1. Introduction

Oceanic equatorial wave modes, such as eastward-propagating oceanic Kelvin waves and westward-propagating oceanic equatorial Rossby waves, have significant influences in modulating the upper-ocean thermal characteristics in tropical ocean basins. Through altering ocean advection, thermocline depth, sea surface height, and other thermal features, they modulate the horizontal and vertical heat distributions of the upper ocean, which feeds back onto important coupled air–sea phenomena in the tropics such as Madden–Julian oscillation (MJO; Madden and Julian 1971, 1972) events (Rydbeck and Jensen 2017; Webber et al. 2010; West et al. 2020) and El Niño–Southern Oscillation (ENSO; Bergman et al. 2001; Harrison and Schopf 1984; Kessler and McPhaden 1995; Lengaigne et al. 2003; McPhaden and Yu 1999; Wyrtki 1977; Zhang and Gottschalck 2002), which in turn impact weather and climate around the world through teleconnections.

Unlike the oceanic equatorial Rossby waves that have a broader meridional scale, oceanic Kelvin waves are tightly confined to the equator, with the maximum amplitude on the equator and meridional scales of 2–300 km. The observed Kelvin waves in the Pacific propagate eastward with a characteristic phase speed of about 2.2–2.8 m s⁻¹ and a period of about 70 days (Cravatte et al. 2003; Eriksen and Richman 1988; Hayes and Halpern 1984; Hendon et al. 1998; Kessler et al. 1995; McPhaden et al. 1988; Shinoda et al. 2008, 2009; Farrar 2008). Such propagation characteristics are consistent with the observed first baroclinic mode intraseasonal wave structure. Observed intraseasonal oceanic Kelvin waves (KWs) are typically forced and triggered by equatorial zonal surface wind stress anomalies (Battisti 1988; Delcroix et al. 1991; Kessler et al. 1995) associated with several atmospheric processes such as the monsoon (e.g., Seiki and Takayabu 2007), single or twin cyclones (Keen 1982; Harrison and Vecchi 1997), convectively coupled equatorial Rossby waves, and the MJO. Recent work found that a large fraction of the anomalous zonal surface stress over the western Pacific Ocean was associated with the MJO (Puy et al. 2016). Anomalous westerly winds that trail MJO convection can weaken easterly trade winds in the equatorial central Pacific and thus drive downwelling KWs that relax the thermocline slope. The forced KWs propagate eastward with anomalous eastward surface currents, deepening the thermocline, and increasing the ocean heat content into the eastern Pacific, which could further contribute to the evolution of ENSO (Kessler et al. 1995; McPhaden and Yu 1999; Zhang and Gottschalck 2002; Puy et al. 2019; Yu and Fedorov 2020). The onset of El Niño events typically follows a series of westerly wind events with forced downwelling KWs (Lybarger and Stan 2018, 2019), which can supplement existing KWs and help maintain the deepening of the thermocline depth in the eastern Pacific. The termination of El Niño events and its phase locking may also be mediated by oceanic KWs (Stuecker et al. 2013).

Therefore, a good representation of oceanic KWs in the current state-of-the-art Earth system models (ESMs) is important for realistic simulations of oceanic thermal characteristics and coupled air–sea feedbacks (e.g., the MJO and ENSO) and for the adequate forecast of global weather phenomena modulated by these phenomena such as tropical cyclone genesis (Shaman and Maloney 2012; Wang et al. 2014), atmospheric rivers (Zhou and Kim 2018), and coastal flooding (Muis et al. 2018) through tropical–extratropical teleconnections. Sea surface temperature (SST) bias in the tropical Pacific in Coupled Model Intercomparison Project (CMIP) multimodel ensembles is linked to the misrepresentation of the thermocline depth (Li and Xie 2012), which may be partially attributed to the misrepresentation of equatorial waves (Li et al. 2015). These model biases in ocean waves have been suggested to be due to the strength and spatial scale of wind stress forcing (e.g., Li and Xie 2014; Richter 2015) and ocean background state such as the ocean vertical stratification (e.g., Kim et al. 2017). For instance, Richter (2015) showed that westerly wind biases on the equator caused by the erroneous southward shift of the ITCZ in models exacerbate the biases near the eastern basin boundary by inducing erroneous oceanic KWs and advection. However, relatively little diagnosis has been done so far to assess CMIP6 models’ ability to simulate oceanic equatorial waves, especially for KWs, because the diagnosis of oceanic KWs requires daily sea surface height (SSH) or daily thermocline depth output, which is not available in previous CMIP archives. The recent availability of the daily-resolved 20°C isotherm depth (T20d; a common proxy for thermocline depth) from several CMIP6 (Eyring et al. 2016) member models effectively removes this diagnostic barrier, making it possible to evaluate the performance of oceanic KWs and identify potential sources of model bias.

