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  • View in gallery

    (a) Standard deviation (STD) distribution of the 8–14-month bandpass filtered MADT derived from AVISO. The green box is the focus region (0°–20°S, 50°–115°E) and the white dotted line is the study section (10°S). (b) Model domain for the LOM. The red square and circles represent the SCSIO and Research Moored Array for African-Asian-Australian Monsoon Analysis and Prediction (RAMA; McPhaden et al. 2009) mooring locations, respectively.

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    (a)–(c) Monthly time series (1993–2011) of the sea level obtained from AVISO and LOM–MR at (10°S, 60°E), (10°S, 75°E), and (10°S, 90°E), respectively. (d) Longitude–latitude diagrams of the correlation coefficients for the sea level between AVISO and LOM–MR. The contour lines represent the STD distribution of filtered sea level of the AVISO as in Fig. 1a. The selected locations for time series in (a)–(c) are highlighted as squares in (d).

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    Longitude–latitude diagrams of the (a),(d) amplitude, (b),(e) phase, and (c),(f) explained variance of the annual harmonic of sea level from 1993 to 2011 derived from (top) AVISO and (bottom) LOM–MR.

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    Normalized vertical structure of the (a) first, (b) second, and (c) third baroclinic modes as derived from the Brunt–Väisälä frequency profiles along 10°S. Longitude–latitude diagrams of nondimensionalized projection coefficients Pn(x, y) of the wind stress for the (d) first, (e) second, and (f) third baroclinic modes calculated from Pn(x,y)=[(HD20/hmix)hmix0ψn(z)dz]/[Hbot0ψn2(z)dz] with realistic mixed layer depth, bottom depth, and mean thermocline depth (mean D20 = 125 m). (g),(h) The standard deviation of the annual and semiannual cycles of wind-induced Ekman pumping velocity in 10−4 cm s−1.

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    Baroclinic mode energy distribution: variability (STD) maps of the first baroclinic mode contribution to sea level for the (a) annual and (b) semiannual frequencies derived from LOM–MR. (c) Variability (STD) along 10°S for the first baroclinic mode contribution. (d)–(f) As in (a)–(c), but for the second baroclinic mode. (g)–(i) As in (a)–(c), but for the third baroclinic mode. (j)–(l) As in (a)–(c), but for the summed superposition of 4–10 baroclinic modes. Units are centimeters.

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    The annual harmonic of the first baroclinic mode contributions to sea level (a) amplitude (cm) and (b) phase derived from LOM–MR. (c) Map of the climatological annual cycle of the first baroclinic mode contributions to sea level along 10°S. The black dotted line indicates the distance of the first baroclinic mode off-equatorial Rossby waves propagating along 10°S with theoretical phase speed of C1x=(ω/k)=(βc12/f02), where c1 = 2.64 m s−1. (d)–(f) As in (a)–(c), but for the second baroclinic mode contributions to sea level, with theoretical phase speed of C2x=(ω/k)=(βc22/f02), where c2 = 1.67 m s−1.

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    The annual harmonic of the (a),(e) first, (b),(f) second, (c),(g) summed superposition of low order (first and second), (d),(h) summed superposition of 1–25 baroclinic modes contributions to (top) dynamic pressure amplitude (hPa) and (bottom) phase derived from LOM–MR along 10°S. The black dotted lines in (c), (d), (g), and (h) show the WKB ray path for the first and second baroclinic mode characteristic speed at the annual period (ω = 2π yr−1).

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    Maps of amplitude (color) and phase (blue lines) of the annual harmonic of superposition of low-order (first and second) baroclinic mode contributions to sea level derived from (a) LOM–DAMP and (b) LOM–REFLECT. (c) STD variability along 10°S for superposition of low-order (first and second) baroclinic mode contribution. (d)–(f) As in (a)–(c), but for superposition of 1–25 baroclinic modes contribution.

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    (a) Ratio of the zonally averaged (70°–100°E) STD of WE and the low-order Rossby wave phase speed cnr as a function of latitude (red line). Zonally averaged (70°–110°E) coherencies of Ekman pumping as a function of latitude (blue line). (b) The spatial structure of zonal coherencies over the STIO (color) and the STD distribution of AVISO ADT (contour). The annual period of the time series was detected based on the 8–14-month bandpass filter.

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    (a) Zonal coherency of Ekman pumping derived from CCMP wind along 6°S (70°–115°E). The vertical axis is the distance from the eastern boundary (km). (d) Longitude–time diagrams of the climatological annual cycle of Ekman pumping derived from CCMP wind along 6°S. (g) Longitude–time diagrams of the climatological annual cycle of dynamic height along 6°S obtained from LOM–DAMP (color) and AVISO (contours). In (g) the magenta solid and dotted lines represent the standard theoretical phase speed for the first and second baroclinic mode of long Rossby waves, respectively. [Here the phase speed is Cn=(ω/k)=(βcn2/f02); c1 = 2.64 m s−1 and c2 = 1.67 m s−1 represent the characteristic speeds of the first and second baroclinic mode]. (b),(e),(h) As in (a), (d), and (g) but for 10°S. (c),(f),(i) As in (a), (d), and (g) but for 16°S.

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Baroclinic Characteristics and Energetics of Annual Rossby Waves in the Southern Tropical Indian Ocean

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  • 1 State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China
  • | 2 Southern Marine Science and Engineering Guangdong Laboratory, Guangzhou, China
  • | 3 Southern Marine Science and Engineering Guangdong Laboratory, Zhuhai, China
  • | 4 Innovation Academy of South China Sea Ecology and Environmental Engineering, Chinese Academy of Sciences, Guangzhou, China
  • | 5 School of Marine Sciences, Sun Yat-sen University, Guangzhou, China
  • | 6 CSIRO Oceans and Atmosphere, Crawley, Western Australia, Australia
  • | 7 Department of Atmospheric and Oceanic Sciences, University of Colorado Boulder, Boulder, Colorado
  • | 8 Institute of Deep-Sea Science and Engineering, Chinese Academy of Sciences, Sanya, China
  • | 9 Center for Ocean Mega-Science, Chinese Academy of Sciences, Qingdao, China
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Abstract

The first baroclinic mode Rossby wave is known to be of critical importance to the annual sea level variability in the southern tropical Indian Ocean (STIO; 0°–20°S, 50°–115°E). In this study, an analysis of continuously stratified linear ocean model reveals that the second baroclinic mode also has significant contribution to the annual sea level variability (as high as 81% of the first baroclinic mode). The contributions of residual high-order modes (3 ≤ n ≤ 25) are much less. The superposition of low-order (first and second) baroclinic Rossby waves (BRWs) primarily contribute to the high energy center of sea level variability at ~10°S in the STIO and the vertical energy penetration below the seasonal thermocline. We have found that 1) the low-order BRWs, having longer zonal wavelengths and weaker damping, can couple more efficiently to the local large-scale wind forcing than the high-order modes and 2) the zonal coherency of the Ekman pumping results in the latitudinal energy maximum of low-order BRWs. Overall, this study extends the traditional analysis to suggest the characteristics of the second baroclinic mode need to be taken into account in interpreting the annual variability in the STIO.

