velocity (cm s−1) on the equator from (c) the OGCM and (d) observations (Johnson et al. 2002).

Citation: Journal of Climate 24, 16; 10.1175/2011JCLI3917.1

Fig. 2.
Fig. 2.

Mean vertical profiles of zonal velocity (cm s−1) from the OGCM and TAO buoy observation on the equator for three locations: (a) 170°, (b) 140°, and (c) 110°W.

Citation: Journal of Climate 24, 16; 10.1175/2011JCLI3917.1

Fig. 3.
Fig. 3.

Mean potential temperature over the Niño-3 region (5°N–5°S, 90°–150°W) for the upper 200 m (a) simulated by OGCM and (b) based on Argo measurements during 2003–07 (the overlapping period between model simulation and Argo float measurement).

Citation: Journal of Climate 24, 16; 10.1175/2011JCLI3917.1

b. Adjoint model and numerical experiment design

Adjoint models are widely used in meteorology and dynamical oceanography, Errico (1997) gave an overview and heuristic explanation of adjoint model application. Marotzke et al. (1999) provided a comprehensive mathematical derivation of adjoint sensitivity (see their section 2) in their study of Atlantic heat transport sensitivity with the MITgcm and its adjoint model. Veneziani et al. (2009) made clear mathematical derivation of adjoint sensitivities to model state and external forcing, and applied adjoint techniques to study the sensitivity of aspects of the California Current System to remote and local atmospheric forcing. The reader is referred to these references for details about adjoint model derivation and construction.

The adjoint model of the MITgcm is generated by the automatic differentiation with the Transformation of Algorithms in FORTRAN (TAF) compiler (Giering and Kaminski 1998; Heimbach et al. 2005). The adjoint model calculates sensitivity (i.e., partial derivative) of the cost function J with respect to control variable υ (). The model dynamics and states are contained in the chain rule of partial derivatives. The control variables can be model state, forcing fields, model parameters, and initial and open boundary conditions. The cost function J is usually chosen as a scalar function of the model output, and as such should be differentiable in principle.

The adjoint model calculates the partial derivatives of a single variable (the cost function) around a selected model trajectory backward to all previous fields. Consequently, the adjoint model solution depends on the chosen period of the forward run that defines the model trajectory. In other words, different forward model states may give different adjoint solutions and the variability in the former can be reflected in the latter. Differences of adjoint sensitivity associated with different ENSO events will be examined in section 3b. Similar differences of adjoint solutions have been discussed in the adjoint passive tracer study by Fukumori et al. (2004).

For our current study, we are interested in the Niño-3 SST variability. In particular we want to understand what controls SST at the peak of one ENSO event. Because of the “phase locking” feature, El Niño events usually reach peaks at the end of the calendar year (e.g., Philander 1990). For ease of explanation, we set t = 0 as the end of December, which is usually the peak time of an event. We run the forward model for 2 yr from −24 months (i.e., 2 yr before the peak) to 0 months. The cost function is defined as the averaged 0–30-m temperature in the Niño-3 region over the last month of a 2-yr model run. Finally the adjoint sensitivity of the Niño-3 surface temperature to control variables going from 0 to −24 months backward in time is calculated.