Based on data availability, we diagnose the spatial distribution and propagation characteristics of oceanic KWs in a subset of CMIP6 models that provide the daily T20d output and explore the potential sources for the misrepresentation of the KWs. We focus on possible ocean sources of bias, as the evaluation of the surface wind stress biases in the Pacific will be separately discussed in Riley Dellaripa et al. (2024). The diagnosis of oceanic KWs will help identify existing biases in ocean processes in the current state-of-the-art models and advance the understanding of how KW biases may influence biases in ENSO variability and the mean atmospheric and ocean states, as well as the implications for SST pattern effects on equilibrium climate sensitivity (e.g., Dong et al. 2019). In this paper, section 2 introduces the observational and reanalysis products and CMIP6 models analyzed in our study and describes the methods and metrics used to analyze KW features. The model performance of the KW amplitude, spatial variance, and propagation speed is evaluated in section 3. In section 4, we interpret biases in model KW characteristics in terms of ocean mean states. A summary is presented in section 5.

2. Data and methods

b. CMIP6 models

KWs can be identified with daily SSH or T20d perturbations because these two variables are linked dynamically and thus highly correlated (Zebiak and Cane 1987). The daily SSH field from AVISO is used for describing the observed KWs; however, since daily SSH is not available in CMIP6 outputs, the daily T20d field is used instead to evaluate the simulated KW features.

The CMIP6 archive currently includes daily T20d in the historical simulations of seven CMIP6 member models, which are IPSL-CM6A-LR, IPSL-CM6A-LR-Interactions with Chemistry and Aerosols (INCA) (IPSL-CM6A-LR-INCA), MPI-ESM-1-2-HAM, MPI-ESM1-2-HR, MPI-ESM1-2-LR, NorESM2-LM, and NorESM2-MM. These CMIP6 models employ different atmosphere components, ocean components, or other optional components. For example, the IPSL-CM6A-LR-INCA adopts an additional atmospheric chemistry/aerosol microphysics model named INCA (Hauglustaine et al. 2014) based on the IPSL-CM6A-LR (Boucher et al. 2020). Similarly, MPI-ESM-1-2-HAM introduces the aerosol component HAM2.3 (Neubauer et al. 2019; Tegen et al. 2019) into the MPI-ESM-1-2-LR (Mauritsen et al. 2019). NorESM2 (Seland et al. 2020) is based on the Community Earth System Model 2.1 (CESM2.1; Danabasoglu et al. 2020) but with some modifications and a different ocean component, the Bergen Layered Ocean Model (BLOM; M. Bentsen et al. 2020, unpublished material). NorESM2-LM and NorESM2-MM are almost identical except for a very limited number of parameter settings (Seland et al. 2020) and the resolution of the atmosphere–land component. The details of the models used in this study are shown in Table 1.

Table 1.

List of the CMIP6 member models included in this study.

View Table

In this study, the daily T20d field from 1990 to 2014 is used for analyzing the model performance of KWs for each model, and the monthly resolved potential temperature and salinity fields for the same period are used to examine the Brunt–Väisälä frequency used to represent the climatology of ocean stratification or ocean stability. Note that only the member “r1i1p1f1” of each CMIP6-member model is analyzed in this study. Here, “r1i1p1f1” refers to the simulation in models with the configuration of the first member of the realization, initialization, physics version, and forcing, which is commonly used to analyze model performances when the spread among all ensemble members is not the focus.

3. Spatial distribution and propagation features of KWs

In this section, we compare KW amplitude and propagation characteristics in models to observations. The horizontal spatial distribution of KW variability for the observations and models is shown in Fig. 1, which is represented by the standard deviation of the 20–180-day and eastward-propagating SSH (observed) and T20d (model) anomalies. The largest KW variability in observations is centered in the equatorial central Pacific, consistent with the first EOF mode of the observed KWs in Rydbeck et al. (2019). However, the CMIP6 model results show large differences from the observations with a large spread in the location of the maximum KW amplitude. For example, while the two IPSL-CM6A models have a better simulation of the spatial distribution among the models, they still exhibit an approximately 20° eastward shift of the variance maximum. For the MPI-ESM1-2 models, the low-resolution simulations (i.e., MPI-ESM1-2-LR and MPI-ESM1-2-HAM simulations) show very different KW amplitude distributions compared to observations, exhibiting two maxima in variability in the western and eastern Pacific instead of the central Pacific. The high-resolution simulation of MPI-ESM1-2-HR also has a maximum located in the western Pacific but not the eastern Pacific. The large KW variability in the western Pacific for these models is consistent with their higher occurrence frequency of western Pacific westerly wind bursts (Riley Dellaripa et al. 2024). The comparison within the three MPI-ESM results implies that model resolution is an important factor affecting the simulated air–sea feedbacks, and the ocean variability may be unrealistically amplified by coarse-resolved dynamic processes. Similar to the IPSL-CM6A models, the NorESM2 models also show eastward-shifted amplitude centers but to a greater extent than the IPSL-CM6A models. The strong KW variability in the eastern Pacific could arise from the greater frequency of westerly wind stress events in the eastern Pacific for those models (Riley Dellaripa et al. 2024).

View Full Size

Horizontal distribution of the standard deviation of space–time-filtered (a) observed SSH anomalies (cm) and (b)–(h) T20d anomalies (m) in models. The top and bottom color bars are for (a) and (b)–(h), respectively. The black boxes show the selected base regions of the western (2°S–2°N, 170°E–180°) and eastern (2°S–2°N, 130°–140°W) Pacific for deriving the Hovmöller diagrams in Fig. 3.

Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-23-0668.1

• Download Figure
• Download figure as PowerPoint slide

It is worth noting that the zonal distributions of KW variability in Fig. 1 closely align with the zonal distributions of the ENSO amplitude measured by SST anomalies shown in Planton et al. (2021) and Fig. S1 in the online supplemental material. The ENSO amplitude is regularly diagnosed as a key metric for assessing the performance of ENSO variability. KWs are an important process for modulating the upper-ocean temperature structure and warm water volume of the equatorial thermocline, on which the ENSO amplitude depends (Meinen and McPhaden 2000). The similarity between the ENSO amplitude and KW variability is evidence of the positive feedback between the two processes and indicates that a good representation of KWs is fundamental for describing the realistic ENSO variability in models.

Figure 2 shows the evolution of KW SSH (observed) and T20d (model) anomalies divided into eight phases following Wheeler and Hendon’s (2004) method for diagnosing the MJO. The phase analysis shows a similar distribution of KW variability to the results in Fig. 1. Positive anomalies for observed KWs maximize in the central Pacific when they are in phases 3–5. However, for the IPSL-CM6A and NorESM2 families, the KW amplitude gradually increases and maximizes in the eastern Pacific two or three phases later than observed. For the MPI-ESM models, the maximum composite KW amplitude occurs in the western Pacific. These features are highly consistent with the zonal distribution of variance in Fig. 1. It is worth mentioning that the distributions of surface wind stress in MPI-ESM models, especially the HAM and LR models, have a westward shift compared to observations (Riley Dellaripa et al. 2024), which could lead to the westward-shifted phase biases in Fig. 2b and the biased variability center located in the western Pacific in Figs. 1d and 1f. Similarly, the stronger biases in wind stresses located in the eastern Pacific in the NorESM family could largely cause the too-strong KWs there as Figs. 1g, 1h and 2c show.

View Full Size

Meridionally averaged (2°S–2°N) KW composites in eight phases for the observed SSH anomalies (black in all panels; cm) and the simulated T20d anomalies (m). The results are derived from (a) the IPSL-CM6A models, (b) the MPI-ESM1-2 models, and (c) the NorESM2 models. The KW propagation speed averaged over all eight phases for each model is listed on the right.

Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-23-0668.1

• Download Figure
• Download figure as PowerPoint slide

We also use the results shown in Fig. 2 to compute the KW propagation speed. Using the lead-lag spatial pattern correlation method described in section 2c(1), the KW propagation speed is computed for each phase (details shown in Fig. S2) and then averaged over all eight phases to determine the overall propagation speed (i.e., “C-speed”), which is shown in the legend of Fig. 2. There is an obvious slow bias in models’ KW propagation speeds. The observed KWs propagate at an eastward speed of about 2.5 m s⁻¹, which is close to the typical propagation speed for the first baroclinic mode oceanic Kelvin waves. However, the model results are around 1.5 m s⁻¹, with a bias of almost 1 m s⁻¹ slower than observed.

Theory and observations indicate that the KW propagation speed is not constant along the equator; it is typically faster in the west and slower in the east (e.g., Eriksen et al. 1983), which results from the shallower thermocline depth in the east following the dispersion relation $c=\sqrt{g^{\prime }H}$, where H is the thermocline depth and g′ is the reduced gravity (Gill 1982). Additionally, sometimes, the KWs are dispersed into higher-order baroclinic modes due to internal ocean processes so that the phase speeds slow down (e.g., Busalacchi and Cane 1988; Giese and Harrison 1990; Iskandar et al. 2005). To determine whether KW propagation in models is consistently slower than observations across the entire basin, we examine the zonal variation in KW phase speeds by constructing Hovmöller diagrams in the western and eastern Pacific. In Fig. 3, the tilting patterns represent the propagating KWs, and the values are the derived phase speeds. Both the observations and model results show that KW propagation speeds are faster in the west and slower in the east. However, for both locations, the simulated KWs propagate more slowly than those in observations, which is consistent with the C-speed shown in Fig. 2. To show the zonal variation and the model biases in the KW phase speeds intuitively, the values for the Hovmöller propagation speed in each region and their averages (i.e., H-speed) within each model are shown in the scatterplot in Fig. 4. Also plotted in Fig. 4 are the C-speeds derived from the phase composites (shown in Fig. 2) for comparison.

View Full Size

(top) Hovmöller diagrams based on the western Pacific (2°S–2°N, 170°–180°E) for the (a) observed SSH anomalies and (b)–(h) T20d anomalies in the CMIP6 models (1). The solid lines show the slopes from day −10 to day 15 to calculate propagation speeds. The brown dashed lines in (b)–(h) show the slopes in the observations for reference, which is the same as the solid line in (a). The derived propagation speeds are shown in the bottom-right corner of each figure. (bottom) As in (top), but based on the eastern Pacific (2°S–2°N, 130°–140°W).

Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-23-0668.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Phase speeds (m s⁻¹) of the KWs in observations and CMIP6 models. They are derived from the phase composite in Fig. 2 (C-speed; marked by asterisks), the single region Hovmöller speed (colored circle markers) shown in Fig. 3, and the averaged Hovmöller propagation speed (H-speed; marked by triangles). The dashed line and dotted line represent the averaged phase speed among all the models for the eastern and western Pacific, respectively.

Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-23-0668.1

• Download Figure
• Download figure as PowerPoint slide

The H-speeds and C-speeds for individual models or the observations are similar for the two calculations; in H-speeds, both the models and observations show slower speeds in the eastern Pacific compared to the west by about 0.3 m s⁻¹. However, the most significant feature in Fig. 4 is that the KW propagation speed in all the CMIP6 models is 1.5 m s⁻¹ on average, which is about 1 m s⁻¹ slower than the observations (2.5 m s⁻¹). A bias of this magnitude could influence ocean dynamical processes, especially the onset and termination of ENSO events. This line of reasoning is supported by results shown in Planton et al. (2021) (and also adapted in Fig. S3), which showed that El Niño onset, which typically occurs in April and May in observations, is generally delayed by several months (i.e., to midsummer) in the CMIP6 models analyzed herein.

To further examine the slow biases in KW phase speeds, wavenumber–frequency spectra are shown in Fig. 5. The spectra are derived with a 2D Fourier transform of the time series of the latitudinal-averaged SSH/T20d in the region of the equatorial Pacific (2°S–2°N, 150°E–90°W). In Fig. 5, the zonal wavenumber refers to the wavenumber within the Pacific basin rather than the global zonal wavenumber. In observations, the SSH peak power is well aligned with the dispersion curve corresponding to the first baroclinic mode, characterized by an eastward speed of about 2.7 m s⁻¹. However, in the CMIP6 model results, the KW power concentrates near the dispersion relation of the second baroclinic mode, whose character speed is about 1.5 m s⁻¹. The observed KW variance is prominent throughout the entire 30–180-day band, while the T20d peak power in models is confined to the longer-period band of more than 50 days. This suggests that KWs tend to have a longer period together with a slower phase speed in the CMIP6 models.

View Full Size

Wavenumber–frequency power spectra for space–time-filtered (a) observed SSH anomalies and (b)–(h) T20d anomalies in the CMIP6 models. The black lines in the positive zonal wavenumber domain for each panel show the theoretical (left) first and (right) second baroclinic modes of the oceanic KWs, with a character speed of 2.7 and 1.5 m s⁻¹ corresponding to equivalent depths of 0.75 and 0.23 m, respectively.

Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-23-0668.1

• Download Figure
• Download figure as PowerPoint slide

The main conclusion from this section is that model KW propagation speeds are notably slower than those seen in observations. This finding is consistent for each of the different methods used to assess KW propagation characteristics. Although the second baroclinic mode KWs also exist in observations detected by the Tropical Atmosphere Ocean (TAO) array of moorings, the first baroclinic mode KWs still have larger variance in the equatorial Pacific (Cravatte et al. 2003) and are the dominant observed signals in this study. Thus, in the next section, we examine possible sources of the slow biases in KW propagation speeds in the models.

Note that, since the H-speeds and C-speeds are similar within each model and observation, the results in the following analysis are comparable using the two measurements. Therefore, the phase speeds below refer to the H-speeds except for further mention.

4. Possible sources of biases in KW propagation in CMIP6 models

By deriving the Brunt–Väisälä frequency N² from temperature and salinity, Fig. 6 shows the observed N² and the differences between the models and the observations. In the observed upper ocean, the local maximum in stability generally follows the thermocline depth due to large vertical temperature gradients in the thermocline. The exception is in the western Pacific (west of 170°E), where the maximum stability and thermocline depth diverge. This is likely attributed to abundant convective precipitation in the warm pool and subsequent freshwater input, resulting in higher salinity-induced stratification above the thermocline.

View Full Size

The meridional averaged (2°S–2°N) Brunt–Väisälä frequency N² (10⁻⁴ s⁻²) for (a) the observation and (b)–(h) the differences between the model results and the observation. The solid curve in each panel shows the climatology of the thermocline depth (represented by the 20°C isotherm depth) for the observation and CMIP6 models, respectively. The dashed gray curves in (b)–(h) show the observed climatology of the thermocline depth as a reference, which is the same as the solid line in (a). The top and bottom color bars are for (a) and (b)–(h), respectively.

Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-23-0668.1

• Download Figure
• Download figure as PowerPoint slide

Most models have a comparable zonal distribution of thermocline depth to observations. However, the MPI-ESM1-2-HR has a shallower thermocline depth with a bias of about 50 m across the tropics, which is also clearly indicated by the Brunt–Väisälä frequency N² profiles in either the western or the eastern Pacific (Fig. 7) and likely associated with the biased warmer upper ocean across the entire Pacific (Fig. S4). For the N², the two IPSL-CM6A and the two low-resolution MPI-ESM models are slightly less stable near the thermocline across the entire Pacific, which may partly explain the slower KW propagation speed due to the proportionality between c and N. Meanwhile, MPI-ESM1-2-HR and the two NorESM2 models show a more stable ocean background near the thermocline depth (Figs. 6 and 7), which should theoretically result in faster KW propagation for a given baroclinic mode KW, thus, slightly faster than observed KW phase speeds should be seen but which is inconsistent with our findings of slower KW propagation speeds in these models. We suspect that the slower KW propagation in the models may be attributed to the biased distribution of baroclinic modes in KW vertical structures in these three model families, which we explore in the remainder of this section.