Current affiliation: University of Hamburg, Hamburg, Germany.

Corresponding authors: Dr. Gengxin Chen, chengengxin@scsio.ac.cn; Dr. Dongxiao Wang, dxwang@mail.sysu.edu.cn

Abstract

The first baroclinic mode Rossby wave is known to be of critical importance to the annual sea level variability in the southern tropical Indian Ocean (STIO; 0°–20°S, 50°–115°E). In this study, an analysis of continuously stratified linear ocean model reveals that the second baroclinic mode also has significant contribution to the annual sea level variability (as high as 81% of the first baroclinic mode). The contributions of residual high-order modes (3 ≤ n ≤ 25) are much less. The superposition of low-order (first and second) baroclinic Rossby waves (BRWs) primarily contribute to the high energy center of sea level variability at ~10°S in the STIO and the vertical energy penetration below the seasonal thermocline. We have found that 1) the low-order BRWs, having longer zonal wavelengths and weaker damping, can couple more efficiently to the local large-scale wind forcing than the high-order modes and 2) the zonal coherency of the Ekman pumping results in the latitudinal energy maximum of low-order BRWs. Overall, this study extends the traditional analysis to suggest the characteristics of the second baroclinic mode need to be taken into account in interpreting the annual variability in the STIO.

Current affiliation: University of Hamburg, Hamburg, Germany.

Corresponding authors: Dr. Gengxin Chen, chengengxin@scsio.ac.cn; Dr. Dongxiao Wang, dxwang@mail.sysu.edu.cn

1. Introduction

The southern tropical Indian Ocean (STIO) is a dynamically and climatically important region; its interaction with the atmosphere plays an important role in influencing climate and ecosystem on both regional and global scales (Trenberth et al. 2000; Yoder and Kennelly 2003; Ma et al. 2014; Liao et al. 2020). In particular, the dynamic response to the atmospheric forcing in the STIO is partly facilitated by pronounced baroclinic Rossby waves (BRWs), which have been noted in sea level, thermocline depth, subsurface temperature, density, velocity, and bottom pressure signals from observations and model simulations (Woodberry et al. 1989; Périgaud and Delecluse 1992; Fu and Smith 1996; Masumoto and Meyers 1998; Potemra 2001; Wang et al. 2001; Wijffels and Meyers 2004; Feng et al. 2010; Johnson 2011; Piecuch and Ponte 2014). The BRWs have large annual variability, with a maximum at ~10°S that stretches across much of the STIO, strongest near 90°E (Fig. 1a) (Périgaud and Delecluse 1992; Masumoto and Meyers 1998; Birol and Morrow 2001). The phase lines for the annual variability in sea level are oriented from west-northwest to east-southeast, as expected for westward propagating off-equatorial planetary waves.

Fig. 1.
Fig. 1.

(a) Standard deviation (STD) distribution of the 8–14-month bandpass filtered MADT derived from AVISO. The green box is the focus region (0°–20°S, 50°–115°E) and the white dotted line is the study section (10°S). (b) Model domain for the LOM. The red square and circles represent the SCSIO and Research Moored Array for African-Asian-Australian Monsoon Analysis and Prediction (RAMA; McPhaden et al. 2009) mooring locations, respectively.

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0294.1

Model simulations showed that the BRWs in the STIO are primarily driven by the seasonal wind, with dominant contribution from the first-mode BRW (Woodberry et al. 1989; Masumoto and Meyers 1998; Wang et al. 2001). More recently, Johnson (2011) and Nagura (2018) observed deep annual signals in density and velocity with upward phase propagation, and further proposed the existence of vertical energetic penetration from the surface to the deep ocean (1600–1900 dbar). While previous studies emphasized on the dominance of the first-mode BRW based on a 1.5-layer reduced-gravity model (Masumoto and Meyers 1998; Wang et al. 2001), the observed vertical structure of the annual BRWs in the STIO cannot be explained by a single baroclinic mode with only one dissipation term (Rayleigh type) accounting for energy loss from the surface to the deep ocean (Gill and Niiler 1973; Gill 1982; McCreary 1984; Moore and McCreary 1990; Shankar et al. 1996; Han et al. 2004). The roles of annual BRWs associated with other baroclinic modes should be taken into account (e.g., Cane 1984; Doi et al. 2010), in order to decipher the observed vertical energy propagation (Dewitte et al. 2008). Yet, the effects of higher-order modes on the observed annual BRWs in STIO have not been examined.

As noted by Masumoto and Meyers (1998) and Wang et al. (2001), one of the interesting features of the annual sea level variability in the STIO is that there is a distinct meridional maximum centered at ~10°S. However, the causes for this meridional maximum were not provided. It has been suggested that the annual BRWs of the STIO generated by the winds and the eastern boundary reflection seem to accumulate the energy near ~10°S (Masumoto and Meyers 1998; Trenary and Han 2012). An immediate question is what physical processes cause the meridional maximum, and why the influence of the wind forcing varies geographically.

Here we use a continuously stratified linear ocean model (LOM) (McCreary 1981) to explore the baroclinic characteristics of annual BRWs in the STIO and explore the mechanisms that cause the maximum sea level variability by assessing the effects of local and remote wind forcing. The study is structured as follows: section 2 introduces the data and the numerical model. Section 3 presents the analysis results, including model validation, the vertical mode decomposition, the propagating features of the baroclinic modes and the forcing mechanisms. Finally, the concluding remarks and discussion are presented in section 4.

2. Data and model

a. Observations and reanalysis

The Maps of Absolute Dynamic Topography (MADT), formerly known as the Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO) products, are referenced in the Copernicus Marine Environment Monitoring Service (CMEMS; http://marine.copernicus.eu) catalogue as “Ocean gridded L3/4 sea surface heights and derived variables reprocessed” products now. The maps used here are from January 1993 to December 2017, which are derive from an optimal interpolation of combined altimeter data through the TOPEX/Poseidon and ERS-1/ERS-2 missions (Le Traon et al. 1998; Pujol et al. 2016) on a 1/4° × 1/4° grid every week. The precision of this product (in terms of root-mean-square error) is around 2–3 cm.