As described above, the adjoint model finds the sensitivity of one output (i.e., cost function) to all inputs (e.g., wind forcing). In addition to the adjoint method, the sensitivity can also be calculated in the forward manner by trial perturbation, though by this way the sensitivity of all outputs to one input is derived. By applying a sufficiently small perturbation to the forcing (e.g., wind forcing), we can derive the sensitivity by comparing the perturbed model trajectory with the control (unperturbed) one. For instance, the zonal wind stress is perturbed uniformly with a magnitude of 0.05 N m−2 for 1 week at some time lead over a 6° × 1.5° box centered at 0°, 162.75°W (the box size and location is chosen by considering the resolution and grid location of zonal wind forcing from the ECMWF Interim Reanalysis). The only difference between the perturbed and control runs is due to a zonal wind stress perturbation for one week over that region. The sensitivity (in this case the partial derivative of the cost function Niño-3 surface temperature with respect to the control variable zonal wind stress) can be approximated by division of the change of Niño-3 surface temperature in December 2006 by the zonal wind stress perturbation (Fig. 4). To obtain a spatial map of such a perturbation-derived sensitivity at that time lead, we need to repeat the perturbation experiment for each grid box of the model domain (or at least for each subregion at coarser horizontal resolution). By repeating the wind perturbation for 1 week for every 4-week period in 2006 for the above region, we can derive time-dependent perturbation-based sensitivity, which matches the adjoint sensitivity well (Fig. 4). Obviously, such a forward perturbation-based sensitivity is computationally expensive, so a grid-by-grid comparison between forward sensitivity and adjoint sensitivity both spatially and temporally is impractical and almost impossible [for spatial comparison at 1 time lead alone, about 80 000 (the number of grid points) perturbation runs are needed]. So like Bugnion and Hill (2006) and Hill et al. (2004), we are only able to compare the adjoint with forward sensitivity fields for some representative locations for validation purposes, and found that they are generally consistent with each other (see, e.g., Fig. 4).

Fig. 4.
Fig. 4.

Comparison of adjoint sensitivity and forward perturbation-based sensitivity for a 6° × 1.5° box centered at 0°N, 162.75°W. The adjoint sensitivity over the region (solid line) is obtained by spatial summation of adjoint sensitivity covering grid points within that box [refer to Eq. (1), which displays similar summation over the Niño-3 region]. Crosses are obtained from perturbation experiments (see section 2 for details).

Citation: Journal of Climate 24, 16; 10.1175/2011JCLI3917.1

We also want to clarify that the purpose of this adjoint sensitivity is to identify physical relationship between control variables (e.g., wind stress) and cost function (e.g., Niño-3 surface temperature), rather than to seek to the optimal perturbation or sensitivity as Mu et al. (2003) and Duan and Mu (2009a), which has usually been applied to check the predictability of numerical weather and climate prediction.

3. Result

a. Example of 2006–07 El Niño event

The adjoint model can calculate the sensitivity of a cost function J to all control variables in a single integration. Nonetheless, in our current study, we mainly focus on the sensitivity of Niño-3 surface temperature to zonal wind stress. Using the 2006–07 El Niño event as an example, Fig. 5 shows the sensitivity of the final cost (i.e., monthly average temperature in the upper 30 m of the Niño-3 region in December 2006) to monthly average wind stress in the preceding months during 2006. The physical interpretation of this sensitivity field is that, if wind stress is perturbed over a single grid box (1/3° × 1/3°) for 1 month with 1 standard unit of wind stress (i.e., 1 N m−2), the monthly average temperature in the upper 30 m of the Niño-3 region in December 2006 (the last month of a 2-yr forward simulation) will change by the given number of degrees Celsius. Positive (negative) sensitivity indicates increase (decrease) of the Niño-3 surface temperature in December 2006 due to a unit positive change in monthly zonal wind stress in previous months.

Fig. 5.
Fig. 5.

Adjoint sensitivity of monthly average temperature in the upper 30 m of the Niño-3 region in December 2006 to monthly zonal wind stress in previous months during 2006.

Citation: Journal of Climate 24, 16; 10.1175/2011JCLI3917.1

At −1 month, there is almost uniform positive sensitivity, mainly limited to the eastern equatorial Pacific (Fig. 5a). There is also a tonguelike shape on the equator extending westward from 150°W. This local positive sensitivity can be understood by the Ekman pumping mechanism. It takes only ~1–2 months for local wind stress to affect the Niño-3 surface temperature. Moreover the Niño-3 surface temperature in December 2006 is most sensitive only to local wind forcing at ~1–2-months lead. At −3 months (Fig. 4b), the maximum of positive sensitivity shifts westward and there is broad positive sensitivity across the tropical Pacific basin. Snapshots of the sensitivity field available at higher temporal resolution clearly show the westward propagation of sensitivity (not shown), with propagation speed in agreement with the phase speed of the first baroclinic mode Kelvin wave. The physical explanation behind this positive sensitivity to remote wind forcing can be understood in this way: a remote westerly wind stress perturbation excites downwelling Kelvin waves, which take ~2–3 months to propagate eastward and finally deepen the thermocline in the Niño-3 region. The deepening of the thermocline is usually associated with a warm SST perturbation, due to the so-called thermocline feedback mechanism (e.g., Jin and An 1999; ZM06; Zhang and McPhaden 2010, hereafter ZM10).