View Full Size

The regional averaged N² profiles (s⁻²) for the (a) western Pacific (2°S–2°N, 170°E–180°) and (b) eastern Pacific (2°S–2°N, 130°–140°W) for the observation and models.

Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-23-0668.1

• Download Figure
• Download figure as PowerPoint slide

Because the daily three-dimensional ocean temperature output is not available for CMIP6 models, it is impossible to directly examine either the development of KW vertical profiles or the decomposition of the KW vertical structure onto ocean baroclinic modes. An alternative way to infer the vertical structure is by computing the ratio of N to c. Figure 8 shows the distribution between the calculated phase speed and the square root of the Brunt–Väisälä frequency N in the western and eastern Pacific, respectively. Here, the N is the average along the T20d curve for models and observations. The slope of the dashed line connecting each point to the origin (0, 0) (not covered in the figure) indicates the vertical wavenumber m for a 1-m-thick layer. The slope is less steep for observations than for models in both the western and eastern Pacific, indicating a higher vertical wavenumber value m per meter depth of ocean for models than observed. These results are also reflected in Fig. 9a, where each point represents the corresponding slope (i.e., the value of the vertical wavenumber m per meter depth) in Fig. 8.

View Full Size

The relationship between the KW speed c (m s⁻¹) and the square root of the mean value of the Brunt–Väisälä frequency N (10⁻² s⁻¹) along the T20d curve in observations and the CMIP6 models for the (a) western Pacific (5°S–5°N, 160°E–170°W) and (b) eastern Pacific (5°S–5°N, 120°–150°W). The colored dashed line represents the slope for individual result points connecting to the origin (0, 0), indicating the ratio of N to c.

Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-23-0668.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

(a) The vertical wavenumber m (10⁻² m⁻¹) per 1-m-thick layer and (b) upper-ocean vertical wavenumber (unitless) in the upper ocean above the thermocline depth in the western (5°S–5°N, 160°E–170°W), eastern (5°S–5°N, 120°–50°W), and entire Pacific (5°S–5°N, 160°E–120°W), respectively, for the observations and CMIP6 models. Different markers represent different regions. The gray dashed line in (b) shows the situation when the upper-ocean vertical wavenumber is 1, which represents the first baroclinic KW mode.

Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-23-0668.1

• Download Figure
• Download figure as PowerPoint slide

To examine KW vertical structures in the upper ocean (i.e., from the surface to the thermocline depth), the upper-ocean vertical wavenumber can be estimated by multiplying the model or observed thermocline depth averaged across the focus region with its corresponding vertical wavenumber m. We refer to this value as the “upper-ocean vertical wavenumber” to indicate the integrated vertical wavenumber in the upper ocean above the thermocline depth and plot the results in Fig. 9b. Here, the upper-ocean vertical wavenumber (m × H) is analogous to the planetary zonal wavenumber used to define the horizontal structure within a full circle around the globe at a given latitude (e.g., Wheeler and Kiladis 1999), where m is similar to the zonal wavenumber in a unit of zonal distance. When the estimated upper-ocean vertical wavenumber is 1, the waves show the first baroclinic mode feature in the upper-ocean layer with the surface and bottom perturbations of opposite signs. The upper-ocean vertical wavenumber is introduced here to describe the vertical baroclinic features of KWs in a conceptual and mathematical way, as we can quantitatively identify whether models have a realistic KW vertical structure by comparing the estimated upper-ocean vertical wavenumber in observations and models.

The observed KWs in the equatorial Pacific have an upper-ocean vertical wavenumber close to 1 in both the western and eastern Pacific (Fig. 9b), which is associated with the first baroclinic mode, consistent with previous studies (e.g., Shinoda et al. 2008). The first baroclinic mode exhibits a downward thermocline depth perturbation when the corresponding SSH anomaly is positive. However, most of the models show higher-order baroclinic vertical features in the upper ocean rather than the first baroclinic feature as observed. For example, the two NorESM2 models show a second baroclinic vertical structure in the upper ocean, which implies that the surface height and the thermocline depth perturbation are in phase (Giese and Harrison 1990) and the phase speed is slower. For NorESM2-MM, the upper-ocean vertical wavenumber in the eastern Pacific is even higher than 2, indicating that higher-order modes may appear there. IPSL-CM6A and MPI-ESM low-resolution models display an upper-ocean vertical wavenumber of about 1.5 in all the regions, indicating that both the first and second baroclinic modes may exist in the models. Together with the less stable ocean in the thermocline (Figs. 6 and 7), the biased incorporation of the second baroclinic mode would imply a slower KW propagation in the Pacific. In Fig. 9b, the upper-ocean vertical wavenumber in the eastern Pacific in MPI-ESM1-2-HR is close to 1, which may result from compensating biases. The positive bias in m (Fig. 9a) may be offset by a positive bias in the ocean stability N at the thermocline depth (Figs. 6e and 7b) to produce an upper-ocean vertical wavenumber near 1. Overall, all the models show biases in the upper-ocean vertical wavenumber with higher-order baroclinic features, which is not suggested by the observations.