The Ocean Reanalysis System 4 (ORAS4) (Mogensen et al. 2012; Balmaseda et al. 2013) monthly dataset from January 1993 to December 2017 is analyzed to 1) validate the ability of LOM in simulating the upper-layer currents and 2) calculate the Brünt–Väisälä frequency through the temperature and salinity profiles in the STIO. The ORAS4 has a horizontal resolution of 1° × 1° and a vertical resolution of 42 levels with 23 levels above 500 m.

The surface wind dataset is derived under the Cross-Calibrated Multi-Platform (CCMP version 2) Ocean Surface Wind Vector Analyses project and contains a value-added 5-day mean ocean surface wind and pseudostress to approximate a satellite-only climatological data record (Atlas et al. 2009). The CCMP dataset includes the cross-calibrated satellite winds derived by Remote Sensing Systems (RSS) from a number of microwave satellite instruments (available from the Physical Oceanography Distributed Active Archive Center, Jet Propulsion Laboratory, National Aeronautics and Space Administration). Here, the dataset is time averaged over monthly period and distributed on a 0.25° cylindrical coordinate grid to derive the L3.5 product.

All monthly time series are performed a 3-month sliding average and linear detrend to remove the influence of the high-frequency signals and long-term trend.

b. Numerical model

The LOM has been previously used to investigate the dynamics of equatorial Indian Ocean currents and northern tropical Indian Ocean variability (McCreary et al. 1996; Shankar et al. 1996; Yuan and Han 2006; Chen et al. 2015; Huang et al. 2018a). This model was linearized with a realistic stratification of the STIO (Fig. S1 in the online supplemental material) and applied here to explain the dynamics of annual sea level variability. The LOM with an idealized Indian Ocean basin (Fig. 1b) is first spun up for 20 years, forced by climatological CCMP winds averaged over 1988–2011. Restarting from the spinup run, the LOM is integrated forward in time using monthly CCMP winds from 1988 to 2011. This solution is referred to as the LOM main run (LOM–MR). A second run is performed with a damper in the eastern basin to efficiently isolate the effects of eastern-boundary reflected waves. We refer to this run as LOM–DAMP. The difference between the two experiments (LOM–MR minus LOM–DAMP, defined as LOM–REFLECT) isolated the effects of Rossby waves generated at the eastern boundary of the equatorial Indian Ocean, which primarily measure remote forcing from the equatorial Indian Ocean. The first 25 baroclinic modes are categorized into two groups: the low-order modes (1 ≤ n ≤ 2) and the high-order modes (3 ≤ n ≤ 25). More details about the linear ocean model is described in (McCreary 1981; McCreary et al. 1996; Han et al. 2004, 2011; Han 2005). Since the ocean surface in the LOM is a rigid lid, the sea level d is defined by using the dynamic pressure pn as

d=1gn=025pnψn(0),

where g is the acceleration due to gravity, pn is the pressure of the nth baroclinic mode, and ψn(0) is the nth baroclinic mode eigenfunctions in the surface layer. Appendix A describes the model in more detail.

Based on the linearized momentum equations of the LOM, the multimode linear forced Rossby wave equation is (referred to Shankar et al. 1996):

pntcrnpnx+(Acn2)pn=cn2WEZnHn=cn2WEPn(x,y),

where crn=(βcn2/f2) is the phase speed of the nth mode BRW, and WE=[(τy/f)x(τx/f)y]/ρ¯ is the Ekman pumping velocity. The term cn is the characteristic speed of the nth baroclinic mode, β is the beta-plane parameter, f is the Coriolis parameter, τx and τx are zonal and meridional wind stress, and ρ¯ is the mean density. The factors Zn=Hbot0(z)ψndz, Hn=Hbot0ψn2dz, and pn(x, y) = Zn/Hn are projection parameters of the wind forcing, determining how strongly the driving wind couples to each baroclinic mode (e.g., Shankar et al. 1996; Doi et al. 2010). The term ψn is the vertical structure function, which is determined from the mean stratification. The term Hbot is the ocean floor. The only tunable parameter of the equation is A, which is defined to be proportional to the vertical viscosity coefficient (McCreary 1981). An analytic solution to this simplified momentum equation with a wind forcing of single frequency ω is illustrated in appendix B.

3. Results

a. Model validation

Figures 2a–c show the monthly time series of sea level in three locations along 10°S from January 1993 to December 2011. The LOM–MR outputs generally agree well with the AVISO, with correlation coefficients of 0.91, 0.78, and 0.89 (>99% significance), respectively. The standard deviations (STDs) of AVISO and LOM–MR are all within a proper range [7.1 and 6.4 cm at (10°S, 60°E); 9.4 and 6.9 cm at (10°S, 75°E); 10.1 and 6.6 cm at (10°S, 90°E)], confirming the validity of this linear simulation, albeit with active oceanic instabilities in this region (e.g., Feng and Wijffels 2002; Yu and Potemra 2006; Li and Han 2015). Figure 2d shows the spatial distributions of the correlation coefficients between AVISO and LOM–MR. Again, the LOM–MR results show similar properties as the AVISO in the latitudinal range of 5°–18°S, with statistically significant correlation (>99%). The comparisons of upper-layer zonal currents (0–100 m) derived from the mooring/ORAS4/LOM–MR (Fig. S2) further reveal the ability of LOM–MR in simulating the zonal current in the STIO.

Fig. 2.
Fig. 2.

(a)–(c) Monthly time series (1993–2011) of the sea level obtained from AVISO and LOM–MR at (10°S, 60°E), (10°S, 75°E), and (10°S, 90°E), respectively. (d) Longitude–latitude diagrams of the correlation coefficients for the sea level between AVISO and LOM–MR. The contour lines represent the STD distribution of filtered sea level of the AVISO as in Fig. 1a. The selected locations for time series in (a)–(c) are highlighted as squares in (d).

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0294.1

Figure 3 shows the observed and simulated sea level variability derived from annual harmonics in the STIO. The strong annual variability both occurs in the latitudinal range of 5°–15°S (Figs. 3a,d). Along 10°S, the strongest amplitudes are located at about 90°E with a peak phase near October (Figs. 3b,e), although the simulated amplitude is weaker. Possible causes for this underestimation include the thermal structure in LOM being fixed spatially, and the discrepancies in the wind forcing. The sea level obtained from AVISO and LOM–MR both propagate westward (Figs. 3b,e). The supplemental material (Fig. S3) further shows the phase speeds derived from annual harmonics of LOM–MR propagate westward as a function of latitude from 4° to 20°S, that are compatible with the observation and theoretical Rossby wave of low-order baroclinic modes. The explained variances derived from the AVISO and LOM–MR both account for more than 45% of the total variance of sea level around the patch maximum and fade to 20% outside the warm-color fillings (Figs. 3c,f). The annual harmonic of AVISO sea level, however, explains up to 60% of total variance along 90°E, illustrating more significant annual variability in these regions than that of LOM–MR.