The local maxima of positive sensitivity on the equator gradually disappear between −3 and −5 months. At −5 months, in the far western Pacific, negative sensitivity begins to appear on the equator with positive sensitivity off the equator (Fig. 5c). At −7 months, negative sensitivity extends to the eastern equatorial Pacific, and positive sensitivity exists symmetrically off the equator (Fig. 5d). Such a feature is consistent with equatorial Rossby wave dynamics: an easterly wind stress perturbation on the equator with a westerly perturbation off the equator efficiently excites downwelling Rossby waves. Equatorial Rossby waves propagate westward, and symmetric modes (especially the first meridional Rossby wave mode) can be reflected into Kelvin waves at the western boundary (WB). It then takes about 3 months for these WB reflection–generated Kelvin waves to reach the eastern equatorial Pacific and finally induce an increase of the Niño-3 surface temperature mainly through depressing the thermocline depth there. A funnel-like feature first appears in the far western Pacific, with its central axis aligned with the equator. There are negative sensitivity inside and positive sensitivity on the edge of the “funnel.” This funnel-like feature propagates eastward, while its westward-facing opening also gets enlarged gradually. This is consistent with equatorial Rossby wave dynamics: the farther away from the WB where the wind forcing perturbation is located, the longer it takes for wind-excited Rossby waves to propagate westward to reach the WB. Similarly, Rossby waves of higher meridional modes have slower phase speeds and need longer times to propagate westward and reach the WB, which leads to the gradual enlarging of funnel opening, which is most obvious in the western Pacific. Beyond 1-yr lead time (i.e., between −24 and −13 months), the adjoint sensitivity steadily weakens (not shown), indicating that the Niño-3 surface temperature at the peak of this El Niño event is most sensitive to the wind stress during the year just before the peak (refer to Fig. 6).

Fig. 6.
Fig. 6.

Adjoint sensitivity of the peak Niño-3 surface temperature in December to monthly local zonal wind stress in the region (5°N–5°S, 90°–150°W) during the 2-yr period before the peak. Simulations for six ENSO events are shown: one strong El Niño event, three intermediate El Niño events, and two La Niña events. A climatological run is shown for reference. The derived adjoint sensitivity from regression coefficients [Eq. (13)] is also plotted for a 1-yr period.

Citation: Journal of Climate 24, 16; 10.1175/2011JCLI3917.1

So in general, the evolution of the adjoint sensitivity field can be understood quite well by considering linear equatorial wave dynamics in a “reverse” manner. Because the adjoint model runs backward in time, we can imagine the “adjoint” Kelvin waves propagate westward and adjointRossby waves propagate eastward. The whole process can be understood in this way: adjoint Kelvin waves propagate westward from the Niño-3 source region toward the WB first, then they are reflected into adjoint Rossby waves of various symmetrical meridional modes, which finally propagate eastward with various phase speeds.

Snapshots in Fig. 5 show the spatial and temporal evolution of adjoint sensitivity to zonal wind stress. Then we also want to know what the sensitivity of the Niño-3 surface temperature is to local (or remote) wind forcing integrally. Numerically such integrative sensitivity can be derived by spatial summation of sensitivity over grid points within the region. For example, the sensitivity to local zonal wind stress in the Niño-3 region can be derived as
e1
where i, j is the zonal and meridional grid point index, respectively; and the summation is over the Niño-3 region. This sensitivity of Niño-3 surface temperature to local zonal wind stress (Fig. 6) measures the change of monthly average Niño-3 surface temperature in the upper 30 m in December 2006, in response of 1 positive unit zonal wind stress perturbation over the Niño-3 region in preceding months during 2005–06. The time series shown in Fig. 6 indicates positive sensitivity for the first several months immediately before the peak of 2006–07 El Niño event, with maximum values above 20°C N−1 m2 at ~−1 to −2 months (see the cyan line in Fig. 6). Then the sensitivity switches to a negative value between −6 and −7 months and remains negative until −24 months.