The higher-order baroclinic structure features are also reflected in the meridional structure for KWs in models. By regressing the SSH/T20d anomalies onto the space–time-filtered SSH/T20d series averaged over the western, eastern, and entire Pacific, respectively, the horizontal pattern of KWs can be obtained for observations and models. The KW meridional profiles are derived by first zonally averaging over the KW region and then by normalizing by the magnitude at the equator, as shown in Fig. 10. In observation and models, the KW magnitudes decay latitudinally away from the equator with a Gaussian distribution, with the maximum on the equator. Following the least squares method in Pujiana and McPhaden (2020), we fitted the KW meridional structure to the theoretical KW solution on an equatorial beta-plane by

$${\psi }_y={\psi }_0{e}^{[(-\beta y^2)/2c]},$$(5)

where β = 2.3 × 10⁻¹¹, y is the latitude, c is the phase speed, and ψ₀ is the KW magnitude at the equator represented by SSH or T20d anomaly. Thus, the fitted meridional structure can be compared to those of the theoretical first two baroclinic modes. In Fig. 10, the meridional structure for the observed KWs is closely aligned with the theoretical first baroclinic mode with a theoretical phase speed of 2.5 m s⁻¹, while the models show meridional structures closer to the second baroclinic mode, indicating a more constrained meridional scale of $L=\sqrt{\beta /c}\approx 255\hspace{0.17em}\text{km}$ than observed. Similar results were obtained when applying the analysis to western and eastern Pacific points, respectively (not shown).

View Full Size

Meridional profiles of SSH/T20d magnitude associated with KWs in the entire Pacific (2°S–2°N, 160°E–120°W). Green dots represent the normalized KW SSH/T20d magnitude in observations and models. Yellow curves are the fitted profiles derived from the KW solution. The solid and dashed orange curves in each panel are the same, representing meridional structures for the theoretical first and second baroclinic KW modes with character speeds of 2.5 and 1.5 m s⁻¹.

Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-23-0668.1

• Download Figure
• Download figure as PowerPoint slide

Since biases in both the KW vertical structure and the background ocean stratification could result in biases in propagation speed, we next explore the relative contribution of these two key factors to KW propagation speed biases. To do so, we return to Eq. (4) and replace either the vertical wavenumber m or the ocean stability N in each model with the observed value and assess which variable produces a propagation speed closer to the observed value for the western and eastern Pacific, respectively. Figures 11a and 11c show the corrected phase speed using the observed ocean stability N to control for the effect of the biased ocean stability in the models. The improvement from the actual (gray) to the corrected (colored) propagation speeds is less than 10% on average, but the correction can exaggerate the biases in some models. Overall, however, the bias in the ocean stability makes only a small contribution to the model phase speed biases. In contrast, the corrected propagation speeds are much improved when m is corrected (Figs. 11b,d), with a 58% (72%) increase on average in the western (eastern) Pacific, which brings the model propagation speeds to within about 10% of observed. This indicates that the biased vertical KW structures in models are largely responsible for the biased KW propagation speed.

View Full Size

The corrected KW phase speeds (colored markers) for the observations and CMIP6 models by using (a),(c) the observed N and (b),(d) the observed vertical wavenumber m as a fixed parameter when applying the relation $c=N/m$ in the western and eastern Pacific. The results in (a) and (c) are calculated using individual m values from each model and a fixed N based on observations. The results in (b) and (d) are calculated using the individual N values in each model and a fixed m value based on observations. The gray cross-filled circles are the actual KW propagation speeds for the observation and models in all panels. The number at the bottom of each column shows the percentage difference between the corrected speed and the actual speed in the corresponding model.

Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-23-0668.1

• Download Figure
• Download figure as PowerPoint slide

To further verify that the ocean mean stratification profile plays little role in the phase speed biases and has little effect on the distributions of wave vertical modes in models, the vertical mode decompositions based on the Brunt–Väisälä frequency N are employed in individual models and observations. Figure 12 shows the decomposed first two baroclinic vertical structures in the upper 220 m of the ocean in the western and eastern Pacific, respectively (normalized to unity at the individual surface). Similar baroclinic mode features are shown in the western and eastern Pacific, with differences in the depth of the zero crossing. For the western Pacific, the observed first baroclinic mode shows positive anomalies in the upper 125 m and negative anomalies below. The zero crossing in the eastern Pacific is shallower by about 25 m. The eastern Pacific upward-shifted structures are also seen in the second baroclinic mode. The vertical shift of baroclinic modes is largely caused by the shallower thermocline depth in the eastern Pacific. The models display similar structures for the first two baroclinic modes as observed for both the western and eastern Pacific and similar upward-shifted structures from the western to the eastern Pacific. Thus, the characteristic phase speeds for the first two baroclinic modes should be similar between models and observations since the eigenvalues (i.e., characteristic speeds) and eigenvectors (i.e., vertical structure functions) are paired with each other. The characteristic speeds for the first baroclinic modes are shown in the top-left corner of each panel in Fig. 12. All the models show comparable characteristic speeds as observed of about 2.2 m s⁻¹ in the western Pacific and 1.84 m s⁻¹ in the east, suggesting the background density profiles, and thus, the first two upper ocean baroclinic modes in models are relatively realistic. However, those derived characteristic speeds are higher than the actual phase speeds for models shown in Fig. 4. This indicates that the slower KW propagation in models is mainly attributed to the biased incorporation of relatively realistic higher-order baroclinic modes in KW structures, rather than biases in the background density profile. It further supports the conclusion drawn from Fig. 11 that the biased mix of higher baroclinic modes in model KWs is largely responsible for the biased KW propagation speed.

View Full Size

Vertical structures (unitless) for the decomposed first and second baroclinic modes in the upper ocean (above 220 m) for the observations and models. The profiles are normalized by the corresponding unity at the surface. The values in the top-left corner of each panel show the characteristic phase speeds for the first baroclinic mode in the western and eastern Pacific, respectively, in each model.

Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-23-0668.1

• Download Figure
• Download figure as PowerPoint slide

It should be mentioned that, in this study, KWs are diagnosed using SSH for observations and T20d for models, as the daily SSH output is not available from the CMIP6 repository. SSH anomalies include both the first and second baroclinic modes, while T20d anomalies tend to reflect the first baroclinic mode more. Theoretically, the baroclinic modes for equatorial waves could be affected by both the thermocline depth (i.e., T20d used in this study) and the thickness of the thermocline. It would be more comprehensive to explore the biases in baroclinic modes by examining variations in both quantities. However, because the daily-resolved ocean temperature profile is not available in the current CMIP6 archives, it is unclear to what extent the misrepresented higher-order baroclinic mode in models arises from the biased thermocline thickness.

Higher-order baroclinic KW features also exist in observations, thus understanding why they appear in the real world may help reveal the biased processes resulting in higher-order KW modes in models. Cravatte et al. (2003) showed the second baroclinic mode KWs in the equatorial Pacific by sea-level variations from the TAO array of moorings. They originally attributed higher-order KW wave modes to low-frequency (∼120 days) wind stress in the western Pacific. However, when testing this idea with the linear Kelvin wave model developed by Kessler and McPhaden (1995) forced by low-frequency wind stress, they found that an equal mix of first and second baroclinic mode Kelvin waves were excited, rather than a predominance of second baroclinic mode Kelvin waves.

Other studies identify internal processes related to ocean stratification as possible causes for the second baroclinic KW mode. For example, Busalacchi and Cane (1988) suggested that energy may be transmitted from the more typical first KW mode to the less typical second Kelvin wave mode when the wave propagates eastward into a region of a shallower thermocline. Similar conclusions were pointed out by Iskandar et al. (2005) that second-mode Kelvin waves are more efficiently excited when the thermocline is thin and sharp, whereas the first mode often occurs in the regions with thicker thermocline. Giese and Harrison (1990) indicated that wave energy may be redistributed to accommodate changes in different stratification environments, suggesting that a misrepresented ocean stratification or mean thermocline depth in models may result in a biased KW vertical wavenumber across the ocean basin. However, because it is not clear which processes most strongly regulate the KW vertical structure in observations it is also unclear which model mean state or wind forcing biases contribute to biases in simulated KW vertical structures. Another possibility from the model configuration perspective is the insufficient vertical resolutions in models. Stewart et al. (2017) suggested that models need 50 levels to resolve the first baroclinic mode, with an additional 25 levels per higher-order mode. However, the models in this study only have 28 levels in the upper 500 m of the ocean on average, so the energies supposed to be in the first baroclinic variation may leak to higher-order baroclinic variations in those models.

Another possibility is that, although the models in this study have a relatively realistic ocean stratification compared to the largely biased upper-ocean vertical number, the subtle differences in stratification still could influence KW propagation features through the erroneous distribution of wind forcing among the baroclinic modes in models (e.g., McCreary 1981). More analysis is needed to fully understand or deny such a hypothesis.

5. Summary and discussion

Eastward-propagating oceanic intraseasonal Kelvin waves (KWs) help modulate the thermocline depth and other upper-ocean thermal characteristics, which can provide feedbacks to important coupled air–sea phenomena in the tropics such as ENSO and the mean Pacific thermocline state. A realistic representation of KWs in current ESMs is essential not only for producing realistic ocean thermal features and air–sea interactions but also for better forecasts of tropical phenomena and their associated teleconnections. The recent availability of the daily-resolved 20°C isothermal depth (T20d) from several CMIP6 models makes it possible to evaluate the performance of equatorial oceanic KWs in models and identify potential sources of model bias.