Fig. 3.
Fig. 3.

Longitude–latitude diagrams of the (a),(d) amplitude, (b),(e) phase, and (c),(f) explained variance of the annual harmonic of sea level from 1993 to 2011 derived from (top) AVISO and (bottom) LOM–MR.

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0294.1

b. Baroclinic mode contributions

A mode decomposition is first sought to interpret the dynamic importance of different baroclinic modes in response to the momentum forcing. The theoretical momentum projection coefficient on a given baroclinic mode can be derived from the mean oceanic stratification. The projection coefficient in nondimensionalized form is Pn(x,y)=(Zn/Hn)=[(HD20/hmix)hmix0ψn(x,y,z)dz/Hbot0ψn2(x,y,z)dz] (details seen in appendix A), where hmix is the realistic mixed layer depth of ORAS4 (calculated as the depth at which the potential density becomes 0.125 kg m−3 larger than the surface density), ψn is the structure function of the nth baroclinic mode derived from the ORAS4, Hbot is the realistic ORAS4 ocean floor depth above 4000 m (assumed no motion below 4000 m) and HD20 is a dimensionalizing depth equal to the mean thermocline depth (20°C isotherm; mean D20 = 125 m) of ORAS4 in the STIO (Dewitte et al. 1999). The structure functions of the first three baroclinic modes (ψ1–3) along 10°S are presented in Figs. 4a–c. Noted that the structure functions change only marginally, however, some differences in the depth of zero crossings of the profiles when considering the undulating ocean floor, especially in the western basin. The projection coefficients (P1–3 for the first three baroclinic modes are displayed in Figs. 4d–f. The results indicate that a significant part of the wind forcing projects on the first and second baroclinic modes in the STIO (Figs. 4d–e), while a weak behavior on the third (Fig. 4f) and higher-order baroclinic modes (not shown here). Based on the Eq. (2), the role of Pn has to be interpreted with the changes of the local Ekman pumping WE, since Pn × WE is the actual forcing term for each baroclinic mode (Han 2005; Dewitte et al. 2008). Herein, Fig. 4g reveals that the large values around the patch maximum of P1 and P2 are associated with the large amplitude of annual harmonic of WE, indicating the significant dynamical oceanic response to the annual wind forcing. However, higher-order baroclinic modes are not favored in the regions between 5° and 15°S due to the smaller projection coefficients. On the other hand, the results also illustrated that an unfavorable pattern of semiannual wind forcing results in an extremely weak baroclinic response in the STIO at semiannual time scale (Fig. 4h).

Fig. 4.
Fig. 4.

Normalized vertical structure of the (a) first, (b) second, and (c) third baroclinic modes as derived from the Brunt–Väisälä frequency profiles along 10°S. Longitude–latitude diagrams of nondimensionalized projection coefficients Pn(x, y) of the wind stress for the (d) first, (e) second, and (f) third baroclinic modes calculated from Pn(x,y)=[(HD20/hmix)hmix0ψn(z)dz]/[Hbot0ψn2(z)dz] with realistic mixed layer depth, bottom depth, and mean thermocline depth (mean D20 = 125 m). (g),(h) The standard deviation of the annual and semiannual cycles of wind-induced Ekman pumping velocity in 10−4 cm s−1.

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0294.1

We further examined the simulated baroclinic modes of the LOM in response to the wind forcing and their contributions to sea level variability (Fig. 5). The LOM–MR used the vertical structure functions derived from the observed mean Brunt–Väisälä frequency profile of the STIO for a water depth of 4000 m. The results of Fig. 5 show that the wind forcing at annual time scale projects predominantly on the sea level of first and second baroclinic modes, with a peak of annual STDs in the inner basin at 85°E for the first mode and 98°E for the second mode. In contrast, the wind-induced sea levels associated with the third and higher-order baroclinic modes are extremely weak (Figs. 5g,j). On average, the STD value of second baroclinic mode along 10°S is about 81% to that of first mode. Whereas the semiannual cycle is less energetic for all the modes in the inner basin, and only displays marked variability near the coast with about one fifth strength to the annual STD. Overall, the second baroclinic mode has as significant contribution as the first baroclinic mode in forming the annual sea level variability in the STIO.

Fig. 5.
Fig. 5.

Baroclinic mode energy distribution: variability (STD) maps of the first baroclinic mode contribution to sea level for the (a) annual and (b) semiannual frequencies derived from LOM–MR. (c) Variability (STD) along 10°S for the first baroclinic mode contribution. (d)–(f) As in (a)–(c), but for the second baroclinic mode. (g)–(i) As in (a)–(c), but for the third baroclinic mode. (j)–(l) As in (a)–(c), but for the summed superposition of 4–10 baroclinic modes. Units are centimeters.

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0294.1

To quantitatively examine the propagating features of the low-order baroclinic modes and their contributions on annual variability in the STIO, we estimated the horizontal and vertical amplitudes of annual harmonic and associated phase propagation at each grid point (Figs. 6 and 7). Results of Figs. 6b and 6e reveal clear offshore propagation of the sea level with the phase lines near parallel to each other over most of the STIO. The seasonality derived from the Hovmöller diagrams along 10°S section vividly reveals the westward propagating signals from the coast with the faster phase speed of the first baroclinic mode than that of the second mode (Figs. 6c,f). On the other hand, the phase of annual variability illustrates a slower speed than that of the theoretical free first mode Rossby waves along 10°S, suggesting the localized response of the forced waves in responsible for the different phase speeds as discussed by Chelton and Schlax (1996), White (1977), Qiu et al. (1997), and Wang et al. (2001). Overall, the features of the low-order baroclinic waves match the estimation from the propagating framework of the off-equatorial Rossby waves.

Fig. 6.
Fig. 6.

The annual harmonic of the first baroclinic mode contributions to sea level (a) amplitude (cm) and (b) phase derived from LOM–MR. (c) Map of the climatological annual cycle of the first baroclinic mode contributions to sea level along 10°S. The black dotted line indicates the distance of the first baroclinic mode off-equatorial Rossby waves propagating along 10°S with theoretical phase speed of C1x=(ω/k)=(βc12/f02), where c1 = 2.64 m s−1. (d)–(f) As in (a)–(c), but for the second baroclinic mode contributions to sea level, with theoretical phase speed of C2x=(ω/k)=(βc22/f02), where c2 = 1.67 m s−1.

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0294.1

Fig. 7.
Fig. 7.

The annual harmonic of the (a),(e) first, (b),(f) second, (c),(g) summed superposition of low order (first and second), (d),(h) summed superposition of 1–25 baroclinic modes contributions to (top) dynamic pressure amplitude (hPa) and (bottom) phase derived from LOM–MR along 10°S. The black dotted lines in (c), (d), (g), and (h) show the WKB ray path for the first and second baroclinic mode characteristic speed at the annual period (ω = 2π yr−1).