Similarly we can also derive the sensitivity of Niño-3 surface temperature to remote wind forcing () by spatial summation similar as Eq. (1) over the western and central equatorial Pacific region (5°N–5°S, 145°E–155°W; see Fig. 7). Positive sensitivity can also be found for the first several months immediately before the peak, with maximum values just above 10°C N−1 m2 at −3 months (see the cyan line in Fig. 7). The switches from positive to negative value between −5 and −6 and remains negative until −24 months.

Fig. 7.
Fig. 7.

As in Fig. 6, but for sensitivity of the peak Niño-3 surface temperature to remote zonal wind stress in the region 5°N–5°S, 145°E–155°W.

Citation: Journal of Climate 24, 16; 10.1175/2011JCLI3917.1

b. Event-to-event differences

It is well known that each ENSO event is unique and so there are significant event-to-event differences (e.g., McPhaden 2004). To examine the event-to-event similarity and difference in adjoint sensitivity, we also apply this adjoint sensitivity experiment to several recent ENSO events after 1994 because high-quality global ECCO and OCCA products providing open-boundary conditions for our regional model are only available after 1993 (see section 2). El Niño events from 1994–95, 1997–98, and 2002–03, as well as 1999–2000 and 2007–08 La Niña events are tested here. For reference, a climatological run is also designed with surface atmospheric forcings and open boundary conditions repeating their monthly climatological cycles for 2 yr. This climatological run can indicate the adjoint sensitivity under a “neutral” background (i.e., no ENSO events, only seasonal cycles remained).

The temporal evolution of adjoint sensitivity during other ENSO events is qualitatively similar to that during the 2005–06 period as discussed in section 3a (figures are not shown). There are some event-to-event differences as expected from the differences of forward model states during different periods (see section 2 for technical explanation). In this section, we will focus on the differences of adjoint sensitivity to local and remote wind forcing during different events.

As explained in section 3a, the positive sensitivity of Niño-3 surface temperature to local zonal wind stress during several months immediately before the event peak, or equivalently the efficiency of local zonal wind stress affecting the Niño-3 surface temperature, is mainly associated with the Ekman pumping mechanism. There are some event-to-event differences (Fig. 6): stronger positive sensitivity (above 40°C N−1 m2) can be found 1 month before the peak of two La Niña events (i.e., 1999–2000 and 2007–08 events), while weaker positive sensitivity [less than 10°C (N m−2)−1] is found before the peak of the strong 1997–98 El Niño event. Intermediate positive sensitivity [~15°–20°C (N m−2)−1] exists before the peak of intermediate El Niño events (i.e., 1994–95, 2002–03, and 2006–07 events). Quantitatively the maximum positive sensitivity during La Niña events can be 4 (2) times of that during strong (intermediate) El Niño events. These event-to-event differences can be explained by invoking the mixed layer heat balance. Historical studies have found that the dominant driving term for ENSO-related SST variations in the eastern equatorial Pacific is vertical heat advection (e.g., Battisti, 1988):
e2
where w is the vertical velocity and is the subsurface vertical temperature gradient. Here w is roughly proportional to zonal wind stress by considering the divergence of meridional Ekman transport:
e3
where γ is the linear proportionality coefficient (ZM06; Jin et al. 2006). Substituting (3) into (2), we can get
e4
The essence of a sensitivity test is to check the change of cost of function in response to a small perturbation of a control variable ( here). Based on Eq. (4), the perturbation () can be associated with perturbation ():
e5
For all six ENSO events, we are applying the same by definition, thus the SST perturbation, through vertical heat advection perturbation, is proportional to vertical temperature gradient . To verify this relationship, we first check the subsurface temperature gradient in December for all ENSO events:
e6
Here () is the averaged 0–30- (50–60-) m temperature in the Niño-3 region. This calculation of is designed to capture the vertical temperature gradient just below the mixed layer. Consistent with historical studies (e.g., Jin et al. 2003), during La Niña (strong El Niño events), anomalously strong (weak) vertical temperature gradients can be found in the subsurface. In Fig. 8, the scatterplot between in December and maximum positive shows a high positive correlation (correlation coefficient 0.99), that is, strong (weak) sensitivity can be found associated with strong (weak) vertical temperature gradient during La Niña (strong El Niño events). Interestingly, both and during the climatological run are closer to those during La Niña events. This reflects the asymmetry in the vertical distribution of thermocline depth. Because the mean thermocline depth is very shallow in the region, the thermocline lifting during La Niña events is usually constrained (the thermocline can only rise as far as the sea surface), while thermocline deepening during El Niño events does not have a similar constraint.
Fig. 8.
Fig. 8.