By using daily T20d fields from seven CMIP6 models (IPSL-CM6A-LR, IPSL-CM6A-LR-INCA, MPI-ESM-1-2-HAM, MPI-ESM1-2-HR, MPI-ESM1-2-LR, NorESM2-LM, and NorESM2-MM) and daily observed SSH from AVISO, the spatial distribution and propagation speed of KWs were compared between observations and models. Most of the models fail to simulate a realistic distribution of KW variability in the tropical Pacific (Figs. 1 and 2). Observed KWs show the largest variability in the central Pacific, while the models have the strongest KW variability in the eastern (IPSL-CM6A and NorESM2 models) or western (MPI-ESM models) Pacific. These distributions of the simulated KW variability closely align with these models’ distributions of the ENSO amplitude shown by Planton et al. (2021), suggesting that a realistic distribution of KW variability is needed for proper simulations of ENSO variability, consistent with a positive feedback between the ocean waves and ENSO development.

For KW propagation, theory suggests that KWs propagate slower in the eastern Pacific relative to the western and central Pacific due to the sharply reduced thermocline depth in the east. This feature is well captured by observations and all the models (Fig. 4), although simulated KWs have a notably slower average propagation (1.5 m s⁻¹) than observations (2.5 m s⁻¹) across the entire basin and in the western and eastern Pacific separately. Wavenumber–frequency spectra (Fig. 5) also suggest slower KW propagation in models relative to observations. In models, KW power more strongly projects onto the dispersion curve corresponding to the second baroclinic mode, while the observational variance is tightly aligned with the dispersion curve of the first baroclinic mode. Higher-order baroclinic modes are also indicated by KW meridional structures (Fig. 10) in models, compared to the observed first baroclinic mode KWs.

The background state, including the thermocline depth and ocean stratification, was analyzed (Figs. 6 and 7) to further understand the potential sources of the slow biases in KWs. Biases in ocean stratification and a large intermodel spread were found. The models both overestimate (NorESM2 models) and underestimate (IPSL and MPI models) the ocean stability near the thermocline depth, which could cause the KWs to propagate faster or slower than observed. The vertical wavenumber m of KWs also regulates the propagation speed through the dispersion relation. Using the ratio between the KW propagation speed c and the background ocean stability N (Figs. 8 and 9) to estimate m, it was found that most models include a second baroclinic mode KW feature in the upper ocean instead of the observed first baroclinic KW feature. Therefore, the propagation speed in models is slower as the higher-order vertical mode becomes a larger part of the signal.

Since biases in the background ocean stratification and the KW vertical structure could both result in a biased propagation speed, we explore the relative contribution of these two key factors to KW propagation speed biases (Fig. 11). Large differences still exist in models when using the observed ocean stability N to control for the effect of the biased ocean stability on the propagation speed (“corrected-N” test). However, when using the observed vertical wavenumber m while controlling for stability biases (“corrected-m” test), the corrected speeds are within 10% of the observed speed. This sensitivity test suggests that the incorporation of higher baroclinic modes in model KW vertical structures is responsible for the biased KW propagation speed, while biased ocean stratification in models is not the main cause for the slow KW propagation biases. One possibility for this bias is that insufficient model vertical resolution may cause wave energy leakages from the first baroclinic mode to higher modes, but other possible causes for this discrepancy may exist and warrant further study.

The results discussed above are based on the entire analysis period (1993–2014) and include both El Niño and La Niña conditions. To explore whether the slow biases in the KW propagation are more prominent in a certain phase of ENSO, the KW phase speeds in models and observations were examined during El Niño and La Niña periods, respectively. The slow biases exist in both ENSO phases and exist in both the western and eastern Pacific in both phases, which implies that the slower KW biases may not result from the misrepresented energy transmission when waves propagate into a region with a shallower thermocline (Busalacchi and Cane 1988) since the thermocline depth is comparable along the equator under El Niño conditions.

This work aims to provide a suite of diagnostics on KW performance in models that will contribute to the model diagnostic task force (MDTF) diagnostic-framework package provided by NOAA (Neelin et al. 2023; Maloney et al. 2019). This study provides strong evidence that a realistic description of the KW vertical structure is essential for a good representation of KWs and for tropical coupled air–sea interactions and subsequent weather extremes and climate phenomena. However, due to the lack of availability of daily-resolved three-dimensional ocean temperature and salinity data, the vertical profile of the simulated KWs could not be diagnosed. Future work will examine the KW vertical structure in historical runs in other ESMs such as CESM2 that provide daily temperature and salinity profiles and further scrutinize whether the vertical wave mode plays a key role in determining KW propagation and how biased ocean structure contributes.

Data availability statement.

The daily SSH field from the Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO) is from https://www.aviso.altimetry.fr/en/data/products/sea-surface-height-products/global.html. The daily ocean temperature and salinity in Global Ocean Reanalysis and Simulations (GLORYS)-12v1 product is available at https://data.marine.copernicus.eu/products. The CMIP6-member model outputs in this study were downloaded from The Earth System Grid Federation at https://esgf-metagrid.cloud.dkrz.de/search/cmip6-dkrz/.