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0294.1

Figure 7 displays the vertical structures of the annual harmonics for dynamic pressure of different baroclinic modes along 10°S. Results indicate that most energy associated with annual variability were projected on the first and second modes, and trapped in the near-surface layers. Otherwise, a large amount of energy could penetrate through the thermocline into the deep ocean (Figs. 7c,d,g,h). As apparent from the LOM–MR, the annual maximum signatures shifted westward with depth, with an initial signal ~92°E at the surface to ~84°E at 500 m (~8° longitude). To understand this shift, the Wentzel–Kramers–Brillouin (WKB) ray path theory is used (as in Han 2005; Dewitte et al. 2008; Ramos et al. 2008; Huang et al. 2018b). Theoretically, the slope α of the ray paths in the (x, z) plane becomes α~[2ωf02/βcnN(z)], where N(z) is the mean Brünt–Väisälä frequency and ω is the frequency. The phase of dynamic pressure should be constant along the ray path and the energy flux should be parallel to it (e.g., Nagura 2018). As can be seen in Figs. 7c, 7g, 7d, and 7h, the shape of vertical signatures matched well with this energy ray, which justified the westward shifting of the maximum annual variability in the STIO.

c. The forcing processes

1) Forced directly by winds

The striking features of sea level obtained from the summed contribution of low-order (first and second) baroclinic modes in the LOM–DAMP are the strong annual signals in the STIO, along with a pattern of double longitudinal maxima amplitudes [previously dynamically revealed by Wang et al. (2001)] and of one latitudinal maximum amplitude ~10°S (Fig. 8a), which are consistent with that of all modes in LOM–DAMP (Fig. 8d). The low-order modes of the LOM–DAMP experiment reproduces a maximum of about 8 cm near 90°E in October along 10°S, a local minimum at 75°E, and a second maximum near 60°E (Fig. 8c), showing consistency with the result of all modes (Fig. 8f). In particular, the basin-scale propagation of wind-forced annual BRWs derived from the LOM–DAMP fits with that of LOM–MR remarkably well (Fig. 3d). In contrast, the LOM–REFLECT, that represents the contributions from the eastern boundary, shows extremely weak annual signals in the inner basin and only significant values near the boundary regions (east of 95°E) (Figs. 8b,e), of which the amplitude is only 23% to that of the LOM–DAMP. These results indicate that the free annual signals emanating from the eastern boundary suffer from strong damping with rapid attenuation when propagating westward. They are also strongly modified by the open ocean Ekman pumping associated with directly wind-forced waves. Consequently, boundary dumping experiments improved the finding of Masumoto and Meyers (1998) that the directly wind-forced low-order (first and second) BRWs play the principal role in dominating the annual sea level variability in the STIO.

Fig. 8.
Fig. 8.

Maps of amplitude (color) and phase (blue lines) of the annual harmonic of superposition of low-order (first and second) baroclinic mode contributions to sea level derived from (a) LOM–DAMP and (b) LOM–REFLECT. (c) STD variability along 10°S for superposition of low-order (first and second) baroclinic mode contribution. (d)–(f) As in (a)–(c), but for superposition of 1–25 baroclinic modes contribution.

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0294.1

2) Preferential excitation of low-order BRWS

The reasons for the wind in preferential exciting the low-order mode BRWs in the STIO are of interest. Notably, the amplitudes of BRWs in response to the wind forcing are related to the wave structure, the effects of friction, and the ability of each mode to couple with the driving winds (Shankar et al. 1996). For the wave structure, its zonal integral with the amplitude of Ekman pumping ϕ(x), ϕ(x)eikxdx (Han 2005), is proportional to the amplitude of the BRWs, where k is the wavenumber of the Rossby wave. Herein, the amplitude is related to the parameter kL, where L is the zonal scale of the wind. If kL ≥ 1, the eikx oscillates within the region of the wind, producing an oscillating integral. If kL ≪ 1, the eikx = 1 and the amplitude can be the possible maximum. For lower-order modes (large cn) in the annual period, the BRWs have a lower wavenumber, as indicated in theoretical frequency–wavenumber dispersion [ω=(βkcn2/f02)]. Consequently, they can be excited by the large-scale winds more efficiently than higher-order waves (small cn and high wavenumber), and thus couple better with the winds.

For the friction and coupled efficiency, a simple idealized wind forcing is used to solve the analytical solutions of the momentum equation (Shankar et al. 1996). Assuming a forcing scale of L = 800 km in the STIO (as seen in Fig. 9b), the amplitude of Ekman pumping is supposed as ϕ(x)=θ[(1/4)L2x2] within the region of the wind, in which −(L/2) ≤ x′ ≤ (L/2) and θ is a step function {here Ekman pumping in a periodic forcing is forming as WE(x,t)=θ[(1/4)L2x2]cos(ωt)}. Due to the derived approximate solution of forced Rossby wave equations (appendix B), the value of In=xexϕ(x,y)exp[(ikn+αn)x]dx integrated from L/2 to −L/2 becomes |In|=(2/|κn|)exp[(αnL/2)][sinh2(αnL/2)cos2(knL/2)+cosh2(αnL/2)sin2(knL/2)]1/2 [referred to Shankar et al. (1996)], where |κn|=(kn2+αn2)1/2, kn = ω/crn is the wavenumber, and αn=A/crncn2 is the e-folding scale of vertical mixing. Based on the above solution, |In| will close to L when |κn| ≪ 2, and close to 1/|κn| when |κn|L > 2. For the annual period (ω = 2π yr−1), the solution for n ≥ 3 is in range of |κn|L > 2 (Table 1), indicating |In| will rapidly decrease with n. Finally, considering the vertical-mode expansions and their coupling coefficient to wind forcing, the strength of an individual mode Pn also decreases rapidly with n (Table 1). Consequently, the combination of the friction and the coupled ability of each mode means the total amplitude γn=|In|Zn/n falls off rapidly with n, just as γ3 is weaker than γ2 by almost one order of magnitude (Table 1). The physical mechanism for such decrease is that these strongly damped and vertical-stratified Rossby waves cannot couple efficiently to the large-scale wind stress curl and thus excited weakly (Doi et al. 2010). It results that the contributions of the n = 1 and n = 2 baroclinic modes dominate the wind-forced response of sea level variability in the STIO (Zhang and Clarke 2015). By contrast, the high-order modes experience strong damping effects and thus their response is local and manly restricted to the forcing region with weak amplitude.

Fig. 9.
Fig. 9.