Scatterplot between the subsurface vertical temperature gradient in the Niño-3 region (x axis) and maximum positive adjoint sensitivity of Niño-3 surface temperature to local zonal wind stress (y axis). The solid line shows the regression curve. The subsurface vertical temperature gradient is defined by Eq. (6).

Citation: Journal of Climate 24, 16; 10.1175/2011JCLI3917.1

Event-to-event differences of can also be found in Fig. 7. The strong 1997–98 El Niño event stands apart, with a maximum positive sensitivity about half of that during other events. Though the magnitude of maximum positive sensitivity in La Niña events is similar to that of intermediate El Niño events, the sensitivity switches from positive to negative ~1–3 months sooner. Since the remote wind forcing affects the Niño-3 surface temperature in an indirect way by exciting equatorial waves and modulating the thermocline depth in the Niño-3 region, no simple relationship has been found analogous to that for . The weaker maximum positive sensitivity during the 1997–98 El Niño event can be qualitatively understood as follows: several months before the peak of this event, a perturbation of the remote wind forcing excites equatorial Kelvin waves, which move the thermocline in the Niño-3 region. The effect of a thermocline depth perturbation on the change of subsurface vertical temperature gradient [Eq. (6)] also depends on the mean thermocline depth. The same magnitude thermocline depth perturbation can induce weaker vertical temperature gradient change for a deeper mean thermocline depth, as happened during the 1997/98 El Niño event, meaning weaker adjoint sensitivity.

Both and are calculated over two regions with the same area (60° × 10°), so they can be directly compared. Maximum positive is always larger than maximum positive for each individual event. However, the maximum is about 4 times maximum during La Niña events (cf. Figs. 6 and 7), in contrast, the former is only slighter larger than the latter during the strong 1997–98 Niño event. Note a value greater than only implies higher sensitivity of Niño-3 surface temperature to local wind forcing than to remote wind forcing. To compare the real impacts of remote and local wind forcing on Niño-3 surface temperature, one must consider both sensitivity to wind forcing and the variability of wind forcing [refer to Eq. (7)]. In fact, the stronger variability of remote wind forcing wins, and it is the remote wind forcing that has the dominant effect on Niño-3 surface temperature variability associated with ENSO events (refer to Fig. 2 of ZM06).

4. Comparison with previous regression-based sensitivity study

ZM08 derived sensitivity of the Niño-3 SST to local zonal wind stress by a different experiment with a different OGCM (model details can be found in their section 2). They ran a perturbed model run with local wind stress anomalies completely removed for the period 1979–2002. The SST impact due to local zonal wind forcing can be quantified by comparing this perturbed run with the control (unperturbed) run. By regression between local zonal wind stress anomalies and the SST difference derived from this “twin” experiments in the Niño-3 region, they derived the sensitivity of Niño-3 SST to local zonal wind stress on interannual time scales, which is ~80°–90°C N−1 m2, ~2–3 times as large as the maximum adjoint sensitivity derived in the current study (Fig. 6). In the following, we reconcile ZM08’s regression analysis and current adjoint sensitivity study.