(a) Ratio of the zonally averaged (70°–100°E) STD of WE and the low-order Rossby wave phase speed cnr as a function of latitude (red line). Zonally averaged (70°–110°E) coherencies of Ekman pumping as a function of latitude (blue line). (b) The spatial structure of zonal coherencies over the STIO (color) and the STD distribution of AVISO ADT (contour). The annual period of the time series was detected based on the 8–14-month bandpass filter.

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0294.1

Table 1.

Parameters associated with various baroclinic modes estimated along 10°S. The Coriolis parameter f is 2.53 × 10−5 s−1. The period is the annual period (annual frequency ω is 2π yr−1). The length scale for the wind stress L is 800 km.

Table 1.

3) Energy latitudinal distribution

What physical processes excite the BRWs to cause the meridional energy maximum near 10°S are of interest. Notably, the wind-forced sea level variability depends not only upon the magnitude of the integrand (WE/cm), but also upon the way the signals generated at different longitudes superimpose, which in turn may be related to the zonal coherency of the forcing (Capotondi et al. 2003). For example, if the Ekman pumping time series at each longitude are completely uncorrelated in time, the resulting wave field would also be spatially incoherent over a large distance. If, on the other hand, the Ekman pumping are coherent over a large distance, the waves generated at different points may superimpose constructively and give rise to a larger response in terms of amplitude (energy). In their analysis in the southern tropical Pacific Ocean, Capotondi et al. (2003) confirmed that the dominant factor in producing the off-equatorial meridional maximum of annual variability is the wind-induced local Ekman pumping and associated spatial coherency. The zonal decorrelation length scales for the Ekman pumping are used to represent the zonal coherency, which is calculated by considering the correlations between WE at each point x and WE at other points x′ along the same latitude in the eastward direction. In this study, the decorrelation length is computed as the minimum distance at which correlations decreased to values lower than 0.75.

Figure 9a shows the latitudinal dependences of the STD for annual variability of WE divided by the low-order baroclinic mode Rossby waves’ phase speed cr(1,2) (Table 1). The meridional WE/cr(1,2) values increase monotonically with maximum at southern boundary ~17°S (red line in Fig. 9a). These results explain the decreasing amplitude of the solution equatorward of 10°S, but could not explain the poleward decrease seen in Fig. 9b (contours). Herein, the zonal coherency of the wind forcing may be important (blue line in Fig. 9a and color in Fig. 9b). The averaged zonal coherencies varied with latitude and peaked at ~10°S, indicating that the meridional maximum in decorrelation length is closely aligned with the meridional maximum in BRWs’ amplitude. Consequently, the spatial coherency of the wind forcing, rather than the local maxima in the amplitude of the forcing, is the key aspect in modulating the energy latitudinal distribution of resulting low-order BRWs at annual time scale.

The explanation of the above result lies in the fact that the presence of annual forcing and associated long spatial scales appears to be crucial for the existence of the center of annual variability at ~10°S. This can be clarified clearer by comparing the zonal coherency of WE, dynamic height, and the wavelength of the oceanic signals at annual period for 6°, 10° and 16°S, respectively (Fig. 10). Results show that the strong zonal coherency of WE at 10°S (Figs. 10b,e) and associated energy accumulation [zonal integral in Eq. (2)] illustrates more important contributions in formulating the maximum of low-order BRW’s amplitude in the STIO than that at 6° and 16°S.

Fig. 10.
Fig. 10.

(a) Zonal coherency of Ekman pumping derived from CCMP wind along 6°S (70°–115°E). The vertical axis is the distance from the eastern boundary (km). (d) Longitude–time diagrams of the climatological annual cycle of Ekman pumping derived from CCMP wind along 6°S. (g) Longitude–time diagrams of the climatological annual cycle of dynamic height along 6°S obtained from LOM–DAMP (color) and AVISO (contours). In (g) the magenta solid and dotted lines represent the standard theoretical phase speed for the first and second baroclinic mode of long Rossby waves, respectively. [Here the phase speed is Cn=(ω/k)=(βcn2/f02); c1 = 2.64 m s−1 and c2 = 1.67 m s−1 represent the characteristic speeds of the first and second baroclinic mode]. (b),(e),(h) As in (a), (d), and (g) but for 10°S. (c),(f),(i) As in (a), (d), and (g) but for 16°S.

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0294.1

4. Conclusions and discussion

In this study, the LOM is used to systematically investigate the annual Rossby waves and associated baroclinic characteristics in response to the wind forcing in the STIO. Although the model does not resolve nonlinearity and eddy activity of the ocean system, it simulates much of the observed mean state and variability of sea level and velocity fields (Figs. 2 and 3 and Fig. S2).

In the frame of the continuously stratified linear theory considered in this paper, our results are consistent with Masumoto and Meyers (1998) that suggested the existence of two local maxima in annual amplitude with the stronger one in the eastern basin (~10°S, 90°E) and the weaker one in the western basin (11°S, 65°E). Masumoto and Meyers (1998), using a reduced gravity model with best fitted values of the phase speed, improved the results of Woodberry et al. (1989) and revealed the importance of Ekman pumping in generating the broad spatial structure of annual westward propagating BRWs. Wang et al. (2001), using a 1.5-layer reduced-gravity model, advanced Masumoto and Meyers’s (1998) finding and further suggested that the two local maxima and midocean minimum simply result from the constructive interference between the localized Ekman pumping response and the continuously forced BRWs. Their frame of reduced gravity model satisfied the hypothesis that wind forcing drives primarily the first baroclinic mode and associated intrinsic dynamics. Our results suggested that the second baroclinic mode also has significant contribution to the annual sea level variability in the STIO (Figs. 4 and 5).

The vertical mode decomposition in the LOM separates the contributions of distinct vertical modes to the annual sea level variability, as evidenced by their amplitude and propagating characteristics. The results reveal that the wind forcing is projected more efficiently to the low-order (first and second) baroclinic modes in the STIO (Fig. 4). The first and second baroclinic modes are the most energetic for annual sea level variability in the STIO (Figs. 5 and 6). In contrast, the higher-order baroclinic modes are weak and mostly forced near the coast with the “local” response. Energy associated with the low-order BRWs propagate downward via the corresponding off-equatorial WKB ray path, which explained the westward shifting and upward phase propagation of the annual variability (Fig. 7).

The forcing processes in causing the different baroclinic responses are investigated. The directly wind-forced low-order BRWs, rather than the boundary-reflected forcing, plays the dominant role in determining the basin-scale structures of the annual sea level variability in the STIO (Fig. 8). The responses of the high-order baroclinic modes are much less because the wavelengths of high-order BRWs are too short to couple efficiently to the large-scale wind forcing. The spatial coherency appears to play a dominant role in producing the latitudinal energy maximum of BRWs. Along each latitude, these BRWs generated at any given longitude is mainly the superposition of the westward propagating signals generated by the forcing to the east of that longitude (Figs. 9 and 10). By comparing to the results of Cane (1984), Doi et al. (2010), and Huang et al. (2019), the relative importance of the first and second baroclinic oceanic response to the wind forcing in the STIO forms a unique broad dynamically spatial structure in the world oceans.

Even though the phase of simulated sea level of LOM agrees well to that of observations, the amplitudes of LOM are weaker (as in Nagura and McPhaden 2010a,b). Note that in linear ocean modeling, many physical processes are ignored, including 1) the coupling of Rossby waves to bottom topography (Killworth and Blundell 1999), 2) the momentum coupling among the different vertical modes (Inoue 1985; Herrmann and Krauss 1989), 3) the effects of vertical shear (de Szoeke and Chelton 1999), and 4) bottom roughness (Tailleux and McWilliams 2001). Our idealistic numerical model does not include these effects thus becomes difficult to interpret the above underestimation of the amplitude. In addition, the presence of zonal phase variations in Ekman pumping field and the inclusion of these phase differences can lead to “quasi-resonance/off-resonance” (White 1985), which may play an important role in exciting a vigorous Rossby wave (Capotondi et al. 2003; Johnson 2011).

Previous studies suggested that the damping effect of the vertical diffusion plays an important role in shaping the spatial structure of the forced BRWs. Wang et al. (2001) checked the damping time scale of vertical diffusion for the upper-layer annual variability and concluded that adding modest dissipation makes the modeled annual variability (especially the wind-forced Rossby waves) much closer to the observed annual variability than in the case with no dissipation in the STIO. Miyama et al. (2006) revealed that in the simulation of equatorial biweekly oscillations, the mixing of the LOM acts to account for the ocean’s selecting the frequency band both in the upper and deep layers. Despite our model’s success in reproducing the broad basic features of the STIO annual variability, there exist considerable differences in the detailed spatial structure, like relative strong amplitude in the western basin and associated broad phase structure. In our LOM, a constant A = 0.000 13 cm2 s−3 for the coefficient ν of vertical diffusion is chosen and ν has realistic values in the upper ocean, varying from a maximum of 5.3 cm2 s−1 in the upper 200 m to a minimum of 0.44 cm2 s−1 in the thermocline (where N2 is a maximum). However, it increases at greater depths to unrealistically large values, as a value of 59.2 cm2 s−1 at 2000 m. Sensitivity simulations are necessary to examine the uncertainty of the damping time scale in simulating the annual variability in the STIO.

Sea level in the STIO displays not only vigorous annual oscillation, but also pronounced variability at interannual and decadal time scales (Masumoto and Meyers 1998; Potemra 2001; Wijffels and Meyers 2004; Han et al. 2006, 2010; Lee and McPhaden 2008; Feng et al. 2010; Trenary and Han 2012; Zhuang et al. 2013; Wang et al. 2015). These spatially nonuniform sea level variations indicate significant changes in the STIO circulations. Under the trend of global warming in recent decades, the surface wind forcing in the STIO suffers from significant decadal variability, with enhanced upward Ekman pumping at 10°–3°S and weakened downward Ekman pumping at 17°–10°S during 1990s relative to 1960s, which act to reduce the sea level and shallow the thermocline (Han et al. 2006). Yet the changes of oceanic stratification from 1960s to 1990s are unimportant in contributing the observed sea level variability with less than 10% amplitudes than those caused by anomalous winds. These phenomena raise a question on how importance of the low- and high-order baroclinic oceanic response to the decadal wind forcing in inducing the decadal features of sea level variability. This is an interesting topic for our future research.

Efficient vertical propagation of forced Rossby wave energy through the thermocline into the deep ocean appears to be an important oceanic process in the STIO (e.g., Johnson 2011; Nagura 2018). One uncertainty in our dynamical interpretation is the lack of realistic damping coefficient and rough topography in the deep ocean, which might be important in exciting the energetic higher baroclinic modes of Rossby waves. More simulations with realistic damping coefficient and rough topography are necessary to examine the roles of higher baroclinic modes in modulating the deep oceanic variability at annual and other time scales.

Acknowledgments

Appreciations go to the anonymous reviewers for their constructive comments that led to improvements of the manuscript. We wish to thank Yukun Qian (SCSIO) and Jinggen Xiao (Tianjin University) for the helpful discussions. The ORAS4 was obtained online (https://www.ecmwf.int/en/research/climate-reanalysis/ocean-reanalysis). The CCMP Ocean Surface Wind Vector was downloaded from the National Aeronautics and Space Administration (NASA), Jet Propulsion Laboratory (JPL), Physical Oceanography Distributed Active Archive Center (PODAAC) (https://podaac.jpl.nasa.gov/Cross-Calibrated_Multi-Platform_OceanSurfaceWindVectorAnalyses). The remote sea level was downloaded from the Copernicus Climate Change Service (C3S) and disseminated by the CMEMS (previously AVISO) (http://marine.copernicus.eu/). This project was supported by the National Key Research and Development Program of China 2017YFC1405100, the Innovative Research Group of National Natural Science Foundation of China 41521005, the China Ocean Mineral Resources R & D Association DY135-E2-4; the Strategic Priority Research Program of Chinese Academy of Sciences XDB42000000 and XDA20060502; the Second Tibetan Plateau Scientific Expedition and Research (STEP) program 2019QZKK0102; NSFCs 91958202, 41706027, 41822602, 41976016, 41676010, and 41676013; 2017YFB0502700, 2016YFC1401401; GML2019ZD0306, ISEE2018PY06, DY135-E2-4-02, LTOZZ2002, ZDRW-XH-2019-2; Youth Innovation Promotion Association CAS 2017397; Guangzhou Science and Technology Foundation 201804010133; and NHXX2018WL0101. Ke Huang was funded by the Opening Project of Key Laboratory of Marine Environmental Information Technology, State Oceanic Administration of China. Qiang Xie was funded by the Visiting Fellowship of State Key Laboratory of Marine Environmental Science, Xiamen University.

APPENDIX A

Projection Coefficient of the Wind Stress

The linear equations of motion including the wind forcing term are

utf0υ+1ρ¯Px=τx(z)+(νuz)z+ν22u,
υt+f0u+1ρ¯Py=τy(z)+(νυz)z+ν22υ,
Pz=ρg,
ux+υy+wz=0,
ρt+wρ¯z=0,

where u, υ, and w are the zonal, meridional, and vertical velocity anomalies, respectively; P and ρ are the pressure and density anomalies; g is the acceleration due to gravity; and f = βy is the Coriolis parameter. The parameter ν is the coefficient of vertical eddy viscosity, and ν2 is the coefficient of Laplacian mixing. Parameters τx and τy are the zonal and meridional wind stress that force the system, and the stress enters the ocean as body force with the vertical structure (z).