The adjoint sensitivity can also be treated as an “impulse response function” or “Green’s function,” which describes the relationship between inputs (control variables) and output (cost function). Therefore we can estimate the SST response in the Niño-3 region associated with local wind change () by multiplying the local adjoint sensitivity with the actual change in wind stress in the past months (here we only consider the past 12 months, refer to Fig. 6). Mathematically, it can be expressed as
e7
where t is the index for monthly time series t = 1, 2, … , N; Δt is 1 month; λ is the time lag index; and λ = 0, −1, … , −11. For simplicity, here we assume an “average” profile of , which does not change with time (this can be achieved by compositing from many events as shown in Fig. 6). Obviously, Eq. (7) is just a discrete form of a linear response model:
e8
where Y (X) is output (input) and H is the linear response function. This response function is the Fourier transform of the transfer function, as applied by MacMynowski and Tziperman (2010). The frequency-dependent transfer function was derived from the Fourier transforms of a pair of variables and linear regression in frequency domain. They used the transfer function to quantify the input and output relationship for key ENSO dynamics simulated by models in comparison with observations.

The similarity of Eqs. (7) and (8) clearly indicates the close connection between adjoint sensitivity and linear response function.

For ease of expression, we represent (7) in matrix terms:
e9
Here T(Nx1) is the column vector representing the monthly time series of . Here the period of January 1979 to August 2002 is considered following ZM08, thus N = 284. The matrix (Nx12) is the wind stress matrix with its columns representing time series of at various time lags, and A(12x1) is the column vector representing the average local sensitivity at 0, −1,  , −11 months lag.
ZM08 derived a regression coefficient between and at each time lag first, then identified the peak value among them. We refer to such regression as single-lag regression (SLR). Mathematically it can be achieved by multiplying Eq. (9) with the transpose of the column vector λ which represents at time lag λ:
e10
Normalizing Eq. (10) by , a scalar equal to N times the variance of gives
e11
where rwλT is the regression coefficient between and at time lag λ, and ρwλw is the autocorrelation vector of . Equation (11) reveals that the regression coefficient derived from SLR (lhs) is equal to the weighted sum of sensitivity coefficients at various time lags (rhs). With ZM08’s monthly and , the SLR analysis indicates a peak regression coefficient 68°C N−1 m2 at −1-month lag (note the number can increase to 93°C N−1 m2 at −2-months lag if low-pass filtering is applied before regression). We also construct matrix ρwλw (with λ = −1) using ZM08’s monthly data. Then we calculate the regression coefficient according to the rhs of Eq. (11) with sensitivity vector A derived from each individual case and find peak magnitudes of about 60°C N−1 m2 during the 2 La Niña events and the climatological run. This peak value is very close to the maximum regression coefficient (68°C N−1 m2) by ZM08.
On the other hand, Eqs. (7) or (9) can also be thought as a multiple linear regression problem (refer to Emery and Thomson 2001), with multiple independent variables at various time lags and dependent variable . We will refer to such regression as multiple-lag regression (MLR). Multiply Eq. (9) with (i.e., the transpose matrix of ), then we can get
e12
where (12x12) calculates the lagged covariance of , while (12x1) calculates the lagged covariance between and . MLR inverts in Eq. (12) to estimate sensitivity A:
e13
Thus, numerically A can be treated as a vector of either sensitivity (impulse response function) or regression coefficient derived from MLR. The estimated A calculated from Eq. (13) with and T constructed using ZM08’s monthly and data falls well within the range of sensitivities calculated by the MITgcm adjoint model and is very close to the sensitivity of the climatological run (see red dashed line in Fig. 6). Calculation of A based on Eq. (13) indicates the peak regression coefficient 32°C N−1 m2 exists at −1-month lag (i.e., leads 1 month; refer to Fig. 6).

A closer examination of Eqs. (13) and (11) reveals that the difference between MLR and SLR can be explained by the covariance matrix . MLR and SLR will produce the same regression coefficient at each time lag, if the covariance matrix is a diagonal matrix (all elements are zeroes except for diagonal ones) which requires the monthly time series is to be totally uncorrelated in time. Nonetheless, because at different lags are not totally independent of each other (i.e., the decorrelation time scale of is usually longer than one month), is rarely a diagonal matrix. There are positive elements immediately off diagonal, which implies that the MLR-based coefficient is generally less than the SLR-based one.