First, we consider the vertical mode decomposition and Eqs. (A3) and (A5) can reduce to

1ρ¯2Ptz=N2w,
N2=gρ¯ρ¯z,

where N is the Brunt–Väisälä frequency, which is calculated from the vertical profile of potential density of the ORAS4 data. Considering the shallow water approximation and vertically continuous stratification, the horizontal velocity and the pressure are then written as

(u,υ,P)=n=0[un(x,y,t),υn(x,y,t),pn(x,y,t)]ψn(z).

Using the continuity equation, we have

wz=uxυy=[un(x,y,t)x+υn(x,y,t)y]ψn(z)=n=0[wn(x,y,t)]ψn(z).

Here assume wn(x,y,t)=[un(x,y,t)/x][υn(x,y,t)/y]. Integrating Eq. (A9), we have

w=n=0[wn(x,y,t)]Sn(z),[Sn(z)z=ψn(z)],

where Sn(z) is the vertical structure functions of vertical velocity. Substitution of Eqs. (A8) and (A10) into Eq. (A6) gives

n=0[1ρ¯2pn(x,y,t)ψn(z)tz]=N2n=0[wn(x,y,t)]Sn(z),
dψn(z)/dzN2Sn(z)=ρ¯wnpn/t=1cn2.

As the first term in Eq. (A12) is a function of z alone and the second term is a function of (x, y, t) alone, for consistency both terms must be equal to a constant. Here we take the “separation constant” to be 1/cn2 as in (A12), and the eigenvalue cn is also known as the characteristic speed for the mode n. Then the vertical structure simplified from Eq. (A11) is given as

1N2dψn(z)dz=1cn2Sn(z).

Taking the z derivative, we have

ddz[1N2dψn(z)dz]+1cn2ψn(z)=0.

The ψn(z) forms a set of orthogonal functions and are normalized by ψn(0) = 1 for all n. We can solve this eigenvalue problem numerically from the vertical density profile of ORAS4.

Second, to find the equations for the wind projection coefficient, we substitute Eq. (A8) into Eqs. (A1) and (A2). Here, to permit the representation of solutions of motion equations as vertical mode expansion, ν is assumed to be inversely proportional to N2 as ν = A/N2 (A is a constant of 0.000 13 cm2 s−3) (McCreary 1981, 1984). Therefore, the vertical diffusion (mixing) term (νuz)z is parameterized as (d/dz){(A/N2)[undψn(z)/dz]}=(A/cn2)unψn with the time scale of A/cn2. Then we obtain

(t+Acn2)unψnf0υnψn+1ρ¯pnxψn=τx(z)+ν2ψn2un,
(t+Acn2)υnψn+f0unψn+1ρ¯pnyψn=τy(z)+ν2ψn2υn.

After Eqs. (A15) and (A16) are divided by ψn(z), we obtain a formula [similar to Eq. (A8)] for the wind forcing as

τnx(x,y,t)=τx(z)ψn(z)[τx(z)=τnx(x,y,t)ψn(z)].

Multiplying both sides by ψn(z) and integrating over the whole water depth, then we have

τnx(x,y,t)=τxPn(x,y)=τxZnHn=τxHbot0(z)ψn(z)dzHbot0ψn2(z)dz.

Here Pn(x,y)=(Zn/Hn)=[Hbot0(z)ψn(z)dz/Hbot0ψn2(z)dz] is called a wind projection coefficient, determining how strongly the driving wind couples to each vertical mode (e.g., Gill 1982). We assume a linear variation of wind stress over the mixed layer thickness hmix {(z)=θ[(zhmix)/hmix]}, and the only constraint on the form of the body force structure (z) equals 1/hmix. Then, Eq. (A18) simplifies to

Pn(x,y)=1hmixhmix0ψn(z)dzHbot0ψn2(z)dz.

Wind stress excites efficiently the vertical mode for which the wind projection coefficient is large. We note that the wind projection coefficients are sometimes calculated simply by

1/Hbot0ψn2(z)dz,

without the actual value of the mixed layer depth (e.g., Du Penhoat and Treguier 1985), because (1/hmix)hmix0ψn(z)dz is almost unity under the condition that ψn(0) = 1 for all n.

As in Lighthill (1969), Dewitte et al. (1999), and Doi et al. (2010), the wind projection coefficients with the unit of per meter (m−1) are dimensionalized with the mean thermocline depth H to satisfy the diagnosis as

Pn(x,y)=Hhmixhmix0ψn(x,y,z)dzHbot0ψn2(x,y,z)dz=HHn(x,y),

where Hn(x, y) is called the equivalent forcing depth with the unit of meters (m) (Gill 1982).

APPENDIX B

Wind-Forced BRWs Solution

To derive the approximate solution of momentum equations for the wind forcing, Shankar et al. (1996) analyzed the nondimensional variables and reduced them to the set

f0υn+xpn=Fn,
f0un+ypn=Gn,
(t+Acn2)pncn2+xun+yυn=0.

For convenience, a factor of 1/ρ¯ is absorbed into the definition of pn, and with this choice pn has units of velocity squared. Here Fn=τxZn/(ρ¯Hn) and Gn=τyZn/(ρ¯Hn). After eliminating un and υn from Eqs. (B1)(B3), the forced Rossby wave equation is obtained:

pntβcn2f02pnx+(Acn2)pn=cn2Znρ¯Hn[(τyf)x(τxf)y],

We sought the solutions to Eq. (B4) when the Ekman pumping is a periodic forcing of the form

WE(x,y,t)=[(τyf)x(τxf)y]ρ¯=ϕ(x,y)eiωt,

where ϕ is a complex function that describes the amplitude and phase of a forcing at frequency ω.

Following Shankar et al. (1996) and Capotondi et al. (2003), with WE given by Eq. (B5), solutions to Eq. (B4) can be represented as

pn(x,y,t)=Pn(x,y)eiωt=[f2ZnβHnxexϕ(x,y)exp(ikn+αn)xdx]eαnxei(knx+ωt)=[f2ZnβHnxexϕ(x,y)e(ikn+αn)xdx]eαnxei(knx+ωt).

The right-hand term in Eq. (B6) is found by integrating along the wave characteristics and describes the generation of forced Rossby waves with an amplitude given by the term in brackets. Their westward propagation has wave speed crn=(βcn2/f02). According to Eq. (B6), Ekman pumping deepens (shallows) the thermocline in regions where it is negative (positive), generating regions of high (low) pressure that subsequently extend westward due to Rossby wave propagation.

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