So the above derivation and argument describe the relationship between adjoint sensitivity, the response function, and the regression coefficient. The peak sensitivity coefficient between local zonal wind stress and its induced Niño-3 surface temperature change derived from our current adjoint model agrees well with ZM08’s study by considering the difference of methodology.

Despite the similarity between adjoint sensitivity and regression, there are some fundamental differences: 1) regression as a statistical tool does not guarantee any underlying physical mechanisms. In contrast, adjoint sensitivity cannot only indicate the relationship but also reveal the underlying mechanisms since full ocean dynamics is kept in the process of derivation of adjoint sensitivity. 2) Regression usually suffers from statistical noise, correlation of the driving inputs, and cannot examine variability in the relationship over the period of interest. In other words, only an average relationship between local wind forcing and Niño-3 surface temperature can be derived over a period covering several ENSO events, but event-to-event differences of the relationship cannot be estimated since it is statistically meaningless to derive such a relationship over a single ENSO event. Such event-to-event differences can nonetheless be successfully detected with the adjoint method and have clear underlying physical mechanism, as discussed in section 3b.

5. Summary and discussion

We calculated adjoint sensitivity of the Niño-3 surface temperature to wind stress by running the MITgcm and its adjoint model. The adjoint model provides an alternative perspective to understand how the peak Niño-3 surface temperature is sensitive to monthly wind stress in the past two years before the peak. Since the ocean dynamics and thermodynamics are contained in the adjoint model, most sensitivity features can be explained by invoking thermodynamics and ocean dynamics (especially the equatorial wave dynamics). For the first ~1–2 months immediately before the peak of an ENSO event, the Niño-3 surface temperature is most sensitive to local wind stress, which should be associated with the Ekman pumping mechanism. The maximum positive sensitivity of Niño-3 surface temperature to local zonal wind stress, or equivalently the efficiency of local zonal wind stress affecting the Niño-3 surface temperature, is found highly correlated with the subsurface vertical temperature gradient and thus depends on the “background” ocean state. Strong (weak) positive sensitivity can be found just before the peak of a La Niña (strong El Niño) event, where the subsurface vertical temperature gradient [Eq. (6)] is anomalously strong (weak). The sensitivity of Niño-3 surface temperature to remote zonal wind stress in the central and western equatorial Pacific also shows some event-to-event differences, which can be qualitatively understood by invoking equatorial wave dynamics and considering event-to-event differences in mean background.

The relatively strong sensitivity of Niño-3 surface temperature to local wind forcing implies that the local wind forcing can in principle change the Niño-3 surface temperature significantly within a few months. In other words, a local wind perturbation is more efficient at affecting the Niño-3 surface temperature than a remote wind perturbation of the same magnitude, especially during La Niña events. Despite the dominant role played by remote wind forcing during ENSO events, local wind variation should also be closely monitored or correctly simulated for better hindcast or forecast of ENSO event evolution.

In the current study, the sensitivity of Niño-3 surface temperature to both remote and local wind forcings have been investigated using a forced OGCM and its adjoint model. Consequently, the current analysis is from an oceanic point view (i.e., the ocean passively responds to the atmospheric forcings). It is well known that extensive air–sea interactions are involved in ENSO event evolution. Thus the sensitivity of remote or local wind forcing to Niño-3 surface temperature is also an interesting question worth further exploring. If such sensitivity is to be derived with an adjoint method, a coupled atmosphere–ocean general circulation model or a forced atmosphere general circulation model with its adjoint model would be necessary, which is not available to us yet. Nonetheless, Lloyd et al. (2009) recently examined such sensitivity and found underestimation in most climate models from phase 3 of the Coupled Model Intercomparison Project (CMIP3).

Vertical heat advection and subsurface vertical temperature gradient are discussed in section 3b to explain the event-to-event differences of maximum positive sensitivity of Niño-3 surface temperature to local wind forcing. Vertical heat advection as expressed in Eq. (2) is often decomposed to various components to clarify impacts from each component in previous studies (e.g., Jin et al. 2003; An and Jin 2004; Kim et al. 2007; ZM10). Mathematically the decomposition can be expressed as
e14
where the overbar denotes mean climatology and the prime denotes anomaly (i.e., deviation from climatology). Jin et al. (2003) and An and Jin (2004) found that the nonlinear term can contribute to surface warming during both El Niño and La Niña events, thus inducing asymmetry of ENSO cycles (maximum positive SST anomalies during warm events are larger than maximum negative values during cold events). Duan and Mu (2006) found that decadal change of ENSO asymmetry is induced by changes in nonlinear vertical advection associated with decadal change of the tropical background sate. Kim et al. (2007) and ZM10 found that nonlinear vertical advection acts as a thermal damping term. The above differences of nonlinear vertical advection impact can be partially due to differences in data sources, data processing, and heat balance formulas (see detailed discussion in section 6c of ZM10). Nonetheless, it was generally agreed that the nonlinear term is not dominant, and instead is the largest term and is often called the “thermocline feedback” (e.g., Jin and An 1999; Jin et al. 2006). Moreover, the nonlinear component is often found to be mainly evident only during strong ENSO events such as the 1997/98 El Niño events (e.g., Jin et al. 2003; An and Jin 2004; ZM06; ZM10). During strong ENSO events, both upwelling velocity and vertical temperature gradient have relatively large anomalies and there may be some correlation between two of them (e.g., Jin et al. 2003; An and Jin 2004; Duan and Mu 2009b).

As discussed in section 2, the adjoint model calculates sensitivity around a selected “background” trajectory. As a result, the anomalous background during individual events (e.g., anomalously deep thermocline and anomalously weak subsurface vertical temperature gradient during 1997–98 El Niño event) has already been “automatically” considered in the solution of the adjoint model. So the traditional decomposition as expressed by Eq. (14) does not apply for analyzing adjoint sensitivity. Mathematically it means the in Eq. (5) is predefined as the background state and held constant when the sensitivity (partial derivative) is derived (refer to the mathematical definition of partial derivative). Physically it can be thought in this way, given a background model state, we are seeking to quantify the Niño-3 surface temperature perturbation in response to sufficiently small local wind stress perturbations as in the perturbation experiment. Obviously, the stronger the subsurface temperature gradient during one event, the larger the response in Niño-3 surface temperature and thus the higher the sensitivity. Furthermore, the maximum positive adjoint sensitivity usually takes place just 1 month before the peak of ENSO events, and the linearity should hold well for such a short period. It may explain why such high correlation can be found between the adjoint sensitivity and subsurface temperature gradient as shown in Fig. 8.

So nonlinearity associated with the nonlinear vertical heat advection term from traditional decomposition techniques should not be a problem for our current application of adjoint sensitivity analysis. Furthermore, Battisti and Hirst (1989), with a coupled intermediate ocean–atmosphere model, found that the nonlinearity acted to constrain the amplitude of ENSO oscillations, but the essential dynamics was linear. Undoubtedly, nonlinearity in general should be cautiously considered in the application of an adjoint model, consequently testing and validation (as shown in Fig. 4) are necessary before any useful conclusions can be confidently drawn (e.g., Errico 1997; Hoteit et al. 2005; Hill et al. 2004; Bugnion and Hill 2006).

Acknowledgments

The authors would like thank Mike McPhaden for helpful discussions and the MITgcm developers and the ECCO project for providing source codes. We also thank Ibrahim Hoteit, Matthew Mazloff, Ganesh Gopalakrishnan, and Caroline Papadopoulos for helping with implementing adjoint sensitivity numerical experiments. The Argo data used here were collected and are made freely available by the International Argo Program and by the national programs that contribute to it. Critical reviews from two anonymous reviewers and the editor greatly improved the manuscript. B. Cornuelle, D. Roemmich, and X. Zhang were supported through NOAA Grant NA17RJ1231 (SIO Joint Institute for Marine Observations). The statements, findings, conclusions, and recommendations herein are those of the authors and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration or the Department of Commerce.

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